All Positive Action Starts with Criticism

All Positive Action Starts with Criticism

Sacha la Bastide-van Gemert

All Positive Action Starts with Criticism Hans Freudenthal and the Didactics of Mathematics

1 3

Sacha la Bastide-van Gemert Garmerwolde The Netherlands Translated by Marianne Vincken and William Third

Translation from the Dutch language editions: “Elke positieve actie begint met critiek”: Hans Freudenthal en de didactiek van de wiskunde by Sacha La Bastide-van Gemert, © Verloren, Hilversum 2006 ISBN 978-94-017-9333-9 ISBN 978-94-017-9334-6 DOI 10.1007/978-94-017-9334-6 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2014959175 © Springer Science+Business Media Dordrecht 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Financing Statement

This English translation published by Springer Science + Business Media was initiated by Dr. B. Greer and coordinated by Dr. A. Bakker on behalf of the Freudenthal Institute for Science and Mathematics at Utrecht University. The project was funded by Springer, the ICMI (International Commission on Mathematics Instruction), the Freudenthal Institute, the Mathematics Institute of Utrecht University and the Nederlandse Vereniging van Wiskundeleraren (NVvW, Dutch Association of Mathematics Teachers).

v

Acknowledgements

This doctorate research project was started in 1996 at the instigation of Prof. dr. K. van Berkel. It was part of the Dutch research programme ‘Ontsluiting wetenschapsarchieven’ (Opening up of science archives), a collaboration of the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO, Netherlands Organization for Scientific Research) and the Centrum voor Wetenschapsarchieven (Centre for Science Archives) of the Rijksarchief in Noord-Holland. NWO was responsible for my appointment as PhD student with the Rijksuniversiteit Groningen which provided the necessary facilities. The Dutch edition of the thesis was also made possible thanks to the support of the Stichting Tijdschrift Janus, the J.E. Jurriaanse Stichting and the Dr. Hendrik Muller’s Vaderlandsch Fonds. A word of thanks goes to the supervisors of my doctorate research project, Prof. dr. K. van Berkel and Prof.dr. J.A. van Maanen. The following people contributed either in writing or during a personal conversation their memories of and their view on Freudenthal: Prof.dr. J. Terwel, Prof.dr. H.J.M. Bos., Prof.dr. E.M. de Jager, Dr. E. de Moor, Dr. M. Kool, Dr. R. Dekker, Dr. C. de Pater, P. Velthuys-Bechthold, LLM B.D. van der Velden, LLM M. Freudenthal, Prof.dr. F. van der Blij, Prof.dr. A. Treffers, Prof.dr. F.J. Keune, Prof.dr. A. van Streun, Prof.dr. A.W. Grootendorst and Dr. L. Streefland. I am grateful to Prof. dr. A. Treffers, Prof. dr. W.E. Krul, Prof.dr. M.J. Goedhart, H. Hamstra, M. Nelissen and J. Hendriksen for their review of the original text. A special word of thanks goes to Jan van Maanen who contributed significantly in meeting all the translation challenges. To conclude I dedicate this book to Mats, Anna, Lotta and last but not least to Steven, without whom neither the thesis, nor this translation would ever have seen the light of day.

Sacha la Bastide-van Gemert

vii

Contents

1

Introduction. “A Way to Master This World”......................................... 1 1.1 Didactics of Mathematics and Hans Freudenthal: Definition of the Problem and Phrasing of the Question...................... 1 1.2 Don Quixote: The Freudenthal Myth?................................................... 3 1.3 Research Method: The Use of Freudenthal’s Personal Archive............ 4 1.4 The Nature of the Study and the Historiography................................... 5 1.5 The Structure of This Book................................................................... 7 References ................................................................................................... 8 2 Mathematics Education in Secondary Schools and Didactics of Mathematics in the Period Between the Two World Wars..................... 11 2.1 Secondary Education in the Period Between the Two World Wars....... 12 2.1.1 The Origin of the School Types in Secondary Education.......... 12 2.1.2 Some School Types.................................................................... 13 2.1.3 The Competition Between HBS and Gymnasium..................... 15 2.2 Discussions on the Mathematics Education at the VHMO................... 16 2.2.1 The Initial Geometry Education and the Foundation of the Journal Euclides............................................ 18 2.2.2 The Beth Committee and the Introduction of Differential and Integral Calculus.............................................. 22 2.2.3 The Controversy About Mechanics........................................... 25 2.2.4 Educating the Mathematics Teacher.......................................... 27 2.2.5 New Insights and the Wiskunde Werkgroep (Mathematics Working Group).................................................. 30 References ................................................................................................... 32 Hans Freudenthal—A Sketch.................................................................... 37 3.1 Hans Freudenthal—An Impression....................................................... 37 3.2 Luckenwalde.......................................................................................... 39 3.3 Berlin ................................................................................................... 41 3.4 Amsterdam............................................................................................ 43 3.5 Utrecht................................................................................................... 51 References ................................................................................................... 56 3

ix

x

Contents

4 Didactics of Arithmetic............................................................................... 61 4.1 Dating of ‘Rekendidaktiek’................................................................... 62 4.2 Cause and Intention............................................................................... 63 4.3 Teaching of Arithmetic in Primary Schools.......................................... 66 4.4 Freudenthal’s ‘Rekendidaktiek’: The Content....................................... 68 4.4.1 Preface........................................................................................ 69 4.4.2 Auxiliary Sciences..................................................................... 69 4.4.3 Aim and Use of Teaching of Arithmetic.................................... 74 4.5 ‘Rekendidaktiek’ (‘Didactics of Arithmetic’): All Positive Action Starts with Criticism.............................................. 76 References ................................................................................................... 79 A New Start................................................................................................. 81 5.1 Educating............................................................................................... 87 5.1.1 Educating at Home..................................................................... 87 5.1.2 ‘Our Task as Present-Day Educators’........................................ 88 5.1.3 ‘Education for Thinking’............................................................ 90 5.1.4 ‘Educating’ in De Groene Amsterdammer................................. 95 5.1.5 ‘The Cooperative Task of the Educator in Forming a Person’...................................................................... 98 5.1.6 Education: A Summary.............................................................. 101 5.2 Higher Education................................................................................... 102 5.2.1 Studium Generale....................................................................... 103 5.2.2 The Teachers Training................................................................ 106 5.2.3 Student Wage............................................................................. 111 5.2.4 Higher Education: A Ramshackle Parthenon or a House in Order?.................................................................. 113 5.3 The Wiskunde Werkgroep (Mathematics Working Group)................... 114 5.3.1 Activities of the Wiskunde Werkgroep...................................... 116 5.3.2 ‘The Algebraic and Analytical view on the Number Concept in Elementary Mathematics’........................................ 118 5.3.3 ‘Mathematics for Non-Mathematical Studies’........................... 124 5.3.4 Freudenthal’s Mathematical Working Group............................. 125 References ................................................................................................... 125 5

From Critical Outsider to True Authority................................................ 133 6.1 Mathematics Education and the Education of the Intellectual Capacity.................................................................... 136 6.2 A Body Under the Floorboards: The Mechanics Education.................. 142 6.3 Preparations for a New Curriculum....................................................... 148 6.4 Probability Theory and Statistics: A Text Book.................................... 155 6.5 Paedagogums, Paeda Magicians and Scientists: The Teacher Training............................................................................. 161 6.6 Freudenthal Internationally................................................................... 170 References ................................................................................................... 172 6

Contents

xi

7

Freudenthal and the Van Hieles’ Level Theory. A Learning Process..... 179 7.1 Introduction: A Special PhD Project...................................................... 179 7.2 Freudenthal as Supervisor..................................................................... 181 7.3 ‘Problems of insight’: Van Hiele’s Level Theory.................................. 185 7.4 Freudenthal and the Theory of the Van Hieles: From ‘level theory’ to ‘guided re-invention’......................................... 189 7.5 Analysis of a Learning Process: Reflection on Reflection......................................................................................... 198 7.6 To Conclude........................................................................................... 199 References ................................................................................................... 200 8 Method Versus Content. New Math and the Modernization of Mathematics Education......................................................................... 205 8.1 Introduction: Time for Modernization................................................... 205 8.2 New Math.............................................................................................. 207 8.2.1 The Gap Between Modern Mathematics and Mathematics Education....................................................... 207 8.2.2 Modernization of the Mathematics Education in the United States.............................................................................. 208 8.3 Royaumont: A Bridge Club With Unforeseen Consequences............... 212 8.3.1 Freudenthal in ‘the group of experts’......................................... 213 8.3.2 Royaumont Without Freudenthal: The Launch of New Math.............................................................................. 224 8.4 Freudenthal on Modern Mathematics and Its Meaning for Mathematics Education............................................... 226 8.4.1 The Nature of Modern Mathematics.......................................... 226 8.4.2 Modern Mathematics for the Public at Large............................ 229 8.4.3 The Mathematician “in der Unterhose auf der Straße” (“In his Underpants on the Street”).................. 230 8.4.4 Fairy Tales and Dead Ends......................................................... 232 8.4.5 Modern Mathematics as the Solution?....................................... 234 8.5 Modernization of Mathematics Education in the Netherlands.............. 236 8.5.1 Initiatives Inside and Outside of the Netherlands...................... 237 8.5.2 Freudenthal: From WW to ‘Cooperate with a view to Adjust’................................................................ 239 8.5.3 The Commissie Modernisering Leerplan Wiskunde.................. 242 8.5.4 A Professional Development Programme for Teachers............. 244 8.5.5 A New Curriculum..................................................................... 245 8.6 Geometry Education.............................................................................. 250 8.6.1 Freudenthal and Geometry Education....................................... 252 8.6.2 Freudenthal on the Initial Geometry Education: Try It and See........................................................... 254 8.6.3 Axiomatizing Instead of Axiomatics—But Not in Geometry........................................................................ 255 8.6.4 Modern Geometry in the Education According to Freudenthal.......................................................... 257

xii

Contents

8.7 Logic�� 258 8.7.1 Exact Logic................................................................................ 259 8.7.2 The Application of Modern Logic in Education........................ 262 8.8 Freudenthal and New Math: Conclusion............................................... 266 8.8.1 A Lonely Opponent of New Math?............................................ 266 8.8.2 Cooperate in Order to Adjust..................................................... 268 8.8.3 Knowledge as a Weapon in the Struggle for a Better Mathematics Education............................ 269 8.8.4 Freudenthal About the Aim of Mathematics Education............. 271 References ................................................................................................... 272 9 Here’s How Freudenthal Saw It................................................................ 283 9.1 Introduction: Changes in the Scene of Action....................................... 283 9.2 Educational Studies in Mathematics..................................................... 286 9.2.1 Not Exactly Bursting with Enthusiasm: The Launch................ 287 9.2.2 Freudenthal as Guardian of the Level........................................ 289 9.3 The Institute for the Development of Mathematics Education............. 291 9.3.1 From CMLW to IOWO.............................................................. 291 9.3.2 Freudenthal and the IOWO........................................................ 293 9.4 Exploring the World from the Paving Bricks to the Moon............................................................................................ 300 9.4.1 Observations as a Father in ‘Rekendidaktiek’........................... 300 9.4.2 Observing as Grandfather Walking with the Grandchildren...... 304 9.4.3 Granddad Hans: A Critical Comment........................................ 306 9.4.4 Walking on the Railway Track: The Mathematics of a 3-Year Old........................................................................... 310 9.4.5 Observing and the IOWO.......................................................... 312 9.5 Observations as a Source....................................................................... 314 9.5.1 Professor or Senile Grandfather?............................................... 314 9.5.2 The Paradigm: The Ultimate Example...................................... 316 9.5.3 Here is How Freudenthal Saw It: Concept of Number and Didactical Phenomenology................................................. 319 9.5.4 The Right to Sound Mathematics for All................................... 327 9.6 Enfant Terrible....................................................................................... 333 9.6.1 Weeding..................................................................................... 333 9.6.2 Drumming on Empty Barrels..................................................... 335 9.6.3 Freudenthal on Piaget: Admiration and Merciless Criticism..... 339 9.7 The Task for the Future......................................................................... 346 References ................................................................................................... 348 10 Epilogue. We Have Come Full Circle .................................................... 361 References�� 365 Bibliography....................................................................................................... 367 Index.................................................................................................................... 367

List of Abbreviations

APS

Algemeen Pedagogisch Studiecentrum

General Pedagogic Study centre

CIEAEM

Commission Internationale pour L’Etude et l’Amélioration de l’Enseignement des Mathématiques

International Committee for the Study and Improvement of Mathematics Education

CIEM (= ICMI)

Commission Internationale pour l’Enseignement Mathématique

International Commission on Mathematics Instruction

CITO

Centraal Instituut voor Toetsontwikkeling

Central Institute for the development of tests

CMLW

Commissie Modernisering Leerplan Wiskunde

Committee for modernization of the mathematics curriculum

COL

Commissie Opleiding Leraren

Committee for Education of Teachers

COLO

Commissie Organisatie Leerplan Ontwikkeling

Committee Organization Curriculum Development

CPS

Centraal Pedagogisch Studiecentrum

Central Pedagogic Study centre

ESM

Educational Studies in Mathematics

HAVO

hoger algemeen vormend onderwijs

Senior General Secondary Education

HBS

hogere burgerschool

Grammar school but without the Classic languages

HTS

hogere technische school

Technical High School

ICM

Innovatiecommmissie Middenschool

Innovation Committee Middenschool

ICME

International Congress on Mathematical Education

ICMI

International Commission on Mathematical Instruction

ICSU

International Council for Science

IEA

International Association for the Evaluation of Educational Achievement

IMC

International Mathematical Commission

xiii

xiv


IMU

International Mathematical Union/ Internationale Wiskundige Unie

IMUK

Internationale Mathematik-Unterrichts Kommission

International Commission for Mathematics Education

IOWO

Instituut voor de Ontwikkeling van het Wiskunde Onderwijs

Institute for the Development of Mathematics Education

KNAW

Koninklijke Nederlandse Akademie van Wetenschappen

Royal Netherlands Academy of Arts and Sciences

KPC

Katholiek Pedagogisch Centrum

Catholic Pedagogic Centre

LBO

lager beroepsonderwijs

Lower vocational education

LIWENAGEL

Leraren in de Wiskunde en Natuurwetenschappen aan Gymnasia en Lycea

Teachers in Mathematics and Science at Gymnasia and Lycea

MAPHA

Mathematisch-Physikalischen Arbeitsgemeinschaft

Mathematical-Physical Working community

MAVO

middelbaar algemeen voortgezet onderwijs

Junior General Secondary Education

MBO

middelbaar beroepsonderwijs

Secondary vocational education

MC

Mathematisch Centrum

Mathematical Centre

MMS

middelbare meisjesschool

Secondary school for girls

MO-AKTEN

middelbaar-onderwijs akten

Certificates to teach at secondary schools

MO-WET

wet op het middelbaar onderwijs (1863)

law on secondary education

MULO

meer uitgebreid lager onderwijs

more extensive primary education

NOCW

Nederlandse Onderwijs Commissie voor Wiskunde

Dutch Education Committee for Mathematics

NOV

Nederlandse Onderwijzers Vereniging

Dutch (primary) School Teachers’ Association

OECD

Organisation for Economic Co-operation and Development

OEEC

Organisation for European Economic Co-operation

OEES

Organisatie voor Europese Economische Samenwerking

Organisation for European Economic Co-operation

OESO

Organisatie voor Economische Samenwerking en Ontwikkeling

Organisation for Economic Cooperation and Development

OK&W

Onderwijs, Kunsten en Wetenschappen

Education, Arts and Sciences

OSTP

Office for Scientific and Technical Personnel

O&W

Onderwijs en Wetenschappen

Education and Sciences

OW&OC

Onderzoek Wiskundeonderwijs en Onderwijscomputercentrum

Mathematics Education Research and Educational Computer Centre

PA

Pedagogische Academie

Pedagogic Academy

PVDA

Partij van de Arbeid

Dutch Labour Party

SLO

Stichting Leerplan Ontwikkeling

Foundation Curriculum Development


xv

SMSG

School Mathematics Study Group

SVO

Stichting voor Onderzoek van het Onderwijs

UICSM

University of Illinois Committee on School Mathematics

UNESCO

United Nations Educational, Scientific and Cultural Organization

VHMO

voorbereidend hoger en middelbaar onderwijs

Preparatory Higher and Secondary Education

VWO

voorbereidend wetenschappelijk onderwijs

pre-university education

VWO

Vereniging van Wetenschappelijke Onderzoekers

Association of Scientific Researchers

WIMECOS

Vereeniging van Leeraren in de Wiskunde, de Mechanica en de Cosmographie aan hoogere burgerscholen met vijfjarigen cursus b, lycea en meisjes-hoogere burgerscholen met 5-/6-jarigen cursus

Association of Teachers in Mathematics, Mechanics and Cosmography at HBS

WISKOBAS

wiskunde in het basisonderwijs

Mathematics in primary education

WISKIVON

wiskunde in voortgezet onderwijs

Mathematics in secondary education

WVO

Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs

Working community for Education and Instruction

WW

Wiskunde Werkgroep

Mathematics Working Group

Foundation for Research of Education

Chapter 1

Introduction A Way to Master This World

En toch probeer ik—ook in mijn geschriften—één en al spontaniteit te zijn. Iedere volzin die ik schrijf, wordt, eer hij geschreven en gedrukt wordt, gewikt, gewogen en gewenteld tot het summum van spontaniteit is bereikt. Het moet erop lijken of al wat ik schrijf me net te binnen is geschoten. […] Het is niet eenvoudig je te tonen zoals je bent, vooral als je helemaal niet zo bent. […] Zijn niet al mijn schrijfsels verkleedpartijen? Nee, ik schrijf gekleed, maar niet verkleed. (And yet I try—in my writings too—to be spontaneity itself. Every sentence I formulate must be pondered, weighed and reshaped, until it seems the essence of spontaneity before being written and printed. Everything I write must look as if I just thought of it. […] It is not easy to show yourself the way you are, particularly if you aren’t like that at all. […] Isn’t all my writing a series of costume balls? No, I write dressed, but not dressed up. Hans Freudenthal in Schrijf dat op, Hans, 1987 [1]

Hans Freudenthal: Professor of mathematics, scientist, literator, but above-all the personification of mathematics education, inextricably linked to the changes in mathematics education and didactics during the second half of the Twentieth century. To this day he is the dominant figure in mathematics education in the Netherlands. No one discussing mathematics education can ignore Freudenthal: the teaching of mathematics in the spirit of Freudenthal serves either to inspire or to evoke reactions. If you are talking about Freudenthal, you are talking about the didactics of mathematics. That alone would be reason enough to dedicate a study to Freudenthal. But there is more.

1.1 Didactics of Mathematics and Hans Freudenthal: Definition of the Problem and Phrasing of the Question In studying didactics of mathematics one will sooner or later run into the question: what exactly is mathematics? Where does mathematics start? Studying the way children learn mathematics can provide an answer to that question. How children © Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_1

1

2

1 Introduction

achieve a notion of mathematics can provide a clue to answering the question ‘What is mathematics?’ From the moment didactics of mathematics became a field of research, questions about the nature and meaning of mathematics in general kept coming up. In the period between the two World Wars the formative value of mathematics was looked upon as one of the greatest merits of mathematics education. By teaching mathematics in a deductive, closed system, the formative value of mathematics would be done full justice. Mathematics was being offered to pupils ‘ready-to-use’. Advocates of this approach considered the pureness and honesty of mathematical thinking of the utmost importance for the development of the intellect and the enhancement of the pupil’s ethical values. Consequently mathematics education, particularly in secondary schools, was rather aimed at the elite: to understand mathematics was a privilege for a mere handful of pupils. Its high level of abstraction and the apparently limited role mathematics played in society made its implementation as a school subject and its justification, after the introduction in the nineteenth century in the Netherlands, far from self-evident. In the nineteenth century mathematics as a school subject regularly had to legitimize and justify itself. This resulted in a certain tradition in thinking about mathematics education. Why and for whom? [2]. After the Second World War when Freudenthal increasingly pursued didactics of mathematics, he walked well-trodden paths. Others before him had already written about didactics of mathematics and discussed it. These discussions occurred regularly and studying didactics of mathematics gained prestige. The changing role of mathematics in society was indirectly responsible for this. After the Second World War when the importance of practical applications of mathematics became clearer in the Netherlands, ‘mathematics as a productivity factor’ increasingly pushed ‘mathematics as a culture factor’ into the background [3]. The practical usefulness of mathematics not only made its presence increasingly felt in the discipline, but also in teaching one could not get around it: ‘mathematics for the sake of mathematics’ increasingly fell short as more people needed mathematics. Precisely at that moment Freudenthal entered the discussion and through his activities he has left his mark on the didactics of mathematics in the Netherlands (and beyond). Mathematics education in the Netherlands has had a very good name internationally for many years and Freudenthal’s reputation as education innovator is one of the causes. Both within and outside the field of didactics of mathematics he is considered as being responsible for the kind of mathematics education in which practicing techniques as separate capabilities moved into the background [4]. Instead of being imposed ready-to-use mathematics pupils were given more opportunities to discover and acquire mathematics in a context relevant to them. Mathematizing reality became the starting point: mathematics is everywhere. Attention was given to interaction between pupils, to cooperation and to looking for different solution-finding strategies. ‘Realistic mathematics education for everyone’ is an appealing characterization of the kind of mathematics education with which Freudenthal’s name is associated.

1.2 Don Quixote: The Freudenthal Myth?

3

This study wants to answer the question: What was Freudenthal’s role in mathematics education? How correct is the view of the key role that he played in the innovations? To be able to answer these questions, firstly important sub-questions have to be dealt with. What were his ideas about mathematics, education and didactics? How did these ideas originate, by what were they influenced and how did they change (if they did)? In what way did they have an influence, and what influence did they have, on the prevailing views at that time? In the description of the changes in mathematics education the view of mathematics as an element of culture will also come up for discussion. Being a versatile scientist and a respected university professor with an urge to unremittingly bring publications into the open, Freudenthal had a particularly important social function in that field. Organizing the first international congress for mathematics education and founding the international journal for mathematics education Educational Studies in Mathematics are to his credit. He was indefatigable in founding didactics of mathematics as a serious academic field. Freudenthal’s vision on mathematics— what is mathematics? Where does mathematics start?—indirectly had a continuous influence on his ideas on mathematics education. To him, this was a guideline in his quest for the right approach to the development of mathematics education. In Freudenthal’s view an essential role for mathematics had been reserved in everybody’s life. In particular in education, the mathematics of everyday reality, as opposed to teachers imposing on pupils a system of pure mathematics, was vital to him. Opposite to this view of mathematics as a sophisticated subject matter, dropped from above to below, is that of mathematics as a natural and social activity which develops according to the growth and the growing needs of the individual in an expanding world. Mathematics is an attitude, a way to master this world cognitively, practically, emotionally. [5]

Mathematics was everywhere and for everybody, and this mathematics should according to Freudenthal be taken as the starting point in developing mathematics education. In the 1970s he saw this vision put into practice through educational developments like those of the collaborators of the Instituut voor de Ontwikkeling van het Wiskunde Onderwijs (IOWO) (Institute for the Development of Mathematics Education), the present-day Freudenthal Institute for science and mathematics education at the University of Utrecht, the Netherlands. His successes in this field alone are enough to justify a closer investigation.

1.2 Don Quixote: The Freudenthal Myth? Freudenthal’s role in the transformation of the mathematics education has traditionally assumed almost mythical forms. A myth further reinforced by the stereotypical image people had of him. His bow tie and pipe made him a somewhat eccentric professor with an established reputation as a mathematician before he became interested in didactics. Freudenthal himself contributed implicitly—and perhaps without being aware of it—to this myth. He described his role in didactics of mathematics extensively in

4

1 Introduction

his autobiography, Schrijf dat op, Hans from 1987 [6]. At the end of the 1980s he sketched his career as an educator, with the spontaneity of the intricate citation at the beginning of this introduction and his characteristic casual nonchalance. In his other articles and books of the 1970s and 1980s he regularly and explicitly looks back on the history of mathematics education in the Netherlands and the part he played in it. In the way he portrays himself as a didactician, a Don Quixote-like person dominates, a lonely fighter against the evil tendencies in mathematics education. He describes himself as an ‘enfant terrible’, proud of his ‘mischiefs’, always rowing upstream, but, eventually, being right. This description corresponds with the common image of Freudenhal the didactician. But there are also other stories. An inquiry among former staff members of the IOWO resulted in conflicting opinions on Freudenthal’s role in the extensive developments in the institute [7]. In addition, it is surprising how different people remember Freudenthal in different and sometimes almost contradictory ways. Speaking with people who either collaborated with him, were his students or met him only casually, it is remarkable that often different images of Freudenthal come up. For some, he was a person who seemed difficult to approach, distant and inaccessible, for others he was a warm and enthusiastic, almost childlike person. On the one hand is the image of a loving grandfather, on the other hand that of an inaccessible professor scaring people off and preferably to be avoided. A social democrat, pig-headed, loyal, straight from the shoulder, but also, often for tactical reasons, conservative and calculating. Extremely critical, as hard as nails and with a persistent judgement, never afraid of voicing criticism and mercilessly reprimanding someone. He was the awe-inspiring mathematician, but at the same time also the down-to-earth educator, observing in a classroom. He might seem to be asleep in the audience, but then suddenly he was not, and a clever remark was made out of nowhere. For some the image of an authoritarian, frankly spoken bad lecturer, lacking in self-control was dominant, for others he was a brilliant inspirer with a heart for pupils, students and student affairs.

1.3 Research Method: The Use of Freudenthal’s Personal Archive As is the case with every view, Freudenthal’s own view on the role that he played in the didactics of mathematics is also coloured, but it is one that gives an extra dimension to this study. Freudenthal speaks to us regularly through his Schrijf dat op, Hans and so contributes to the reconstruction of his work, which also allows room for his version of what happened. However, his autobiography and the image he sketches there of his career in education, in no way serves as a guideline for this reconstruction. To make my own interpretation I deliberately took Freudenthal’s personal archive as the most important starting point for this study. The reconstruction of events and activities is primarily based on documents found in the 16 m long archive [8]. Apart from a list that Freudenthal had composed, of more than 200 of his publications in

1.4 The Nature of the Study and the Historiography

5

the field of mathematics education, this archive appeared to contain a cornucopia of (un)published material on or closely related to the field of education, teaching and didactics of mathematics. In particular Freudenthal’s carefully kept correspondence turned out to be an important source. Letting the texts speak immediately implies that there is a distance between me and Freudenthal and because of it I have a less outspoken opinion on Freudenthal as a person. This was necessary to keep a certain distance so that the Freudenthal myth could be held up to the light—or perhaps up to another, different, light. The result is a reconstruction of mainly written data answering the question of the extent to which the ‘Freudenthal myth’ corresponds to reality—or anyway to the reality occurring in the sources used. Using Freudenthal’s personal archive also brought some difficulties with it. Freudenthal’s hugely varied and extensive fields of interest resulted in a rich and intriguing collection and in exploring it there is great risk of being side tracked into writing a biography. No one could incorporate newly acquired knowledge in his own ideas so smoothly, mostly without direct reference to its source, as Freudenthal could. To be able to record the development of these ideas required a lot of work in the archive. It was easy to be carried away by Freudenthal’s personality and his points of view. The ease with which he wrote was one of the causes. This is what he says about his style of writing: Ik hoef me geen geleerd lijkende taal aan te meten om diepte voor te spiegelen. Ik heb geen carrière voor me waarin zo’n taal tot aanbeveling zou kunnen strekken, en ik stel er geen prijs op het nageslacht aan het werk te zetten om dieptes van een onleesbare tekst te doorgronden. Aan mijn geestelijk portret valt niets meer te wijzigen, op mijn denkbeeld van wetenschappelijkheid hoef ik geen correctie aan te brengen. Simplex sigillum veri—vertaal ik als: wat waar is mag je onomwonden zeggen. (I do not have to avail myself of highbrow language to conjure up depth. I do not see before me a career where such language could be in my favour and I do not want to put future generations at work scrutinizing the depths of an unreadable text. My intellectual portrait can no longer be altered, I do not need to correct my notion of a scientific nature. Simplex sigillum veri—my translation: the truth may be told in no uncertain terms.) [9]

When studying his writings it has been tempting to let the plain and seemingly modest style of writing he used in Schrijf dat op, Hans influence the tone in this study. The challenge was to adopt a middle course between a business-like distant description of Freudenthal’s work and a romantic-biographic essay. Looking beyond the texts and the correspondence was a way to avoid the first pitfall. The second one had to be avoided by adopting a rather critical reserve to using oral history and by not indiscriminately following Freudenthal’s own reconstruction of his career.

1.4 The Nature of the Study and the Historiography This study is neither a didactical nor a mathematical study, although it can be relevant in both respects. Testing Freudenthal’s ideas for their didactical value or using them as a source for the development of educational methods was not

6

1 Introduction

the aim. Neither was the writing of a biography. The study has biographical elements but only if these relate to Freudenthal as a didactician. Consequently, and unfortunately, numerous aspects of Freudenthal’s life and work do not appear in this study. The main aim of this study was to make an historical analysis. Freudenthal’s didactical ideas are submitted to an historical analysis with his personal archive as a starting point. It is not a complete report of ‘how things went’ but distils from the manuscript a possible way of looking at Freudenthal and his work. The resulting image of Freudenthal will be tested against that of the ‘Freudenthal myth’. Quite a bit has already been written about Freudenthal’s didactical work and the history of mathematics education. This study fits in between two lines of approach. On the one hand it can be placed in the line of the history of mathematics education in the Netherlands. Klaas van Berkel wrote about mathematics education in the period before the Second World War in his biography of E.J. Dijksterhuis [10]. In his thesis E.W.A. de Moor described the history of teaching geometry in the Netherlands, H.J. Smid wrote about mathematics education at schools which taught in French and Latin during the nineteenth century [11]. The history of teaching of arithmetic is dealt with in a review article by Jo Nelissen and a compilation on mathematics education in the last century Honderd jaar wiskundeonderwijs (A century of mathematics education) was published [12]. Henk Klomp’s thesis also pays much attention to the developments in the field of mathematics and physics education between the two World Wars and emphasizes the democratization thererof [13]. Finally, a great deal about the changing role of science (and scientists) in mathematics in the Netherlands after the Second World War can be found in the theses of Gerard Alberts and Leo Molenaar [14]. The second line in which this study fits is the biographical line (even though biographical aspects certainly are present in studies of the first line). First of all, there is Freudenthal’s autobiography Schrijf dat op, Hans, in which he also analyses his own didactical work. Aspects of Freudenthal’s didactical and historical studies are discussed in The legacy of Hans Freudenthal [15]. Shortly after his death several other articles on Freudenthal’s life and work, such as those written by J. Adda, A. Bishop, J. van Dormolen, W.T. van Est and J. de Lange, appeared [16]. An analysis of a part of Freudenthal’s didactical mental legacy is made in a number of publications including the theses of De Moor, A. van Streun and J. Terwel [17], who along with K. Gravemeijer also devoted an article to the didactician Freudenthal [18]. In Van Berkel’s book Dijksterhuis. Een biografie (Dijksterhuis. A Biography) the didactician Freudenthal appears as one of Dijksterhuis’ opponents. And finally in Dirk van Dalen’s biography, L.E.J. Brouwer Een biografie (L.E.J. Brouwer, A biography) aspects of Freudenthal’s life and work in Amsterdam are discussed [19]. Short biographies of Freudenthal are also found in Molenaar’s thesis and in the compilations Honderd jaar wiskundeonderwijs and Kopstukken van het laagland (Leaders of the low countries) [20].

1.5 The Structure of This Book

7

1.5 The Structure of This Book This book is outlined chronologically. The next chapter sketches the situation in (mathematics) education between the two World Wars, with emphasis on pre-university education. For a full understanding of the nature and background to the discussion about mathematics education, first the various school types that existed at that time in secondary and pre-university education (VHMO) are discussed, while elucidating their origins, the underlying motives and their aims. Apart from the school types, the disputes within mathematics education and some of the important players in that field are also reviewed. Next, a biographical chapter describes Freudenthal’s life in broad outlines. The emphasis lies on the period before the Second World War, the later period is treated more broadly. Relevant biographical aspects of these later periods will come up in the appropriate places in the rest of the book. During the last years of the Second World War Freudenthal writes a didactical manuscript, the unpublished ‘Rekendidactiek’ (‘Didactics of arithmetic’), his first publication in that field. Chapter 4 describes this manuscript which, in view of the development of his ideas in the 1970s, offers much of interest. ‘A new start’ is the title of the next chapter describing the period just after the Second World War. In these years Freudenthal took the first steps in the field of mathematics education and didactics of mathematics and he published his first manuscripts. It is the period in which he finds his niche in Utrecht and becomes a prominent and active member of the Wiskunde Werkgroep (Mathematics Working Group). Chapter 6 covers the period from 1950 to 1957 when Freudenthal’s national and international reputation as an ‘educator’ gradually became established because of his involvements in committees, but also by his publications for instance in De Groene Amsterdammer (a Dutch news weekly). His mathematical-didactical ideas receive an important new impulse by the work of Mr. and Mrs. van Hiele at the end of the 1950s. Freudenthal, being Pierre van Hiele’s thesis-supervisor, is closely involved and finds in Van Hiele’s level theory an inspiring new didactical approach. Chapter 7 contains an analysis of this theory and the way it is reflected in Freudenthal’s mental legacy. The chapter ‘Method versus content’ deals with the period around 1960 when a wave of modernization called ‘New Math’ from the United States arose in mathematics education. In those years Freudenthal plays the role that lingers in tradition as that of ‘the lonesome opponent of New Math’. The final chapter entitled ‘Here’s how Freudenthal saw it’, describes the years around 1970. The central issue in this period, apart from, of course, a leading role for the IOWO (the predecessor of the current Freudenthal Institute), is the consolidation of Freudenthal’s philosophy on (mathematics) education. The construction phase of Freudenthal’s didactics is closed. This is where the study ends. But the story of Freudenthal’s didactics of mathematics is by no means finished. After all, Freudenthal worked indefatigably on the development of mathematics

8

1 Introduction

education and didactics of mathematics till his death in 1990. But at the moment this study stops, the basis of an important part of Freudenthal’s didactical ideas has been defined. The direction in which his work will develop is then charted in broad outlines. Finally, an epilogue looks ahead and gives a retrospective.

References 1. Hans Freudenthal, Schrijf dat op, Hans. Knipsels uit een leven (Amsterdam 1987) 11. 2. Something comparable happened to the classical languages as school subjects which lost their legitimization in the nineteenth century. The discussion on the legitimization of the subject raised more attention for the didactics with an emphasis on its formative value; Arie Jan van Duyvendijk, De motivering van de klassieke vorming: een historisch-paedagogische studie over twee eeuwen (Groningen 1955); W. Hoving en J. van IJzeren, Onderzoek naar de wenselijkheid en de mogelijkheid om bij het onderwijs in de oude talen op gymnasia en lycea de lectuur met na-klassieke schrijvers uit te breiden beoordeeld in verband met de betekenis van het klassieke onderwijs in zijn geheel (Groningen 1940); Pim Verhoeven, Tekstbegrip in het onderwijs Klassieke Talen. Een onderzoek naar de verwezenlijking van nieuwe doelstellingen in het schoolonderzoek Grieks en Latijn 1997. 3. This terminology has been borrowed from the work of Gerard Alberts, Jaren van berekening. Toepassingsgerichte initiatieven in de Nederlandse wiskundebeoefening 1945–1960 (Amsterdam 1998). 4. An example of this view of Freudenthal can be read in: A. Freeman-Smulders, ‘Lezen als ongedeelde vaardigheid. Een Freudenthal voor het leesonderwijs gezocht’, Leesgoed: tijdschrift voor kinderboeken 24 (1997) 30–33. 5. Hans Freudenthal, ‘New Math or new education’, Prospects ix (1979) 321–331, q.v. 322. 6. Freudenthal, Schrijf dat op, Hans; Apart from the casual remarks to be found throughout the book, he devoted a complete chapter to this: ‘Werken aan onderwijs. Op weg’; ibid., 335–363. 7. E.W.A. de Moor, Van vormleer naar realistische meetkunde. Een historisch-didactisch onderzoek van het meetkundeonderwijs aan kinderen van vier tot veertien jaar in Nederland gedurende de negentiende en twintigste eeuw (Utrecht 1999) 484–485. 8. P.J.M. Velthuys-Bechthold, Inventory of the papers of Hans Freudenthal (1905–1990), mathematician. 1906–1990 (Haarlem 1999). 9. Hans Freudenthal, Didactische fenomenologie van wiskundige structuren (Utrecht 1984) 191–192. 10. Klaas van Berkel, Dijksterhuis. Een biografie (Amsterdam 1996). 11. De Moor, Van vormleer naar realistische meetkunde; H.J. Smid, Een onbekookte nieuwigheid? Invoering, omvang, inhoud en betekenis van het wiskundeonderwijs op de Franse en Latijnse scholen 1815–1863 (Delft 1997). 12. Jo Nelissen, ‘Eeuwenlang rekenen. Historische ontwikkelingen in het rekenonderwijs’, Willem Bartjens 18 (1998/99) 36–39; Fred Goffree, Martinus van Hoorn, Bert Zwaneveld, eds., Honderd jaar wiskundeonderwijs—een jubileumboek (Leusden 2000). 13. Henk A. Klomp, Relativiteitstheorie in Nederland. Breekijzer voor democratisering in het interbellum (Utrecht 1997). 14. Alberts, Jaren van berekening; Leo Molenaar, ‘Wij kunnen het niet langer aan de politici overlaten.’ De geschiedenis van het Verbond van Wetenschappelijke Onderzoekers 1946– 1980 (Delft 1994).

References

9

15. These are the articles from the ‘In memoriam HF (’90)’-copy of Educational Studies in Mathematics door Josette Adda, Fred Goffree, Henk J.M. Bos, W.T. van Est, A. Treffers, L. Streefland, J. de Lange, Christine Keitel; Leen Streefland, ed., The Legacy of Hans Freudenthal (Dordrecht 1993). 16. J. Adda, ‘Freudenthal cursief. Hans Freudenthal (1905–1990)’, Nieuwe Wiskrant. Tijdschrift voor Nederlands Wiskundeonderwijs 10 (1991) 26; A. Bishop, ‘Freudenthal cursief. HF—an inspiration in the search for truth’, Nieuwe Wiskrant 10 (1991) 13; J. van Dormolen, ‘Freudenthal cursief. Hoe didactiek een deel van wiskunde kan worden’, Nieuwe Wiskrant 10 (1991) 19; W.T. van Est, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Nieuw Archief voor Wiskunde 9 (1991) 131–136; W.T. van Est, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Koninklijke Nederlandse Akademie van Wetenschappen. Jaarboek 1991 (Amsterdam 1991) 139–142; J. de Lange, ‘In memoriam Hans Freudenthal’, Nieuwe wiskrant 10 (1990) 3. 17. De Moor, Van vormleer naar realistische meetkunde; A. van Streun, Heuristisch wiskundeonderwijs. Verslag van een onderwijsexperiment (Groningen 1989); J. Terwel, Onderwijs maken (Harlingen 1984). 18. K. Gravemeijer en J. Terwel, ‘Hans Freudenthal: a mathematician on didactics and curriculum theory’, Journal of curriculum studies 32 (2000) 777–796. 19. Dirk van Dalen, L.E.J. Brouwer. Een biografie. Het heldere licht van de wiskunde (Amsterdam 2001). 20. Martijn van Calmthout, ‘Hans Freudenthal. Kampioen van de alledaagse wiskunde’, in: P. Brill, red., Kopstukken van het laagland. Een eeuw Nederland in honderd portretten (Amsterdam 1999).

Chapter 2

Mathematics Education in Secondary Schools and Didactics of Mathematics in the Period Between the Two World Wars

Om mijn verhaal goed in te kaderen, moet ik nog veel verder teruggaan en dan tot 1924—jaren voor ik in Nederland kwam—het soort annus mirabilis van de wiskundedidactiek in Nederland. (To embed my story well, I have to go back much further than 1924—years before I came to the Netherlands—a kind of annus mirabilis in the didactics of mathematics in the Netherlands.) Hans Freudenthal on the Dutch mathematics education in Schrijf dat op, Hans, 1987 [1]

Hans Freudenthal’s first introduction to the Dutch educational system took place at the end of 1930 when, at the age of 26, he was employed by the University of Amsterdam. His image of the Dutch student was that of a diligent, not particularly self-reliant pupil who on top of that was also slow-witted. It was remarkable that Freudenthal blamed the (rote) learning of many foreign languages for the passive attitude of the students. He wrote in a letter in 1931: Das Niveau des hollaendischen Studenten ist geringer als das des deutschen; das haengt mit dem etwas schwerfaelligen, traetigen Volkskarakter zusammen. Der hollaendische Student arbeitet sicher fleissich, begreift aber sehr langsam. Selbstaendigen produktives Arbeiten findet man bei ihm sehr selten. Schuld daran ist vielleicht auch das Viellernenmuessen, das von dem Zwang Sprachen zu beherrschen, der bei einem kleinen Volk verstaendlich ist, herkommt […] Leider habe ich mit den Studenten noch nicht viel Kontakt bekommen und sehe sicherlich manche Dinge noch sehr vom Gesichtspunkt des Dozenten. Die Studenten sind sehr schuechtern, stellen weder in der Vorlesung noch nachher Fragen, man kommt sich waehrend des Vortrages wie ein Prediger in der Wueste vor. (The level of education of the Dutch student is lower than that of the German student; that is related to the somewhat ponderous, obtuse character of the people. The Dutch student certainly works very diligently, but understands very slowly. You will rarely find him working independently and productively. Perhaps having to learn such a great deal, because of the obligation to learn foreign languages, as is understandable for a small nation, is to blame. Unfortunately, I have not yet been much in contact with students and it is certain that there are a number of things I only know from the viewpoint of a teacher. The students are very shy, do not ask questions during or after lectures, and as a teacher you feel like a preacher in the wilderness when lecturing.) [2] © Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_2

11

12

2 Mathematics Education in Secondary Schools and Didactics …

Freudenthal was doing nothing more than expressing his astonishment about the differences between Dutch and German students. In the 1930s he spent most of his time working as the assistant of the mathematician L.E.J. Brouwer. During the Second World War his interest was aroused by the teaching of arithmetic and the accompanying didactics of arithmetic at primary schools and he was engaged in a literature study on the subject [3]. It was only after the war that Freudenthal became actively involved in mathematics education and didactics of mathematics in the Netherlands and would go public with his ideas. In the 1940s and 1950s he particularly focussed on mathematics education at secondary schools (VHMO, Voorbereidend Hoger en Middelbaar Onderwijs, Prepatory Higher and Secondary Education). It was not the frontiers of knowledge he was studying. Quite a lot of discussions had already been held in that field before Freudenthal became involved. To get an idea of how Freudenthal fitted into this mathematical-didactical landscape, a rough outline of mathematics education in the period between the two World Wars is needed first. What did secondary schools (VHMO) and the way mathematics was taught look like at that time? Which discussions were held? And who were the leading participants in these discussions? [4].

2.1 Secondary Education in the Period Between the Two World Wars 2.1.1 The Origin of the School Types in Secondary Education Around 1918 secondary education in the Netherlands was governed by the 1863 law on secondary education, the MO or Middelbaar Onderwijs law, the first concerning secondary education. Various types of school such as the ULO—Uitgebreid Lager Onderwijs—for extended primary education, and the MULO—Meer Uitgebreid Lager Onderwijs—for further extended primary education, were in the primary education category. However, the Gymnasium (comparable to Grammar school in the UK) was part of higher education. The author of the MO-law, liberal Johan Rudolf Thorbecke (1798–1872), clearly had a dual function of education, directly related to the hierarchy of the class society, in view. First of all there was the kind of school type that would offer a training course directly aimed at the needs of society. In Thorbecke’s view this education was meant for the ‘core of the society’, the large middle class (consisting of the ‘humble middle class’ and the ‘upper middle class’). This group needed a modern, nonvocational yet practice-oriented type of education. This would prepare the children from the middle classes for their future tasks in society, for example working in trade and industry and offices. To this aim a new school type was designed for the middle class (HBS: Hogere Burger School, Grammar school but without the Classic

2.1 Secondary Education in the Period Between the Two World Wars

13

languages). The Polytechnische School (Polytechnic School) in Delft was defined as an extension for some of the HBS students. Along with the secondary schools for girls (MMS) these school types formed the secondary education. The HBS and the Polytechnische School would prove to be the most successful types [5]. In addition, Thorbecke distinguished higher education as comprising the Gymnasium and the University. The Gymnasium was the only school that offered the Classic languages, education that was reserved for the children of the learned classes, and the only possible path to a university study [6]. Thorbecke, advocate of a strict distinction between ‘mind’ (higher education) and ‘matter’ (secondary education), thus maintained the Gymnasium as a separate institute with a higher status.

2.1.2 Some School Types 2.1.2.1 The HBS One of the characteristics of the Dutch educational system of that time was the freedom of choice that the schools had, at least at first. The MO-law did not lay down requirements for the final examination, neither for the subjects chosen nor for their subject matter. To guarantee that the matriculants had achieved a certain level of education, a state exam with rules concerning the design, the content and the evaluation of the final exam, was soon drawn up [7]. Unfortunately, this regulation resulted in an enormous overburdening for the HBS and too many subjects. There were also complaints about the outdated subject matter that was taught. Over a long period, ministers and school inspectors tried to produce new order and drafted several reforms but this never resulted in a full-scale reorganisation of the secondary education system. Nevertheless, the HBS developed into the most successful type of secondary education among those proposed in the law of 1863. Being the counterpart of the Gymnasium with its prevalent Classic languages education the HBS accentuated modern languages, science and mathematics. On average HBS pupils put one third of their time into science and mathematics. As with other school types, classroom teaching was the most widespread method used in the HBS. Which subject matter was dealt with was determined by the requirements set by the final exam. There were two main streams in the exam programme: HBS-A, mainly focussing on languages, and HBS-B, on mathematics and sciences. As far as mathematics was concerned the final exam for HBS-B consisted of arithmetic and algebra, trigonometry, stereometry and descriptive geometry. A mathematics class followed the pattern ‘explanation, demonstration, imitation and practicing endlessly’. Pierre van Hiele, mathematics teacher and didactician, described the attitude at that time with respect to the mathematics that had to be taught as follows [8]: Geen leerling betwijfelde de noodzaak van bewijzen in de ‘wiskunde’. Er waren er natuurlijk genoeg die daar moeite mee hadden. Als je wiskunde per se nodig had, deed je je uiterste best om er ten minste zoveel van te begrijpen dat je later niet in de moeilijkheden kwam. In het andere geval kon je je werpen op de vraagstukken om zo hoog mogelijk te

14

2 Mathematics Education in Secondary Schools and Didactics … scoren. De leraren speelden daarop in: zij beijverden zich om de stof zo exact mogelijk voor te dragen, maar zij hielpen de leerlingen met de uitwijkmogelijkheid van de vraagstukken. Per slot van rekening nam de bewijsvoering op het eindexamen niet zo’n grote plaats in. (There was no pupil who did not doubt the need of proofs in mathematics. Enough of them, of course, found proving theorems difficult. If you absolutely needed mathematics you did your utmost to understand at least enough to avoid difficulties later. If not, you could launch into solving problems to score as many points as possible. Teachers anticipated this: they tried their hardest to present the subject matter as accurately as possible, but also helped the pupils with the fall-back arrangement of solving problems. After all, proving theorems did not figure largely in the final exams.) [9]

Despite its great success there was also a lot of criticism of the HBS school type. E. Jensema, director of the Rijks-HBS (State-HBS) in Groningen in 1927, distinguished three points of general criticism. The education in the HBS would be too mathematical and too many pupils would leave the school prematurely. In addition, the school would be too dualistic by trying to offer both final and preparatory (higher) education at the same time [10]. 2.1.2.2 The Gymnasium The first Gymnasium was the result of a merger of the long-standing Latin School and the municipal primary school “of the upper class” in The Hague [11]. The Gymnasium had two sections: the first one, a continuation of the Latin School, emphasized teaching Latin and Greek as a preparation for a university study. The emphasis in the second one was on mathematics, physics and modern languages. This section met the society’s growing need for education. The Gymnasium kept its formal and social status after the adoption of the MOlaw of 1863. From 1878 Latin schools were also classified as Gymnasia. Since the higher education act of 1876 both the programme and the timetable of the classes had been laid down precisely. Mathematics was one of the compulsory subjects. The freedom in interpretation of the educational programme such as existed at the HBS was out of the question here. As the practical use in society of studying the Classic languages appeared to diminish, the antique argument of the formative value of the Classic languages was used to preserve the teaching of the Classics. But the competition with the HBS was felt more and more clearly and this led in 1876 to a division of the Gymnasia in a α-department/division (with a language-oriented programme) and a β-department/division (where apart from the Classic languages a great deal of attention was devoted to science and mathematics). The written final exam in mathematics for the β-department consisted of algebra, analytical geometry, stereometry and goniometry. 2.1.2.3 The MMS The secondary school for girls (Middelbare Meisjes School, MMS) was one of the types of school which could freely develop and hold its ground. It offered an alternative to the HBS which was mainly attended by boys. There were no statutory

2.1 Secondary Education in the Period Between the Two World Wars

15

regulations to lay down content and form of the education. It was not until 1935 that some rights for further study were laid down for this type of school. Mathematics was not part of the final exam, which offered teachers and authors of the school books a great deal of freedom. Usually mathematics was only taught in the first 3 years of the MMS [12]. 2.1.2.4 The Lyceum The first Lyceum was established in The Hague in 1909 by headmaster R. Casimir (1877–1957) who later became the first professor of pedagogy in the Netherlands. The Lyceum, consisting of a Gymnasium, an HBS and sometimes an MMS too under one roof, originally aimed at postponing the choice for a specific school type by offering education in the first few years in combined classes. After all, when pupils had to choose between Gymnasium and HBS at the age of 12, it directly determined their options for further education. On top of this it was more efficient and less expensive to found three schools simultaneously rather than three separate ones. The influential didactician and textbook writer Wansink wrote on the importance of the Lycea for the educational reform: De vernieuwende tendensen op pedagogisch-didactisch terrein zijn bij de lycea sterker geweest dan bij de meer starre vormen van hogereburgerschool en gymnasium in de V.H.M.O.-periode. Bij het lyceum werd in onze schoolorganisatie voor het eerst rekening gehouden met het kind als medebepalend element. (The innovative tendencies in the pedagogic-didactic field have been stronger in Lycea compared to the more uncompromising school types such as HBS and Gymnasium in the VHMO period. The Lyceum was the first school in the Dutch school organisation in which the child was considered a contributory element.) [13]

In a way the Lyceum can be seen as the predecessor of the modern ‘Scholengemeenschap’ (combined school). It was not until 1968 that Lycea were legally established as an existing school type. Until that time the Lyceum existed merely due to the grace which the law allowed for deviations in the organisation of Gymnasia [14].

2.1.3 The Competition between HBS and Gymnasium From the moment the MO-law was adopted the view started to take root that the HBS would prepare pupils just as well (if not better!) for specific university studies (in particular medicine, science and mathematics). Increasingly, HBS-graduates continued their education at universities via a detour (for example by attending testimony classes in Greek, Latin and logic, and taking university entrance exams). From 1865 onwards the Education Ministry began granting exemption for these exams in exceptional cases. As from 1878, holders of an HBS diploma could also study medicine by doing exams that paralleled the academic exams. There was an increasing number of such regulations allowing HBS-graduates to enter universities. This finally induced J. Limburg, member of parliament, to put forward a bill in parliament in 1917. His bill which, among other things, qualified

16


HBS-graduates for doing exams at the science and mathematics faculty, was passed by a majority vote. This definitively ended the matter-mind dichotomy which Thorbecke advocated. Apart from that it also implied people were no longer convinced that schooling in the Classics was a prerequisite for an academic study and it implicitly emphasized the appreciation for science and mathematics. It was now evident that the original plan for the HBS as a complete and final education for the whole society had failed. An attempt was made to rectify this deficiency by dividing the HBS in two, HBS-A (modern languages and economy) and HBS-B (with the emphasis on science and mathematics). The assignment of the right to study at a university to HBS-B graduates (and later for some academic studies to HBS-A graduates as well) considerably weakened the monopoly position of the Gymnasium of course. To slightly increase the attraction of the Gymnasium for pupils wanting to study science and mathematics it was decided by Royal Decree to modernize it in 1919. The programme offered more space for science and mathematics. New, modern subjects such as differential calculus and integral calculus now became part of the algebra that was taught. Thus the Gymnasium was still one step ahead of the HBS. However this new difficult Gymnasium-β, combined with the traditional status of the Gymnasium, made it the most demanding school, reserved for a small elite only.

2.2 Discussions on the Mathematics Education at the VHMO As a result of the industrialization in the Netherlands of the end of the nineteenth century, thanks to its practical value and its social usefulness, science enjoyed increasing prestige. After the First World War, however, a revolution in the appreciation for science and mathematics took place and these subjects consequently lost a lot of their prestige. The mathematician Dijksterhuis called this a spirit of the times hostile to the Mathesis, that influenced many ideas on the reform of the educational system [15]. After studying mathematics in Groningen E.J. Dijksterhuis (1892–1965) was for many years a mathematics teacher at the Rijks-HBS (State-HBS) in Tilburg and during his career he worked for various universities as a science historian (Fig. 2.1). Being an advocate of mathematics education organised on a logical-deductive basis he played an important role in the discussions about the mathematics education between the two World Wars. His demands for the strictness of the mathematics education were very stringent and he was deeply convinced that such a mathematics education would certainly emanate formative value. The formative value of mathematics was an argument that to some extent kept coming back in the discussions about mathematics education. People were convinced that mathematics would have a positive influence on reasoning power in general and logical reasoning in particular—the so-called ‘stimulation of the mind’ [16]. The underlying idea was that transfer was possible: being able to apply skills

2.2 Discussions on the Mathematics Education at the VHMO

17

Fig. 2.1 E.J. Dijksterhuis

in a subject field different to the one in which they are acquired. In the literature a distinction is made between horizontal transfer (between tasks of similar order) and vertical transfer (from a task of a lower order to one of a higher order) [17]. It is called a positive transfer when the first task learned facilitates learning the second one. Inherent in the discussion on the formative value is the question whether ‘learning to think’ can be taught [18]. At the end of the Nineteenth century the consensus was that mathematics education had formative value. But in the United States of America at the beginning of the Twentieth century people started to use the argument of formative value in an opposite sense [19]. Psychologists like E.L. Thorndike published research results that suggested that there was hardly any transfer, if at all. This caused a shift in emphasis on the aim of the American arithmetic and mathematics education towards practical applications of these subjects [20]. In the United States the tendency turned in the 1930s, when new studies apparently showed that mathematics education aimed at ‘understanding’, in particular, did indeed produce transfer to a certain extent. In the Netherlands similar tendencies were little noticed in the period between the two World Wars. Teachers still considered the formative value of mathematics an established fact and it was seen as an important goal of mathematics education. Pedagogues also agreed that mathematics education (and Classic languages too) served the purpose of learning to think (logically). The few critical comments, such as those of the Amsterdam University professor G. Mannoury in 1921 and later those of D. van Dantzig, did not have much influence [21]. According to Dijksterhuis mathematics education should “sharpen the intellect of the pupil and give him insight in the aspiration of the human mind that goes beyond the material from which science has arisen” [22]. The educational value of mathematics was not under discussion, but there was a lot of criticism of the overly mathematical nature of the HBS. The mathematician

18


H.J.E. Beth (1880–1952), director of the Rijks-HBS in Deventer, wrote in 1924 in the Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde: De uitdrukkingswijze “meer en meer wiskundig karakter der HBS” is reeds zoo ingeburgerd […], dat menigeen, buiten het wiskundig kamp opgesteld (zelfs vele wiskundige collega’s zijn het praatje gaan gelooven), zich met verbazing zal afvragen, of het dan niet waar is. Welnu, op deze vraag kunnen we antwoorden met één woord: nonsens! (The expression the “increasingly mathematical character of the HBS” has already been established to such an extent […] that many a person outside the mathematical field (even many colleague mathematicians started to believe the rumour) will wonder whether it is true or not. Well, there is only one answer to this: nonsense!) [23]

According to Beth the mathematical character of the HBS had even been “dubiously decreasing” for a couple of years. Compared to 25 years before there were indeed rather fewer hours reserved for mathematics [24]. The reduced appreciation for mathematics at the HBS mainly resulted from the disappointment of the many drop-outs for whom mathematics would have been a stumbling block. Beth also mentioned much-heard arguments such as “mathematics is of no use to me” and “not everybody is made for mathematics” [25]. However, in addition to these voices a plea was also to be heard in favour of a further modernization of mathematics at the HBS. In the years between the two World Wars there were three important discussions in the field of mathematics education at the HBS. First, there was the discussion about the organisation of the initial geometry education. In addition there were attempts to modernize mathematics education at the HBS, centred around the curriculum design of the Beth committee and the discussion about the introduction of differential calculus and integral calculus. Finally there was the controversy around the subject of mechanics: where did that belong and the teacher of which discipline should teach it? These three discussions revealed the thoughts and motives of the various advocates and opponents and provided a good picture of the prevailing views regarding how and why mathematics is taught [26].

2.2.1 The Initial Geometry Education and the Foundation of the Journal Euclides In 1924 mathematician Tatiana Ehrenfest-Afanassjewa (1876–1964) wrote the brochure Wat kan en moet het Meetkunde-onderwijs aan een niet-wiskundige geven? ( What can and should geometry education offer a non-mathematician? [27]) Mrs. Ehrenfest was originally from Russia and completed her study in mathematics and physics in Saint Petersburg (Fig. 2.2). After that she studied in Göttingen with mathematicians Felix Klein and David Hilbert. In addition to her scientific research (in the field of theoretical physics and in other fields) she taught mathematics until 1908. She lived In Leyden for a long time from 1912 with her husband Paul Ehrenfest (1880–1933). Mrs. Ehrenfest was greatly interested in teaching and education and lived up to it by organising monthly mathematical-didactical colloquia for teachers at her house. Discussions about the, in her view fossilized, mathematics education in the Netherlands were brisk [28].


19

Fig. 2.2 T. Ehrenfest-Afanassjewa

The above-mentioned Dijksterhuis wrote a reaction to Mrs. Ehrenfest’s brochure with the title Moet het Meetkunde-onderwijs gewijzigd worden? (Should the geometry curriculum be changed?), which he sent to his publisher Noordhoff. The publisher suggested P. Wijdenes (1872–1972) to include the reaction in Nieuw Tijdschrift voor de Wiskunde (New Journal for Mathematics). Apart from being a mathematics teacher in Amsterdam, Wijdenes was also a prominent author of text books to be used in mathematics education. This eventually led to the foundation of the periodical Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde gewijd aan Onderwijsbelangen (Supplement to the New Journal of Mathematics devoted to Educational Interests), of which Wijdenes along with J.H. Schogt (also a mathematics teacher in Amsterdam) would be the editors (Fig. 2.3). Three years later the Bijvoegsel would be renamed Euclides. Tijdschrift voor de Didactiek der Exacte Vakken (Journal for the Didactics of Science and Mathematics), and still later the journal would become the bulletin of the associations of teachers, WIMECOS and LIWENAGEL. WIMECOS, the “Vereeniging van Leeraren in de Wiskunde, de Mechanica en de Cosmographie aan Hoogere Burgerscholen met vijfjarigen cursus B, Lycea en Meisjes-Hoogere Burgerscholen met 5-/6- jarigen cursus” (Association of teachers in mathematics, mechanics and cosmography at the 5-year HBS-B, Lycea and the 5 or 6 year HBS-B for girls), was founded in 1925. LIWENAGEL—“Leraren in de Wiskunde en Natuurwetenschappen aan Gymnasia en Lycea” (Teachers in Mathematics and Science at Gymnasia and Lycea)—had been established 4 years before that. All important discussions, modernizations and plans in the field of mathematics education would be brought forward in Euclides in the years to come [29]. What had Mrs. Ehrenfest written that provoked such a reaction? In her brochure she raised the question about the initial geometry education. The final aim she had in mind with this kind of education did not differ that much from Dijksterhuis’: the lead-up to an Euclidian geometry built on axioms. However, the routes they proposed for getting there immediately made it clear that they differed fundamentally in opinion.


20 Fig. 2.3 The first two volumes of Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde gewijd aan Onderwijsbelangen, later renamed Euclides. Tijdschrift voor de Didactiek der Exacte Vakken

Mrs. Ehrenfest argued in favour of preparing the pupils better for the logics of the axiomatically built geometry by first and foremost paying attention to developing their intuition and three-dimensional imagination. She wanted to realise this with an introductory course in which graphics and concrete action would take up a central position. In this course, at any rate in the initial education, the pupils would not have to prove directly the theorems that were the most evident to them. Later she would develop such an introductory course in a more practical sense in her book Uebungensammlung zu einer geometrischen Propädeuse (Exercise collection for an Introduction in Geometry). In this book she described close on 200 practical exercises to be used in the classroom. According to her the most important result a teacher could achieve if he used these exercises was: Man wird imstande sein die Schüler vor solche Probleme zu stellen, wovon der geometrische Inhalt ihnen vollkommen zugänglich sein wird, wo sie aber merken werden, dass für dessen sichere Beantwortung eine kritische Untersuchung nötig ist: sie werden den Impuls bekommen um Sätze zu beweisen! (You will be able to present the pupils with problems


21

with a geometrical content that is completely accessible to them, but, as they will notice, they will need a critical investigation to solve them: they will be incited to prove theorems!) [30]

Such an approach based on graphics was in Dijksterhuis’ view an attack on what he called … wat steeds als een der meest kostelijke vruchten van het wiskundeonderwijs heeft gegolden: op de zuiverheid en eerlijkheid van het mathematische denken en spreken, op de geestelijke tucht, orde en reinheid, die de mathesis nastreeft. (… something that has been considered to be one of the most precious fruits of mathematics education: On the pureness and honesty of mathematical thinking and speaking, on mental discipline, order and purity, that the Mathesis is striving for.) [31]

According to Dijksterhuis getting to know a world such as that of mathematics was of the utmost importance for the development of the intellectual capacities of the pupils. In his view the goal of geometry education was to teach the pupils to think and speak accurately. It was not in the least necessary to discourage spontaneous conjectures or insights, but in his view nothing in Euclidean geometry should be accepted without a strict logical proof. As a canon of geometry education Dijksterhuis quoted the German number theorist R. Dedekind (1831–1916) in Was sind und was sollen die Zahlen (1888) (What are numbers and what should they be (1888)): Was beweisbar ist, soll in der Wissenschaft nicht ohne Beweis geglaubt werden. (What can be proved, should in science not be believed without proof.) [32]

Dijksterhuis doubted very much the feasibility of the course that Mrs. Ehrenfest proposed, in particular for the pupils with little talent for mathematics (without an adequate power of imagination): Men kan juist den niet wiskundig aangelegden leerling geen sterkeren moreelen steun geven, dan wanneer men hem de overtuiging weet bij te brengen, dat alles wat hij op de H.B.S. van Wiskunde heeft te leeren (de vaak zo gevreesde Beschrijvende Meetkunde niet uitgesloten) voor hem bereikbaar is door zuiver logisch redeneeren en dat een goed voorstellingsvermogen weliswaar voor hem, die het bezit, een machtig hulpmiddel vormt, maar dat het gemis aan dat vermogen nooit een onoverkomelijk struikelblok kan zijn. (In particular the pupil without talent for mathematics cannot be given stronger moral support than when you succeed in convincing him that everything that he has to learn about Mathematics at the HBS (including the often dreaded Descriptive Geometry) can be learned by purely logical reasoning and that a good imagination for those who have it, can be a powerful aid, but for those who don’t, it can never be an insurmountable stumbling block.) [33]

In this can already be recognized the basic thought of what Dijksterhuis would present as epistemic mathematics education in an address before the Vereeniging voor Paedagogisch Onderwijs (Association for Pedagogic Education) at the Rijksuniversiteit in Groningen on 12 February 1934. In naming this mathematics education he relied on Plato’s antithesis between εμπειρια (knowledge based on experience) and επιοτημη (knowledge in which mutual coherence of those facts and the basis of that knowledge are taken into account). The basic thought in this approach was that at any given moment the pupil should be able to render account of the mutual coherence of what he has learned. An important aid in such an education would be the history of the development of a subject matter, as this would help the pupils to better understand it [34]. However, in 1924 Dijksterhuis had not yet arrived at that conclusion and he ended his line of reasoning by inviting Mrs. Ehrenfest to continue

22


to express “her very interesting thoughts about mathematics education that were also of value to her opponents”. In her reaction Mrs. Ehrenfest once again defended the need to enhance the threedimensional power of imagination by means of an introductory course to elicit the pupils’ awareness of the necessity of a good systematic geometry curriculum: De heer D. vraagt, hoe ik verlangen kan, dat de leerlingen zich de ruimtelijke figuraties althans qualitatief juist voorstellen, vóór ze geleerd hebben, daarover logisch bevredigende stellingen uit te spreken. Ik moet daar de vraag tegenover stellen: hoe kunnen ze iets anders doen, dan die stellingen napraten, vóór ze geleerd hebben, zich de dingen, waarover ze spreken, goed voor te stellen? (Mr. D. wonders how I can ask the pupils to conceive threedimensional figurations, at least qualitatively correctly, before they have learned to express logically satisfying theorems about them. I have to ask the opposite question: How can they do anything else than repeat these theorems before they have learned to properly imagine the things they are speaking about?) [35]

She ended with the hope that many teachers, who until then only used to think of the methods formerly used, would be provoked by the discussion to test out their ideal against the method. In his final reaction Dijksterhuis once again disputed the proposition that a clear representation would be necessary for understanding geometry. He wrote that it would be better to develop the pupil’s imaginative power during the systematic curriculum of geometry. To prove theorems a child found trivial did not matter and Dijksterhuis simply ignored such complaints. Een kind moet tenslotte ook leeren, zich tot gedachtenconcentratie te dwingen en het is niet altijd noodig, dat het het waarom beseft van de dingen, die men het laat doen. (After all a child should learn to force himself to concentrate his thoughts and it is not always necessary that he realises why he is supposed to do things.) [36]

This ended the discussion between Mrs. Ehrenfest and Dijksterhuis in the Bijvoegsel and a period in which the journal functioned as a medium for the various opinions and discussions on mathematics education started. Dijksterhuis, who, in any case as far as the education for the advanced pupils was concerned, represented the common opinion, had the final say in the discussion. Mathematics teacher B. Coster referring to the above-mentioned discussion wrote: … en ik geloof met één der beoordeelaars van het Tijdschrift, dat de meeste collega’s, die aan gevorderde leerlingen stereometrie hebben te onderwijzen, zich in hun overtuiging zullen plaatsen aan de zijde van den heer D. (… and I, together with one of the reviewers of the Journal, believe that most colleagues that have to teach stereometry to advanced pupils will be convinced and take sides with Mr. D.) [37]

2.2.2 The Beth Committee and the Introduction of Differential and Integral Calculus When in 1919 modernization of the content of mathematics education at the Gymnasium was accomplished, the HBS had to follow suit of course. An important issue in reforming the mathematics curriculum was whether or not to introduce differen-


23

tial calculus and integral calculus, which were considered part of ‘higher mathematics’ at secondary schools. Advocates of introduction were inspired by the German mathematician Felix Klein (1849–1925), the originator of reform in mathematics education in his country. Klein was known for the emphasis that he put on developing mathematical insight and learning to think independently in pupils, instead of the rote learning of technical skills then common. In the “Reform Bewegung” (“Reform movement”) of the early Twentieth century (also called ‘Klein’sche Reform’) that was strongly influenced by Klein, the pivotal point in the mathematics education was the fundamental role that the concept of function should get. In Germany this led to the introduction of analytical geometry and infinitesimal calculus in mathematics education in 1925, with similar movements taking place in England and France. In the Netherlands from the beginning of the Twentieth century more and more people had been in favour of the introduction of differential calculus and integral calculus, the concept of function and analytical geometry and this led in 1919 to a partial success when the curriculum of the Gymnasium was revised [38]. In 1925 inspector Bolkestein initiated the formation of a committee under chairman Beth that aimed to submit proposals for a new mathematics curriculum for the HBS. Dijksterhuis was the secretary of the committee and the other members were J. van Andel (director of the Christelijke HBS in Den Haag) and P. Cramer (teacher at the Roman Catholic HBS in Rolduc). In 1926 the committee published its first draft of “een leerplan voor het onderwijs in de wiskunde, mechanica en kosmographie op de H.B. Scholen met vijfjarigen cursus” (“curriculum for the education in mathematics, mechanics and cosmography at the HBS with a five-year course”) in the Bijvoegsel, followed by a draft of the accompanying programme for the final examination. In an introduction the committee firstly explained its points of departure. It considered the aim of the HBS mathematics education not to be primarily transfer of knowledge but rather ensuring … dat ze [de HBS] hun geest in voldoende mate vormt en ontwikkelt, om hen tot het verwerven der kennis, die ieder op zijn eigen terrein zal behoeven, in staat te stellen. De Commissie meent zich ontslagen te mogen achten van de taak, te betoogen, dat, bij huldiging van deze opvatting, een intensief onderwijs in wiskunde en aanverwante vakken voor alle leerlingen onmisbaar is. (… that they [the HBS] cultivate and develop their minds sufficiently so as to enable them to acquire the knowledge that each of them will need in his own field. The Committee holds the opinion that they, if this idea of an intensive education in mathematics and similar subjects is indispensable for all pupils is accepted, may consider themselves dismissed from the task to assert that opinion.) [39]

The committee apparently considered the formative value and the contribution to mental training which mathematics education would bring, of greater importance than the practical use that knowledge of mathematics could have. In this notion Dijksterhuis’ conviction as it appeared in the discussion with Mrs. Ehrenfest could be recognized. In its description of the curriculum the committee did not strictly prescribe the teaching methods, but explicitly gave the teachers a free hand. However, they advised that science and mathematics were to be taught by one and the same teacher

24


to ascertain that the cohesion between the science subjects would be expressed better. One of the most important reforms of the curriculum was the introduction of differential calculus and integral calculus already mentioned. The most important argument of the committee was … het streven, om het wiskundeonderwijs dienstbaar te maken aan de ontwikkeling van het functioneele denken en daarbij gebruik te maken van het hulpmiddel der graphische voorstelling? (… striving to make the mathematics education instrumental in the development of functional thinking and use graphical representation as a tool.) [40]

Another argument was the indispensability of differential calculus and integral calculus for mechanics education (where it was already treated in some veiled manner) and the wish to contribute by way of mathematics education at the HBS (and in particular by teaching infinitesimal calculus) to “the dissemination of better ideas on the essence of the Mathesis and the key role it has played and still plays in the mental development of mankind” [41]. In this way infinitesimal calculus would become the same common intellectual property as the fundamentals of the Euclidian geometry had been for many decades [42]. Its introduction would also considerably facilitate the transition from secondary education to higher education for new mathematics students (but also for new chemistry, biology and medicine students). As could have been expected, the publication of the committee’s proposals evoked various teachers’ reactions in the next issues of the Bijvoegsel. These reactions were for the greater part positive. There was, in particular, much appreciation for the thoroughness with which the committee put the out-of-date mathematics education in order and rooted out many deeply held convictions. It was remarkable that most people accepted the principles as described by the Beth committee, even though not everyone agreed with the details. Among these reactions is one of David van Dantzig (1900–1959), remarkable because he raised a question that was rather unusual at the time: how to organise mathematics education so it can be of value for society? This approach, not so popular at the time, would gain more influence after the Second World War. Van Dantzig had studied mathematics in Amsterdam after taking the KI and Kv certificates to teach at secondary schools. He was a mathematician with a broad interest, which reached from topological algebra to statistics and he foresaw for mathematics and mathematical thinking an important role to be played in society. After the war he would give concrete shape to his wish to make mathematics instrumental in society by actively taking part in the foundation of the Mathematical Centre (MC) [43]. At the time of the Beth committee Van Dantzig was practically on his own in the world of mathematics education with his conviction to deny mathematics education formative value and his wish to include social and societal aspects in the discussion about mathematics education [44]. There were also other less favourable reactions to the proposal of the Beth committee. The ‘Vereeniging van Directeuren van Hoogere Burgerscholen met 5- jarigen cursus’ (‘Association of Directors of HBS with a five-year course’) concluded, on the basis of an otherwise very concise enquiry, that the programme would be overloaded and not fit for the average pupil. The association reported this to the


25

inspectors of the Ministry of Education. This report was also published in the Bijvoegsel [45]. In that same issue of the Bijvoegsel the committee ignored this criticism, since its members were not convinced of the inquiry’s quality and thought that a lot of the critical remarks made could be considered not valid, were the programme to be read more precisely. Furthermore, the report by the Vereeniging did not offer much to go on as far as concrete improvements in the programme were concerned and that was, after all, what the committee had explicitly asked for. At the inspector’s request WIMECOS had also made a report on the curriculum and that was considerably more positive and advised actually introducing it. The committee (which had written a response in answer to all the comments in the Bijvoegsel) incorporated the criticism and advice and in 1927 published a somewhat altered curriculum in which, integral calculus, among other things, was to be treated less elaborately. But even this adapted curriculum did not result in the official modernization of mathematics education. The only result was that the textbooks had been improved [46]. In 1931 the members of WIMECOS voted unanimously in favour of introducing the new curriculum, but it took until 1937, when inspector Jensema was succeeded by Van Andel, before the mathematics curriculum at the HBS was revised and differential calculus and integral calculus were introduced. However, the corresponding requirements of the final examination were not adapted and thus the introduction was only keeping up appearances. In the textbook Nieuwe schoolalgebra ( New algebra for school) written by Wijdenes and Beth differential calculus and integral calculus were, for instance, introduced at the end of the book after the chapter with ‘Vraagstukken van het staatsexamen’ (‘Problems of the final state examination’) [47]. Due to the outbreak of the Second World War it took until the 1950s before new developments in this field could be reported.

2.2.3 The Controversy About Mechanics Parallel to the discussion about the introduction of the Beth curriculum there was another discussion about the future of mechanics as a school subject at the HBS. This subject was considered to be part of mathematics and in keeping with that view it was taught as axiom-based theoretical mechanics. Starting from the early 1920s more and more people from the physics side wanted to involve mechanics in physics education in order to better express the experimental character of this branch of science. In 1926 the Bestuur van de Nederlandsche Natuurkundige Vereeniging (Executive Committee of the Netherlands Physical Society) set up a committee chaired by physicist A.D. Fokker (1887–1972) with the aim of examining the physics curricula at the HBS and Gymnasium. After consulting the Beth committee (who in their report originally argued in favour of maintaining mechanics as an independent subject) a compromise was reached; the classes were to be divided between physicists and mathematicians. In that way the inductive-experimental base of mechanics could be part of physics education and the mathematical presentation and the theoretical foundation could be postponed until the higher grades [48].

26


However, even before the consequences of this compromise could be discussed, a proposal—on which neither the teachers, nor the Beth committee were consulted—to completely incorporate mechanics as a part of physics (supported by physicists from Utrecht among whom L.S. Ornstein (1880–1941) and M.C.J. Minnaert (1893–1970) was accepted at a meeting of the Natuurkundige Vereeniging. On 11 July 1928 the minister of Onderwijs, Kunsten en Wetenschappen received a plea to discontinue mechanics as an independent subject. In 1928 Euclides devoted a special issue to the resulting consternation and discussion. Beth started the discussion with a dispute in which he once again explained the motives of the Beth Committee and the Fokker Committee. He expressed his feelings about the decision taken at the meeting of the Natuurkundige Vereeniging as follows: Met groote teleurstelling heb ik dan ook kennis genomen van het besluit van de op 19 Mei te Amsterdam door de Nederlandsche Natuurkundige Vereeniging gehouden vergadering, waarin de doodstraf voor de rationeele mechanica wordt geëischt. (I was therefore deeply disappointed when I heard about the decision taken at the meeting of the Nederlandsche Natuurkundige Vereeniging of 19th May in Amsterdam in which the death sentence for rational mechanics has been demanded.) [49]

Furthermore he argued in favour of introductory inductively-empirically organised education but this was certainly not to be the final goal. He warned against overestimating the value of the experiment in mechanics. According to Beth the organisation of mechanics education was not so much a didactical, but rather a social issue, and in addressing that the decisive factor should be the expectation people had of the HBS. Wil de HBS echter blijven, wat zij m.i. steeds heeft trachten te zijn, dan moet zij de grondslagen van de vakken, die zij onderwijst, met de bereikbare wetenschappelijke nauwgezetheid behandelen, en de bovenste grens van wat zij bereiken kan, zal bepaald worden door het hoogste niveau, waarop zij, van den grond af degelijk opbouwend, komen kan; zij mag geen genoegen nemen met het oppervlakkige succes, dat de belangstelling slechts prikkelt en niet verzadigt. (If the HBS wants to continue to be what it, according to me, always tried to be, then it has to treat the fundamentals of the subjects to be taught with the best attainable scientific accuracy, and the upper limit of what can be attained will be determined by the highest level it can reach, starting thoroughly from the ground level; it cannot settle for superficial success that only stimulates interest and does not satisfy.) [50]

In an article in the same issue, Dijksterhuis challenged one of the most important arguments that the Natuurkundige Vereeniging had for incorporating mechanics into physics: the historical argument that mechanics in education too, should be what it always and everywhere had been, that is a part of physics, and therefore its character should be experimental. By means of historical examples Dijksterhuis showed that experiments played a very limited role in the historical development of mechanics. If there were experiments, they were performed afterwards “as an illustration, to confirm, to check the truth that had been acquired by rationalizing” [51]. In the same issue of Euclides Schogt argued that the support that mechanics as a school subject offered mathematics education was vital. A subject such as mechanics, with its relation to mathematics, was an excellent means for the pupils to practice the characteristically mathematically correct language and to acquire a critical


27

attitude of mind [52]. This clearly illustrates the fundamental idea of mathematics education as formulated by the Beth committee. To conclude, Van Dantzig’s article pointed out that in the mechanics discussion a distinction should be made between experiments in physics as an essential foundation of a theory and the role experiments played in schools: “an aid to illustrate education” [53]. So the proposal of the physicists to do more experiments in teaching mechanics with the goal of aligning the subject with the scientific study of physics was not justified. There was no fundamental difference between physics and mathematics, since both started from the idea of acquiring richer experiences by formalizing or mathematizing language: Samenvattende: niet de intuitief [sic] geordende, maar de in scherp omgrensde woorden uitdrukbare en in een gemathematiseerd taalsysteem samengevatte ervaring is inhaerent aan de physica. (In summary: not the intuitively ordered experience, but rather the experience that can be expressed in sharply limited words and can be summarized in a mathematized language system, is inherent to physics.) [54]

A year later, the concern and consternation illustrated by the reactions described above had already been superseded: a Koninklijk Besluit (Royal Decree) of 8th June 1929 even made the independent subject mechanics a compulsory subject of the final examination [55]. In 1930 Dijksterhuis once again returned to the subject, when the Nieuw Leerboek der Natuurkunde ( New Textbook for Physics) written by W. Reindersma and T. van Lohuizen, both members of the Fokker committee, was published. This book treated mechanics on an experimental basis. Dijksterhuis admitted, after reading it, that he now had a better appreciation of the value of experiments as an aid in mechanics education, but he still couldn’t work up the slightest appreciation for what, in his view, was the inexact and careless manner of presenting mechanics education in this book [56]. It was not the experimental method itself that caused his opposition, he argued, but the manner in which it was brought into practice by its supporters. He ended his article with: Het laatste woord over de mechanica als leervak op de middelbare school is blijkbaar nog niet gesproken. (It appears that we haven’t heard the last word on mechanics as a subject at secondary schools yet.) [57]

There was still a long way to go before agreement could be reached on the desired approach to mechanics education. It was not until 1962 that mechanics was abolished as a separate subject at secondary schools to be incorporated as an experiment-based discipline in physics [58].

2.2.4 Educating the Mathematics Teacher There was no specific education for teachers in the Netherlands until well into the Twentieth century. The prevailing idea was that things one learned well, one could also teach without a problem. So there was little time and attention for a didacticpedagogic component in the education of those who were to become teachers. However, since 1828 there had been compulsory pedagogic education at the university

28


for prospective teachers consisting of lectures on the general theory of education, the methods used in teaching disciplines and practical exercises in the art of teaching. But nothing was actually put in place, so that in practice the arrangement had hardly any effect. From 1863 onwards, when the MO-law became effective, engineers also qualified as teachers in secondary schools in those subjects in which they had been taught themselves (this also included mathematics). In this law a legal distinction was made between the various secondary-school teaching certificates and the holders of these certificates were required to follow a pedagogic-didactic preparation (among other things they needed to know about the various study methods, their advantages and disadvantages and their background). But once again, in practice there was nothing beyond these fine words. A real education for teachers was never established. A consequence of this lack of pedagogic-didactic education for teachers was that Dutch teachers were not particularly “education-minded” for a long time, as Wansink stated [59]. Gradually, this situation changed and the awareness grew that a more pedagogicdidactic schooling of teachers was necessary. In 1918 Casimir was appointed professor at a special chair in Educational Sciences in Leyden and a year later he was followed by Ph.A. Kohnstamm in Amsterdam. In 1921 after a revision of the higher education act it was finally possible to do academic pedagogy exams [60]. In 1926 Dijksterhuis, on behalf of the Beth committee, published an article about the training of mathematics teachers in the Bijvoegsel [61]. He argued that universities should indeed prepare students to become teachers (in academic circles there was a fair amount of opposition to this), since there was a scientific interest in performing that job well. This should, of course, be performed without the risk of the university losing its scientific character. To facilitate the transfer of the ‘higher mathematics’ at the university to the ‘lower mathematics’ (which the teacher to be was expected to teach), Dijksterhuis in the first place argued that more attention should be given to the ‘lower mathematics seen from a higher point of view’, a phrasing that clearly revealed Klein’s influence. In addition, a historic perspective of the development of mathematics and physics was in his view indispensable for prospective teachers. It would not only be useful to pass on this historical knowledge to their pupils, but would also help prospective teachers to better imagine the problems their pupils might have (because of their knowledge of the problems that people had been struggling with historically). This avoided what Dijksterhuis called a methodical error: “… because not understanding that you are not understood is an error” [62]. To conclude, Dijksterhuis mentions the lack of “philosophical, in particular epistemological information on the essence and the methods in mathematics and physics”. This information would give a teacher an indispensable insight into the fundamentals of these disciplines. Above all, Dijksterhuis stressed once again, that one should be careful that preparing for a job as teacher would not be at the expense of a lowering of the scientific level of the university education to become a teacher: Men kan zoo volhouden, dat men (…) geen Euclidische meetkunde kan doceeren, zonder de niet-Euclidische te kennen, geen drie-dimensionale, als men niet in de vierdimensionale


29

eenigermate thuis is. (Thus one can insist, that one (…) cannot teach Euclidian geometry without knowing the non-Euclidian geometry, nor three-dimensional geometry without to some extent knowing about the four-dimensional geometry.) [63]

However, Dijksterhuis did not say a word about a possible pedagogic-didactic component and the advisability of gaining practical experience was only mentioned casually. In a later article on the non-academic schooling to become a teacher he did not mention that aspect either [64]. In 1926 a committee chaired by D.P.A. Verrijp (mathematics teacher at the Gymnasium in Arnhem) was appointed to investigate the education for teachers in mathematics and similar disciplines. Dijksterhuis was appointed secretary of this committee and the content of the above-mentioned article was the committee’s point of departure. The response of the various universities to the report of the Beth committee induced this committee to bring forward a proposal for a separate examination for prospective mathematics teachers (in addition to that to obtain the master’s degree in mathematics) that would confer teaching qualification on them [65]. It was striking that the only protest response came from the faculty of mathematics and physics of the University of Amsterdam who objected to the emphasis the committee put on making the education ‘stricter’ in order to improve it. According to the authors this demonstrated …van een volkomen miskenning van de psychologische en paedagogische gezichtspunten (…), die bij de ontwikkeling van het wiskundig denken in het oog gehouden dienen te worden. (…a complete denial of the psychological and pedagogical points of view (…), that should be kept in mind when developing mathematical thinking.) [66]

In addition, this faculty put emphasis on the advisability of students participating in the teachers education programme to gain practical educational experience. In its final recommendations the Verrijp committee once again stressed the importance of a knowledge of lower mathematics, the epistemological development of mathematics, the history of mathematics and the benefit of methodological and didactical schooling. A few years later, at the request of the Verrijp committee and others, a committee chaired by professor B. Sijmons from Groningen was appointed to investigate the education of teachers in general. In 1929 the Sijmons committee published its report [67]. Dijksterhuis, as one of the representatives of the Algemeene Vereeniging van Leeraren bij het Middelbaar Onderwijs (General Association of Secondary-School Teachers) was also involved. When the report was published he wrote in Euclides: Het is een bewijs, hoe onder de leeraren bij het Gymnasiaal en Middelbaar Onderwijs, die in groote meerderheid jaren lang met volstrekte onverschilligheid tegenover het vraagstuk van de opleiding hebben gestaan, het besef begint te groeien, dat het belang van het onderwijs eischt, dat volgende generaties van docenten niet even onvoorbereid hun moeilijke en verantwoordelijke betrekking zullen aanvaarden, als zij dat indertijd zelf hebben moeten doen. (It proves that, among teachers of Gymnasium and Secondary Schools, who for the greater part have been completely indifferent towards the question of schooling, there is a growing awareness that it is in the interest of education that the next generations of teachers should not have to accept their difficult and responsible tasks equally unprepared as they themselves had to do at the time.) [68]

30


In its report, the Sijmons committee proposed resorting to splitting the teachers’ education in a pedagogical and a didactical part. The pedagogic part would consist of general pedagogy classes, given by a pedagogy professor, and would theoretically prepare the students for the teaching profession. The didactical part, that would prepare students for the practice of teaching, should be part of these professors’ lectures and newly appointed university lecturers (who should also have a teaching job at a Gymnasium or an HBS). This didactical education (leaving enough scope for the history of mathematics!) should be considered a necessary subsidiary/minor for the master’s examination. The last stage of the didactical schooling was to gain practical experience in teaching. To conclude, Dijksterhuis mentioned briefly the other forms of training for the teaching profession; he proposed bringing the requirements for secondary-school certificates and the training for engineers to the same level as the new plans for a university schooling for teachers that was to be implemented [69]. However, the implementation of these plans would not start for several decades, though in 1939 the State did appoint lecturers in didactics of all secondary-school disciplines at all universities. In the end it would take until 1952 before requirements were set by Royal Decree for the teaching qualification of Masters of Science [70].

2.2.5 New Insights and the Wiskunde Werkgroep (Mathematics Working Group) All in all it can be concluded that the formative value of mathematics played a central role in the discussions about the mathematics education in the period between the two World Wars. To express the formative value of mathematics as well as possible, advocates of the strict logical-deductive approach such as Dijksterhuis preferred to present mathematics in such a way that no, or hardly any, concessions were made to strictness or exact formulations. Most mathematics teachers, however, had a more moderate traditional attitude, but broadly speaking could agree to the logical-deductive approach—be it less strictly. The work of the Beth, Fokker and several other committees about mathematics education have, despite the fact that no concrete results were achieved, contributed significantly to establishing self-confidence and self-esteem, qualities that the mathematics teachers of the early 1920s apparently missed. In addition to the logical-deductive approach as represented by Beth, Schogt and Dijksterhuis, other approaches towards mathematics education could be distinguished. Mrs. Ehrenfest, for instance, argued in favour of a more psychological approach in which consideration was shown for the adolescent pupils. And the Lycea, more inclined to modernization, took the initiative to reform mathematics education. W. Reindersma (mathematics and physics teacher at the Nederlandsch Lyceum in The Hague) for instance, supported a less formal approach in mathematics and physics education [71]. This was in real contrast with the above-mentioned


31

members of the Beth committee as transpired from a letter Wijdenes wrote to Dijksterhuis in 1925. In it he characterized Gunning (extraordinary professor pedagogy), Reindersma, Mannoury, Casimir and Kohnstamm as opposites who “have had more than enough authority in The Hague” [72]. After his university education and his PhD in physics Philip Abraham Kohnstamm (1875–1951) shifted his attention more and more to educational and social matters [73]. He was one of the general managers of the Maatschappij tot Nut van ’t Algemeen (an association to promote well-being of individuals and society) and was appointed professor in pedagogy at the University of Amsterdam in 1919. Kohnstamm stood for ‘personality pedagogy’, in which a person’s dignity served as a starting point. He argued in favour of a more democratic school system, that allowed every citizen to frequent a school matching his personal character and would open up the intellectual culture to more sections of the population. He criticized the selective and elite character of the HBS, a school type with many drop-outs. The value of an education was, in his opinion, found in its possible use in concrete situations in life. His ideas contradicted the view of people such as for example Dijksterhuis, to whom the educational value of the HBS (and there notably that of the mathematics education) was the primary matter of importance in didactics. After the Second World War many of Kohnstamm’s ideas were realised in the on-going modernization of the Dutch educational system [74]. There were for example more people arguing in favour of a mathematics education that would take the child and not the mathematical system as a starting point. In 1936 the Wiskunde Werkgroep (WW) (Mathematics Working Group) was established as part of the Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs (WVO) (Cooperative for Modernization of Upbringing and Education). This Cooperative (led by educational reformer Kees Boeke (1884–1966) was officially appointed in that year as the Dutch department of an international organisation “New Education Fellowship” (NEF), an organisation for ‘Nieuwe Opvoeding’ (‘New Education’) established by several pedagogues in 1929 [75]. This organisation aimed at promoting worldwide pedagogical and educational improvements and reforms in which an important motto was ‘helping every child to become what it is’. The WW was primarily occupied with organising periodical meetings during which questions related to didactics and methods of mathematics education were discussed. These meetings were held at the home of Mrs. Ehrenfest and thus were more or less a continuation of her didactical colloquia. In the period before the Second World War the members of the WW concentrated mainly on geometry education and the problems of the transition from primary to secondary education. In 1938, after consultation with Kohnstamm, a committee of WW members and some employees of the Nutsseminarium voor Pedagogiek in Amsterdam was established and tackled these connection problems. It was a mixed crowd of people: among others the committee consisted of Mrs. Ehrenfest, E.W. Beth and Dijksterhuis. In addition, smaller study groups were composed to tackle various sub-problems. The WW was special in the sense that both teachers and non-teachers could discuss problems on equal terms; everyone interested in the improvement of


32

mathematics education (in the spirit of the WVO and NEF) could become a member of the working group. As opposed to the more structured associations of teachers the atmosphere in the WW in which educational reformers such as Mr. and Mrs. Ehrenfest and Kohnstamm brought up their modernizing ideas, was informal. There were, via Mrs. Ehrenfest, contacts with Casimir and Reindersma of the Nederlandsch Lyceum. Thus a cooperation between both professors, and academic lecturers, representatives of the Nutsseminarium, novice and experienced teachers came into being in the WW. In 1953 Wansink concluded an article about the WW (that appeared on the occasion of the realisation of the concept curriculum for mathematics education at Gymnasium-β and HBS-B) as follows: De wiskundewerkgroep van de W.V.O. (Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs) is een groep van theoretiserende en practiserende, enthousiaste didactici, die het m.i. verdienen, dat er van hun werk enige nota wordt genomen. (The Wiskunde Werkgroep of the WVO (Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs) is a group of theorizing and practising, enthusiastic didacticians whose work deserves to be noticed in some way, in my view.) [76]

In the years after the Second World War the WW would take the lead from the pre-war innovative movement and thus would play lead violin in the on-going discussions about the reform of the mathematics education in the Netherlands. It was in this post-war working group that Freudenthal became engaged in mathematics education for the first time and made a name for himself in this field. At the summer conference of the WVO in 1945 he gave his first lecture on education and pedagogy. How coincidental were the mathematician Freudental’s choices for the WW and the WVO? How did they result from Freudenthal’s conviction and interests in that period? What was the role of his family—and notably that of his wife? Who was Hans Freudenthal?

References 1. Hans Freudenthal in Schrijf dat op, Hans, 339. 2. Freudenthal to Darlehenkasse des Deutschen Studentenwerks E.V., 30 May 1931; Rijksarchief in Noord-Holland, Papers of Hans Freudenthal (1905–1990), mathematician, 1906– 1990 (from now on: RANH, Hans Freudenthal Papers), inv.nrs. 21, 141. 3. More about this literature study can be found in chapter 4. 4. In composing the following sections data have been derived from the following studies: A. Bartels, Een eeuw middelbaar onderwijs 1863–1963 (Groningen 1963); P. Th.F.M. Boekholt and E.P. de Booy, Geschiedenis van de school in Nederland vanaf de middeleeuwen tot aan de huidige tijd (Assen/Maastricht 1987); N.L. Dodde, Het Nederlandse onderwijs verandert (Muiderberg 1983); De Moor, Van vormleer naar realistische meetkunde; Smid, Een onbekookte nieuwigheid?; Joh. H. Wansink, Didactische oriëntatie voor wiskundeleraren. Delen i, ii, iii (Groningen 1971, second edition). 5. Ph.J. Idenburg, J. Karsemeijer en H.H. Janssen, Een eeuw middelbaar onderwijs herdacht (Groningen 1963) 5–6. 6. Ibid., 6. 7. Ph.J. Idenburg, Schets van het Nederlandse schoolwezen (Groningen 1964) 40–41. 8. More about Pierre van Hiele and his mathematical-didactical work can be found in chapter 7.

References

33

9. Pierre van Hiele, ‘De illusie van het streng redeneren’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 71–82, q.v. 72. 10. E. Jensema, Het middelbaar onderwijs en de critiek, die daarop uitgeoefend wordt (Groningen 1927) 6–7; see also section 2.1.3. 11. Smid, Een onbekookte nieuwigheid? 41, 143. 12. N. Blom and F. Meester, ‘Hanneke koos toch wiskunde’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 281–296, q.v. 286. 13. See also: Idenburg, Schets van het Nederlandse schoolwezen 372; Wansink, Didactische oriëntatie voor wiskundeleraren. Deel i 80. 14. Gymnasium, HBS en Lyceum together constituted what was called Voorbereidend Hoger en Middelbaar Onderwijs (VHMO) (Preparatory Higher and Secondary Education). 15. Van Berkel, Dijksterhuis 96. 16. For the definition and the discussion about the formative value of mathematics in the period between the two World Wars, see also: Van Berkel, Dijksterhuis 90–93; De Moor, Van vormleer naar realistische meetkunde 313–340. 17. H.J. Klausmeier, J.K. Davis, ‘Transfer of learning’, in: R.L. Ebel et al. ed., Encyclopedia of educational research (London 1969) q.v. 1483–1493. 18. On Freudenthal and learning to think, see also sections 5.1.3 en 6.1. 19. S.S. Willoughby, ‘Mathematics’, in: R.L. Ebel et al. ed., Encyclopedia of educational research (London 1969) 766–777, q.v. 769. 20. Kohnstamm criticized these results since he thought that the experiments on which they were based were wrong; Kohnstamm, ‘Over de “vormende waarde” der leervakken’, in: Keur uit het didactisch werk van Prof.dr. Ph. Kohnstamm (Groningen 1952) 1–31, originally published in 1929- 1930. 21. Van Berkel, Dijksterhuis 92; De Moor, Van vormleer naar realistische meetkunde 318. 22. As cited in: Van Berkel, Dijksterhuis 91. 23. H.J.E. Beth, ‘Het “meer en meer wiskundig” karakter der H.B. School met 5-jarigen cursus’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924/25) 90–100, q.v. 92–93. 24. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel I 99. In 1900 in total for all classes it was 29 hours per week, in 1920 it was 26. 25. Beth, ‘Het “meer en meer wiskundig” karakter der H.B. School met 5-jarigen cursus’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924/25) 90–100, q.v. 97–100. 26. In composing the following sections data are derived from the following studies: Van Berkel, Dijksterhuis; Wansink, Didactische oriëntatie voor wiskundeleraren. Parts i, ii, iii and the volumes of Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1–3, and Euclides. Tijdschrift voor de Didactiek der Exacte Vakken, 4–20. 27. In agreement with the contemporary style of writing (and to distinguish her from her husband Paul Ehrenfest) she will from now on be referred to as ‘Mrs. Ehrenfest’. 28. D. Struik, ‘Schoolwiskunde voor de Eerste Wereldoorlog’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 37–41, q.v. 40. 29. More about the development of geometry education in this period in general and on the discussion between Mrs. Ehrenfest and Dijksterhuis in particular can be found in: De Moor, Van vormleer naar realistische meetkunde. 30. T. Ehrenfest-Afanassjewa, Uebungensammlung zu einer geometrischen Propädeuse (Den Haag 1931) 10. 31. E.J. Dijksterhuis, ‘Moet het Meetkunde-onderwijs gewijzigd worden? Opmerkingen naar aanleiding van een brochure van Mevr. Ehrenfest-Afanassjewa’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924/25) 1–26, q.v. 11. 32. Ibid., 14. 33. Ibid., 21. 34. E.J. Dijksterhuis, ‘Epistemisch wiskundeonderwijs’, Euclides 10 (1933/34) 165–213, q.v. 173. 35. T. Ehrenfest-Afanassjewa, ‘Moet het Meetkunde-onderwijs gewijzigd worden? Een antwoord aan den heer E.J. Dijksterhuis’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924/25) 47–59, q.v. 55.

34


36. E.J. Dijksterhuis, ‘Antwoord aan mevrouw Ehrenfest-Afanassjewa’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924/25) 60–68, q.v. 67. 37. B. Coster, ‘De ontwikkeling van het ruimteinzicht’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 3 (1926/27) 143–154, q.v. 143. 38. See paragraph 2.1.3. 39. H.J.E. Beth, J. van Andel, P. Cramer and E.J. Dijksterhuis, ‘Ontwerp van een leerplan voor het onderwijs in wiskunde, mechanica en kosmographie op de H.B. Scholen met vijfjarigen cursus’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 2 (1925/26) 113–139, q.v. 113. 40. Ibid., 125. 41. Ibid., 126. 42. Ibid., 128. 43. During the Second World War Van Dantzig lived in Amsterdam where he regularly met Freudenthal. More about Van Dantzig's involvement in mathematics education can be found in: H.J. Smid, ‘David van Dantzig en het onderwijs in de wiskunde’, in: Gerard Alberts and Hendrik Blauwendraat, red., Uitbeelden in wiskunde. Proceedings van het symposium Van Dantzig 2000, gehouden op 22 september 2000 ter gelegenheid van de 100e geboortedag van David van Dantzig op 23 september 2000 (Amsterdam 2000); Gerard Alberts, Twee geesten van de wiskunde. Biografie van David van Dantzig (Amsterdam 2000). See chapter 5 for the foundation of the Mathematisch Centrum (Mathematical Centre). 44. D. van Dantzig, ‘Over de maatschappelijke waarde van onderwijs in wiskunde’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 3 (1926/27) 186–196. 45. ‘Verslag, door de Vereeniging van Directeuren van Hoogere Burgerscholen met 5-jarigen cursus, in zake het rapport der Commissie-Beth, uitgebracht aan den Inspecteur E. Jensema’, Bijvoegsel 3 (1926/27) 69–72. 46. E.J. Dijksterhuis, ‘Epistemisch wiskundeonderwijs’, Euclides 10 (1933/34) 165–213, q.v. 169. 47. H.J. Smid also published about this: H.J. Smid, ‘Dien onvergetelijken stap vooruit’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 297–313; P. Wijdenes en H.J.E. Beth, Nieuwe school-algebra. Derde deel (Amsterdam 1947). 48. H.J.E. Beth, ‘Het experimenteel georiënteerde onderwijs in mechanica’, Euclides 5 (1928/29) 49–60, q.v. 50. 49. Ibid., 50–51. 50. Ibid., 59. 51. E.J. Dijksterhuis, ‘De aanslag op de mechanica’, Euclides 5 (1928/29) 61–69, q.v. 64. Dijksterhuis knew what he was talking about as he had a year prior to that moment already published in an article in Euclides in which he set forth his approach to mechanics education: E.J. Dijksterhuis, ‘De historische behandelingswijze van de axiomatia der mechanica van Newton’, Euclides 4 (1927/28) 245–255. 52. J.H. Schogt, ‘De mechanica als afzonderlijk leervak op de hoogere burgerschool’, Euclides 5 (1928/29) 70–79, q.v. 73. 53. D. van Dantzig, ‘Woord en werktuig. De strijd om het mechanica-onderwijs’, Euclides 5 (1928/29) 86- 103, q.v. 91–92. 54. Ibid., 90. J. Scheltens would dispute this later in an article. According to him there certainly was a difference: physics, contrary to mathematics, returned to the experiment. However, he counted mechanics as belonging to mathematics; J. Scheltens, ‘Natuurkunde en wiskunde’, Euclides 5 (1928/29) 268–269, q.v. 268–269. 55. H.J.E. Beth, ‘Mechanica opnieuw examenvak’, Euclides 6 (1929/30) 44–48. 56. E.J. Dijksterhuis, ‘Opmerkingen over het onderwijs in mechanica als onderdeel der physica’, Euclides 7 (1930/31) 76–94, q.v. 78. 57. Ibid., 94. 58. See also paragraph 6.2. 59. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel I. 60. W.J. Brandenburg, Modernisering van het wiskundeonderwijs (Groningen 1968) 104.

References

35

61. E.J. Dijksterhuis, ‘Beschouwingen over de universitaire opleiding tot leeraar in wisen natuurkunde’, Bijvoegsel 2 (1925/26) 81–95. 62. Ibid., 88. 63. Ibid., 94. 64. E.J. Dijksterhuis, ‘Nadere beschouwingen over de universitaire opleiding tot leeraar in wisen natuurkundige vakken’, Bijvoegsel 2 (1925/26) 146–157. 65. ‘De universitaire opleiding tot leeraar in wiskunde en aanverwante vakken’, Euclides 4 (1927/28) 41-57. 66. Ibid., 45–46. 67. W.J. Brandenburg, Modernisering van het wiskundeonderwijs (Groningen 1968) 104. 68. E.J. Dijksterhuis, ‘De opleiding tot leeraar in wisen natuurkunde volgens de plannen van de commissie- Sijmons’, Euclides 6 (1929/30) 185–203, q.v. 185. 69. Ibid., 203. 70. W.J. Brandenburg, Modernisering van het wiskundeonderwijs (Groningen 1968) 104. 71. E. de Moor, ‘Het ‘gelijk’ van Tatiana Ehrenfest-Afanassjewa’, Nieuwe Wiskrant 12 (1993) 15–24, q.v. 16. More about Reindersma in: De Moor, Van vormleer naar realistische meetkunde 245–250. 72. Van Berkel, Dijksterhuis 134–135. 73. On Kohnstamm's development from physicist to democrat see Klomp, De relativiteitstheorie in Nederland. 74. Ibid., 198. 75. The following data are derived from: Joh. H. Wansink, ‘De wiskunde werkgroep van de W.V.O.’, Euclides 28 (1952/53) 197–205, E. de Moor, ‘Didactische pioniers’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 193–206; J.D. Imelman and W.A.J. Meijer, De nieuwe school gisteren en vandaag (Amsterdam 1986). 76. Wansink, ‘De wiskunde werkgroep van de W.V.O.’, Euclides 28 (1952/53) 197–205, q.v 204.

Chapter 3

Hans Freudenthal—A Sketch

3.1 Hans Freudenthal—An Impression “…all positive action starts with criticism” Freudenthal wrote in his ‘Rekendidaktiek’ in 1944 [1]. His whole life he would stick to this principle. He was critical both to himself: Mijn hele leven ben ik een slecht leermeester geweest. Ik was me er ook van bewust en om er het beste van te maken ben ik al vroeg over onderwijs gaan nadenken. Aan mijn leraren heb ik me niet gespiegeld, evenmin als, wat opvoeding betreft, aan mijn opvoeders. Of wel, maar dan in de zin van: het moest beter. Aan mezelf heb ik me aldoor gespiegeld. Ook: het moest beter. (My whole life I have been a bad teacher. I was aware of this and to make the best of it I started thinking about education at an early stage in my life. I did not follow my teachers, nor did I follow my educators as far as education was concerned. Or actually I did, but then, more in the sense of: it must be better. I have taken warnings from myself continually. In addition: It must be better.) [2]

… and to the work of others: Son intelligence était particulièrement aigüe et son esprit critique acéré. Mieux valait pour vous que vos travaux lui plaisent car ses remarques pouvaient être des flèches redoutables! […] Arracher les mauvaises herbes et semer […], là était son devoir [3]. (His intelligence was particularly keen and his critical spirit biting. It would be better for you if your works please him because his remarks could be fearsome weapons! […] Pull out the weeds and sow […], that he deemed his duty.) [3]

…but always with the positive intention of helping people: However, this critical stance regarding knowledge did not necessarily mean that he was also critical of the person. He was, to my eyes, a very supportive and sympathetic colleague, and he saw another of his major responsibilities as being that of helping people. [4]

Freudenthal presented himself a modest man: Zelfs wat ik wiskundig heb gewrocht is, voor zover het zijn nut heeft bewezen, al in de kringloop opgenomen en van het minst onbelangrijke dat ooit aan mijn wiskundig brein is ontsproten, kent haast niemand de bron. (Even the mathematics I produced, as far as it has proven its use, has been recycled and nobody knows the source of the least unimportant things that have originated from my mathematical brain.) [5] © Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_3

37

38

3 Hans Freudenthal—A Sketch

…but a man who did not let himself be walked on: Don’t shout at me, because I can shout as loud as you can and in more languages! [6].

A man who inspired awe: Nu was Freudenthal in die tijd niet iemand waarbij je veilig ging vragen wat hij precies bedoelde. We waren eigenlijk een beetje bang voor hem en begrepen niet, dat diens scherpe kritieken helemaal niet betekenden dat hij je dom vond. Jaren later leerde ik dat hij het juist in iemand waardeerde als die met hem in debat wilde gaan, maar als jong leraar begon je daar niet aan. (At the time Freudenthal was not a man to whom you turned safely to ask what he precisely meant. We were, in fact, a little afraid of him and did not understand that his biting criticism did not mean that he considered you to be ignorant. Years later I learned that he rather appreciated it if someone wanted to debate with him, but being a young teacher you did not think of debating with him.) [7]

…with the characteristic looks of a professor, with a pipe and bow tie, whose reputation meant something: Een magische naam. Die magie werd onmiddellijk bevestigd door de eerste ontmoeting: deze professor zag er precies uit zoals een professor eruit diende te zien. Bovendien was zijn gedrag geheel in overeenstemming met zijn uiterlijk. (A magical reputation. Already when we first met that magic was confirmed: This professor looked exactly like a professor should. And what was more, his behaviour agreed completely with his looks.) [8] Fig. 3.1 Little Hans with his mother (about 1906)

3.2 Luckenwalde

39

Freudenthal was not the type of mathematician who let himself to be locked up in his proverbial ivory tower. His knowledge and interest went beyond a narrow scientific area. Er zijn wetenschappers en er zijn geleerden. Hans Freudenthal was een geleerde. Er zijn er wier werkzaamheid zich geheel afspeelt in de beslotenheid van de wetenschappelijke wereld en er zijn er wier activiteiten de grenzen hiervan ver overschrijden. Tot deze laatste categorie behoorde Hans Freudenthal. Als weinigen is hij in staat geweest een breed terrein van kennis te integreren tot een geheel met de wiskunde als dominante noot. (You have scientists and scholars. Hans Freudenthal was a scholar. You have scholars who work completely isolated in the scientific world and those whose activities reach far beyond its borders. Hans Freudenthal belonged to this last category. He was one of the very few who were able to integrate a wide field of knowledge with mathematics as the dominant aspect.) [9]

Who was this remarkable personality? Where did he come from? What happened during his life that was crucial to his development?

3.2 Luckenwalde Hans Freudenthal was born on 17 September 1905 in Luckenwalde, a small town not far from Berlin [10]. His Hebrew name was Jitschak (Isaac), but at home he was called “Hans” or “Hansing”. He was the oldest son of Joseph Freudenthal (1879– 1967) and Elsbeth Ehmann (1882–1929) (Fig. 3.1). Joseph Freudenthal was teacher of religion and cantor of the Jewish congregation of Luckenwalde. Freudenthal described his mother as the “most beautiful woman” in town, his father as the most cheerful man. Mijn moeder was sterk van lichaam en geest, slim handig, hartstochtelijk, mateloos in haar doen en laten, in haat en liefde, een diepe zee onder een bruisend oppervlak. Mijn vader was—ik vertelde het al—een vrolijkerd, iemand die elke keer dat hij de deur uit was geweest en thuiskwam, een verhaal kon vertellen, terwijl zijn zoon—ook een beetje een verteller—zijn verhalen uit de krant, uit andermans verhalen of moeizaam uit de diepte van zijn geheugen moet knippen. (My mother was strong in body and mind, smart and deft, passionate, extravagant in her doings, in hating and loving, a deep sea beneath the seething waves. My father—I already told you—was a cheerful man, someone who, every time when he had been out and returned home, could tell a story, while his son—a bit of a storyteller himself—has to extract with difficulty his stories from the papers, from other persons’ stories or from the depth of his memory.) [11]

Freudenthal’s ancestors had lived in Luckenwalde since 1740. Freudenthal spent his childhood in Luckenwalde where his only sister Lotte was born in 1909 (Fig. 3.2). “A troublemaker, a howler and a mischief to his sister who was three-and-a half years younger”, as Freudenthal described himself as a child, but: “when visiting with uncles and aunts a child who behaved exemplary” [12]. From 1911 Freudenthal attended the Vorschule (primary school) of the ReformRealgymnasium, the Friedrichsschule. By his own account he was “a good pupil and very willing to show that he was top of the class”, despite the fact that because of his violent asthma attacks he was often ill and had to stay at home [13]. Reading

40


Fig. 3.2 Hans and Lotte (around 1920)

and playing chess were two of his favourite hobbies in that time. His field of interest ranged from the theory of relativity and philosophy to film, theatre and architecture, the subjects he wrote about in the school paper of the Friedrichsschule. Almost at a daily basis Freudenthal reported about his activities in his diaries in the years from 1918 to 1923 [14]. Reading these diaries again when he wrote his autobiography, he wrote: Wat me wel verrast heeft, is dat ik me, vijftien à zestien jaar oud, volkomen doorzie, weet wie en wat ik ben, wat ik kan, waartoe ik (niet) in staat ben. Ik doorzie mezelf, dat wil zeggen precies zoals ik me nu doorzie, en ondertussen heeft mijn leven en werk voor mijzelf bevestigd dat ik mezelf toen goed heb doorzien. (What has surprised me is that I, at the age of fifteen or sixteen, perfectly see who and what I am, what I can and what I can’t do. I can read myself like a book, that means exactly like I do now, and in the meantime my life and work has confirmed that I saw it correctly at the time.) [15]

3.3 Berlin

41

In 1923 Freudenthal took his finals and subsequently decided to start studying mathematics at the University of Berlin. He had no idea whatsoever of what lay ahead of him. Geen ogenblik vatte ik het idee op dat wiskunde iets was om zelf voort te brengen, maar dit had alles te maken met de wijze waarop je met wiskunde—op school en buitenschools— werd geconfronteerd. Daarbij stond al vanaf, zeg, mijn negende levensjaar vast dat ik eens wiskunde zou gaan studeren. Maar in het stadje en ook elders kwam ik met niemand in aanraking die me had kunnen vertellen wat wiskunde nu echt was, behalve dan wat er uit de boeken op me af kwam. (Not even for a moment I had the idea that mathematics was something that you could produce yourself, but that had everything to do with the way in which you were—in and out of school—confronted with mathematics. What is more, it had been certain from the age of nine that I would some day start studying mathematics. But in town and elsewhere I met no one who could have told me what mathematics, apart from what reached me from the books, really was.) [16]

On 19 April 1923 Freudenthal officially started his study mathematics. The first semester he continued to live at his parents’ house in Luckenwalde and everyday travelled by train to Berlin. Thanks to his extra earnings as a private tutor and later as Hilfsassistent (help assistant) in mathematics and editor’s assistant with the Jahrbuch über die Fortschritte der Mathematik (Annual on the progress of mathematics) he could soon afford to rent a furnished room in Berlin [17].

3.3 Berlin Freudenthal immensely enjoyed his student years in Berlin, a city he already knew very well of the many times he went there on cultural trips, first with his parents and later on his own. Above all he enjoyed the atmosphere at the university, in particular at the mathematics department, that he described as “ein buntscheckige Gesellschaft […] wo jeder seiner ihm zugestandenen Narrenfreiheit frönen konnte”. (“a colourful group of people […] where everyone could live up to the fool’s freedom he was allowed”) [18]. Freudenthal made the most of the possibilities the university offered. He attended lectures with mathematicians such as Erhard Schmidt (Hilbert spaces), Richard von Mises (projective geometry), Issai Schur (abstract algebra), Ludwig Bieberbach (analytical geometry), Karl Löwner (differential calculus and integral calculus), John von Neumann (foundations and functional analysis) and Heinz Hopf (topology). He also attended lectures in physics (with Max Planck (1858–1947) and Albert Einstein (1879–1955)), history, Greek, Russian and criminal law. In 1927 he studied a summer semester in Paris, where he took lectures with Jacques Hadamard, Gaston Julia and Emile Picard. To Freudenthal Paris was an introduction to a completely new world: “glorious, but intensely old-fashioned”. Mathematics in Paris fell very much short of his expectations: De meeste hoogleraren waren een generatie ouder dan mijn Berlijnse leermeesters. Hun colleges waren traditioneel, tientallen jaren achter bij de wiskunde die ik thuis geleerd had. […] Van mijn generatie wiskundigen, die na de Tweede Wereldoorlog de toon zouden

42

3 Hans Freudenthal—A Sketch aangeven, zag ik er haast geen. (Most of the professors were a generation older than my Berlin tutors. Their lectures were very traditional, decades behind the mathematics I had learned at home. […] I hardly met anyone of my own generation that would set the pace after the Second World War.) [19]

Back in Berlin, Freudenthal attended the guest lectures of the Dutch mathematician L.E.J. Brouwer. He was very interested in intuitionism, the theory of foundation Brouwer was working on at the time. He wrote: Der Intuitionismus war, wie ich bald lernte, in Berlin das Tagesgespräch. War es Sympathie für den Aufruhr im Grundlagenstreit, oder betrachteten die Berliner den Holländer Brouwer als einer der ihren im Gegensatz zum Göttinger Hilbert? Keine Revolution, sondern ein Putsch—so hatte Hilbert den Intuitionismus verurteilt, und “Putschist” wurde der Ehrenname, den die Brouwer-Supporter annahmen… (Intuitionism was, I soon found out, the talk of the town in Berlin. Was it sympathy for the tumult in the foundational debate or did the Berliners consider Brouwer one of their own in the conflict with Hilbert who came from Göttingen? Not a revolution, but a coup—that was how Hilbert condemned intuitionism, and “Putchist” was the proud nickname Brouwer’s advocates adopted…) [20]

Bertus Brouwer (1881–1966) was from 1912 onwards (extraordinary) professor in Amsterdam. A great mathematician because of his topological work he was a man of distinction, but first and foremost he was famous as the founding father of intuitionism. The foundational debate Freudenthal referred to was the one between the prominent mathematician David Hilbert (1862–1943) of the University of Göttingen and Brouwer, who supported a different philosophy [21]. Brouwer’s lectures in Berlin were the first time that Freudenthal as a student, who prepared well for these meetings, met him: Durch Lektüre und Arbeitszirkel auf die Vorlesungen wohl vorbereitet, habe ich damals— schriftlich einzureichende—Diskussionsfragen gestellt, denen ich Separate und meine spätere Assistentur in Amsterdam verdanken sollte. (Well prepared for the lectures by reading literature and attending study groups, I had at the time asked questions during the discussion—to be handed in in written form—and to those I owe perhaps offprints and my later assistantship in Amsterdam.) [22]

In the years following these lectures Brouwer regularly sent his mathematical texts to Freudenthal so that he could study them. In 1929 Freudenthal wrote his mathematical thesis about the ends of topological spaces en topological groups, which he finished in 1930. After an oral exam on 20 February 1930 he received the title of ‘doctor philosophiae’ [23]. The thesis was approved by Hopf and Bieberbach and published in 1931 with the title ‘Über die Enden topologischer Räume und Gruppen’ (About the ends of topological spaces and groups) [24]. Later Hopf, among others, would extend the theory as described by Freudenthal. After that, on 3 August 1930, Brouwer asks Freudenthal whether he fancied the idea to: … zum naechsten Winter eine (die Habilitation mit einschliessende) Assistentenstelle bei mir zu uebernehmen Ihre wesentlichste Aufgabe wuerde darin bestehen, mich bei der Herausgabe der neuen Zeitschrift “Compositio Mathematica” zu unterstuetzen Ueberdies wuerden Sie eine einstuendige Vorlesung ueber einen von Ihnen selbst zu waehlenden Gegenstand zu halten und dann und wann einem Studenten bei der Vorbereitung eines Seminarvortrags oder anderweitig bei einer Schwierigkeit zu helfen haben Ihr Gehalt wuerde 3000 bis 3500 holl. Gulden Betragen. (… come and work with me as an assistant during the next winter). Your main task would be to support me in publishing the new

3.4 Amsterdam

43

journal “Compositio Mathematica”. Apart from this you would have to give a one-hour lecture on a subject of your choice and now and then help a student preparing his seminar lecture or with any other problem. Your salary will be 3000–3500 Dutch guilders.) [25]

On 15 November 1930 Freudenthal left for Amsterdam.

3.4 Amsterdam After “Berlin bubbling with youth, culture and science” Freudenthal experienced Amsterdam as an anti-climax [26]. After living for a few months in an Amsterdam guesthouse, he stayed, on request of Brouwer—who preferred to come to Amsterdam as few times as possible—for a few months in Laren, not far from where Brouwer lived himself. In February 1931 Freudenthal met student Susanna (Suus) Lutter, who together with fellow students attended a carnival party in hotel Hamdorff in Laren. ‘The winter night that would not spring, did me far stranger to you bring’ is how he described this acquaintance 55 years later (Fig. 3.3) [27]. Susanna Johanna Catharina Lutter was born on 2 May 1908 as the youngest daughter of an Amsterdam working-class family. Contrary to common practice and against all expectations of a girl from such a social class she eventually was allowed to attend secondary school. After seven classes in primary school she attended a HBS with a 3-year course. In 1924, through the mediation of her German language teacher, she succeeded in convincing her parents to allow her to switch to the Barlaeus Gymnasium. In 1928 she took her finals and started with a study German literature and language at the University of Amsterdam. She would graduate in 1934.

Fig. 3.3 Susanna Freudenthal-Lutter (1930s)

44


Fig. 3.4 Freudenthal’s visiting card. The family lived in the Molenbeekstraat from June 1936 to May 1939

Autumn 1931 Freudenthal moved back from Laren to Amsterdam where he would continue to live, later with his wife, until after the Second World War (Fig. 3.4) [28]. On 20 July 1932 they were married. The couple had four children: Jedidja (Didi 1935); Matthijs (Thijs 1937); Thomas (Tom 1939) and Mirjam (1942). These years in Amsterdam were very productive for Freudenthal’s mathematical work. It was a period in which Dutch mathematicians conquered a prominent position in the international mathematical world. Brouwer’s role was considerable. In the period before the Second World War he created new stimuli (topology and foundations of mathematics) and ensured the recruitment of young promising mathematicians. His ideal was to create in that way ‘a second Göttingen in Amsterdam’. One of these promising young mathematicians was Freudenthal [29]. On 28 may 1931, when he was appointed university lecturer without salary increase at the University of Amsterdam, Freudenthal gave an inaugural lecture entitled: ‘Qualität und Quantität in der Mathematik’ (Quality and Quantity in Mathematics.). He lectured about the interaction between analysis, geometry and combinatorics [30]. Until 1936, when he left for the United States, the Pole Witold Hurewicz, also one of Brouwer’s assistents, was Freudenthal’s immediate colleague. According to Freudenthal it was, among other things, because of Hurewicz’s influence that he did not “end up in his grave with all his scientific knowledge” in those years [31]. They both worked on a fundamental theory of homotopy groups. Freudenthal worked on subjects from topology, topological groups, foundations of mathematics and intuitionism, almost periodic functions and functional analysis [32]. This resulted in some tens of papers (written in German). A striking feature of Freudenthal’s work was the emphasis he put on the connection and interaction between the various disciplines such as topology, algebra and analysis. In his later work too he contributed much to the connection between the disciplines. Van der Blij wrote in 1975: “The geometrician Freudenthal taught analysis in Amsterdam and algebra in Utrecht” [33]. From the start of his appointment Freudenthal criticized the outdated subject matter and educational methods of the mathematics education at the university. In their caput-lecture he and Hurewicz introduced the most modern mathematical

3.4 Amsterdam

45

subjects, outside the scope of the standard subject matter (subjects such as elementary theory of fields, group theory, measure theory, set theory, theory of complex functions, topology, elementary mathematics seen from a higher point of view and theory of linear operators). They also introduced the phenomenon exercise class-, during which students made exercises to practice with the subject matter. On 3 September 1937 Freudenthal was appointed conservator of the Mathematical Institute with the promise, not in writing, that he would be appointed lecturer with proper salary as soon as the financial situation would allow it. He started giving the compulsory lectures in analysis in a more exact way than was hitherto customary: De nieuwe leerstof die ik invoerde was ‘Bourbaki-avant-la-lettre’—goede stof, maar te hoog gegrepen, hoewel door de oefeningen verzacht. Nee, ‘Bourbaki-avant-la-lettre’—dat is weer overdreven. Het was hoog gegrepen, maar toch wel overwogen didactisch. Niemand die mij hierbij terzijde [sic] stond. Ik was de eerste die moderne wiskunde in Nederlandse collegezalen bracht en dan zijn fouten onvermijdelijk. (The new subject matter that I introduced was ‘Bourbaki-avant-la-lettre’—the subject matter was good, but it aimed at too high a level, even though the exercises eased it. No, ‘Bourbaki-avant-la-lettre’—that is overdoing it a bit. It aimed highl, but still considerately didactical. Nobody stood by me in this. I was first in bringing modern mathematics into the Dutch lecture-halls and then it is inevitable that you make mistakes.) [34]

His lectures were difficult to follow by less gifted students and consequently Freudenthal was not known as a good lecturer. W.T. van Est who attended his lectures in analysis, remembers Freudenthal’s didactical qualities: Als het erom gaat in te zien hoe het stuk speelt, dan heb ik aan hem altijd een geweldige leraar gehad. Als je wilde nagaan in hoeverre allerlei details netjes op elkaar aansloten en zonder feilen waren, dan bleek dat niet altijd 100 p te zijn. Hij liet impliciet veel aan de student over. Het was een opvoedende ervaring voor iedere student, om de gaten in zo’n college voor zichzelf te dichten. In dat opzicht vond ik het eigenlijk wel goed, maar niet iedereen was het daarmee eens. (As far as getting insight in the gist of a subject, he was for me a great teacher. But when you wanted to check if and how much all kinds of details were connected, then it appeared to be not 100 i correct. Implicitly he left a lot to the students themselves. It provided students with a pedagogical experience when they themselves had to fill in the gaps in such a lecture. In that sense I actually found it a good way of lecturing, but not everybody agreed to that.) [35]

Another student from that time, P.M. van Hiele, wrote about Freudenthal’s lectures in the 1930s: Ofschoon hij toen vanzelfsprekend nogal wat moeite had met de Nederlandse taal en ook niet altijd de draad kon vasthouden, waren deze kolleges [sic] een meer dan voldoende basis voor wat ik later van deze vakken moest weten. Al heel spoedig volgde er een colloquium met als onderwerp de didaktiek der wiskunde en men kan zich voorstellen, dat het er daarbij soms heel levendig toeging. (It goes without saying that he had some trouble with the Dutch language and often floundered, but still these lectures were a more than substantial base for what I later on had to know of these subjects. Very soon after that a colloquium on didactics of mathematics was held and you can imagine that the discussions were sometimes very lively.) [36]

During this didactics colloquium Freudenthal’s students had to explain academic subjects to each other. In that way he not only increased their understanding of the

46


subject, he also let them master the art of explaining. Forty years later Freudenthal would react full of surprise when he heard of the didactics colloquium Van Hiele mentions here. He had completely forgotten about this didactical action (and please note that it was his first ever!) [37]. On speaking Dutch he wrote to German friends in May 1931: Als Deutscher lernt man hollaendisch sehr schnell; obwohl ich kaum in Grammatiken geschaut habe und fuer intuitives Sprachenlernen keine Begabung besitze, spreche ich doch schon seit einem Vierteljahr nach meiner Ankunft fliessend hollaendisch [38]. (As a German you learn Dutch very fast; even though I hardly opened grammar books and do not have a natural flair for languages, I am fluent in Dutch since three months after I arrived here.) [38]

From the day in 1935 when Jedidja was born, Hans and Suus only spoke Dutch at home. It would only be after the war that Freudenthal started to publish in Dutch more regularly. In addition to Hebrew, French, Latin, English (languages he learned in secondary school), Russian and Greek (learned at the University in Berlin) Freudenthal also read texts in Slavonic and Romanic languages, and in Danish, Norwegian and Swedish. And he taught himself Akkadian (cuneiform script). In 1948 he finally would acquire the Dutch nationality. The outbreak of the Second World War implied that Freudenthal’s scientific career in Amsterdam was interrupted. Being a Jewish employee he was suspended in November 1940 [39]. Mathematician and physicist Evert Marie Bruins (1909– 1990) took over his lectures in January 1941. Through his wife Freudenthal still had access to all (also the non-public) libraries in the Netherlands (Figs. 3.5, 3.6) [40]. This enabled him to work at home. He did not devote much time to his mathematical work during these years. But he supervised his two students J. de Groot and A. van Hemert when they were writing their theses. Both would defend their thesis (in 1942 and 1943 respectively) with G. Schaake in Groningen [41]. They and many others supported Freudenthal’s family (for example by providing them with packages of provisions such as butter, bread, potatoes and onions). Freudenthal still taught mathematics: he was appointed by the Centrale Commissie voor het Joodse Onderwijs (Central Committee for Jewish Education) of Amsterdam (later it would become the Joodse Raad voor Amsterdam, afdeling Onderwijs (Jewish Council of Amsterdam, department Education)) lecturer in mathematics at the institute that prepared for the KI and Kv certificates which were required for teachers in secondary schools. This type of education was intended for Jewish students who were no longer allowed to study at the university. This implied that Freudenthal taught four hours ‘general mathematics’ (calculating areas, arc lengths and volumes) and ‘mathematical capita’ (power series and integral theory). These classes together consisted of only ten students. Despite the close contact he had with his pupils, Freudenthal was not satisfied with the results. On 3 may 1943 he wrote to the Council: Ik denk voorlopig voort te gaan op de ingeslagen weg, de behandelde stof onophoudelijk met responsies te repeteeren; dat schijnt mij de eenige manier, om onder de tegenwoordige omstandigheden den leerlingen zoo niet veel dan toch iets van blijvende waarde te leeren.

3.4 Amsterdam

47

Fig. 3.5 Carbon copy of Freudenthal’s letter to the Regeeringscommissaris (representative of the government) for Amsterdam, in which he inquires about using the library of the Mathematisch Instituut (Mathematical Institute) (12 April 1941) (For the time being I plan to continue the path I have chosen, continually testing the subject matter I have treated in tutorials; that seems to me in the present circumstances the only way to teach the students perhaps not much but at least something of lasting value.) [42]

Because of the deportations of Jews this teaching stopped after only 1 year. Freudenthal was, for the time being, sufficiently protected from deportation by his marriage to a woman who was not Jewish. During the war Freudenthal regularly helped his elder sons with reading and arithmetic. “Playing school” he called it. This aroused his interest in the didactics of arithmetic and he commenced an extensive study of all publications on this subject on which he could lay his hands. He shared this interest in didactics and pedagogy with his wife who had, since the birth of her children, been looking for a school that could live up to all her ideals. She wrote about this: “wie Odysseus trieb ich mich ziellos in unbefriedigster Suche auf dem Pfad” (“like Odyssey I wandered around unsatisfyingly looking but not finding a path”) [43]. Freudenthal’s extensive literature study eventually resulted in the unpublished document called ‘Rekendidaktiek’ (Didactics of arithmetic) (1944) [44]. In this period Freudenthal started to bury himself in Dutch history, the history of science and in particular the history of mathematics. He wrote short stories about,

48


Fig. 3.6 Answer from the Regeeringscommissaris for Amsterdam. Freudenthal is not allowed to use the library (21 May 1941)

among others, Newton, Huygens, Leibniz, Chasles, Maupertuis, Legendre, Abel, Jacobi, the Bernoullis, complex numbers and the Möbius Strip, which he later described as “historical and philosophical novellas, with a mathematical content and a

3.4 Amsterdam

49

literary form” [45]. Even more literary inspiration resulted during the war in poems, novellas, plays and novels, with which he even managed to win prizes (under the pseudonym V. Sirolf) [46]. The novels ‘De schuldenaar’ (The debtor) and ‘Viersprong der grote wegen’ (Crossing of large roads) were published after the war. In a letter from 1945 Freudenthal wrote to Hopf about writing literature: Ich setze das fort. Es ist meine eigentliche Begabung [47]. (I will continue doing that. It is my real talent [47].)

However, after the war Freudenthal was taken up by other work and for the time being he put ‘literature’ aside. But in the columns and articles that he wrote among others for De Groene Amsterdammer (1946–1971) (a Dutch opinion weekly) and NRC/Handelsblad (1974–1989) (a Dutch newspaper) he would find an outlet for his literary (and other) statements [48]. In 1984 Freudenthal received the Gouden Ganzenveer (Golden Quill, a prestigious Dutch prize, annually awarded to someone of literary importance) of the Koninklijke Nederlandse Uitgeversbond (The Royal Dutch Publishers Association) (Fig. 3.9). Despite the ‘mixed’ marriage the occupying forces did not leave the family alone. On 2 February 1942 Freudenthal was arrested because his passport did not contain the for him mandatory “J”. He spent 6 weeks in the Huis van Bewaring (the house of detention) at the Weteringschans. On 15 may 1944, when he was medically examined, he was, despite his asthma, unexpectedly declared fit and set to work at the airfield of Havelte which was under construction; there he had to do all kinds of heavy physical work [49]. On 5 September—Dolle Dinsdag (Mad Tuesday)—he managed to escape and after a lot of trouble travelled back to Amsterdam by train. Thijs and Didi (his elder sons) stayed from February 1945 until after the liberation in the countryside in Friesland. During the same time Tom stayed for a few months in North Holland. By that arrangement the family managed to survive the Hunger Winter (Dutch Famine of 1944) unharmed. Looking back at the 5 years of occupation Freudenthal wrote in a letter to Hopf on 28 July 1945 (Figs. 3.7, 3.8): Was mich selbst betrifft, so hielten wir das Risiko des Entdeckt werdens nach dem Untertauchen für größer als das Risiko von Maßnahmen gegen die Mischehen. Aber es hätte auch anders gehen können. Ich hatte jedenfalls meine falsche Identitätskarte für alle Fälle. (As far as I am concerned, we thought the risk of being discovered when in hiding was larger than the risk of measures taken against mixed marriages. But that could also have gone otherwise. Anyway, I had my forged identity card, just in case.) [50]

He was at the time still very optimistic about returning to the Amsterdam University. In a letter he wrote to his sister Lotte on 25 August 1945 (Figs. 3.7, 3.8): De universiteit begint ook weer geleidelijk. Al mijn wiskundige collega’s en mijn opvolger zijn geschorst. Ik ben voorlopig de enige. Maar officieel benoemd ben ik nog niet en het Mathematisch Instituut kan ik ook niet openen. Ik hoop dat het nu gauw in orde komt. (The university is gradually starting again. All of my mathematical colleagues and my successor are suspended. For the time being I am on my own. But I am not yet appointed officially and I cannot open the Mathematical Institute. I hope everything will soon be sorted out.) [51]

Very soon after that Freudenthal could give his lectures in the way he already started giving them before the war. In November 1945 he wrote to Hopf:

50


Fig. 3.7 Freudenthal’s forged identity card in the name of mathematical advisor ‘Johann Ferdinand Terlep’

Fig. 3.8 Freudenthal’s forged identity card in the name of mathematical advisor ‘Johann Ferdinand Terlep’ Ich darf nun, wie Sie wohl schon wissen, mit Erlaubnis meines Nachfolgers meine Kollegs geben. Wir haben dreimal so viel Studenten wie sonst. Ich halte hauptsächlich Übungen ab (und zwar so daß die Studenten Aufgaben behandeln, während ich dabei rumlaufe und nachsehe, was sie machen). Es macht den Studenten und mir Spaß, und ich hoffe, daß auch daß Übrige noch geregelt werden wird [52]. (I am now allowed by my successor, as you may already know, to give my lectures. We have three times as many students as before. I mainly give them exercises (and in such a way that the students discuss the exercises and I walk around and look what they are doing). Both I and the students are pleased with this and I hope that all the other things will soon be taken care of.) [52]

But this optimism was soon eclipsed. It was evident that Freudenthal was reappointed only as conservator (7 May 1945) and as an unpaid lecturer private teacher in Analysis, Group Theory and Topology (13 November 1945) and not in a postion

3.5 Utrecht

51

Fig. 3.9 Freudenthal receives the Gouden Ganzeveer (Golden Quill, a prestigious Dutch prize, annually awarded to someone of literary meaning) from the minister of science and education Mr. W.J. Deetman (1984)

with a salary. His replacement during the occupation, Bruins, was appointed lecturer and was allowed to continue in that role also after the war [53]. Freudenthal could not accept this situation. But the many protests of Freudenthal and a petition of his students and ex-students did not have an effect [54]. An invitation of the University of Utrecht eventually offered the way out. With a heavy heart Freudenthal moved with his family from Amsterdam to Utrecht in 1947 [55].

3.5 Utrecht By Royal Decree of 4 October 1946 Freudenthal was appointed ‘Professor of the Faculty of Mathematics and Physics at this university to teach pure and applied mathematics and the foundations of mathematics’ [56]. On 9 December 1946, at the inauguration to the office of professor of the University of Utrecht, Freudenthal delivered his lecture entitled ‘5000 years of international science’, in which he spoke of the history of mathematics with an emphasis on the pre-Greek mathematics [57]. Later, in an interview in Vrij Nederland (a Dutch opinion magazine) he said about his move to Utrecht. Ik was veel liever in Amsterdam gebleven, waar ik al mijn studenten had. Het heeft mij jaren gekost om daar overheen te komen maar ik heb hier nu mijn eigen existentie geschapen. (I would have preferred to stay in Amsterdam where all my students were. That took years to overcome but now I have created my own existence here.) [58]

52


That existence would mainly consist of setting up a mathematical institute and improving the quality of the research and education in Utrecht. Utrecht mathematician Frederik van der Blij (1923) remembers professor Freudenthal as the binding factor in the Mathematical Institute [59]. Every day, from dawn till dusk, Freudenthal was present in the institute, assisted in all his work by his secretary Mrs. Breughel. He partly credited her with his large productivity during the Utrecht years: If you wonder how anybody could travel, lecture, edit, publish so much at a time, my explanation is that for 25 years I had a secretary, Mrs. Breughel, who was unsurpassable. If I die early enough to get an obituary, her name should not be forgotten. [60]

As his students remembered he was, in Utrecht too, not a gifted teacher. His social clumsiness made him distant and difficult to approach for most students. Students affairs, however, affected him. Together with Van der Blij he kept a finger on the students’ pulses. All second-year students were invited by the two of them to discuss their progress. From the 1950s onwards all first-year students were discussed at what was known as ‘De Vergadering’, (‘The Meeting’) [61]. The way in which he, when the student revolution of the end of the 1960s came, gave up his chair in that meeting to a student was typical for Freudenthal. Apart from working at the institute Freudenthal’s Utrecht years were characterized by a growing interest in (mathematical) educational affairs in general. The seeds for this interest were, of course, already be sown during the occupation and were also nourished by his wife’s activities in the field of education and pedagogy. The fact is, Mrs. Freudenthal’s search for the ideal school for her children had introduced her to the WVO [62]. After having searched for years for a school that would live up to her ideals, she found it in 1952 in the educational philosophy of the German educator Peter Petersen (1884–1952), the founder and inventor of the Jenaplan system. She wrote: Die Odyssee war zu Ende. Die Hoffnung diese Schule einmal in den Niederlanden zu verwirklichen war geweckt [63]. (The odyssey had come to an end. The hope of one time realizing such a school in the Netherlands was raised.) [63]

The following years she would be intensively engaged in spreading and promoting this educational philiosophy [64]. In 1959 the Werkgroep Jenaplan (Working Group Jenaplan) as part of the WVO was established, followed in 1969 by the Stichting Jenaplan (Jenaplan Foundation) [65]. According to Imelman and Meijer the flourishing of the Jenaplan school in the 1980s in the Netherlands was for the greater part the result of her activities [66]: Nu staat Petersens opvoedingsfilosofie volgens ons alleen nog maar overeind dank zij de enorme werk-en werfkracht van de Nederlandse voortrekster Freudenthal-Lutter. Haar energieke initiatieven op het terrein van de onderwijsvernieuwing hebben inmiddels een bloeiende zogenaamde Jenaplanpraktijk opgeleverd. (We think that the fact that Petersen’s educational philosophy still stands is due to the enormous working power and appeal of the Dutch pioneer Freudenthal-Lutter. Her energetic initiatives in the field of educational reform have in the meantime resulted in a flourishing Jenaplan practice.) [67]

Besides this Working Group Jenaplan the WVO consisted of various other working groups. One was the WW founded in 1936 [68]. Freudenthal also got involved in the activities of the WVO. In August 1945 at a conference of the WVO he delivered

3.5 Utrecht

53

a lecture with the title ‘Opvoeding tot denken’ (Educating for thinking). In a letter to his sister dating from that same year he wrote about this): Ondertussen heb ik een vierdaagse pedagogische conferentie [van de WVO op De Steeg] meegemaakt op een kasteel hier in de buurt van Arnhem. Ik heb een lezing gehouden, met veel succes, over pedagogische vraagstukken waarover ik de laatste jaren heb nagedacht. Het was daar echt fijn. Ik was er weer vier dagen buiten. (In the meantime I have taken part in a four-day conference [organized by the WVO in De Steeg] in a castle close to Arnhem. I have, with great success, delivered a lecture on educational questions that I have been thinking about a great deal these last few years. It really was pleasant to be there. I spent four days outside there.) [69]

Starting from early 1947 Freudenthal regularly attended the WW meetings. He became chairman in February 1950. An important task that kept Freudenthal busy in the WW was ‘clearing out’ the mathematics education of the VHMO. This work would eventually lead to a change in the curriculum in 1958, in which a new mathematics curriculum along the lines of the WW became effective [70]. Van Hiele wrote about the flourishing of the WW after the war: “In brief, it was a glorious time, in which didactics was made” [71]. These were productive years for Freudenthal during which he could develop his ideas in the field of mathematics education and didactics of mathematics and test them against the feedback he received in this working group. The WW was to him an “environment of mathematicaldidactical awakening”, as he described it [72]. In (pure) mathematics Freudenthal remained active as well. In the years after the war an “excellent mathematics (sub)faculty”, in Van Est's words, had come into being in Utrecht [73]. In 1951 the Mathematisch Instituut (Mathematical Institute) in Utrecht was opened. In that time Freudenthal directed his attention in his mathematical work to the relation between geometry and topological groups (Lie groups). In his foundation day speech entitled ‘De wereld in de spiegel’ (‘The world in the mirror’) in 1964, the year that he was rector of the Utrecht University, Freudenthal would speak about the orientation of space, which was related to his research work [74]. This was only one of the many of Freudenthal’s fields of interest. A random pick from his publications produces apart from mathematical subjects from topology, statistics and logic, a large diversity of subjects, varying from history of mathematics and science, the distribution of sizes of ladies’ wear, communicating with extraterrestrial intelligence to physics and astronomy [75]. The common factor in many of his publications was always his ambition to illuminate mathematics in a broader context and to propagate its versatility and applicability. In total Freudenthal would receive five honorary doctorates for his scientific work [76]. Van Est wrote: Zeer terecht is in de bul van het eredoctoraat dat de Humboldt Universität te Berlijn hem in 1960 verleende als considerans vermeld: ‘…in Würdigung seiner hervorragenden Verdienste um die Entwicklung der Theorie der topologischen und der Lieschen Gruppen und der algebraïschen Topologie sowie in Anerkennung seiner vielseitigen Bemühungen, Probleme der modernen Kultur mathematisch zu durchdringen…’ (The degree certificate of the honorary doctorate the Humboldt Universität in Berlin bestowed on him in 1960 rightly states as consideration: ‘…to acknowledge his outstanding merit to develop theories on topological and Lie Groups and the algebra topology, just as in recognition of his wide involvement in problems in modern culture in a mathematical way…’) [77]

54


During the academic year 1960/1961 Freudenthal lived with his wife and daughter for a year in the United States, where he spent his sabbatical at Yale University (New haven, Connecticut). Back in Utrecht, he was during the academic year 1963/1964 rector of the university, a function that asked a lot of his time and energy, so that he could hardly give any attention to educational affairs or the WW. Freudenthal shifted his attention in the 1960s to widely spreading the ideas that the WW generated and he developed into a mathematics didactician with an international reputation [78]. In 1954, for example, Freudenthal had succeeded Beth as chairman of the Nederlandse Onderwijs Commissie voor Wiskunde (NOCW) (Dutch Education Committee for Mathematics), a committee appointed by the Wiskundig Genootschap (Mathematical Society) that at the same time was the Dutch sub-committee of the International Commission on Mathematical Instruction (ICMI) [79]. Four years later he became ‘Member at large’, from 1966 to 1970 he was president of the ICMI. Freudenthal, however, remained ‘l’enfant terrible’, as he described himself proudly, holding on to his own ideas (even when they were not as common and accepted as ever) and criticizing keenly everything he considered scientific or didactical ‘weed’ [80]. In 1968 he established the first international journal for mathematics education, Educational Studies in Mathematics [81]. He was also the organizer of the first International Congress on Mathematical Education (ICME) in Lyon in 1969. On 26 January 1971 in Utrecht the Instituut voor de Ontwikkeling van het Wiskunde Onderwijs (IOWO) (Institute for the Development of Mathematics Education) was established as a professional institute linked to the (governemental) Commissie Modernisering Leerplan Wiskunde (CMLW) (Committee for the Reform of the Mathematics Curriculum) [82]. Freudenthal became professor-director. IOWO employee Jan de Lange wrote about cooperating with Freudenthal at the time: Alhoewel iedereen het erover eens was dat HF de laatste jaren veel milder was geworden, blonken zijn bijdragen altijd uit door hun scherpte. Veel mensen stoorden zich daaraan, met name zijn slachtoffers. Ze realiseerden zich misschien niet altijd dat hij ook binnen de Vakgroep en het IOWO zijn kritiek op dezelfde heldere wijze uitte. (Even though everybody agreed that HF had become milder over the last few years, his contributions always stood out by their sharpness. Many people, notably his victims, were annoyed by this. Perhaps they did not always realize that he also used this same clear manner of criticizing in the Department and in the IOWO.) [83]

The official task of the IOWO was to develop mathematics curricula for pupils from 4 to 18 years of age, but apart from that, IOWO was also involved in professional development for mathematics teachers, improving educational methods and developing computer education in secondary schools [84]. This institute would prove to be a means for Freudenthal to test and develop his ideas and theories in the practice of the classroom and to further spread these ideas and theories. When he retired as professor-director in 1976, Freudenthal wrote about the way of working of the IOWO [85]: IOWO is no research institute; its members do not regard themselves as researchers but as producers of instruction, as engineers in the educational field, curriculum developers. [86]

3.5 Utrecht

55

During his IOWO-time Freudenthal wrote his magnum opus, Mathematics as an Educational Task (1973), a bulky book that contained the greater part of his didactical articles that had been published until then [87]. Five years later this was followed by Weeding and sowing. Preface to a science of mathematical education (1978), in which he sketched the path and approach that had to be taken to come to a science of mathematics education [88]. After his retirement as professor(-director) in 1976 Freudenthal remained in close contact with the work of the institute. He could be seen there almost every day. However, in 1981 the IOWO as such was discontinued to be swallowed up into the general educational structure so wanted by politics. A small group of employees continued the IOWO research in the university department Onderzoek Wiskundeonderwijs en Onderwijscomputercentrum (OW&OC) (Mathematics Education Research and Educational Computer Centre) of the faculty of Mathematics and Informatics in Utrecht, from 1991 onwards called the ‘Freudenthal Institute’. In 1983 Freudenthal wrote Didactical phenomenology of mathematical structures, one volume of which was published a year later in Dutch with the title Didactische fenomenologie van wiskundige structuren [89]. In the introduction he wrote: De didactische draagwijdte van mentale objecten en activiteiten en de aanzet van begripsvorming waar die didactisch mogelijk is, zijn het hoofdonderwerp van deze fenomenologie. Ze werd geschreven in de stimulerende sfeer van het iowo en wordt derhalve opgedragen aan de nagedachtenis van die instelling die vermoord en ter ziele is, en aan haar medewerkers die doorgaan met in haar geest te werken. (The didactical implications of mental objects and activities and the instigation to generate understanding wherever didactically possible, are the main subjects of this phenomenology. It was written in the stimulating atmosphere of the IOWO and is therefore dedicated to the memory of that institute which is murdered, dead and gone, and to its employees who continue their work in its spirit.) [90]

Early in the 1980s Freudenthal started clearing out and ordering all his papers in preparation for writing an autobiographical book. On 24 September 1986 his wife died at the age of 78. Freudenthal said in his funeral oration: Wat ben jij altijd flink geweest! Flink als voorbeeld voor mij en om me uit de put te helpen. Ik weet dat dat je laatste zorg is geweest: hoe zal Hans het zonder mij klaarspelen? Wees gerust, meisje, je hebt me geleerd je flinke jongen te zijn. (How brave you have always been! Brave as an example for me and to cheer me up when I was down. I know that this has been your last worry: how will Hans manage without me? Rest assured, my girl, you have taught me to be your brave boy.) [91]

One year later Freudenthal’s Schrijf dat op, Hans. Knipsels uit een leven, (Write that down, Hans. Cuttings from a life), a collection of anecdotes, stories and reports from his life, was published. “An eye witness account from a turbulent era” he called it [92]. Ik vertel verhalen, omdat ik hoop dat anderen ze onthouden en aan het nageslacht doorgeven—een ijdele hoop. Verhalen over mezelf, maar meestal over anderen. En dan gebeurt het telkens weer dat iemand zegt ‘Schrijf dat op, Hans!’, blijkbaar omdat hij het, net als ik, zonde vindt, dat zo’n verhaal in vergetelheid raakt. Daar doe ik nu verslag van, van wat ik gezien en gehoord heb. (I tell stories because I hope that others remember them and pass them on to their offspring—a vain hope. Stories about myself, but more often about others. And then, again and again it happens that someone says ‘Write that down, Hans!’, apparently because, just like I do, he fears that such a story will be forgotten. That is what I report about here, about what I have seen and heard.) [93]

56


Freudenthal remained actively involved in the development of mathematics education. At the end of the 1980s he even, in his words, “converted to the computer” [94]. A wonderful invention, he thought: not only was working with it “just as fast as you think or speak”, by his own account he now could write letters, he would not have written otherwise, because the only people who could read his scribbling had been retired a long time ago [95]. In his old age Freudenthal still had many ambitious plans. In Schrijf dat op, Hans he said: Laat ik geen bescheidenheid veinzen. Ik doe nog nuttig werk. Ik kan het nog aan en doe het redelijk goed. Het lijkt veel, maar het is toch meer herhalen van eerder gezegde, eerder gedane zaken. Stokpaardjes berijden? Ik pak ook nieuwe ideeën op. Ideeën van anderen, wel te verstaan, om er ruchtbaarheid aan te geven. De anderen—daar ga je uiteraard selectief te werk: je vult je eigen weten en kunnen aan met wat erbij past en je ziet scherper wat er strijdig mee is. Standvastig en toch flexibel zijn is—althans in de goede verhouding —geen ouderdomskwaal, maar niet te erkennen dat er niettemin aan deze activiteit eens een eind moet komen, zou het wel kunnen zijn. (I will not pretend to be modest. I still do useful work. I am still up to it and do it reasonably well. It looks much, but yet it rather is repeating things that have been said or done before. Go off on hobby horses? I also pick up new ideas. Other people’s ideas, that is, to spread them. Other people’s—that is where you act selectively: you complement your own knowledge and ability with matching things and you see sharper what is inconsistent. To be steadfast and yet flexible—at least in the right proportion—is not an illness of old age, but not recognizing that there nevertheless has to be an end to this activity, could be one.) [96]

In 1987 after a tour of China where he lectured on his didactical ideas, Freudenthal once again felt the need to streamline his ideas and put them on paper. This resulted in the book Revisiting mathematics education. China lectures, that would turn out to be his last one [97]. One week after he had gone over the manuscript once again, Hans Freudenthal died on 13 October 1990.

References 1. Hans Freudenthal, ‘Rekendidactiek’ (unpublished 1944) 91; RANH, Hans Freudenthal Papers, inv.nr.465. See also chapter 4. 2. Freudenthal, Schrijf dat op, Hans 335–336. 3. J. Adda, ‘Freudenthal cursief. Hans Freudenthal (1905–1990)’, Nieuwe Wiskrant 10 (1991) 26. 4. A. Bishop, ‘Freudenthal cursief. HF-an inspiration in the search for truth’, Nieuwe Wiskrant 10 (1991) 13. 5. Freudenthal, Schrijf dat op, Hans 13. 6. Freudenthal said to Dieudonné in a discussion; J. Adda, ‘Une lumière s’est éteinte. H. Freudenthal—Homo Universalis’, in: Streefland, ed., The legacy of Hans Freudenthal 9–19, q.v.18. 7. J. van Dormolen, ‘Freudenthal cursief. Hoe didactiek een deel van wiskunde kan worden’, Nieuwe Wiskrant 10 (1991) 19. 8. Jan de Lange, ‘Nieuwlichters’, in: ‘Kijk op Hans’, a collection of essays by IOWO-employees, presented to prof.dr. H. Freudenthal at the occasion of his seventy-fifth anniversary (unpublished 1980) 37–38, q.v. 37–38. 9. W.T. van Est, ‘Hans Freudenthal.17 september 1905–13 oktober 1990’, Nieuw Archief voor Wiskunde 9 (1991) 131–136; q.v. 131.

References

57

10. For this biography data are taken from the following books: Freudenthal, Schrijf dat op, Hans; Hans Freudenthal, Berlin 1923–1930. Studienerinnerungen von Hans Freudenthal (Berlin 1987); Van Dalen, L.E.J. Brouwer; Fred Goffree, ‘HF: Working on mathematics education’, in: Streefland, ed., The legacy of Hans Freudenthal. 11. Freudenthal, Schrijf dat op, Hans 63. 12. Ibid., 67. 13. Ibid., 67. 14. Diaries Hans Freudenthal 1931–1988; RANH, Hans Freudenthal Papers, inv. nr.91–139. 15. Freudenthal, Schrijf dat op, Hans 150. 16. Ibid., 91. 17. Freudenthal, Studienerinnerungen 4. Georg Feigl (1890–1945), much admired by Freudenthal when he was a student, was from 1925 onwards editor of the Jahrbuch über die Fortschritte der Mathematik. The Jahrbuch was for a long time the only mathematical journal in which mathematical publications were discussed and was published from 1868 to 1944; Jahrbuch über die Fortschritte der Mathematik 51 (1925) 54 (1928). 18. Freudenthal, Studienerinnerungen 18. 19. Freudenthal, Schrijf dat op, Hans 115–116. 20. Freudenthal, Studienerinnerungen 10. 21. Besides logicism and formalism, intuitionism (Brouwer's ‘school’) was a third new movement in the philosophies in the foundations of mathematics. Until that time formalism—as advocated by Hilbert—was the most common philosophy. In it mathematics was not considered an existing entity, but a collection of formulas with accompanying rules of the game, without a direct meaning or value. Brouwer's intuitionism was based on the lack of the law of excluded middle (“all mathematical statements are either true or false”). The starting point of this philosophy was that only those mathematical objects that could be constructed from natural numbers in a finite number of steps had a meaning. This starting point made many mathematical theories unprovable. The debate on the foundations of mathematics, in which Hermann Weyl (1885–1955) also played an important role, is described extensively in: Van Dalen, L.E.J. Brouwer; P.J. Davis en R. Hersch, The mathematical experience (Boston 1984). 22. Freudenthal, Studienerinnerungen 10. 23. Hans Freudenthal, ‘Über die Enden topologischer Räume und Gruppen’ (Berlin 1931). The official ‘Tag der Promotion’ (day of defending the thesis) as stated on the thesis was 6 October 1931. The thesis was published in Mathematische Zeitschrift 1931 (33) 692–713 (Verlagsbuchhandlung Julius Springer in Berlin). 24. A few years later L. Bieberbach would declare to be a convinced National Socialist and would distinguish ‘Aryan mathematics' and ‘Jewish mathematics’: the latter would only elaborate on existing mathematics, while, by contrast, Aryan mathematicians were creative; Van Dalen, L.E.J. Brouwer 361–365. 25. Brouwer to Freudenthal, 3 August 1930; RANH, Hans Freudenthal Papers, inv.nr. 87. 26. Freudenthal, Schrijf dat op, Hans 117. 27. Freudenthal in the funeral oration for his wife: ibid. 365, Suus Freudenthal-Lutter died on 24 September 1986. 28. Brouwer eventually agreed to this move but insisted that Freudenthal would acquire a telephone connection; Van Dalen L.E.J. Brouwer, 393; Letter from Brouwer, 8 July 1932; RANH Hans Freudenthal Papers, inv. nr 87. 29. H.J.A. Duparc and A.W. Grootendorst, ‘Historical Survey’, in: E.M.J. Bertin, H.J.M. Bos, A.W. Grootendorst, ed., Two decades of mathematics in the Netherlands 1920–1940. A retrospection on the occasion of the bicentennial of the Wiskundig Genootschap. Part 1 (Amsterdam 1978) v-xxiv. 30. Hans Freudenthal, ‘Qualität und Quantität in der Mathematik’, Euclides 8 (1932) 89–98. 31. Freudenthal, Schrijf dat op, Hans 117. 32. W.T. van Est, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Koninklijke Nederlandse Akademie van Wetenschappen. Jaarboek 1991 139–142; W.T. van Est, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Nieuw Archief voor Wiskunde 9 (1991), 131–136.

58


33. F. van der Blij, ‘Professor Freudenthal zeventig jaar’, Euclides 51 (1975) 3–7, q.v. 4. 34. Freudenthal, Schrijf dat op, Hans 336–337. On the phenomenon Bourbaki and its influence on mathematics education, see section 8.2.1. 35. G. Alberts and I. Moerdijk, ‘Interview W.T. van Est. De geometer’, Nieuw archief voor wiskunde 5/3 (2001) 52–56, q.v. 52. 36. P.M. van Hiele, ‘Freudenthal en de didaktiek der wiskunde’, Euclides 51 (1975) 8–10, q.v. 8 37. Freudenthal to Howson, 19 July 1983; RANH, Hans Freudenthal Papers, inv.nr. 38. In the preface of his book Mathematics as an educational task from 1973 he still attributed the origin of his didactical interest to his wife; Hans Freudenthal, Mathematics as an educational task (Dordrecht 1973). 38. RANH, Hans Freudenthal Papers, inv.nrs. 21, 141. 39. RANH, Hans Freudenthal Papers, inv.nr. 321. 40. Rijksarchief in Noord-Holland, Papers of Susanna Freudenthal-Lutter (1908–1986), educationalist, 1914–1986, inv.nr. 37. 41. Van Est, ‘Hans Freudenthal’, Nieuw Archief voor Wiskunde 9 (1991) 131–136, q.v. 134 42. RANH, Hans Freudenthal Papers, inv.nrs. 43, 322. 43. RANH, Susanna Freudenthal-Lutter Papers, inv.nr. 37 (from: ‘Die Entwicklung der Jenaplan-bewegung in den Niederlanden’, handwritten document meant for publication in Gesamtschulinformationen, 1985). 44. RANH, Hans Freudenthal Papers, inv.nr. 465. See also chapter 4. 45. RANH, Hans Freudenthal Papers, inv.nrs. 282, 438 and 464; Freudenthal, Schrijf dat op, Hans 314. 46. RANH, Hans Freudenthal Papers, inv.nrs. 295, 1858. Floris V, reading ‘V. Sirolf’ backwards, was the subject of Freudenthal's first play from 1941; he researched extensively for this play. Both for the novel De schuldenaar and for Viersprong der grote wegen Freudenthal came out first in competitions of publishing house Gottmer. In 1947 he received the Letterkundeprijs (Literary prize) of the city council of Amsterdam. 47. Freudenthal to Hopf, 1945; RANH, Hans Freudenthal Papers, inv.nr. 88. 48. RANH, Hans Freudenthal Papers, inv.nrs. 1405–1577, 1627–1700. 49. In 1945 Freudenthal wrote extensively to his sister Lotte about the time in Havelte; Freudenthal, Schrijf dat op, Hans 154–227. 50. Freudenthal to Hopf, 28 July 1945; RANH, Hans Freudenthal Papers, inv.nr. 88. 51. Freudenthal, Schrijf dat op, Hans 225. 52. Freudenthal to Hopf, November 1945; RANH, Hans Freudenthal Papers, inv.nr. 88. 53. RANH, Hans Freudenthal Papers, inv.nrs. 90, 323; P.J. Knegtmans, Een kwetsbaar centrum van de geest. De Universiteit van Amsterdam tussen 1935 en 1950 (Amsterdam 1998) 249. 54. More about this question and about Freudenthal's appointment in Utrecht can be found in Chapter 5. 55. Since the relevant biographical information on Freudenthal's mental legacy will as a matter of course be treated more extensively in the following chapters, the period 1945–1990 will be described less elaborately in the remaining part of this chapter. 56. RANH, Hans Freudenthal Papers, inv.nr. 324. 57. Hans Freudenthal, ‘5000 jaren internationale wetenschap’, Euclides 22 (1946) 222–241. 58. ‘H. Freudenthal’, Vrij Nederland (1976) 232–247, q.v. 233. 59. Typical was the weekly tea hour (sometimes with cakes!) at the institute where purchase requests for the library were considered. Freudenthal turned out to be a great storyteller—in particular in a small party; conversation with Van der Blij, 20 May 1999. 60. Freudenthal to Howson, 19 July 1983; RANH, Hans Freudenthal Papers, inv.nr. 38. 61. Conversation with Van der Blij, 20 May 1999. 62. See also section 2.2.5. 63. RANH, Susanna Freudenthal-Lutter Papers, inv.nr. 37. 64. More about Mrs. Freudenthal's educational work in paragraph 8.5.2. 65. RANH, Susanna Freudenthal-Lutter Papers, inv.nr. 87.

References

59

66. In 1982 Mrs. Freudenthal would receive an honorary doctorate of the University of Giessen in recognition of her merits in the field of education; RANH Susanna Freudenthal-Lutter papers, inv nr 006. 67. J.D. Imelman and W.A.J. Meijer, De nieuwe school gisteren en vandaag (Amsterdam 1986) 107. 68. See also paragraph 2.2.5. 69. Freudenthal, Schrijf dat op, Hans 224. 70. More about this in paragraph 6.3. 71. Van Hiele, ‘Freudenthal en de didaktiek der wiskunde’, Euclides 51 (1975) 8–10, q.v. 9. 72. Freudenthal, Schrijf dat op, Hans 343. 73. W.T. van Est, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Koninklijke Nederlandse Akademie van Wetenschappen. Jaarboek 1991 (1991) 139–142, q.v. 140. 74. Hans Freudenthal, ‘De wereld in de spiegel’, Jaarboek Rijksuniversiteit Utrecht (Utrecht 1964); Van der Blij, ‘Professor Freudenthal zeventig jaar’, Euclides 51 (1975) 3–7, q.v. 6. 75. Hans Freudenthal and J. Sittig, De juiste maat (Leiden 1951); Hans Freudenthal, LincosDesign of a language for cosmic intercourse. Part I (Amsterdam 1960). 76. These five were the Humboldt-Universität (Berlin) in 1960, the Friedrich-Alexander Universität (Erlangen-Nürnberg) in 1972, York University (Toronto) in 1974, the Vrije Universiteit (Brussel) in 1974 and the Universiteit van Amsterdam in 1977; RANH, Hans Freudenthal Papers, inv.nr. 1858. 77. W.T. van Est, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Koninklijke Nederlandse Akademie van Wetenschappen. Jaarboek 1991 (1991) 139–142, q.v. 142. 78. For a more elaborate description of these developments, see chapters 8 and 9. 79. See also paragraph 6.6. 80. Freudenthal, Schrijf dat op, Hans 348. 81. See also paragraph 9.2. 82. More about the establishment of the IOWO in chapter 9. 83. J. de Lange, ‘In memoriam Hans Freudenthal’, Nieuwe wiskrant 10 (1990) 3. 84. Educational Studies in Mathematics 7 (1976) 351. 85. Van der Blij succeeded Freudenthal as professor-director of the IOWO. 86. Hans Freudenthal, ‘Preface’, Educational Studies in Mathematics 7 (1976) 189. 87. Hans Freudenthal, Mathematics as an educational task. 88. Hans Freudenthal, Weeding and sowing. Preface to a science of mathematical education (Dordrecht 1978). 89. Hans Freudenthal, Didactical phenomenology of mathematical structures (Dordrecht 1983); Hans Freudenthal, Didactische fenomenologie van wiskundige structuren (Utrecht 1984). 90. Freudenthal, Didactische fenomenologie van wiskundige structuren 10. 91. Freudenthal, Schrijf dat op, Hans 366. 92. Ibid., 12. 93. Ibid., 13. 94. Freudenthal to Hans Georg Steiner, 20 September 1990; RANH, Hans Freudenthal Papers, inv.nr. 71. 95. Ibid. 96. Freudenthal, Schrijf dat op, Hans 330. 97. Hans Freudenthal, Revisiting mathematics education. China lectures (Dordrecht 1991).

Chapter 4

Didactics of Arithmetic

Iets anders dat ik heb neergeschreven en niet gepubliceerd— trouwens niet eens voltooid—was een rekendidactiek. Het manuscript—103 bladzijden folio getypt en wat niet-getypte die zoek zijn—moet uit 1942 dateren. […] Ik zei het al: het manuscript is niet voltooid—ik zou zelfs kunnen zeggen dat ik aan de eigenlijke rekendidactiek niet ben toegekomen. (Something else that I wrote down and neither published— nor even completed was a didactic of arithmetic. The manuscript—103 typed folio pages and some handwritten pages that have gone missing—must date from 1942. […] As I mentioned earlier: the manuscript is unfinished—I could even say that I did not get down to the actual didactics of arithmetic.) Hans Freudenthal in Schrijf dat op, Hans, 1987 [1]

Unfinished, unpublished and in the end not even real didactics of arithmetic, that is how Freudenthal looked back upon the manuscript ‘Rekendidaktiek (Didactics of Arithmetic)’. Nevertheless, there are sufficient reasons to devote attention to this manuscript. The most important being that this manuscript is the first of Freudenthal’s documents on didactics of arithmetic to have been preserved. His own description of his activities in this field even justified the conclusion that it was actually the first manuscript he wrote on didactics [2]. At the time, it was the start of a new interest for him and a first reflection of his ideas and thoughts on the teaching (of arithmetic). It was—he concluded sometimes to his own surprise—a real start and many of his later ideas and method of working were based on it. Ik heb het manuscript van 1942 geboeid herlezen, soms, zoals ik al zei, verbaasd dat ik dit of dat toen al heb gezegd, soms teleurgesteld door het gevoel: wat heb ik er eigenlijk bijge leerd? (I reread the 1942 manuscript enthralled, sometimes, as I already mentioned, even being surprised that I had already said this or that at that time, which occasionally made me feel disappointed: what had I actually added to my knowledge since then?) [3]

That seems more than enough reason to expound on the appearance, content and function of the ‘Rekendidaktiek’ manuscript.

© Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_4

61

62

4 Didactics of Arithmetic

Referred to as ‘Rekendidaktiek’ in this publication, the manuscript has 103 numbered pages. The Rijksarchief in Haarlem possesses two copies: a typed copy and its carbon copy. At the top of the first page of the typed copy written in black ink (most probably by Freudenthal himself) is: Hans Freudenthal Fragment Rekendidaktiek plm 1942 Woord Vooraf (Hans Freudenthal Fragment ‘Rekendidaktiek’ about 1942 Preface)

On the carbon copy in a similar place it states: Onvoltooid rekendidaktiek ±1942 Getypt (Unfinished didactics of arithmetic, about 1942. Typed)

Both copies feature corrections in black ink of typing errors and spelling mistakes, and words and phrases crossed out or added. The manuscript can be read sequentially according to the page numbers, with the exception of pages 70–72, which contain an addition to the text on page 66. The running text contains footnotes, accompanied by, in writing or typing, the commentary ‘footnote’, in general separated from the running text with lines over and under the text of the footnote and typed with closer line spacing. Quotations indent and are also closer-spaced. The margins occasionally contain instructions for the printer, such as “—preferably written in capitals only” (p. 13), “petit” (p. 15), or “no new section” (p. 72). As far as has been checked the handwritten corrections and commentaries seem to be identical in both copies. For the following review the first (typed) copy of the manuscript has been used.

4.1 Dating of ‘Rekendidaktiek’ As mentioned before, Freudenthal himself dated both copies ‘about’ 1942. The fact that he added ‘about’ makes it likely that this date was added afterwards with some uncertainty, maybe even as late as the 1980s when he wrote Schrijf dat op, Hans. A wartime notebook also suggests this. This notebook contains Freudenthal’s notes on about 370 books and articles on teaching (arithmetic), pedagogy, psychology and didactics [4]. In addition it contains some notes of lessons in arithmetic he gave his children and of conversations with them, accompanied by the children’s ages. From these it can be deduced that the notes date from 1943. In the text of ‘Rekendidaktiek’ Freudenthal also refers to this literature study. De konklusies, die ik uit mijn waarnemingen trok, waren niet altijd in overeenstemming met de gedachten, waardoor de auteurs van de rekenboekjes, die ik toevallig in mijn lessen gebruikte, zich hebben moeten laten leiden. Er was een zeker verschil van inzicht, en allengs gevoelde ik de behoefte, om door nauwkeurigere literatuurstudie vast te stel-

4.2 Cause and Intention

63

len, “wie gelijk had”. […] Ik heb tenslotte, terwijl het plan van dit boek rijpte, alle voor mij bereikbare publikaties op het gebied van het rekenonderwijs bestudeerd. (Conclusions drawn from my own observations were not always consistent with the ideas of the authors of the arithmetic textbooks that I happened to use in my lessons. There was some difference of opinion and gradually I felt the need to assess “who was right” by doing a more thorough literature study. […] In the end, while the plan to write this book matured, I studied every publication on teaching arithmetic that I could lay my hands on.) [5]

His reference to this notebook can also be deduced from the fact that he mentions it in Schrijf dat op, Hans. Ik heb toen—in de oorlogsjaren—Didi en Thijs rekenen en lezen geleerd, en voor het rekenonderwijs heb ik de hele literatuur die ik te pakken kon krijgen bestudeerd; het boekje aantekeningen van de literatuurstudies is er nog. (At that time—during the war—I taught Didi and Thijs arithmetic and reading, and for the arithmetic education I studied every book and article that I could lay my hands on; the booklet with the notes of this literature study still exists.) [6]

Practically all of the literature references in Freudenthal’s ‘Rekendidaktiek’ can be found in this notebook. For that matter, this notebook is one of the few with notes on didactics to be found in Freudenthal’s personal archive. A sound conclusion is that Freudenthal wrote ‘Rekendidaktiek’ not in 1942, but later. A rather precise dating can be deduced from his 1940–1944 diaries. From 16 March 1943 onwards various notes referring to a literature study for ‘Rekendidaktiek’ can be found [7]. On 1 January 1 1944 Freudenthal wrote in his diary: Mijn voorstudie tot het rekenpaed. boek nadert langzamerhand zijn eind. Weken of maanden niets anders gedaan. (My preparatory study for the book on pedagogics of arithmetic (rekenpaed.) is slowly nearing its end. I haven’t done anything else for weeks or months.) [8]

On January 19 he started writing the manuscript proper: “started with the first seven pages of ‘rekenpaed.’” the diary mentions, and the entries on the following days contain phrases such as “typed ‘rekenpaed.’” or “continued typing”. On April 14 Freudenthal noted “Reached page 100 of method of arithmetic”. In the end the manuscript would not run to many more pages. The manuscript thus dates from 1944.

4.2 Cause and Intention The Second World War and the accompanying enforced idleness induced some scholars to mark time and to follow new leads in their theoretical work [9]. Being forced to stay at home also affected Freudenthal in the same way. The academic world (as far as there were activities in that field, of course) was to a large extent out of reach. This gave Freudenthal the opportunity to devote himself to interests that had hitherto been dormant. Until then he was fully occupied with his mathematical work and university matters, but now he suddenly had time to write stories and novels, to lose himself in history and—first and foremost—to closely follow the cognitive development of his children.

64


Fig. 4.1 Fragment from ‘Rekendidaktiek’ (1944)

As early as May 1942, probably in connection with the preparation of his lectures for the Jewish Council, Freudenthal had already asked his wife to bring him textbooks for the Kv certificate from the library [10]. It was a year before he began to study literature for ‘Rekendidaktiek’ so these lessons do not seem the immediate cause of his interest in didactics. Freudenthal had never published his views on this subject prior to writing ‘Rekendidaktiek’. In his ‘Preface’ he felt the need to explain to the audience what he had in mind and where the unexpected interest had originated. He wrote: Iedereen, die zoekende of wandelende ooit op gebieden terecht kwam, die hij bij zijn eerste stappen niet zag—laat staan overzag, zal mij geloven, wanneer ik verzeker, dat ook ik tot hen behoor, die een geschrift als het nu verschenen niet van mij hebben verwacht. (Anyone, who while on a quest has ever ended up back in fields, he overlooked—let alone looked over when he took his first step, will believe me when I assure you that I myself belong to that category of people who would not have expected me to write a book like the one now published.) (Fig. 4.1) [11]

The immediate cause of this interest was teaching his children arithmetic “sometimes deliberately, sometimes casually and more coincidentally”, without any “theoretical ulterior motives” [12]. In his diaries he reports on such activities from 1941 onwards. Freudenthal regularly notes these “sometimes casual” conversations with his sons. He also made a note of the more intentional (arithmetic) lessons he gave them. “Now I always spent half of my days working with Didi. Does his sums well” he wrote on 1 December 1941 [13]. And some months later: “done the usual sums, reading and writing with the boys” [14]. He continued this as best as he could during the war: “every day I play school with Didi and Thijs,” he wrote on 23 April 1943. In his ‘Preface’ Freudenthal revealed yet another fundamental reason for writing ‘Rekendidaktiek’.

4.2 Cause and Intention

65

Voor den lezer, die dit boek graag psychologisch zou willen verklaren, wil ik de andere grondslag, waar het boek op rust, niet verheimelijken: ik bedoel een bepaalde instelling van den auteur, een bepaald toegerust zijn op de behandeling van theoretische vragen en op het opwerpen en analyseren van problemen, een geestelijke instelling, die weliswaar niet op paedagogisch terrein is verworven, maar die—één keer op welk wetenschappelijk gebied dan ook verworven—niet zo licht verloochend kan worden. (For those readers who should want to explain this book psychologically, I will not hide the second fundament for this book: I refer to a certain disposition of the author, in a sense being equipped to treat theoretical questions and to raise and analyse problems, a mental attitude that indeed has not been acquired in the pedagogical field, but that—once acquired on whichever scientific area—cannot be denied easily.) [15]

This scientific attitude was according to Freudenthal essential to notice and analyse especially those “little incidents in domestic teaching” that someone lacking this attitude would perhaps experience as irrelevant. Above all this background made it possible for Freudenthal to do “experiments”. Ik deed dus, wat elke onderzoeker doet, die ertoe in staat wordt gesteld, waar een vraag meer dan één antwoord toeliet, doordat de ervaringsstof onvoldoende was, trachtte ik in de werkelijkheid de voorwaarden van de vraag te scheppen, om de werkelijkheid te dwingen, mij het antwoord op mijn vraag te geven. (I did what every researcher who is in this position does; when, by lack of observational subject matter, a question has more than one answer, I tried to create in reality the conditions for the question, to force reality to give me the answer to my question.) [16]

With this Freudenthal applied the ‘transfer’ principle to his own capacity as a researcher. After all, ‘transfer’ (or the comparable Dutch term ‘vormende waarde’ (formative value)) of a discipline meant being able to apply capabilities in another discipline than that in which these were acquired. ‘Transfer’ would turn out to be a regularly returning subject in the discussions about didactics of mathematics [17]. The manuscript was clearly meant as the start of something that eventually ought to have become a book on the didactics of arithmetic, probably intended to be a theoretical base for (future) teachers and authors of methods of arithmetic. For example, Freudenthal regularly spoke of “this book” and he referred in various places to sections that he had not yet written. In addition, the already mentioned notes to the printer indicated that this manuscript was certainly not only meant as a writing exercise or a fabrication. It remains a matter of speculation as to how bulky the book eventually would have become. From the references to subsequent, eventually unwritten chapters, it could be understood anyway that it had been Freudenthal’s intention to write more about the concept of numbers, about the practical requirements the teaching of arithmetic would have to meet, about the formative value or transfer of mathematics, about the positive sides of psychology, pedagogics, general didactics and methodology as auxiliary sciences for didactics of arithmetic, about the systematic American research under C. Washburne’s leadership, about test psychology, about proportionality about the requirements society sets for teaching of arithmetic and about ‘learning to think’ as a result of teaching of arithmetic [18]. As he himself had already noticed in Schrijf dat op, Hans he had in any case written some pages that

66


got lost in the course of time [19]. The manuscript ended abruptly at the foot of the page with the following unfinished sentence on page 103: En ik breid dit oordeel uit tot in die grensgebieden van het rekenen, die al enigszins aan wiskunde doen denken; want de innerlijke waarde van de wiskunde begint. (And I extend this judgement to those fringes of arithmetic, that somewhat remind one of mathematics; because the innate value of mathematics starts.) [20]

But it is improbable that many more than a few pages got lost. A month after he had written the first 100 pages Freudenthal was forced to work in Havelte. When he came home in September a long, difficult winter of starvation was imminent. The book that he envisioned when he studied the literature and typed the first part would eventually never come to anything. But whom was he talking to in his ‘Rekendidaktiek’? What did people think and write about teaching of arithmetic at the time? What in fact did teaching of arithmetic in practice look like?

4.3 Teaching of Arithmetic in Primary Schools In 1900 the compulsory education law was enacted [21]. From that moment on children from the age of 7 were obliged to go to school for 6 years in total. Around 1905 something well over 95 % of all children aged 6–12 attended primary schools. In the first 25 years of the last century not much changed in primary education. For most pupils primary education was their first and final education: only a small percentage of the pupils was being prepared for higher education. The influence of the rising Montessori and Jena Plan educational systems, with their freer learning and working methods, was still small in the period between the wars. Education in primary schools was mainly by class teaching, no wonder considering the often large number of children per teacher. Fifty or more children, divided over two or more classes in one schoolroom were no exception. A teacher spoke about her teaching experiences around 1920: Vertelvakken in een gekombineerde klas vertel je aan alle drie tegelijk. Schrijven als het kan ook alle drie tegelijk, soms afwisselend bespreken en dan weer laten maken, en dan weer een andere groep iets uitleggen enzovoort. Zo ging het ook met rekenen. De kinderen wennen aan dit systeem en het voordeel is dat ze veel zelfstandiger leren werken. (Narrative subjects are taught to all three classes in a group at the same time. Writing can also be done simultaneously by all groups, sometimes you have to discuss the results and then let them write an exercise, and explain something to another group, etc. The same thing with arithmetic. Children get used to this system and the advantage is that they learn to work much more independently.) [22]

The emphasis in primary education was on reading, writing and arithmetic, with attention also being paid to language, geography, history, biology, gymnastics and singing. Arithmetic had been a regulated compulsory subject since 1806.

4.3 Teaching of Arithmetic in Primary Schools

67

It is obvious that in teaching arithmetic the emphasis in a situation as sketched above was often on mechanically acquiring arithmetic skills. The motto was explaining, demonstrating, copying and exercising a great deal. This was reflected in the arithmetic methods of the time. A 1917 arithmetic textbook for teacher training distinguished in the teaching of arithmetic between “arithmetic with visual aid” (with objects and drawings available), mental arithmetic and written arithmetic (including making calculations and solving problems) [23]. Textual problems, the so-called arithmetic word problems, with a high puzzle content were notorious. An influential arithmetic method of P.J. Bouman and J.C. van Zelm from the 1930s contained the following ‘arithmetic word problem’: Aan een gemeenschappelijke dis neemt de eerste gast 1/3 van een rondgaand gerecht en iedere volgende gast neemt 1/3 van wat hij nog op de schotel vindt. Als nu de vierde gast 47,5 gram van dat gerecht minder krijgt dan de eerste, hoeveel gram liet hij dan nog voor de overige gasten over? (During a communal meal the first guest takes one third of a dish passed around and each following guest takes one third of what he finds on the dish. If the fourth guest gets 47.5 grams less than the first guest, how many grams are left over for the remaining guests?) [24]

The text book for teacher training already mentioned attributed the value of arithmetic in the first place to the material (practical) value: “life requires that we are able to calculate, that means decomposing and composing quantities, or, in brief, can work with quantities” [25]. In the second place according to the author the value of teaching arithmetic was to be found in cultivating good habits of thinking. Elke werkzaamheid van den geest geschiedt des te vlugger en nauwkeuriger naarmate zij vaker wordt herhaald. Het rekenen geeft veel aanleiding tot oordeelen en besluiten en leert daardoor logisch denken over onderwerpen, die tot zijn gebied behooren. Het kweekt verder, door het nauwkeurig redeneeren en de telkens noodige omzichtigheid in het maken van gevolgtrekkingen, goede denkgewoonten, die ook op ander gebied nuttig zijn. (As they are repeated more often all activities of the mind go so much faster and accurately. Arithmetic leads to judging and deciding and therefore trains logical thinking on related subjects. Furthermore, by accurately reasoning and repeatedly concluding with the necessary caution, it cultivates good habits of thinking, that can be of use in other fields as well.) [26]

In addition, the reward of the pupil’s continuing exertion to deliver good work would render a certain moral value to the teaching of arithmetic. Such starting points seemed generally accepted in the teaching of arithmetic. In spite of all that the emphasis in practice was mainly on acquiring and automating tricks and learning to recognize the correct answering strategy for a certain type of problems. In particular arithmetic word problems—even though these were chosen because of their formative value—induced this. In Germany after the First World War, however, great changes were taking place in the teaching of arithmetic initiated by Johannes Kühnel. Indirectly these followed from the Klein’sche Reform [27]. Kühnel challenged in his ‘Neubau des Rechenunterrichts’ (‘Reconstructing the teaching of arithmetic’) (1916) the nonpsychological buildup of teaching of arithmetic, the insufficient use of visual aids, the early abstraction and mechanization and the neglect of calculating related to

68


reality. According to him, the development of the concept of numbers in the child would have to be taken as a starting point in teaching arithmetic. In the Netherlands of the Nineteenth century schoolteacher Jan Versluys (1845– 1920) had already been arousing the discussion of a more heuristically (‘self-induced’) and graphically organized teaching of arithmetic and mathematics [28]. Publication of his Methoden bij het onderwijs in de wiskunde en bij de wetenschappelijke behandeling van dat vak ( Methods for teaching mathematics and for the scientific treatment of that subject) in 1874 is considered the origin of teaching methodology in the Netherlands [29]. However, his ideas hardly influenced the practice of the teaching of arithmetic. The strict and ‘bare’ arithmetic methods of Bouman and Van Zelm were more influential in the period between the two World Wars. It was only in the 1930s that a certain modernization tendency in teaching of arithmetic came up. J. Waterink had been professor in pedagogics at the Free University in Amsterdam since 1926 and he regularly published about didactics of arithmetic. In that same era Kohnstamm stimulated the development of arithmetic teaching by paying attention to the psychological basis for didactics (of arithmetic), something he found in the psychology of thought from Otto Selz. Both Waterink and Kohnstamm also paid a lot of attention to the improvement of the connection between primary and secondary school types. After the method Fundamenteel rekenen ( Fundamental arithmetic) of P.A. Diels and J. Nauta was published in 1936 more attention was being paid to restricting the subject matter and returning to the basis of the teaching of arithmetic: teaching the fundamental concepts. Freudenthal’s literature study of didactics of arithmetic during the Second World War consisted of more than 370 titles. The only criterion that he appeared to have had in selecting this literature was the availability of the book in question. He literally read everything he could lay his hands on. Practically all of the textbooks, methods and publications in the field of pedagogics described above (and more) figured in his bibliography. Hence, Freudenthal started writing his ‘Rekendidaktiek’ from a sound theoretical basis. However, there is no trace to be found of any correspondence or contact between him and didacticians and pedagogues in this period. Freudenthal had, in the tiniest possible script, filled the more than 300 pages in his notebook with notes on the books he had studied. Most of these notes were quotations or short summaries of the book in question, only incidentally was he tempted to judge (or condemn). From this relatively neutral notebook to the content of ‘Rekendidaktiek’ is a large mental leap, that cannot be traced back to his writings.

4.4 Freudenthal’s ‘Rekendidaktiek’: The Content The manuscript consisted of three chapters, notably ‘Woord vooraf’ (‘Preface’, pp. 1–11), ‘Hulpwetenschappen’ (‘Auxiliary sciences’, pp. 11–91), and the incomplete chapter ‘Doel en nut van het rekenonderwijs’ (‘Aim and use of teaching arithmetic’, pp. 91–103).

4.4 Freudenthal’s ‘Rekendidaktiek’: The Content

69

4.4.1 Preface In ‘Preface’ Freudenthal explained to the reader how he, by observing his own children when they were learning arithmetic, ended up writing this work. Another foundation of this work was his own ability to treat theoretical questions in a scientific way, to analyse problems and to do ‘experiments’. He warned his readers not to expect too many ready-made solutions. The most important motive for this modesty was that Freudenthal thus wanted to “stimulate others to do their own research, where his’ had shortcomings” [30]. The psychological data that Freudenthal had collected by observing his own children should not be overrated. For this kind of observations (found often in the literature) only concern a few children from an intellectual environment and that was not a good sample. He wrote about this: Ik zou gaarne, op kollektieve schaal willen inhalen, wat ik, helaas, moest verzuimen, maar ondertussen gaapt hier een klove, die ik, node, anders heb trachten te overbruggen. (I would, on a larger scale, prefer to make up for what I had to omit, but in the meantime there is a gap that I reluctantly have tried to bridge in another way.) [31]

This bridging was to him an extensive literature study, which formed the theoretical foundation for his conclusions. Unfortunately, because of the war Freudenthal was unable to study all the American literature himself. A great deal of his knowledge was not straight from the horse’s mouth: …ik hoop, dat ik deze boeken, boekjes en artikels, die ik niet kon bemachtigen, onder gunstiger tijdsomstandigheden nog met vrucht zal kunnen doorbladeren, zodat ik ook op dit punt een verzuim zou kunnen inhalen. (…in more favourable circumstances I hope to be able to productively leaf through these studies, booklets and articles that I could not lay my hands on, and so would be able to make up for this omission as well.) [32]

According to Freudenthal the validity of a theory for teaching arithmetic was much less nationally restricted than that for other subjects. The differences existing internationally were mainly caused by tradition. When studying the literature Freudenthal hardly encountered any publications that turned out to be completely useless. He mainly read English, German, Dutch and some French literature, and one or two Russian booklets. He valued American literature most. Het meest heb ik geprofiteert [sic] van de haast overstelpende Amerikaanse literatuur—ik zeg dat met des te meer klem, naarmate mij steeds duidelijker blijkt, dat er nog bij veel belangstellenden het besef ontbreekt van de grote betekenis der Amerikaanse theoretische en praktische opvoedkunde. (I profited most from the almost overwhelming American literature—I say this with all the more emphasis as it became increasingly clear to me that many interested parties are not aware of the great significance of the American theoretical and practical pedagogy.) [33]

4.4.2 Auxiliary Sciences In this chapter Freudenthal took a close look at a collection of sciences individually that could function as an auxiliary science for the didactics of arithmetic.

70


The first auxiliary science was philosophy. According to Freudenthal philosophical considerations were often only given at the start of a scientific study as an ‘ornament’ and were not used in the rest by the author. An example of this in writings about teaching of arithmetic were the misplaced considerations on the number concept which, according to him, did not contribute in a meaningful way to a didactics of arithmetic. Instead, considering ‘understanding numbers’ would be much more appropriate As an example of such useless philosophical additions he quoted, among others, Kühnel [34] Philosophy as auxiliary science used in this way did not, if at all, contribute to the quality of a method to teach arithmetic and the same could, for instance, be said of religion as an auxiliary science. Sociology and economy as auxiliary sciences could only provide data for the practical requirements that education ought to meet. Statistics could, according to Freudenthal, only in a purely technical sense render services, as it did to all other sciences when numerical data had to be conveniently arranged. Mathematics and arithmetic were treated when the question was raised how much a future school teacher should know about them. It could be useful that a schoolteacher acquired more skills in arithmetic and a deeper insight in the subject matter. The latter could unfortunately not be achieved during the teacher’s training. The methods of learning arithmetic used in the teacher’s training at the time were according to Freudenthal “worthless”: Haast alle bewijzen, die erin staan, zijn schijnbewijzen, en het is sneu, dat de kwekeling, die ze zichzelf heeft ingepompt, zich misschien wonderen verbeeldt over het geestelijk bezit, dat hij meent te hebben verworven. (Almost all the proofs in it are fake and it is a pity that the teacher trainees who had drilled themselves perhaps imagine miracles about the mental property they think they have acquired.) [35]

The influence of mathematics could better be restricted to an indirect influence: Wie een rekenmethodiek schrijft, kan die invloed doorgeven, maar kan niet van den lezer eisen, dat hijzelf die wiskundige begrippen verwerft, waar zulk een invloed op berust. (An author who writes a method of teaching arithmetic can pass on that influence, but cannot demand the reader to acquire those mathematical notions on which such an influence is based.) [36]

The ‘main dish’ of the auxiliary sciences consisted of psychology, pedagogy, general didactics and methodology. Freudenthal only discussed the negative aspect, the positive ones would follow later [37]. Hands-on pedagogy, according to Freudenthal, had never bothered much about theoretical psychology. Until recently this theoretical psychology was nothing more than the description of a number of regularities that were never actually indicated or ‘discovered’. But even now psychology as a science left much to be desired. Met een periode van lege programma’s, van strenge wetten, die er moesten zijn, maar niet waren, van uitgebreide theorieën zonder grondslag in de ervaring is ook de psychologie begonnen, en zij is haar kindsheid nog niet geheel ontgroeid; er verschijnen nog steeds boeken—en die zijn volstrekt niet door de eerste de beste psychologen geschreven— boeken, waarin een berg programma een muisje “uitvoering van dat programma” verstikt, de uitkomst van een kleine proefneming op een onderdeel van het zielsleven wordt graag met reusachtige speculaties aangelengd tot een verklaring van alles, wat er in de ziel


71

omgaat. (Psychology has started with a period of empty programmes, of strict laws, that should exist, but did not, of extended theories without experimental foundations, and it has not yet left its childhood; books are still published—and these are not written by just some psychologist—books in which the mountain (the programme) chokes the mouse (the execution of that programme), the result of a small experiment on part of the spiritual life is gladly diluted into an explanation of everything that is going on in the spirit). [38]

The psychology (of thought) that had come into being in the meantime, was still not applicable in practice. Yet, psychological literature was quoted, whether it was relevant or not, to give more ‘weight’ to methods of teaching arithmetic, to the great annoyance of Freudenthal. Het staat iedereen vrij, de gehele psychologie aan zijn laars te lappen; maar het staat hem dan niet meer vrij, zich tegelijkertijd te beroepen op wat hij aan zijn laars heeft gelapt. Ik noemde zo iets ergerlijk, en ik vindt het dubbel ergerlijk, als het niet één keer, maar systematisch gebeurt. (Everybody can choose to take not the slightest notice of the entire psychology, but they cannot at the same time refer to what they choose to ignore. I called such things annoying and I find it doubly so if it happens not once, but systematically.) [39]

He found various examples of this ‘repeatedly ignoring’ in the instruction book that went with Fundamenteel rekenen ( Fundamental arithmetic) of P.A. Diels and J. Nauta, where various pedagogues were quoted haphazardly or wrongly. In general, Freudenthal’s conclusion was that arithmetic textbooks on their own neither benefitted nor suffered because of it. The psychology that schoolteachers had to learn in training was mainly of a theoretical nature and as inexperienced teachers they could not process that knowledge in a practical sense. That theoretical psychology would only be of use if they knew the practice of it. The psychology that could be applicable as an aid and that would justify in advance to draw every conclusion did not yet exist. Pasklaar levert geen psychologie mij wat ik haar vraag—ik herhaal deze oude formu leringen—; om een theorie toe te passen, is weer een hele nieuwe theorie nodig; de oude ervaringen kunnen het opdoen van nieuwe vergemakkelijken, maar niet vervangen. (No psychology delivers me ready-made what I ask of it—I repeat these old formulations—; to apply a theory, a completely new theory is needed; old experiences can facilitate gaining new ones, but not replace them.) [40]

Child psychology as an auxiliary science also did not deliver many results. In a curriculum one often and gladly referred to the psychological or logical order of the subject matter, but according to Freudenthal it was more important to take the general development of a child at a certain moment in time into account. En zolang we op dit punt niets weten, ontbreekt ons een belangrijk werktuig, dat de kinderpsychologie voor de praktische pedagoog zou kunnen vervaardigen, en zolang zal veel van het kenmerkend kinderlijke aan het elementaire onderwijs moeten ontbreken. […] Zolang dat onderzoek ontbreekt, is de kinderpsychologie tenslotte, al haar bekoorlijkheid ten spijt, een dorre statistiek over wat kinderen op een bepaalde leeftijd doen. (And as long as we do not know a great deal as far as this point is concerned, we lack an important tool that child psychology could construct for the practical pedagogue, and for the time being elementary education will lack many of the childlike characteristics as a result. […] For as long as that research is lacking child psychology is in the end, despite all its charm, a barren statistic about what children do at a certain age.) [41]

72


More knowledge of the psychology of school teachers, or: ‘teaching, how is it actually done?’ would be very beneficial to education. Preferably this research would have to be developed by third parties and not by the school teachers themselves. The psychology of the ‘anomalous’ could also supply a lot of information about the usefulness of the various methods. These children, in particular, are very sensitive to the quality of a method. Een goede maag slikt alles; of het voedsel deugdelijk is, daarover kan ons iemand met een gevoelige maag beter inlichten, en zolang er voor onderwijsmethoden geen keuringsinstantie met zo iets als reageerbuisjes en reagentia bestaat, om onderwijsmethoden goed-en af te keuren, zal de reactie van het achterlijke kind zeker een uitstekend criterium blijven. (A good stomach swallows everything; whether our food is good or not is something that a sensitive stomach is better equipped to tell us, and as long as there is no inspection service for teaching methods using things like test tubes and reacting agents to approve or disapprove of teaching methods, the reaction of the mentally retarded child will certainly remain an excellent criterion.) [42]

One of the most ready-made psychological methods was to be found in the area of the American test research. In the Netherlands, such research had been rejected as an aid in teaching, because it would not be sufficiently applicable to the Dutch situation. Unfortunately, this possible applicability had neither been investigated systematically, nor was it used to gain experience: Het vruchtbaarste psychologische gebied—dat van het testonderzoek—waar alles wat de klok slaat praktijk heet, waar het praktische als het ware voor het grijpen ligt, is voor de Nederlandse onderwijzer tenslotte weer niets anders dan een dorre woestijn, althans zolang niemand de moeite heeft genomen, om wat daar ligt te glinsteren, op te rapen en voor onze doeleinden te verwerken. (The most fertile psychological area—that of test research— where the order of the day is practice, where practice is as it were for the asking, is to the Dutch school teacher in the end nothing but a barren desert, at least as long as nobody has taken the trouble to pick up what is sparkling there and use it for our purposes.) [43]

The last possible auxiliary sciences were general didactics and methodology. According to Freudenthal these two had actually never existed as independent sciences. An example from the history of didactics of systemizing education was the Socratic interview between teacher and pupil. Freudenthal described his own (unsuccessful) attempt to conduct a Socratic interview with one of his children and concluded [44]: Leermeesters zijn meestal even hardnekkig in het ondervragen als leerlingen in het niet weten (of niet willen weten), maar zoveel er ook aan de socratische methode prijzenswaardig is—toch zal de leermeester soms beter doen, om minder hardnekkig socratisch te zijn […]. (Instructors usually are just as persistent in questioning as pupils are in not knowing (or not wanting to know), but as much of the Socratic method is praiseworthy—the instructor nevertheless will do better if he is a bit less persistent in his Socratic manner […].) [45]

Another example from the pedagogy of the Nineteenth century were the formal steps of instruction of Herbart-Ziller, which in fact were nothing more than a systemization of education in natural history. An incidental circumstance that turned out positively for this teaching method was that teachers who tried to apply this theory to unsuitable subject matter were in any case forced to think it through well. Requirements such as ‘repetition of subject matter is necessary’ and ‘subject matter must be taught in a captivating way’ were according to Freudenthal part of


73

general didactics and should be realized with the educational system as universal requirements. There existed a general methodology and didactics in that sense: the theory of educational systems. It was difficult to make people aware of the traditionally developed educational system which was therefore also difficult to transfer. New concepts such as those of Montessori and Dalton did not have that problem. Eventually practice proved whether a pedagogical method was useful or not, such as had happened with the successful kindergarten education of Fröbel compared to the failure of Pestalozzi’s concept. In that context there ought to be an attempt to investigate the success of, for example, the Montessori schools in the Netherlands, if these were to become a success: Men heeft niet bepaald een unieke kans voorbij laten gaan, maar men heeft toch veel verzuimd, en tot nu toe is het oordeel over het stelsel-Montessori beperkt tot de particuliere meningsuitingen van Mevrouw Janssen, die in geestdrift geraakt is over de riante keukentjes, die ze bij het Montessori-vlo hebben, of van mevrouw Pietersen, die zich over Pietje Janssen ergert, en die zich gekwetst voelt, als de Montessori-leerling onuitstaanbaar neuswijs—naar het haar schijnt—en met de zelfbewustheid van een pedagogisch vakman praat over het “oranje materiaal”, waarmee hij op de Montessorischool aan het werken was, en over doos F van het werkwoord, die hij morgen “bereikt” zal hebben. (It is not so much a unique opportunity that has been missed, but still a lot has been neglected, and until now the judgement on the Montessori concept has been limited to the private expressions of opinions of Mrs. Janssen who has become enthusiastic about the charming kitchens of the Montessori kindergarten schools, or those of Mrs. Pietersen, annoyed by Pietje Janssen, and hurt when the Montessori pupil in an—it seems to her—exasperating cheeky way talks like a pedagogic expert about the “orange material” he was working with in Montessori school and about box F of the verb which he will have “reached” tomorrow.) [46]

There was a lot of justified criticism of this type of education, but yet it was the only well thought-out educational system in the Netherlands (as good or bad as it would turn out to be). The significance of the Montessori education was found in the ingenious way in which one teacher could teach individual pupils in a collective. The whole-class concept in its proper form was no longer useful according to Freudenthal. He argued in favour of smaller groups in the class: Het klassikale stelsel zal misschien blijven, maar het starre klassikale schema zal allengs worden verlaten. Het is volstrekt niet nodig, dat het onderwijs er individualistischer door wordt. Wanneer de klas als lerend lichaam niet door individuen wordt vervangen, maar door de groep—de groep van 3–6 kinderen—, zoals in sommige stelsels is geschied, dan zal er meer collectivisme in het onderwijs zijn of kunnen zijn, dan in het klassikale onderwijs— geen klascollectivisme, maar groepscollectivisme, dat naar gelang van de omstandigheden intensiever zou kunnen zijn, dan het klascollectivisme. (The (whole-)class system will probably be maintained, but the strict class schedule will gradually be abandoned. It is absolutely unnecessary for the educational system to become more individualistic because of this. When a class as a learning body is not replaced by individuals, but by a group—a group of three to six children—, such as has been done in some concepts, there will be, or can be, more collectivism in the educational system than in the (whole-)class system—not class collectivism, but group collectivism, that could be more intense than class collectivism depending on the circumstances.) [47]

The Montessori concept was the only concept that supplied the (prospective) teacher with a method and in that way equipped the inexperienced teacher with ‘tools’. Even though all that the teacher trainees had to learn at that moment was better

74


founded than the Montessori theory, it still missed these necessary tools to be used in practice. Ultimately Freudenthal had rejected all auxiliary sciences: one was impracticable, the other could not yet be applied ready-made, a third was completely lacking. This negative chapter seemed superfluous, but yet according to him it was useful to establish it and use it as an onset to answering the question of how things should be done. Things had to change, that was his firm conviction: Voor den schrijver rest tenslotte de taak, de lezer te overtuigen, dat ook wat een grapje leek, ernstig bedoeld was, en dat overal, waar ontsporingen gecritiseerd werden, een stelsel gecritiseerd werd, dat niet enkel meer een ontsporing, maar een weg is met diepe karresporen, die dringend uitnodigen, er opnieuw in te rijden. Het stelsel om jonge mensen, die een taak gevonden hebben en zich voor een taak willen laten opleiden—het stelsel, om deze mensen van alles en nog wat in te stampen, onbeschrijflijke, ongeschifte hoeveelheden stof uit tal van wetenschappen, hoeveelheden stof, die gekend moeten worden, eer de poort tot het beroep opengaat, en vergeten moeten zijn, eer die poort achter de candidaat zich sluit—dit stelsel is geen ontsporing, maar een geheiligde instelling, geheiligd niet alleen door de traditie, maar—wat veel erger is—door de namen van talloze hervormers, die er tenslotte voor moesten dienen, om als etiket op weer een onderdeel van dat geheel van onwetenswaardigheden te worden vastgeplakt. (Finally the author is left with the task to convince the reader that, what seemed a joke, was seriously meant, and everywhere where derailments were criticized, where a concept was criticized, it was not just a derailment but a path with deep tracks that urgently invites one to take it again. The system for young people who have found a task and want to be trained for it—the system to drum in all kinds of everything, indescribable, unsorted quantities of subject matter from numerous sciences, quantities of subject matter that have to be known before the door to the profession opens, and must be forgotten before the door closes behind the candidate—this system is not a derailment, but a sacred institution, sacred not only by tradition, but—much worse—by the names of numerous reformers, who in the end served only as a label to be put on pieces of information not worth knowing.) [48]

Freudenthal ended this chapter with an appeal to his readers: Wie over dit hoofdstuk ontevreden is, late het niet de schrijver voelen, die er evenmin op gesteld is, maar richte zijn critiek tegen allen, die de schrijver critiseren wilde. Dat is misschien voldoende. En als het niet voldoende is, dan is het toch noodzakelijk, want elke positieve actie begint met critiek. (Those who are not satisfied with this chapter, please do not let the author, who also doesn’t like it too much, feel this, but direct your criticism to those who the author wanted to criticize. That is perhaps sufficient. And if it isn’t sufficient, yet it is necessary, for all positive action starts with criticism.) [49]

4.4.3 Aim and Use of Teaching of Arithmetic In the course of time teaching of arithmetic, apart from ‘learning to do sums’ also served higher purposes. In the Nineteenth century, for instance, a religious or moral value was attached to teaching arithmetic, and the world of numbers was an example of an ‘honest world’. What people hoped to find, according to Freudenthal, was a function that was specific to teaching arithmetic.


75

To be able to take up his place in the society the child had to learn to calculate, both in the interest of the society and in the interest of the individual. The ‘world reformer’ Pestalozzi wanted to teach the children first to ‘calculate by thinking’ and later to ‘think by calculating’ in order to protect them against exploitation and delusion. Just like Pestalozzi Montessori attached too much importance to the ideal of ‘reforming the world by education’. According to Freudenthal this resulted in the development of a type of teaching of arithmetic which exactly did not meet societal requirements. In the Nineteenth century ‘learning to think’ was the goal of teaching of arithmetic. ‘Learning to think’ as a goal in itself was not a problem, provided that it was clearly specified and not seen too narrowly, but in combination with the subject. Freudenthal warned against wanting to describe ‘thinking’ as one logical outline or process, something which in his opinion was still done by many arithmetic teachers. De electricien, die een schakelaar moet repareren, de coupeur, die uit een lap stof een pak moet knippen, de zuinige huisvrouw, die met één overstapje zo veel mogelijk boodschappen wil afdoen—ze denken na, en iedereen, die voor een taak gesteld wordt, die hij niet zuiver werktuigelijk af kan, denkt na of tracht na te denken. De manier van denken, die in ons voorbeeld de electricien [er] op na houdt, of die van de coupeur, of die van de plannen makende huisvrouw gelijken weinig op elkaar, en geen van alle gelijkt op de manier te denken bij de sluitreden “alle mensen zijn sterfelijk en [sic]” of bij het vormen van dieren plantkundige “begrippen” of bij het oplossen en beredeneren van een rekenvraagstuk. Dat het denken van de zogenaamde formele logica en het reken-denken een bijzonder eenvoudige, doorzichtige structuur bezit, maakt dit soort van denken geenszins tot een representatief voorbeeld voor het denken in het geheel, en de bewering, dat men door te leren rekenen kan leren denken, vindt in de structuur van de onderscheiden denkfuncties in elk geval geen steun. (The electrician, who has to repair a switch, the tailor who has to cut a suit from a piece of fabric, the thrifty housewife, who wants to do as many errands as possible without changing buses—they all think, and everyone who finds themselves confronted with a task he cannot perform purely mechanically, will think or will try to think. The ways of thinking of the electrician, the tailor or the planning housewife in our example are very different and none of them resembles the way of thinking in a syllogistic way “all men are mortal” or by forming of zoologist or botanist ‘concepts’ or by resolving and reasoning of an arithmetic problem. The fact that thinking of the so-called formal logic and arithmetical thinking have a particularly simple structure, does not make this form of thinking at all a representative example of thinking as a whole, and the statement that one can learn to think by learning to calculate, is in any case not supported by the structure of the different thinking functions.) [50]

Freudenthal believed that it certainly was possible to teach people to think deliberately, provided that at the start of the process the what and why of it was formulated exactly. Arithmetical methods could according to him certainly teach other forms of thinking than only arithmetical thinking, for example thinking logically. However, this could be achieved more efficiently within the native-language teaching (parsing). Apart from arithmetical thinking there were according to Freudenthal no important consequences of the teaching of arithmetic, that is how the manuscript ended.

76


4.5 ‘Rekendidaktiek’ (‘Didactics of Arithmetic’): All Positive Action Starts with Criticism Freudenthal’s reason to immerse himself in the didactics of arithmetic was a practical one: Observing his own children when they learned to calculate. As a scientist he buried himself in the available literature, developed ideas and subsequently tested these against the literature and in practice. Correspondence with others about this subject could not be found. The quest for didactics of arithmetic was an isolated and solo affair for Freudenthal. From the moment early in 1944 when he started writing ‘Rekendidaktiek’ he already had the design and content of this book in mind. It had to become a didactical handbook, based on scientific experiments and research, purged of false pretences of the ‘scientific character’ he encountered in existing studies. However, owing to circumstances he did not get any further than the first three chapters, which are filled with references to the chapters that would follow [51]. Strikingly, what he did write in the end, was mainly the criticism of the chapter ‘Auxiliary sciences’ (80 pages). In this he tested the usefulness of various sciences as auxiliary science for the didactics of arithmetic. One by one he rejected them for different reasons. Incidentally he pointed out that several authors of arithmetic books and pedagogic studies misused one or more of such auxiliary sciences, notably psychology. Remarkably, Freudenthal neither addressed representatives of an individual movement, nor consciously sided with one of the movements. He targeted both studies coming from the field of (test)psychology and from Reform pedagogy. The arithmetic textbooks of Bouman and Van Zelm did not escape his criticism, nor did those of the more modernizing (but mechanistic) Diels and Nauta. Pedagogues such as Montessori, Pestalozzi, Waterink and Kühnel, and the American psychologist Dewey and the test psychologist Luning Prak were all critically analysed by Freudenthal for using correctly the auxiliary sciences in their studies and were found ‘guilty’. This did not automatically imply that he would not agree with the remaining content of the study in question. It looks easy, only to criticize and leave it at that. However, he simply had not found the time for a more positive evaluation of other people’s work when he started with ‘Rekendidaktiek’. The overly negative line of approach was therefore not necessarily a conscious choice of Freudenthal. Freudenthal made one rare—since it was positive—note in his notebook on the arithmetic method of L. Lockefeer and G. Dieleman, De nieuwe koers. Rekenmethode voor de l.s. ( The new course. Arithmetic method for the primary school) from 1939. Uncoincidentally it was the method used by the school of Didi and Thijs [52]. So Freudenthal was familiar with this method. With reference to another arithmetic textbook, Enkele praktische sommen, ( Some practical exercises) by H. Nijhof and B.J. Douwes he noted down:

4.5 ‘Rekendidaktiek’ (‘Didactics of Arithmetic’)

77

Waarom enkel pract. sommen? Rekenen leren kan men met alle soort sommen rekenen. Leren toepassen liefst met toepasselijke. Een kind van 8–9 jaar sommen te laten rekenen, die het misschien over 6 jaar eens zou kunnen toepassen, is een dwaling. Niet practisch, maar de belangensfeer van het kind toebehooren (zie Lockefeer en Dieleman) is de goede keuze. Ook L&D zondigen soms, maar alles fleuriger, concreter, niet het onophoudelijke monotone thema van Nijhof Douwes. (Why only practical exercises? Learning arithmetic can be done with all kinds of exercises. Learning to apply preferably with applicable ones. To let a child of the age of eight or nine do exercises that he or she would perhaps apply six years later is a mistake. Not practical, but belonging to the child’s sphere of interest (see Lockefeer and Dieleman) is the correct choice. L&D sometimes sin too, but all in all are more cheerful, more concrete, not the continuously monotonous theme of Nijhof Douwes.) [53]

This is an important note because it shows a glimpse of ‘how it should be done’ according to Freudenthal in 1943. Statements such as ‘learning to apply’ and ‘belonging to the sphere of interest of the child’ seemed to stem from Freudenthal’s ideas in the 1970s [54] However, this note stood on its own. He did not incorporate it in his ‘Rekendidaktiek’ since he never got down to discussing the aspects concerning the content of the teaching of arithmetic. However, in ‘Rekendidaktiek’ Freudenthal did go beyond criticism of the auxiliary sciences. While rejecting them one by one, he described not only the reasons for it, he also gave a direction on how to find the use that these auxiliary sciences can have for didactics of arithmetic. The great value he attached to doing ‘real’ science emerged strongly in this. He rejected, for instance, the quasi-scientific use of empty footnotes and emphasized the importance of drawing conclusions based on representative samples. In this respect ‘Rekendidaktiek’ resembled his later book Weeding and Sowing. Preface to a science of mathematical education from 1978. Here he described how he, via an analysis of the concepts “science”, “education” and “science of education”, came to a “science of mathematical education”. Just as he did in ‘Rekendidaktiek’ he criticized in Weeding and Sowing the role that psychology and didactics had played as auxiliary sciences in the development of a science of education. 10 years later in Schrijf dat op, Hans he wrote about this criticism: Op mijn scepticisme ten aanzien van wat psychologie en algemene didactiek en methodenleer tot de rekendidactiek konden bijdragen, hoef ik heden nog geen fundamentele correctie aan te brengen, van het voordeel van de twijfel dat ik toen [in ‘Rekendidaktiek’] nog aan de toetsontwikkeling wilde toekennen, is alleen de twijfel overgebleven. (There is no need to report a fundamental correction on my scepticism with respect to what psychology, general didactics and methodology could contribute to didactics of arithmetic, from the benefit of the doubt that I at that time wanted to give to the development of testing, only the doubt remains.) [55]

A great difference was, however, that while in 1943 Freudenthal still overconfidently had the intention to write a complete didactic of arithmetic, in 1978 he knew beforehand that he could not go beyond a ‘preface to a science of mathematical education’ and only gave hints how to build up such a science [56]. Another aspect, also to be found in his later work, was the value he attached to observations. He wrote, for instance, about the usefulness of observing experienced teachers in class. Another phenomenon that he would continue to use were

78


his descriptions of the conversations with his children. Even in his book Mathematics as an educational task from 1973 he still referred to the conversations with his children during the war that he had noted down in the aforementioned notebook of 1943 [57]. Later Freudenthal would also elaborate the plea for using small groups in teaching. With hindsight he was surprised that he had already argued in favour of smaller groups in ‘Rekendidaktiek’, without knowing the motives for it which he did formulate in later studies: Ik beschouwde het toen bij voorbeeld al als een averechts idee het klassikaal onderwijs te vervangen door individueel onderwijs zoals de Montessorianen deden; ik had toen al mijn zinnen gezet op onderwijs in kleine groepen; waarom weet ik niet, want het karakter van de groep, het sociaal en eventueel heterogeen karakter, komt niet uit de verf. (At the time I considered it, for example, a counterproductive idea to replace whole-class education by individual teaching like teachers in the Montessori schools did; I had already set my heart on teaching in small groups; why I don’t know, since the nature of the group, the social and perhaps heterogeneous character, is not expressed well.) [58]

Finally, there were two other subjects in ‘Rekendidaktiek’ that he would return to in articles and lectures in the years after the war: ‘learning to think’ by arithmetic and mathematics education (and the supposed formative value of it) and the curriculum of the training for teachers in primary and secondary schools [59]. ‘What is it that I have added to my knowledge since then?’ a disappointed Freudenthal wondered after rereading ‘Rekendidaktiek’ 40 years later, and immediately concluded: Het valt mee: ik heb allereerst geleerd gedachten aan te scherpen, maar dat heeft dan ook weer veel te maken met nieuwe stromingen in het wiskundeonderwijs, die tegen mijn ideeën ingingen. […] Het lijkt of ik toen vooruitliep op mijn kritiek op Piaget […], maar ook op mijn kritiek op New Math. (It is not too bad: the first thing I have learned is to sharpen my mind, but that also has a great deal to do with new movements in mathematics education that went against my ideas. […] It seems as if at the time I anticipated my criticism of Piaget […], but also that of New Math.) [60]

And indeed, one of the most important spearheads of Freudenthal’s later criticism of Piaget could already be found in ‘Rekendidaktiek’: falsely deriving a ‘scientific nature’ from an auxiliary science [61]. Something else that Freudenthal had added to his knowledge—and that he passed over in Schrijf dat op, Hans—was the value of the observation of a single child. In ‘Rekendidaktiek’ he still warned the reader not to attach too much value to such a single observation. In the 1970s, however, he would discover that such observations on the contrary were a productive source of insights. The need to “catch up on a collective scale” as he called this in ‘Rekendidaktiek’ where observations were concerned, had completely disappeared 30 years later [62]. The difficult times, in particular during the last years of the war, and all the new developments that would happen after the liberation made it probable that ‘Rekendidaktiek’ would remain an intermezzo within Freudenthal’s work. Nothing was further from the truth. “all positive action starts with criticism” he wrote, and seen in that light ‘Rekendidaktiek’ seems a worthy start of Freudenthal’s efforts in the field of mathematics education and didactics of mathematics.

References

79

References 1. Freudenthal, Schrijf dat op, Hans 337. As stated in section 4.1, the manuscript dated from 1944. 2. Diaries; RANHRANH, Hans Freudenthal Papers, inv. nr. 147; Hans Freudenthal in Schrijf dat op, Hans 335–363. 3. Freudenthal, Schrijf dat op, Hans 338. 4. Notebook containing notes of a literature study on didactics of arithmetic accompanied by a separate index of subjects; RANHRANH, Hans Freudental Papers, inv. nr. 440. 5. Hans Freudenthal, ‘Rekendidaktiek’ (unpublished 1944) 6–7; RANH, Hans Freudenthal Papers, inv.nr. 465. 6. Freudenthal in Schrijf dat op, Hans 337. 7. Diaries 1940–1945; RANH, Hans Freudenthal Papers, inv. nr. 147. 8. Ibid. 9. It was in that period that Dijksterhuis wrote the first concept of his Mechanisering van het wereldbeeld (The mechanization of the world picture) (Amsterdam 1950) and that astronomers in Leyden and Utrecht prepared to break new ground; Klaas van Berkel, ‘Kwaliteit en zuinigheid in het wetenschappelijk onderzoek’, in: Kees Schuyt and Ed Taverne, ed., 1950. Welvaart in zwart-wit (Den Haag 2000) 331–354, page 333. 10. Diaries 1940–1945; RANH, Hans Freudenthal Papers, inv. nr. 147. 11. Freudenthal, ‘Rekendidaktiek’ 2; RANH, Hans Freudenthal Papers, inv.nr. 465. 12. Ibid. 13. Diaries 1940–1945,. 1 December 1941; RANH, Hans Freudenthal Papers, inv. nr. 147. 14. Diaries 1940–1945, 1 April 1942; RANH, Hans Freudenthal Papers, inv. nr. 147. 15. Freudenthal, ‘Rekendidaktiek’ 3; RANH, Hans Freudenthal Papers, inv.nr. 465. 16. Ibid., 4. 17. Freudenthal would return to the phenomenon of transfer within the teaching of mathematics extensively in his lecture ‘Opvoeding tot denken’ (Educating for thinking) of 1945. More about this can be found in paragraph 5.1.3. 18. Freudenthal, ‘Rekendidaktiek’ 19, 21, 23, 26, 34, 58, 65, 95, 101 and 103; RANH, Hans Freudenthal Papers, inv.nr. 465. 19. See also the quotation on page 69. 20. Freudenthal, ‘Rekendidaktiek’ 103; RANH, Hans Freudenthal Papers, inv.nr. 465. 21. The following studies were used to derive data for this paragraph: M. du Bois- Reymond, J. Lamboo, M. Smits, T. Walaardt-Sacré, Onderwijzersleven (Nijmegen 1981); Dodde, Het Nederlandse onderwijs verandert; A. Leen, De ontwikkeling van het rekenonderwijs op de lagere school in de 19e en het begin van de 20e eeuw (Groningen 1961); Nelissen, ‘Eeuwenlang rekenen’ (‘Ages of arithmetic’), Willem Bartjens 18 (1998/99) 36–39. 22. M. du Bois-Reymond, J. Lamboo, M. Smits, T. Walaardt-Sacré, Onderwijzersleven 102. 23. P.H. Heijnen, Methodiek van het rekenen in de lagere school, ten dienste van normaallessen en kweekscholen (Groningen 1917) 7. 24. Quoted in: Leen, De ontwikkeling van het rekenonderwijs op de lagere school 133–134. 25. Heijnen, Methodiek van het rekenen in de lagere school 1. 26. Heijnen, Methodiek van het rekenen in de lagere school 2. 27. See also paragraph 2.2.2. 28. More about Jan Versluys and didactics of mathematics in: De Moor, Van vormleer naar de realistische meetkunde 120 and next. 29. Ibid., 120. 30. Freudenthal, ‘Rekendidaktiek’ 1; RANH, Hans Freudenthal Papers, inv.nr. 465. 31. Ibid., 6. 32. Ibid., 7. 33. Ibid., 10. 34. Ibid., 15. Freudenthal talked about Kühnels Neuaufbau des Rechenunterrichts from 1919.

80


35. Ibid., 23. 36. Ibid., 24. 37. However, Freudenthal would not get round to discussing those ‘positive aspects’ of psychology, pedagogy and general didactics in ‘Rekendidaktiek’. 38. Freudenthal, ‘Rekendidaktiek’ 29; RANH, Hans Freudenthal Papers, inv.nr. 465. 39. Ibid., 35–36. 40. Ibid., 47. 41. Ibid., 51. 42. Ibid., 56. 43. Ibid., 60. 44. This interview will be discussed in paragraph 9.4.1. 45. Freudenthal, ‘Rekendidaktiek’ 71; RANH, Hans Freudenthal Papers, inv.nr. 465. 46. Ibid., 81. 47. Ibid., 85. 48. Ibid., 89–90. 49. Ibid., 91. 50. Ibid., 101–102. 51. See also paragraph 4.2. 52. Diaries 1940–1945, 20 April 1943; RANH, Hans Freudenthal Papers, inv. nr. 147. 53. Notebook from 1943; RANH, Hans Freudenthal Papers, inv.nr. 440. 54. See chapter 9. 55. Freudenthal, Schrijf dat op, Hans 338. 56. Freudenthal, Weeding and Sowing. 57. Freudenthal, Mathematics as an educational task 141. 58. Hans Freudenthal, Schrijf dat op, Hans 338. 59. About ‘learning to think’ and ‘formative value’, see for instance: Hans Freudenthal, ‘Opvoeding tot denken’ (unpublished 1945) (a lecture held at a conference of the WVO): RANH, Hans Freudenthal Papers, inv.nr. 466; Hans Freudenthal and T. Ehrenfest-Afanassjewa, Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? Discussie tussen T. Ehrenfest-Afanassjewa en Hans Freudenthal (Purmerend 1951); Hans Freudenthal, ‘Erziehung des Denkvermögens (Diskussionsbeitrag)’, Archimedes 6 (1954) 87–89. On the teacher’s training see for instance: Hans Freudenthal, ‘Oude en nieuwe universiteiten’ (unpublished 1945): RANH, Hans Freudenthal Papers, inv.nr. 467; Hans Freudenthal, ‘Tekort aan leraren’, De Groene Amsterdammer (3 juli 1948); Hans Freudenthal, ‘Vakgeleerde of paedagoog?’, Christelijk Schoolblad 2132, 2132 and 2133 (1953). 60. Freudenthal, Schrijf dat op, Hans 338. 61. About Freudenthal’s critique on Piaget’s work, see paragraph 9.6.3. 62. More about Freudenthal and the importance he attached to observing children in chapter 9.

Chapter 5

A New Start

Ik ben geen doorzetter, al zullen anderen zeggen van wel. In elk geval geen doordrammer. Maar het niet doorzetten werd me ook makkelijk gemaakt door al het werk dat er na de bevrijding vóór me lag en dat ik op me nam. Werk in het onderwijs volgens ideeën die ik in de tijd van ‘lediggang’ had ontwikkeld, werk aan de universiteit, in de faculteit, in commissies en commités, in verenigingen en besturen, maatschappelijk werk—om van het wetenschappelijke maar geheel te zwijgen. Ik had bergen te verzetten—althans datkon ik me verbeelden. (I am not overly persistent, even though others will say I am. In any case I am not dogmatic. But not persevering has been made easy for me by all the work that I had in prospect after the liberation, which I (gladly) took on. Work in the educational system following the ideas I developed during the ‘idle’ times, work at the university, in the faculty, in commissions and committees, in associations and on boards, social work—not to mention the scientific work. I had mountains to move—at least, that is what I envisaged.) Hans Freudenthal in Schrijf dat op, Hans, 1987 [1]

For Freudenthal, the liberation of May 1945 meant the end of years of forced unemployment and ‘idleness’ [2]. Forty years later he recalled the liberation and the mathematician J.F. Koksma from Amsterdam: Hij zag me bij de intocht van de Canadezen op een tank zittend voorbijkomen en zei later daarover: ‘Toen ik Freudenthal op die tank zag zitten wist ik pas zeker dat de oorlog afgelopen was.’ (He saw me passing by, sitting on a tank when the Canadians entered, and later said: ‘When I saw Freudenthal sitting on that tank I knew for sure that the war was over.’) [3]

Full of optimism and in good spirits Freudenthal expected to be able to go back to his work at the University of Amsterdam. However, his reappointment as curator on 7 May marked the beginning of a laborious struggle for his rehabilitation. After a short suspension and a reprimand by the College van Herstel en Zuivering (Board


81

82

5 A New Start

of Recovery and Purification) Bruins kept the function of lecturer. In Freudenthal’s opinion he had been promised this position before the outbreak of the war [4]. But Bruins, as advised by the Board of Recovery and Purification, refused to resign. Brouwer supported Bruins in this and did nothing to contribute to Freudenthal’s rehabilitation. For Freudenthal the resulting situation was an unacceptable insult to him. He wrote to the Mayor and Aldermen of Amsterdam: Ik verzoek Burgemeester en Wethouders met de meeste eerbied, het mij niet euvel te willen duiden, wanneer ik mij beroep op 15 jaar diensttijd aan deze universiteit, op de positie, die ik in de wetenschappelijke wereld meen in te nemen, en op de verwachtingen van mijn leerlingen, die aan de Amsterdamse Universiteit verder zullen studeeren, examens doen of een proefschrift indienen, en wanneer ik aan mijn verzoek om herplaatsing de hoop verbind, dat de universitaire en gemeentelijke instanties gezamenlijk een weg zullen mogen vinden om mijn herplaatsing te doen geschieden op een wijze, waarbij ik royaal word gerehabiliteerd na de disqualificatie, die voor elk buitenstaander moest blijken uit de carrière van mijn opvolger, die als natuurkundige op wiskundig-wetenschappelijk gebied een vreemdeling was en bleef en desondanks na één jaar diensttijd benoemd werd tot lector—een benoeming, die mij na haast tien jaar diensttijd ondanks talrijke beloften nog niet ten deel was gevallen. (I respectfully appeal to Mayor and Aldermen, not to impugn my motives when in this request, I refer to my 15 years of employment at this university, to the position I think I hold in the scientific world, and to the expectations of my students, who, at the University of Amsterdam, will continue their studies, do exams or defend their theses, and when I connect to my request for reappointment the hope that academic and municipal authorities together may find a way to bring about my reappointment in a manner in which I will be generously rehabilitated after the disqualification, that had to become clear to every outsider from the career of my successor, who, as a physicist, was and remained a stranger in the field of the science of mathematics and nevertheless was appointed lecturer after only one year of employment—an appointment that, despite numerous promises, was not bestowed on me after almost ten years of employment.) [5]

This letter was the first of an extensive, but futile exchange of letters between Freudenthal, Mayor and Aldermen, Bruins and others, which ended on 8 November 1945 when Freudenthal resigns from his function as curator at the University of Amsterdam to be able to accept a chair in Utrecht. A bitter disappointment: Amsterdam-Utrecht was geen reis, maar een langdurig proces waarvan de pijnlijke wonden veel tijd nodig hadden om te helen. (From Amsterdam to Utrecht was not a journey, but a long process with painful wounds that took ages to heal.) [6]

…but also a positive new start. At the end of his inaugural lecture in Utrecht on 9 December 1946 Freudenthal said goodbye to ‘his’ Amsterdam: Er resten mij nog enkele woorden te zeggen. Zes moeilijke jaren zijn aan dit plechtige uur vooraf gegaan. Ik ben ze met mijn gezin te boven gekomen, en ik heb mij zelfs nog enigszins aan mijn studiën kunnen wijden, dank zij vele goede vrienden, en zij vooral hebben het mij lastig gemaakt, om van Amsterdam afscheid te nemen. Ik kan geen woorden van dank op dit ogenblik met groter hartelijkheid uitspreken dan die voor de materiële en morele steun, die ik van hen heb ondervonden. Velen hunner, die ik bedanken wil, zijn aanwezig; zij mogen zich door deze woorden getroffen voelen. Anderen ontbreken, maar deze zaal zou te klein zijn, om plaats te bieden aan allen, die ik bedoel: gans het Nederlandse volk. (Only a few words remain for me to say. Six difficult years preceded this solemn hour. I managed to get through them along with my family, and I was even able to devote myself a little to my studies, thanks to good friends, and they in particular have made it difficult

A New Start

83

for me to say goodbye to Amsterdam. There are at this moment no words of gratitude that I can express with greater sincerity than those for the material and moral support that they gave to me. Many of those I want to thank are present; they may feel moved by these words. Others are absent, but this hall would be too small to accommodate all those I mean: the entire Dutch people.) [7]

During the summer of 1947 Freudenthal finally moved to Utrecht with his family. This disappointingly tough start after the war did not keep Freudenthal from enthusiastically and convincingly disseminating the results of the work and studies he conducted during the war. More clearly than before the war he operated as a socially interested and unconventional person geared to modernization. Supported by the regard and prestige he had already enjoyed in his field of study (boosted in 1946 by his position as a professor in Utrecht), he did not recoil from going against the pre-war establishment. His attitude fitted the spirit of post-war innovation and reconstruction in the Netherlands, a period of a certain gloominess and pessimism, but at the same time one of “disciplined energy” as Kossmann described it [8]. It also matched the changing position that science, and in particular natural sciences and mathematics, would take up in society. More emphasis on and demand for practical applications forced the scientists to come down from their ‘ivory tower’ and practice science less isolatedly: less isolated from each other, but also less isolated from society. ‘Teamwork’ and the development of applied sciences were on the rise. In addition the conviction prevailed that the Netherlands after 5 years of occupation had to make up arrears, both in the field of pure and applied sciences. This resulted in founding such institutions as the Centraal Planbureau (CPB Netherlands Bureau for Economic Policy Analysis, 1945), Fundamenteel Onderzoek van de Materie (Foundation for Fundamental Research on Matter, 1946) and the Vereniging voor Statistiek (Netherlands Society for Statistics and Operations Research, 1945). The establishment of the Mathematisch Centrum (Mathematical Centre, MC) on 11 February 1946 was also a good example. Its founding aim was “to establish a centre for scientific-mathematical activities and at the same time to devise means to make closer contact between pure mathematics and its applications in other fields” [9]. Alberts called the Mathematisch Centrum “the outcome of a contemplation of the social role of mathematics” [10]. The Second World War had shown the public at large what the consequences of the ‘planning’ of science could be: the development of atomic weapons, the discovery of penicillin and the development of large calculators in America, the predecessors of the computer, were examples. In addition to causing high hopes for the role of science in the future it also engendered a certain fear for the consequences of these opportunities. The far-reaching consequences that scientific work could have for society forced the scientists to reflect on the position they took up in that society and the responsibility involved in it. The foundation of the Verbond van Wetenschappelijke Onderzoekers (Association of Scientific Researchers, VWO) in 1946 was a direct consequence of this awareness. It was also an expression of the wish to bridge the gap between science, politics and public and, by deepening the sense of social responsibility of researchers, to achieve the highest possible output for man and society’ [11].

84

5 A New Start

Freudenthal remained anything but aloof from these developments. His regular visits to general meetings of the PvdA (Partij van de Arbeid, the Dutch Labour Party) betrayed his political preference. This was also reflected in the choice of the magazines in which he published, such as the humanistic paper De Nieuwe Stem, Maandblad voor Cultuur en Politiek (The New Voice. Monthly for culture and politics) and the originally communistic but by then more generally left-oriented paper De Vrije Katheder (The Free Lectern). Freudenthal became a member of the VWO in 1947, and from 1950 onwards was a member of the general committee, for a few years even being vice-chairman. He also contributed to informing the public at large about the latest developments in science by his reports of the meetings of the Koninklijke Academie van Wetenschappen (Royal Academy of Sciences) in De Groene Amsterdammer [12]. Freudenthal was not directly involved in the establishment of the MC but he mainly played the role of a critical observer. His criticism of the functioning of the MC mainly boiled down to the fact that its activities were too much directed at Amsterdam and its surroundings, which resulted in the national aspect of the Centre not being sufficiently expressed. Alberts says about the fact that Freudenthal and others were not involved in the establishment of the MC: Niet aan de beraadslagingen neemt deel een aantal aankomende wiskundigen van naam als Freudenthal, Haantjes, Heyting en Kloosterman. Dat zij de volgende generatie zouden gaan vormen was ook toen al zonneklaar. […] Om te begrijpen dat ze in het geheel niet betrokken waren in de voorbereidende beraadslagingen voor het mc, moeten we bedenken dat, in die tijd ‘een hoogleraar nog een hoogleraar was’. (A number of prospective mathematicians of standing among whom Freudenthal, Haantjes, Heyting and Kloosterman are not participating in the consultations. That they would constitute the next generation was also obvious at that time. […] In order to understand that they were not at all involved in the preparatory consultations for the MC, we have to remind ourselves that at the time ‘a professor was still a professor’.) [13]

The fact that several chairs still had to be assigned played a role as well. In October 1945 the Groningen professor G. van der Leeuw, Minister van Onderwijs, Kunsten en Wetenschappen (Minister of Education, Art and Science) had appointed a ‘Commissie tot Coördinatie van het Hoger Onderwijs in de Wiskunde in Nederland’ (Committee for the Coordination of the Higher Education in Mathematics in the Netherlands). Chairman of this committee was Professor J.G. van der Corput, also from Groningen, but the members also included Van Dantzig, Koksma and the Utrecht astronomer Minnaert—in the absence of a Utrecht mathematics professor. The main task of the committee was to prepare the establishment of the MC. In addition this committee was given a coordinating and advising task in assigning vacant chairs [14]. In Utrecht, after the war, two chairs were still vacant: one because of the retirement of J.A. Barrau in 1943 and one caused by the death of Professor J. Wolff who died in a concentration camp in 1945. By the agency of the committee members Van Dantzig and Van der Corput a complicated and dubious ‘game of musical chairs’ followed, in which Van Dantzig, shortly before his appointment in Amsterdam, induced the committee to appoint him Professor in Utrecht from

A New Start

85

Fig. 5.1 A group picture on the occasion of Freudenthal’s appointment in Utrecht (1946), with (among others) on the front row from left to right: Mrs. Van Dantzig, Koksma and his wife, Mrs. Van der Corput, Barrau and his wife, Freudenthal and his wife (with seated on the floor their sons Didi and Thijs), and next to them Van der Corput. Second from the left in the back row is De Iongh, to the left and behind Barrau is Van Dantzig, behind Mrs. Barrau in the back row is Minnaert and behind Mrs. Freudenthal in the back row is Popken

March to May 1946. Brouwer, who wanted to remove Freudenthal from Amsterdam, then approached Minnaert about an appointment of Freudenthal in Utrecht [15]. Some of his students, among whom Van Est, campaigned to plea to the alderman in Amsterdam on behalf of Freudenthal, but this didn’t help much. When it became clear that he could not expect rehabilitation in Amsterdam and that Van Dantzig and Van der Corput were not offering a lot of support, Freudenthal finally accepted the geometry chair in Utrecht (Fig. 5.1 and Fig. 5.2). J. Popken filled the vacancy in analysis, previously the chair of Wolff. H.B.A. Bockwinkel was at the time lecturer ‘first-year mathematics’ in Utrecht. Despite these complications Freudenthal was an active supporter of the aims of the MC. He gave, for instance, lectures during the MC summer courses for teachers in the years from 1947 to 1950 and on his own initiative he conducted a course ‘Numerieke en graphische methoden’ (‘Numerical and graphical methods’) in Utrecht for which he requested cooperation from the MC. For the accompanying exercises he borrowed calculators from the Life Insurance Company ‘Utrecht’ [16]. Forty years later he said: Altijd heb ik me zo opgesteld dat ik, ook als ik het ergens niet mee eens was, toch zakelijk meewerkte. Maar hoewel ik ook van binnen uit heb getracht bij het mc invloed uit te oefenen, heeft men nooit ergens naar geluisterd. Nu had ik natuurlijk ook mijn ambities, wilde bepaalde dingen bereiken en misschien dachten ze wel: ‘Wat moeten we met zo’n

86

5 A New Start lastpak als Freudenthal?’ (I have always adopted the attitude of cooperating in a businesslike manner, even when I do not agree with something. But, even though I tried to influence the MC from the inside, they have never listened to anything I said. Of course I had my ambitions, I wanted to achieve certain things and maybe they thought: ‘What are we supposed to do with a troublemaker like Freudenthal?’) [17]

A very practical example of Freudenthal’s contribution to the realization of the pursuit of ‘mathematics in service of society’ was his cooperation with J. Sittig, who worked with the Adviesbureau voor Toegepaste Statistiek (Consultancy for Applied Statistics) and cofounded the Vereniging voor Statistiek. Commissioned by De Bijenkorf (a Dutch department store) they developed a sizing system for ready-towear clothes in 1948 [18]. This too was an example of the variety in the interests and the apparently inexhaustible drive that Freudenthal showed in these areas in addition to his day to day work as a professor.

Fig. 5.2 Freudental in Euclides (1947)

5.1 Educating

87

Initially that day to day work involved, in addition to his mathematical work, many mainly practical problems as well. There was for instance the problem of housing the Mathematisch Instituut. In 1947 the building in the Boothstraat was rented for this purpose. When they finally moved into this building the increasing numbers of students made it too small from the start, and in the 1950s several extensions and annexes followed. “In the intervals between lectures we had to run from one institute to another,” Freudenthal recalls [19]. Another of Freudenthal’s causes was the struggle to recapture some of the time that mathematics students had to devote to the physics labs. He wanted to free this time for mathematics labs similar to those he had organized in Amsterdam [20]. There were also discussions about the content of the curriculum with the physicists, for instance about the position of linear algebra within physics. In 1947 Freudenthal was able to appoint a first assistant, Willem Titus van Est (1921–2002). Freudenthal knew Van Est from the time he was a physics student in Amsterdam. He was Freudenthal’s first PhD student and obtained a permanent appointment in Utrecht in 1951 [21]. But what about Freudenthal’s ideas about didactics of arithmetic? How was it possible, for such a busy professor as Freudenthal was, to go from ‘Rekendidactiek’ during the war to ‘the Wiskunde Werkgroep of Freudenthal’?

5.1 Educating 5.1.1 Educating at Home After the liberation no one could reproach Freudenthal for having a wait-and-see attitude. He seemed to be in a hurry to publish his pedagogic-didactical ideas from the war period [22]. Initially he chose as his public the innovative members of the Werkgemeenschap voor Opvoeding en Onderwijs (Cooperative for Modernization of Upbringing and Education), with which he was already familiar through the journal Vernieuwing van Opvoeding en Onderwijs (Modernization of Upbringing and Education). His wife already subscribed to it before the war [23]. In the family she was the one most expressly occupied with education in general and with the upbringing of their four children in particular. It was her quest for the ideal school for their children that stirred her interest for modernization of the educational system. While fetching books for her husband from the library during the war, she stumbled on a series of outdated volumes of The New Era in Home and School, in which she found many of her own ideas and opinions reflected [24]. However, no single type of education seemed to bring universal happiness. In the end the eldest son attends a public school, the three younger children go to a Montessori kindergarten, followed by various primary schools and lycea (including a Montessori school, a protestant school and the Werkplaats (Workshop Community) in Bilthoven [25]. Both parents closely and critically kept up with the school results of the children. Neither Freudenthal nor his wife hesitated to notify the school

88

5 A New Start

management when the results disappointed them or when they did not approve of the educational methods [26]. In 1952 she would find in the ideas of Petersen the educational method that lived up to her expectation [27]. In her book Naar de school van morgen (Towards tomorrow’s school) she wrote that in the upbringing of her own children she was mainly led by the ideas of the teacher-pedagogist Jan Ligthart (1859–1916) [28]. Freudenthal’s involvement in the upbringing of his children primarily revealed itself during the war. The daily walks, ‘playing school’ with his three sons and the many notes from their conversations dating from that time are evidence of it [29]. His personal correspondence from this time too showed his commitment to the development and the achievements of his children. From time to time he noted the weight, height and report marks of his children in his pocket diaries [30]. He would later use many of these notes as illustrations in his publications about education. On the difference between the upbringing of children in the Netherlands and that in Germany he remarked: Nederlanders vond ik indertijd, toen ik hier kwam, sympathieker dan de meeste Duitsers waar je in die tijd mee in aanraking kwam. De manier waarop hier kinderen werden opgevoed was mij liever—rustiger—dan het slaan en het geweld dat in Duitsland gebruikelijk was. (At the time I arrived here I liked the Dutch better than most of the Germans you would meet then. The way in which children were raised here suited me better—it was calmer—compared to the beating and the violence common in Germany.) [31]

His appointment in Utrecht and the amount of work he took on after the war, however, took all his time. Little time and attention was left for his children with the youngest only a toddler. After 5 years of enforced idleness at home it was quite a transition for Freudenthal. He wrote to a friend of the family in Jerusalem about this other side of his new appointment: Aber man muß arbeiten wie ein Sklave und hat gar keine Zeit, sich um seine Familie zu kümmern. Im Kriege bin ich noch viel mit meinen Kindern spazieren gegangen, aber nun muß ich sie eigenen Wege gehen lassen. (But you have to work like a slave and don’t have time to worry about your family.)During the war I often went for walks with my children, but now I will have to let them discover their own paths.) [32]

5.1.2 ‘Our Task as Present-Day Educators’ In July 1945 on the occasion of the imminent summer conference Freudenthal wrote a letter to J.R. Janssen, the secretary of the Central Workgroup of the WVO. After a description of his war-time study in the field of “pedagogy of arithmetic and mathematics, and education for thinking in the broadest sense”, he continued with (Fig. 5.3): Nu de omstandigheden het voor mij weer mogelijk maken, zal ik spoedig mijn onderzoekingen bekend maken. Daar ik mij vooral aan de moderne stromingen in de Engels sprekende landen georiënteerd heb, lijkt mij Uw conferentie een geschikte plaats om over het onderwerp “Opvoeding tot Denken” iets uitvoeriger te spreken dan in een discussie mogelijk is. Zou het misschien mogelijk zijn, een soort inleiding over dit onderwerp te houden? Zo ja, hoeveel spreektijd zoudt U voor mij uit kunnen trekken? Met het oog op

5.1 Educating

89

Fig. 5.3 Letter to J.R. Janssen of the WVO in which Freudenthal proposes to speak about ‘Education for thinking’ during the summer conference of the WVO (31 July 1945) de nodige voorbereiding zie ik Uw mededeling gaarne tegemoet. (Now that circumstances allow me I will soon publish my investigations. Since I explored primarily modern movements in English-speaking countries your conference seems the appropriate place to speak about the subject “Education for thinking” in a bit more detail than is possible in a discussion. Would it perhaps be possible to give an introductory lecture about the subject? If your answer is positive, how much time could you allow me? In view of the necessary preparation I look forward to hearing from you.) [33]

The conference organizers gladly accepted this offer. Janssen, who was also chairman of the Wiskunde Werkgroep (WW), could not refrain from drawing Freudenthal’s attention to the existence of this working group. In his answer to Freudenthal he wrote: Misschien weet U niet, dat er voor de oorlog een Wiskunde Werkgroep van de W.V.O. bestond […] We hopen, dat deze werkgroepen binnenkort ook weer aan het werk zullen kunnen gaan en zullen dan gaarne profijt trekken van de studie, die U reeds van dit onderwerp gemaakt heeft. (Perhaps you are not aware of it but before the war there was a Wiskunde

90

5 A New Start Werkgroep (Mathematics Working Group) of the WVO […] We hope that these working groups will be able to resume their studies soon and we will gladly benefit from the study that you have already done concerning this subject.) [34]

It would take until January 1947 before Freudenthal first attended a meeting of the Wiskunde Werkgroep [35]. The 1945 summer conference of the WVO was entitled ‘Onze taak als opvoeders nu’ (‘Our task as present-day educators’) and was held from 11 to 14 August on the Rhederoord estate in De Steeg. Despite the many (travel) problems confronting the organizers, the conference was a great success. The participants arrived “from near and far in the country, on bicycles without tyres, with ration cards and little bags full of potatoes” [36]. Now that the war was over, the demand for modernization of the educational system increased with great intensity. At the conference, Kees Boeke, chairman of the Centrale Werkgroep (Central Working Group), launched the plan to make an inventory of all modernization attempts in the Netherlands and in that way come to a concerted action plan [37]. Freudenthal’s lecture ‘Education for thinking’ matched very well with the within the WVO popular outlooks such as ‘education is guiding a child in its natural growth’ and ‘formation and education in the service of peace’. With his striving for change and modernization of education he had found his niche within the WVO [38]. The view of most of the modernizers represented in the WVO was that the school was more than just a place to acquire knowledge. It was a place where you learned by being educated, where children were helped ‘to become what they are’, the way it was in Montessori and Dalton schools. Modernizing society to prevent a new war started with modernizing the educational system. Hence the how and why of the ‘formative value’ and ‘learning to think’ from Freudenthal’s lecture were in this context interesting and important points.

5.1.3 ‘Education for Thinking’ In his lecture ‘Education for thinking’ Freudenthal mainly emphasized the practical approach to ‘learning to think’ in general [39]. He said that he owed it to studies such as professor Kohnstamm’s school based on the psychology of thinking and American influences, that he had gained the insight that you could teach something without knowing what that something was. He did not further elaborate on Kohnstamm’s work in particular, but, as will become clear, he borrowed much from his articles and agreed readily with his insights. Logic as the science of the forms of learning to think was not of much use according to Freudenthal: it was too rigid, had nothing to do with the reality of thinking and was not suited to describe conclusions drawn from daily life [40]. With this he endorsed Kohnstamm who had arrived at such conclusions in his article ‘De formele logica en het kinderlijk denken’ (‘Formal logic and childlike thinking’) which Freudenthal had read in 1943 for his literature study [41]. In this article Kohnstamm discussed various intelligence studies among which psychologist Hans Paul Roloff’s study in which children were asked to define concepts [42]. The

5.1 Educating

91

Fig. 5.4 Fragment from Freudenthal’s ‘Rekendidaktiek’ notebook (1943)

starting point of this study was that there would be only one correct way to define notions and that was the genus proximum and differentia specifica approach, which Kohnstamm rather questioned in his article. Freudenthal did not mention this study but implicitly refers to it [43]. Elke pedagoog zal het ook af en toe proberen, en toen ik eens een van mijn jongens—hij was toen vijf—liet definiëren, wat een ding is, kreeg ik het antwoord ‘een ding is, als je aan iets denkt en je weet niet hoe het heet’, een prachtdefinitie, maar afgrijselijk onder de loep van de aan de genus proximum en differentia specifica verknochte logicus. (Every pedagogue will once in a while try it and when I asked one of my boys—he was five at the time—to define a thing, he answered ‘a thing is if you think of something but do not know its name,’ a beautiful definition but a horrible one to view through a looking glass for a logician attached to genus proximum and differentia specifica) (Fig. 5.4.) [44]

It made no sense to teach ‘empty’ logical forms and schemes to learn to think. Such forms should not be presented empty, but ‘filled with life’s content’. We should not trouble students with our greater knowledge about thinking, instead we should use that to separate, order and summarize the subject matter according to difficulty. As an example Freudenthal mentioned the study by C.W. Washburne, from which it appeared among other things that ‘fractions’ should not be part of teaching arithmetic in primary schools [45]. To Freudenthal, however, so-called arithmetic word problems were of greater social importance than fractions and long divisions. Dat er aan het onderwijs in redactiesommen iets hapert, weten we, maar waar het hem precies in zit en hoe wij het kunnen verhelpen, dat zou het onderwerp moeten zijn van een zeer gedetailleerd en zeer nauwkeurig voorbereid onderzoek. (We know that there is something wrong with teaching arithmetic word problems, but we don’t know exactly what nor how to remedy it, and that should be the subject of a very detailed and carefully prepared study.) [46]

There was no need at all for a transfer between two different formulations of one and the same arithmetic problem, according to him. Freudenthal also believed that American studies into transfer showed us that practice in a specific area was of benefit only to very adjacent areas. Which areas and how this happened should be investigated. Freudenthal proposed making a selection from the many thousands of arithmetic word problems one thousand essential ones, to order these according to age and to investigate how many were needed to get enough practice. These problems were to be used as a basis to construct the method for arithmetic word problems. Enkelen zullen verbaasd opkijken, wanneer ze horen, dat het oplossen van dergelijke sommen iets met denken te maken heeft. Toch is het zo. Met veel ervaring brengt men het ook op dit gebied tot werktuiglijkheid, tenminste zolang de vraagstukken van een vrij eenvoudig type zijn. Maar met het moeilijker en onaanschouwelijker worden van het type komt er ook bij de meest geoefenden een grens aan de werktuiglijkheid, en dan blijft er niets anders over dan te denken. (Some will be surprised when they hear that solving such problems has

92

5 A New Start something to do with thinking. Yet it has. A lot of experience, also in this field, results in a rote response at least as long as the problems are of a relatively simple nature. But when the nature of the sums becomes more difficult and less illustrative even the most experienced student’s rote response is limited and thinking is the only remedy left.) [47]

Freudenthal implicitly distinguished between a mechanical approach and what he called ‘thinking’. ‘Thinking’ to him contained at least a certain amount of original action and creativity on the part of the thinker. As soon as a problem could be solved mechanically by ‘rote’, then in his definition ‘thinking’ was no longer involved. According to Freudenthal the four fundamental arithmetical operations could best be learned by rote. Er bestaat geen reden, om de leerlingen deze bewerkingen anders dan werktuiglijk te leren, evenmin als er aanleiding bestaat om iemand eer hij een schakelaar om mag draaien, de hele theorie van de electriciteit uit te leggen. De kinderen maken zeer bepaalde fouten bij de bewerkingen (originaliteit is in deze geheel zoek), maar er is geen aanwijzing, dat deze fouten door een beredenering van de werktuiglijke behandeling voorkomen zou kunen worden. Het is zelfs heel waarschijnlijk, dat de beredenering schade sticht, door de werktuiglijkheid tegen te werken. (There is no reason to teach students these fundamental operations other than by rote learning, just as there is also no reason to explain the complete theory of electricity to someone before he is allowed to flick a switch. Children make very particular errors in these operations (originality is completely lacking here), but nothing indicates that these errors could be prevented by explaining the rationale behind the rote response. It is even probable that reasoning causes harm by hindering the rote response.) [48]

Contrary to the four fundamental operations thinking was indispensable in solving arithmetic word problems. But as long as we knew so little about thinking it was the only thing we could do: practicing meaningful problems. By meaningful Freudenthal meant the type of problems that a child would need as a future member of society. So much for arithmetical thinking, which according to Freudenthal, despite its social indispensability for the student, only took up a very modest position within thinking and certainly should not be overestimated in the larger context of thinking (or learning to think). The most important aid for children in (unintentionally) acquiring mental performances was reading, according to Freudenthal: a child who reads twice as much, learns to think twice as much. Freudenthal said that in recent years grammatical exercises had turned into puzzles. This happened under the influence of the intelligence tests as developed by the Americans. These were perhaps useful for research purposes, but not for exercises: every type of puzzle offered only practice for one way of thinking (which was, furthermore, an artificial way). Freudenthal regretted that the Dutch educational system, instead of investigating their use (for example as a criterion for entering secondary or higher education), had rejected such tests as aids. Freudenthal was an advocate of the use of exercises in thinking with a dual character: not only as a test but also as an exercise in ways of thinking, having the appearance of ‘stories taken from real life’ that end in a question. These exercises had to appeal to the self-motivation and as such should indicate a task. Freudenthal did not give an example of such an exercise. He said that such a scheme could be

5.1 Educating

93

found with the American educational philosopher J. Dewey and his followers who had organized the Project Method in education and teaching. But Freudenthal was by no means enthusiastic about this. De boeken van voorstanders zijn gekenmerkt door een—soms belachelijke—naïviteit en door een vervreemding van het leven, die weinig bij hun algemene denkbeelden past. De realiteit van hun plannen is namelijk topzwaar door een overtollig totaliteitsaspect; om één enkel project heen wordt het gehele onderwijs gegroepeerd—iets dat tot hypertrophieën aan de ene en schromelijke tekorten aan de andere kant moet leiden, en dus een behoorlijke organisatie van het onderwijs tegenwerkt. (The advocates’ books are characterized by a— sometimes ridiculous—naïveté and by an alienation from life which does not match their ideas. The fact is that the reality of their plans is top-heavy, caused by a superfluous aspect of totality; the whole of the education is clustered around a single project—and this had to lead on the one hand to hypertrophies and on the other to gross deficits, and therefore opposes a good organization of the educational system.) [49]

Freudenthal himself could not yet give a ready-made solution for teaching thinking: Met de uitwerking van dit programma [uit het leven gegrepen verhaaltjes die aan zelfwerkzaamheid appelleerden] zou men de eerste stappen doen in de richting naar een denkonderwijs in het raam van onze volksopvoeding. De eerste stappen—zeg ik, want wat het denkonderwijs zou moeten behelzen, zullen wij pas doende en proefondervindelijk uit alle fouten lerend, beseffen. (In drawing up this programme [stories taken from real life appealing to self-motivation] one could take the first steps towards an education in thinking in the framework of educating our people. I deliberately say ‘the first steps’, since what education in thinking should amount to we will only realize while we are experiencing it in practice and are learning from our mistakes.) [50]

Freudenthal did have an idea what arithmetic word problems should look like: he preferred the synthetic type, which required one great effort of the child, to the analytical type, in which you so to speak only had to pull the right string to thrash out the solution [51]. Research into transfer training taught us, according to Freudenthal, that secondary mathematics education only sharpened mathematical ingenuity. It was for that reason not justified to hold on to a programme that required pupils without talent to do mathematics. Freudenthal argued in favour of setting up a type of secondary school that would not include mathematics, not even as an optional subject. What a pupil learned should, in his view, depend on his capabilities not on a choice. It so happened that in secondary schools rote learning rather than thinking was chosen as a concession to the masses of mathematically untalented. Maar het is toch niet aan [sic] de haak, dat vrijwel alle eerstejaarsstudenten wiskunde hun studie beginnen met een algehele onervarenheid in het wiskundig denken. (It is not quite right that practically all first-year mathematics students start with their study totally inexperienced in mathematical thinking.) [52]

Since the patterns in mathematical thinking were, according to Freudenthal, more abstract, more ‘form’ than those in daily life, mathematical thinking was easier to learn than arithmetical thinking, which was more related to daily life. At the university Freudenthal argued in favour of practice in the form of tutorials, such as he gave before the war. In that way the contact between lecturer and student was the ideal form for university education.

94

5 A New Start

Apart from thinking in general,—arithmetical thinking and mathematical thinking, Freudenthal also mentioned manual thinking in his lectures, about which hardly anything was known according to him. There was a large shortage of what he called ‘all-round-manual-thinkers’: (prospective) craftsmen who were educated without being focussed on a single specific craft. This shortage should be filled up by, for example, a secondary schooling around a core of manual skill and manual thinking. Eventually Freudenthal concluded that the problem of education of the masses was still unsolved. The class teaching system should also be scrutinized. The practice accompanying Freudenthal’s theory depended on this, since according to him, a whole nation had to be taught thinking. Freudenthal’s experiences in the concentration camp in Havelte led him to conclude that thinking among “the masses of the people” was in a deplorable state. They could not understand newspapers, their conversations remained “of a primitive character” and in many working-class families books were not to be found. Intelligent pupils remained agricultural labourers or worked in industry and according to Freudenthal we could not tolerate this. Ons volk heeft in deze oorlog grote verliezen geleden. En percentagegewijs zal juist hier te lande het verlies aan intelligentie bijzonder groot zijn. Er is een tekort aan geschoolde krachten in elk vak op komst. Een noodlottige bedreiging van onze beschaving! Een noodlottige bedreiging van onze beschaving is een ongeschoolde massa, die niet de bekwaamheden bezit, om haar levensonderhoud te verdienen, en die toch haar eisen doet gelden. (Our people have sustained great losses during the war. In particular in this country the loss of intelligence in terms of percentage will be exceptionally large. A shortage of skilled labour is looming for every trade. A disastrous threat to our civilization! A disastrous threat to our civilization is the unskilled masses who do not have the capabilities to earn their keep and still assert their rights.) [53]

Freudenthal ended his lecture by once again drawing attention to the means to train these masses: printing and distributing books. The great social involvement emerging from Freudenthal’s lecture was remarkable. He did not only speak of didactics (of arithmetic), the issue was educating the whole of the Dutch nation and teaching them to think. For as long as the knowledge about learning to think was limited, the little knowledge that was available had to be used in education as best as one could, for instance by developing a well thought-out system of arithmetic word problems and developing the right type of problems. In passing, Freudenthal declined the formative value of mathematics education and arithmetic education. According to him you could not expect much more than practising mathematical thinking and arithmetical thinking from these subjects. In the course of his lecture he sketched some requirements that the (secondary) education should satisfy in order to educate the Dutch people. He argued for instance in favour of a secondary school type without the obligatory mathematics. For people who were no good at mathematics there was no reason to learn it. In this way, the people who were good at it (and used it in their follow-up studies) could be taught to think really mathematically, instead of reverting to the rote learning based method that was at the time commonly used in secondary education. In his body of ideas ‘mathematics for all’ in a literary sense, and certainly not at all costs, was as yet out of the question. But learning to do arithmetic certainly was necessary

5.1 Educating

95

for everyone and according to him it was no shame to revert to rote learning for the fundamental operations. Finally, Freudenthal conceived a secondary school type in which developing manual thinking was pivotal. All this was necessary to fully utilize the capacities of the Dutch people and in addition educate the masses and teach them to think. Freudenthal was pessimistic regarding this goal: if it were not reached, he foresaw the decline of the Dutch civilization. His solution was simple: reading was the means for everybody to learn to think unintentionally. Freudenthal’s lecture was, according to him, received “with great success” during the conference [54]. In Vernieuwing van Opvoeding en Onderwijs he devoted another short piece to this “glorious conference” [55]. In this he spoke highly of the spirit of “being in complete agreement” that prevailed among the conference participants, even though there were two camps to be distinguished (the reformers who wanted to modernize the classical school and the radicals who condemned it). To Freudenthal it was especially important that a gap would not arise between the two camps and “that a gap would not arise between an educational system for children with parents who were, for mental or material reasons, able to choose the pedagogic best for their children, and an educational system for the masses” [56]. He would consider it a detrimental development if the VWO were to degenerate into a platform for the progressive. He incited the participants not to leave behind “the spirit that has inspired us 4 days long at Rhederoord” with “the wet ashes of the campfire” and not to renounce it in society [57]. Too much emphasis on the more radical side of the modernization movements would indeed be the reason that the WVO and notably its journal Vernieuwing van Opvoeding en Onderwijs fell on deaf ears in the wider circle of Dutch school teachers in the years after the war [58].

5.1.4 ‘Educating’ in De Groene Amsterdammer According to Freudenthal, reading was the best way for people to learn to think—“a child that reads twice as much, learns to think twice as much”—and books were the ultimate means to educate the unskilled masses. Freudenthal implicitly contributed to this, for example by the publication of the novels he had written during the war, De schuldenaar ( The Debtor) and Viersprong der grote wegen (Crossroads of main highways). The ‘educational element’ was even more obviously recognizable in his contributions to De Groene Amsterdammer. Freudenthal’s studies during the years of ‘idleness’ had yielded a wealth of notes, manuscripts and articles. Many of these would become the basis for future publications. In February 1946, for example, he sent an article with the title ‘Rekenmachines winnen den oorlog’ (‘Computers win the war’) to the editors of De Groene Amsterdammer (Fig. 5.5) [59]. The editors expressed a definite interest “in your article about the fantastic computer” [60]. It was published on 16 March. In this article, according to Alberts the first in the Dutch press to inform the public about this subject, Freudenthal spoke of the enormous computer that was built during the war for the Harvard University [61].

96

5 A New Start

Fig. 5.5 The note to the Groene Amsterdammer accompanying Freudenthal’s article ‘Rekenmacines winnen de oorlog’ (‘Computers win the war’ 1946)

He not only wanted to inform the public about the possibilities and applications of this invention, he also wanted to warn about the fact that Europe seriously trailed behind American science because of the war: Maar zij [de rekenmachine] lijkt een fantastisch symbool voor het Amerikaanse organisatietalent, dat den tweeden wereldoorlog gewonnen heeft, en dat nog meer zal winnen, als Europa geen kans krijgt, om dien voorsprong in te halen. (But it [the computer] seems to be a fantastic symbol for the American organizational talent which has won the Second World War and which will win more if Europe does not get the chance to make up for this headstart.) [62]

This style would become the characteristic one in the many articles that Freudenthal would write for De Groene Amsterdammer in the following years. In easily comprehensible language he told his public about all sorts of things more or less taken from current events, varying from Brouwer’s intuitionism to the use of tranquillizers during the final exams [63]. The common factor often remained mathematics and in particular its history (Fig. 5.6). That he was interested in the history of mathematics was also obvious in Freudenthal’s inaugural lecture entitled 5000 jaren internationale wetenschap (5000 years of international science) in which he sketched 5000 years of mathematics in a nutshell [64]. Publishing in De Groene Amsterdammer was to Freudenthal not only a means of continuing his literary work, he also reached a wide audience and thereby responded to his own call for action: education of the masses by bringing science and current events out into the open in an accessible manner.

5.1 Educating

97

Fig. 5.6 a, b and c Cuttings Freudenthal kept of his articles in De Groene Amsterdammer, being: ‘Rekenmachines winnen den oorlog’, (Calculating machines win the war, 1946) ‘En nu… de wiskunde’ (And now … mathematics, 1950), ‘Eikenhout is eikenhout’ (Oakwood is oakwood, 1948)

Laten we terugkeren tot de tekst van Sjadoeppoem met de rechthoekige driehoeken. U zult misschien vreemd opkijken, maar ik ga u nu een les geven in het lezen van spijkerschrift. U protesteert, maar ten onrechte. Wanneer ik klaar ben, zult u toegeven: een kind kan de was doen. (Let’s go back to the text of Sjadoeppoem with the right-angled triangles. You are perhaps surprised, but I will now teach you a lesson in reading cuneiform script. You protest, but you shouldn’t. When I have finished, you will admit: it is child’s play.) [65]

Without becoming pedantic he found a humorous style of writing that for the readers turned mathematics into something more understandable, less mysterious and less ‘unworldly’. He wrote for example about the rules of logic that in daily life strike one as almost absurd: Uit één valse stelling kun je alles afleiden. De ware en de onware beweringen, zonder onderscheid. En bovendien nog zonder enige moeite. De wiskunde wordt dan wel heel erg gemakkelijk, en de aardigheid is er af. Daarom haten wij de tegenstrijdigheden en wij blijven zweren bij “Eikenhout is Eikenhout”. (You can deduct everything from one false theorem. True and untrue assertions, without distinction. And, in addition, also without effort. Mathematics becomes very easy then and the fun wears off. That is why we hate contradictions and stick to “Oak-wood is Oak-wood” (or “Eggs is eggs” (equivalent English saying))) [66]

98

5 A New Start

Fig. 5.6 b (continued)

The short pieces in De Groene were also a means to pass and publish his own criticism and commentary on on-going discussions. A recurring subject was Freudenthal’s view on higher education, teachers’ training and mathematics education. Freudenthal would continue to contribute to De Groene until the 1970s, a total of more than 170 articles and short pieces [67].

5.1.5 ‘The Cooperative Task of the Educator in Forming a Person’ Four years after “Rhederoord’ Freudenthal lectured once more about education at a WVO summer conference. This was at the so-called ‘Varende Conferentie’

5.1 Educating

99

Fig. 5.6c (continued)

(‘Sailing Conference’) of 1949, a conference held on a ship that took the participants in 4 days along four cities in the Netherlands and Belgium. The central theme of this conference was ‘forming a person’. ‘Forming a person’ was a much-heard buzz word in the ideas of the educational modernizers of the time. The real problem was not only realizing the changes within the educational system, it was forming a person. In an announcement of the ‘Sailing Conference’ in Vernieuwing Boeke wrote: Slechts als wij de waarde van de persoonlijkheid in zijn betrekkelijkheid erkennen zullen wij het evenwicht vinden, dat zal kunnen helpen om een afgrijselijke wereldcatastrophe te vermijden. Naar “mensvorming” moeten wij heen. Tot sommigen zou ik hierbij willen voegen: “maar die mensvorming moeten wij zien in zijn sociaal evenals in zijn individueel aspect”. (Only if we recognize the value of a person in his relativity will we find the balance that can help to avoid a horrible world catastrophe. We have to aim for “forming a person”. For some I would add to this: “but we have to consider ‘forming a person’ in its social as well as its individual aspects”.) [68]

In a lecture during the conference Boeke again mentioned the importance of forming a personality as part of the struggle against what was experienced as a crisis at

100

5 A New Start

the time: the tendency towards the mass. The aim of forming a person was to reach a “natural self-consciousness, a confidence about their own possibilities” [69]. The moment that had been achieved, there would no longer be “hate and the will to exercise power” according to Boeke. The second lecture by a certain Ms. Dol emphasized once again an important pedagogical reforming starting point: the modest role the educator ought to play in the child’s development process. The child had to be the starting point, be it under the inspiring but certainly not interfering guidance of the adult. The task of the educator was according to her “to support the child by our belief in the spontaneous forces present in every child” [70]. In his lecture entitled ‘De medewerkende taak van de opvoeder bij de mensvorming’ (The cooperative task of the educator in forming a person) Freudenthal addressed the question whether the educator could contribute to forming a person, and if so, how. The extent to which a person was ‘formed’ could according to him be measured from the extent to which the person had assimilated everything they encountered. ‘Forming a person’ was in that case opening up the person so that they can assimilate a variety, as rich as possible, of what we as educators can offer them and what they can find in themselves. If there was something person-forming in the relation between the educator and the educated, then it had to have the form and purpose of the unintentional (and therefore the unobtrusive): Wat wij het krachtigste hebben geassimileerd, maakt op onze medemensen de sterkste indruk. Waarin wij het meest mens zijn, daarin werken we het ingrijpendst mensvormend. Het onbewuste is het geweldigste reservoir van ons menszijn. (The things that we have assimilated the most powerfully, make the deepest impression on our fellow men. We work with the most far-reaching consequences in forming a person when the aspects in which we are persons ourselves are concerned. The subconscious is the most colossal reservoir of our being a human being.) [71]

The intentional element in education of course remained indispensable, especially in teaching. But for forming a person it not only served no purpose, it was also dangerous as the start of ‘violating a person’, as Freudenthal described it. In this context he argued in favour of a more global approach in the schools, a more universal approach in the Higher Education and against specialism in whatever form. Learning to read was not achieved by gluing characters together, learning a language was not achieved by a detour of rules and words. He did not mention arithmetic or mathematics education in this lecture. In view of the emphasis he put on ‘learning to think by reading’ 4 years before this, and the importance of the unintentional in particular, this was logical. In the larger context of education, here too, Freudenthal only attributed a modest contribution to the teaching of arithmetic. The ‘global education’ point of view could be seen as an indication of a more explicit view on arithmetic and mathematics education that Freudenthal would formulate in the 1960s and 1970s: ‘mathematics education from and by reality’. Freudenthal ended his lecture with a call for modesty among educators. As a human being the educator had to realize that he was just as incomplete and still developing as the person he had to educate: the completeness of being a person that he wanted to convey to his pupils, should be his own completeness. It was impossible

5.1 Educating

101

to convey his own being and image of a person to the other, since his past was not the same as that of the person to be educated. The better the unification of the educator had been accomplished, the more he could convey this automatically and unintentionally. Freudenthal ended his lecture as follows: Mensvorming? Ja. En hoe? Door af te blijven van de mens! De mens is niet, zoals primitieven de schepping zich voorstelden, geformeerd uit klei, die je met je hand kneedt. De goddelijkheid van de taak van de opvoeder is dezelfde, die Flaubert toeschrijft aan de dichter, wanneer hij zich manifesteert in de schepping van zijn verbeelding: onzichtbaar en alom tegenwoordig. (Forming a person? Yes. And how? By leaving the person alone! A human being is not, as the primitives imagined the Creation, formed from clay moulded by hand. The divinity of the educator’s task is the same that Flaubert attributes to the poet when he manifests himself in the creation of his imagination: invisible and all present.) [72]

5.1.6 Education: A Summary Freudenthal’s attitude and views on upbringing, education and their purposes were definitely not new or sensational in WVO circles. But it was remarkable that, even though he did not have direct contacts with primary and secondary education, his opinion as a mathematician on general educational affairs was apparently taken seriously and was respected by the WVO audience that consisted for the larger part of pedagogues, educational philosophers and school teachers. The standing of his professorship probably contributed to this as well. On first inspection Freudenthal’s activities within the WVO did not seem to be connected to his other interests and spheres of activity. Yet they were part of a logical whole and complemented his more general ideas on education and his conviction that the solution of social problems was to be found in education. In an article in De Vrije Katheder in January 1949 he wrote [73]: Ik heb opzettelijk alleen gesproken over het lager onderwijs, dat voor 90 % van ons volk het enige onderwijs is, dat ze ooit volgen. Met het meest fundamentele moet men beginnen. Schoolplicht tot 18 jaar zoals in Engeland, middelbaar onderwijs voor het overgrote deel der jeugdige bevolking, zoals in Amerika—dat zijn utopieën. Heden utopieën, morgen eisen, overmorgen werkelijkheid. Onder voorwaarde, dat men het niet langer als een schande beschouwt, culturele kwesties in hun maatschappelijke verband te zien. (I deliberately only spoke of primary education which for 90 % of our nation is the only education they will ever receive. One has to start with the most fundamental aspect. Compulsory school attendance to 18 years as in England, secondary education as is common for the greater part of the youth in America—those are utopias. Today utopias, tomorrow requirements, the day after tomorrow reality. On the condition that it is no longer considered shameful to view cultural questions in coherence with society.) [74]

The next paragraph will show that the educational components of Freudenthal’s vision on higher education, in particular, are very similar to the ideas sketched here. Freudenthal would particularly use his belief in the power of the unintentional, the emphasis on reading as an educational method, the importance of more universal education, the danger of specialisms and the urge to do more (and especially reliable) research in the field of education, as arguments in the discussion on the modernization of the university education.

102

5 A New Start

5.2 Higher Education The end of the war was the starting signal for higher education to make up for lost ground [75]. It was not only catching up arrears (especially in the field of the natural sciences) with respect to other countries such as the United States, education also had to be provided for the students who had fallen behind with their studies because of the war. In Amsterdam in 1945 the number of mathematics students was, according to Freudenthal, thrice the normal number [76]. Immediately after the war various groups of professors announced ideas about the modernization of the universities. For example the Groningen leaflet De vernieuwing der universiteit (The modernization of the university), with cooperation of Minister Van der Leeuw and among others Van der Corput, was published. Similar voices were heard from other universities such as Leyden and Utrecht. A great deal of the criticisms originated from the fact that universities had failed to resist the occupying forces. The causes for this failing were mainly found in the increasingly popular specialist subject fields and the lack of a close-knit academic community of professors and students. The solution for this problem was according to the leaflet to be found in a cultural approach: the introduction of philosophy lectures, Studium Generale (a series of lectures on topics of general interest) or ‘Science and society’ as compulsory study components should lead to the necessary transfer of culture. In this way the university could take the necessary lead, both as a cultural authority and in the field of science, not only for its students, but also in the end for society. Another point of criticism that received a lot of interest in the WVO was the lopsided student population, both in class origin and gender. A more democratic admittance policy should see to it that everyone who could go to a university was offered the opportunity. A specific problem for the mathematics education were the newly arising career opportunities for the graduates. In addition to the formerly obvious prospect of a job as teacher, society increasingly demanded research mathematicians, organizational mathematicians and mathematical researchers. A reconsideration of the contents of the curricula was therefore required. The new government, with the Groningen Professor Van der Leeuw as minister of OK&W, ranked the reform of the university high in its programme. In the ‘recovery and renewal’ framework the university system was scrutinized extensively. On 11 April 1946 the Staatscommissie tot Reorganisatie van het Hoger Onderwijs (Governmental committee for the Reorganization of Higher Education) (the Reinink II committee) was appointed. The resulting Higher Education Act would only be promulgated in 1960 for that matter. Freudenthal also wanted to intervene in this topical discussion: “after all a discussion is a brew, that is not impoverished by the number of participants, but enriched and every voice heard in the discussion can resonate even when it only stimulates considering and arguing” [77]. For this he wrote a 73-page leaflet with the title ‘Oude en nieuwe universiteiten’ (‘Old and new universities’), which (as he said himself) was mainly a direct reaction to the proposals written down by the Groningen group. The article was what he would later call the ‘main dish’ of his contribution to

5.2 Higher Education

103

the discussion [78]. Before making an effort to get the article published, Freudenthal presented it—accompanied by an extremely carefully phrased letter—to van der Corput, one of the six authors of De vernieuwing van de universiteit [79]. Ik heb namelijk op de plannen van de Groningsche docenten ook critiek geoefend, en al weet ik, dat wij wiskundigen wlkaar [sic] geen critiek kwalijk nemen en al heb ik het gevoel, dat ik met mijn critiek altijd binnen de perken van het meest fatsoenlijke ben gebleven, toch zou ik niets willen doen, wat in mijn nadeel zou kunnen worden geïnterpreteerd—ik zou dan ook mijn eigen plannen en de uitvoering in de weg gaan staan. (I also criticize the plans of the Groningen lecturers, and even though I know, that we mathematicians do not blame each other for criticizing each other and even though I have the feeling that my criticism does not exceed the boundaries of decency, I still would not want to do anything that could be interpreted to my disadvantage—I would be standing in the way of my own plans and their execution.) [80]

Van der Corput’s reaction to Freudenthal’s leaflet was positive. He not only agreed with Freudenthal’s criticism of the Groningen leaflet, but he valued the contribution as a stimulant to the discussion, because “our word is—and should not be—the last word spoken on this subject” [81]. Freudenthal’s efforts to get the leaflet published would lead to nothing, but he would later work many of his ideas into his short pieces in De Groene Amsterdammer [82]. What were the main points of Freudenthal’s proposals with respect to the university reforms? How were these connected to his ideas on education and teaching (mathematics)? And how did they affect these ideas?

5.2.1 Studium Generale One of the main points of Freudenthal’s ideas on university reform was the Studium Generale. In the battle against specialization within the disciplines, considered to be bad, Studium Generale was presented as an important aid. In the leaflet of the Groningen professors the solution was seen in a more central role for the modern humanities such as philosophy and psychology to give the university an intellectual foundation. Freudenthal did not believe in the generally formative or educational value of compulsory Studium Generale lectures. In his opinion the criticism on the “mental enchainment of the youth in their discipline” should first and foremost affect the lecturers. It was their task to help the students to find connections between the disciplines. The culture critics who—particularly in the field of the natural sciences—warned against specialization and lopsided intellectualism, forgot that anti-intellectualism had been one of the few rare theoretical dogmas of National Socialism and that intellect had won the war. Intellectualism had failed only where it not had been developed enough and thus could not have prevented the war [83]. It was time to realize, according to Freudenthal, that the university was no longer the cultural centre, it was not even the centre of scientific civilization. The power of the lecture would always be subordinate to that of the book. Instead of complaining about that, we had better look for a more fruitful manner of teaching.

104

5 A New Start

Maar als wij in onze colleges den strijd met de boeken zullen aanbinden, zullen we verslagen zijn, nog eer die strijd begonnen is. We moeten onze mogelijkheden kennen, om onze taak te kunnen vervullen. Zijn de perken, waarbinnen wij onze ambities als leeraar [sic] en opvoeder kunnen bevredigen, te nauw? We hebben niet het recht, daarover te klagen, zoolang we niet binnen die perken ons best hebben gedaan. (However, if we in our lectures will engage in a battle with books, we will be defeated, even before that battle has started. We should know what we can do to be able to fulfil our task. Are the limits within which we as teachers and educators can fulfil our ambitions too narrow? We have no right to complain if we have not done our utmost within these limits.) [84]

Exploiting the personal element—the contact between student and lecturer—was according to Freudenthal what should make oral education stand out over books. In particular where Studium Generale lectures were concerned he warned not to overestimate the possibilities: mass lectures, mass exams and ‘swotting’ for general civilization were, according to Freudenthal, a pedagogical blunder. Studium Generale should not become a regulated institution but “something ordinary and yet pleasant” [85]. General civilization not in addition to, but via studying the discipline itself was according to Freudenthal the solution. And how? By showing that as lecturers we are exponents of society and by “giving the students what we own ourselves” of our general civilization [86]. In addition to this, the university should create time and space for students to meet each other, to attend general-interest lectures, to attend each other’s lectures and to read in a large reading room, where you could actually walk along the shelves with books in order to “—often in passing—find what you were not looking for” [87]. It was obvious that Freudenthal in sketching this idealistic situation remembered his own youth, the many books he read on a diversity of subjects, his time as a student in Berlin, when apart from his mathematics lectures he also attended other lectures. But in particular he remembered the wonderful ‘discipline society’ he found in Berlin at the time, in which the binding social function of the students’ corps, the Mapha (the Mathematisch-Physikalischen Arbeitsgemeinschaft, the MathematicalPhysical Working community), played an important role (Fig. 5.7) [88]: Ikzelf heb in mijn studietijd in zulk een vakstudie-maatschappij geleefd, werkelijk gelééfd, mijn heele studie-leven doorgebracht in die universiteit en in dat instituut, waar die maatschappij huisvestte—men vergeve mij, dat ik mij zoo vaak bezin op een universiteit, die al in puin lag, jaren voordat de dynamietbommen er het laatste schepje oplegden. Het was een zeer hechte maatschappij, een wonderbaarlijke cel van geestelijke hygiene [sic] temidden van een politiek reeds door en door verpest studenten-leven. In mijn wetenschappelijke, algemeen-geestelijke en menschelijke vorming heeft die maatschappij een groot aandeel gehad. (In my student days I have lived in such a ‘discipline society’, really lived, spent all my student days in that university and that institute where the society had found accommodation—please forgive me that I reflect so much on a university that was already in ruins years before the dynamite bombs added the last bit. It was a closely-knit society, a miraculous cell of mental hygiene, in the midst of a politically already thoroughly ruined student life. That society has played a large part in my scientific, general mental and human development.) [89]

Without wanting to lock the student up in an artificial university society (that was at best unsuitable as a preparation for society at large) Freudenthal saw potential for a general education of the student in the closely-knit societal power of the science


105

Fig. 5.7 Group photo of a hike by students’ corps ‘Mapha’ (the Mathematisch-Physikalischen Arbeitsgemeinschaft, the Mathematical-Physical Working community). Freudenthal is second on the right, kneeling (about 1926)

discipline. Lecturers too had to contribute actively. It was not meant to become a formal association, but it should start from what Freudenthal called the prevailing spirit of the discipline: Maar als die sfeer gezellige bijeenkomsten, excursies en fietstochten baart, dan hooren die er even goed bij. Dat is de radicaalste oplossing, en men moet af en toe de vrees, zich belachelijk te maken, overwinnen, om radicaal te zijn. (However, if that atmosphere brought forth meetings, excursions and bicycle trips, these are no less part of it. That is the most radical solution and one should from time to time conquer the fear to make oneself ridiculous, to be radical.) [90]

It was obvious that he was thinking of trips, excursions and hikes such as the Mapha organized in the 1920s. It would not lack attention from Freudenthal: in his inaugural lecture when he was appointed professor he promised his future students a lot of ‘Studium Generale’ in the sense of the word described by him: Er wordt thans veel gesproken over studium generale. Dit is een pakkend woord en te mooi, om te vergooien. Het is mijn overtuiging, dat studium generale noch een door plechtige welsprekendheid gewijde zaak mag zijn noch een nieuw studievak, waarin tentamens behoren te worden afgelegd. Zo licht mogen wij het niet opvatten. Ik gevoel het aldus: studium generale niet naast, maar door de vakstudie; studium generale in onze gehele activiteit; studium generale als verbondenheid met al wat menselijk is. Daarvan zal ik blijk geven in een afzonderlijk college over de geschiedenis der wiskunde, dat ik mij voorstel te geven,

106

5 A New Start

maar evenzeer in elk uur, dat ik in en naast mijn colleges aan Uw opleiding besteed, en ik reken hierbij op Uw aller medewerking. (At the moment there are many discussions about ‘studium generale’. It is a catchy name and too beautiful to throw away. I am convinced that ‘studium generale’ should neither be a thing, sacred by ceremonious eloquence, nor a new discipline with compulsory examinations. We must not take it too lightly. My feeling is: ‘studium generale’ not in addition to, but within the discipline; ‘studium generale’ in the whole of our activities; ‘studium generale’ as a connection to all that is human. I will show this in a separate lecture on the history of mathematics that I propose to give, but also in every hour that I spend on your instruction, during but also outside of my lectures, and I count on the cooperation of all of you.) [91]

These statements already indicated Freudenthal’s ideas on ‘forming a person’ and ‘the power of the unintentional’ which he explained a few years later in his lecture during the Sailing Conference of the WVO [92]. He wanted to create for his students the same “marvellous cell of mental hygiene” as he had known in Berlin. Characteristic for that situation was that a political or mental foundation was missing, the disciplinary link between students and lecturers was, according to Freudenthal, enough basis to form an working community, a ‘professional society’. Implicitly he started from the idealistic proposition that such a disciplinary foundation was more or less the same, or at least compatible, to every student, lecturer and researcher. It was from such a community that he expected an unprecedented general formative value for the student. More general forming and civilization than any compulsory Studium Generale lecture could possibly offer.

5.2.2 The Teachers Training A second important matter that Freudenthal raised in his ‘Oude en nieuwe universiteiten’ was the schooling of teachers at the university. Even before the Second World War attempts had been made to revise the training for secondary-school teachers at the university [93]. The wish to prepare students more explicitly for the teaching profession had in the meantime become generally accepted. For many it was an important condition that this should not be done at the cost of the scientific education of the prospective teachers and their prestige. An extra didactic-pedagogic element in the training was welcome, but should not be introduced at the expense of the academic forming of the mathematician. Now that a closer look was being taken at higher education in the post-war period, the teachers training was looked at critically as well. For mathematics teachers the courses provided by the MC were a novelty, they were optional and in particular intended to inform active teachers of the newest developments in mathematics. Van der Corput, for whom propagating mathematics had cultural value and was of paramount importance, was the driving force behind these courses [94]. Apart from these developments there was also a renewed attention for the didactical element of the training. In Utrecht Minnaert was one of the first who argued in favour of promoting didactical elements in the training of teachers for science and mathematics. Marcel Minnaert, a Flemish activist, had come to the Netherlands after the First World War and worked since 1918 in Utrecht with the physicists W.H. Julius


107

and L.S. Ornstein [95]. In 1937 Minnaert was appointed extraordinary professor in astronomy, and immediately after the war full professor. Molenaar describes him as the pioneer of the didactics of physics in the Netherlands, involved with educators such as Ligthart and Mrs. Ehrenfest. As early as the end of the 1920s he had been appointed as private lecturer in didactics and methodology of physics. And he was also actively involved in the Wiskunde Werkgroep from its foundation. Apart from their didactical interest Freudenthal and Minnaert also shared their politically left-oriented humanistic background. They were both members of the VWO and were at the end of the 1940s passionate advocates of a ‘student wage’ [96]. And what is more, they also hit it off socially. In his inaugural lecture his friend Minnaert was the first of his new Utrecht colleagues Freudenthal thanked: Wat mij gestreeld en hier naar toe gelokt heeft, is vooral de hartelijkheid van Uw taal geweest, Waarde Minnaert. Ik heb mij niet bedrogen: deze hartelijkheid is niet alleen een persoonlijke trek van Uw wezen, maar zij is de genius loci gebleken, aan wie ik zonder te aarzelen in Uw midden het verschuldigde tribuut zal betalen. (It has been in particular your friendly language that has delighted me and enticed me to come here, Dear Minnaert. I am not deceived: This friendliness is not only a personal feature of your nature, but it has turned out to be the genus loci, to which I would, without hesitation, pay due tribute in your midst.) [97]

In the arrival of a new colleague in the Faculty of Mathematics and Science Minnaert saw a possible ally for his attempts to inject new life into didactics at the faculty. On 7 January 1947 he wrote a letter to Freudenthal: Amice, Al lang had ik je willen spreken over het onderwijs in de didaktiek van de wiskunde, dat dringend verzorgd moet worden aan onze universiteit. Ik heb zelf jarenlang een college gegeven “didaktiek der natuurkunde” en heb nog altijd een warm hart voor deze aangelegenheid. Tot hiertoe is in Utrecht niets gebeurd voor de didaktiek der wiskunde, terwijl dit vak juist zo zorgvuldig behandeld moet worden bij het onderwijs. Dit kwam door de afkeer van elke didaktische hervorming, welke Barrau en Wolff aan de dag legden. (My dear friend, For a long time I have wanted to speak to you about education in didactics of mathematics, which our university has to provide urgently. I have given a lecture “didactics of physics” for many years and I still am very keen on this matter. Until now nothing has happened in Utrecht with respect to didactics of mathematics, whereas teaching this subject, in particular, has to be treated very carefully. This was caused by the aversion to all didactic reforms Barrau and Wolff brought forth.) [98]

Minnaert could count on Freudenthal’s interest in this question evidenced by the fact that Freudenthal would attend a meeting of the Wiskunde Werkgroep just 4 days later. But even before Minnaert made him aware of the lack of a didactic component in the training for mathematics teachers, Freudenthal had developed strong views on the question. Just as on the subject “Studium Generale” the core of Freudenthal’s ideas on the teachers training were to be found in his leaflet ‘Oude en nieuwe universiteiten’ [99]. Even though increasing numbers of mathematics students found employment in industry, the faculty of mathematics and science mainly educated teachers. According to Freudenthal the university ought to be better aware of this task and adapt its instruction accordingly. Adding to the theoretical education, by establishing lectures on methodology, didactics, pedagogy, and in particular the psychology of puberty, was not enough on its own. It was more important to let prospective teachers gain

108

5 A New Start

practical experience. An extra exam would no longer be necessary: self-criticism would eliminate the really unsuitable student automatically at that stage [100]. Prospective teachers also had to be trained for their educational task. Youth movements (scouting) and youth care provided a “magnificent opportunity” to be working as an educator according to Freudenthal. Participating in such activities should be considered a matter of course without incentives for the students. It is evident that, in the educational picture sketched here, Freudenthal started from a prototype of the ideal student: one who without being forced and supervised, and with the gift of self-reflection and self-criticism and a certain passion for education and children in general, could be released to train into a responsible pedagogical teacher. The mathematical subject matter that the prospective teacher had to learn, should be motivated by the goal that had to be achieved. This meant, according to Freudenthal, that the curriculum of the secondary school had to be reconsidered and that the subject matter had to be treated from a higher point of view, and in this Klein’s influence on Freudenthal’s ideas resounded. This starting point of Freudenthal (‘lower mathematics from a higher standpoint’) linked to those from the pre-war Verrijp committee [101]. However, Freudenthal not only proposed limiting the subject matter to that of the secondary school, which was out of the question in the Verrijp committee report, but argued in favour of a more conscious and more systematic training in mathematical thinking, whilst an overview should be given of what was going on in modern mathematics [102]. Thinking competences could not according to Freudenthal be acquired in the same way as ready knowledge, so not by purely passively re-thinking of the subject matter treated. Thinking was a goal-oriented process. ‘Learning to think’ should be based on presenting the student problems, important enough to stimulate him to search for a solution and easy enough not to discourage him. The help the lecturer had to offer consisted of organizing such problems, systematically teaching answering methods (using specially prepared problems and not abstract schemes) and stimulating well-organized and consciously generated self-activity [103]. Activities in the laboratory and group work were according to Freudenthal the most suitable form for the mathematics classes at the university [104]. In practical terms, Freudenthal proposed the following: divide the mathematics education into a pedagogic-research and a scientific-research version. The prospective teacher could, of course, also participate in the education to become a scientific researcher, but Freudenthal did not want to make that compulsory. With this proposal he flatly contradicted the Verrijp committee, which proposed an optional extra examination to get the qualification to teach in addition to the mathematical master’s exam. As mentioned above Freudenthal’s ideas from ‘Oude en nieuwe universiteiten’ have never been published. But at the end of 1947 he did write a discussion paper with the title ‘Voorstellen tot reorganisatie van het Hoger Onderwijs in de Wiskunde’ (‘Proposals for reorganizing the higher education in mathematics) (probably intended for internal use in Utrecht) [105]. In this he argued frankly in favour of dividing the university education into the two parts he had earlier described in ‘Oude en nieuwe universiteiten’. He gave three reasons for this:


109

Firstly, there was the changed position of the mathematician in society and the related need for research mathematicians educated in applied mathematics. There was also the gap between the mathematics taught in secondary schools and the more modern university mathematics: Door de op de universiteit gedoceerde stof aan de stand van de wetenschap aan te passen, hebben wij de a.s. leraar genoodzaakt, overwegend dingen te leren, die hij niet nuttig kan toepassen, terwijl desondanks de opleiding geheel ontoereikend is om aan de behoeften van modern research te kunnen voldoen. (By adapting the subject matter taught at the university to the state of the art in science, we have forced the prospective teacher to primarily learn things that he cannot apply usefully, while at the same time the education is wholly inadequate to be able to meet the needs of modern research.) [106]

Secondly, there was the scarcity of (qualified) teachers in secondary education and the accompanying dangers of deploying unqualified teachers or the establishment of an extra-university crash course for teachers [107]. Ik kom dus tot de conclusie, dat het nodig is, de eisen aan de a.s. wiskundeleraar belangrijk lager te stellen (wat de wetenschappelijke vorming aangaat) en hierdoor de duur der opleiding te verkorten. Leerkrachten, die een (zij het dan verkorte) verantwoorde universitaire opleiding hebben genoten, moeten worden verkozen boven geheel onbevoegden. (Consequently, I reach the conclusion that it is necessary to significantly lower the demands that the prospective mathematics teachers have to meet (as far as the scientific education is concerned) and to shorten the training accordingly. Teachers who have had a sound university education (albeit a shortened one) have to be preferred over totally unqualified teachers.) [108]

As a final reason Freudenthal mentioned the overloaded curriculum at the university that resulted in students having neither time left to acquire further cultural skills outside of their discipline nor for qualifying pedagogically in their discipline. The curriculum had to be pruned to make room for pedagogic and general cultural training (Freudenthal’s ‘Studium Generale’). In a practical sense Freudenthal’s proposal implied the following curriculum: the first 2 years consisted of maximum 24 h of education per week, devoted to the scientific programme, followed, for prospective teachers by 2 years of teaching “supervised by an elderly pedagogue”. During these 2 years the student took a course of 1 day/week at the university in order to “conserve his mathematical capabilities”. To finish there was one and a half year of education at the university during which both extending and deepening of the first 2 years’ subject matter and continuation of the cultural and pedagogic training received attention. After his education the teacher kept up to date with the latest developments by attending summer courses [109]. In Freudenthal’s proposal the research version of the programme was to become, after the first 2 years, a more demanding education with a lot of emphasis on applied disciplines. He did not mention a work placement or any other type of practical preparation for a possible job in the business world in his proposal. What exactly has happened to these proposals is unclear. They have probably been used as a discussion paper in the Department of Mathematics and Science in Utrecht. In March 1948 Freudenthal wrote a supplement to the text as a result of a conversation with Minnaert about this, but for the rest he stuck to what he had written before [110].

110

5 A New Start

Perhaps this proposal was Freudenthal’s reaction to Minnaert’s call for attention to didactics of mathematics within the university. As with many other things Freudenthal’s solution was a drastic change of the system and not the introduction of a single lecture ‘didactics of mathematics’. In De Groene Amsterdammer too Freudenthal explained his plea for an education specifically aimed at teaching, and he urged that there should be no let up, particularly in the desire for innovation. En de bedoeling van dit artikel is, vanuit het kabbelen in de commissies een nieuwe golfslag in het openbaar te doen ontketenen. (And the meaning of this article is to unleash from the babbling committees a new avalanche in public.) [111]

One of the committees Freudenthal aimed at was the Reinink II committee. In 1949 this committee presented a report. This report too proposed a division of the study into a part for prospective teachers and one for prospective researchers. The proposals in the report of the committee’s department M (Mathematics and Science), envisaged a division into two different masters examinations after 3 years of both branches following an identical bachelors examination [112]. Admittedly, this proposal was less drastic than Freudenthal’s, but still too drastic for the teachers who consequently rejected it. Their most important counter-argument was the fear of a social devaluation of the position of the teacher if the education were to contain less mathematics than that of a ‘research-mathematician’. Freudenthal refuted this argument: Voor de vormende waarde (voor zover die er is) is niet de hoeveelheid stof beslissend, en van wetenschappelijke didactiek en paedagogiek kan evenzeer vormende invloed uitgaan. En wie de student niet de tijd laat om tijdens de studie ook wat algemene beschaving op te doen, schaadt de standing van de leraar op de meest efficiënte wijze. (It is not the amount of subject matter that is decisive for the formative value (as far as that exists), and scientific didactics and pedagogy can also contribute to this formative value. Those who do not leave students enough time to acquire some general refinement during their studies, damage the reputation of the teacher most effectively.) [113]

Freudenthal also explained his proposal for splitting up the master’s examination during the weekend conference organized by the Wiskunde Werkgroep in 1948. During the discussion that erupted he was sharply criticized by those present. One of them was mathematics teacher Joh. H. Wansink (1894–1985) [114]. After obtaining the KI and Kv certificates, Wansink had, as a great exception been allowed to study mathematics without an HBS certificate, based on his excellent results, at Utrecht where he also defended his thesis in 1931. In addition to his teaching he wrote text books, regularly published articles in Euclides and would later write the didactical study Didactische oriëntatie voor wiskundeleraren (Didactical orientation for mathematics teachers) [115]. Being a member of both the WW and WIMECOS he often played the role of mediator in the 1950s [116]. However, discussions had not yet reached that point by 1948. In a reaction to Freudenthal Wansink advanced the argument that, following their different curricula, research mathematicians and teachers would no longer understand each other. According to Freudenthal many students could become good mathematics teachers provided that they did not have to learn too much mathematics at university.


111

H. Turkstra said that he only wanted to proceed with the discussion after the pedagogic-didactic curriculum had been brought to the same level as that of the scientific-mathematical curriculum. Freudenthal believed that this would happen quickly enough, provided that a start was made immediately. Subsequently he explained his plan in curriculum years and now he required one and a half years for practice instead of the 2 years he had previously mentioned. However, this proposal did not seem open to discussion for the time being. Ending the report of the discussion in Euclides the point of view of the teachers (explained by Wansink) was represented as follows: Tot slot houdt Dr Wansink een vurig pleidooi voor het standpunt van de leraren. Hij merkt op, dat alles afhangt van de sociale waardering voor het leraarsambt. Hij vraagt zich af, waarom een arts na zijn doctoraal niet bevoegd is zijn practijk uit te oefenen, terwijl de leraar met het doctoraal examen de lesbevoegdheid krijgt. Hij stelt voor een “leraarsexamen” in te stellen bv. 1/2 jaar na het doctoraal. We willen echter niet worden gedéclassificeerd en we willen volkomen met de anderen samen werken. De autoriteiten willen ons echter de weg met de minste weerstand laten afleggen naar het leraarsambt. Het zou zeer bedroevend zijn, aldus de Hr Wansink, als de leraren in hun verzet de steun van de Universiteit zouden missen. (Applaus). (To end the discussion Dr. Wansink fervently argues in favour of the teachers’ point of view. He observes that everything depends on the appreciation of the society for the profession of teacher. He wonders why a physician is not qualified to practice after his masters’ exam, whereas a teacher gets his qualification with his masters degree. He proposes introducing a “teachers’ examination”, for instance 1/2 year after the masters’ examination. But we do not want to be declassified and we want to cooperate completely with the others. The authorities, however, want us to choose the line of least resistance to the profession of teacher. It would be very saddening, said Mr. Wansink, if the teachers would, in their resistance, lack the support of the university. (Applause)) [117]

It would take until 1952 before, by Royal Decree, teachers were obliged to attend at least 1 year of courses in pedagogy, psychology of adolescents and general didactics. At that time attending lectures in teaching methodology and doing one’s teaching practice also became obligatory.

5.2.3 Student Wage It was not only because of their interest in teaching methodology but also in their pursuit of a more honest admissions policy for higher education that Freudenthal and Minnaert found kindred spirits in each other. Freudenthal had already maintained in ‘Oude en nieuwe universiteiten’ that in the future more rather than fewer academics would be needed. Onze maatschappij heeft academisch gevormden noodig, en alle sombere voorspellingen ten spijt zou ik willen beweren, dat wij (een behoorlijke herziening van zekere maatschappelijke gewoonten verondersteld) best ook het dubbele aantal afgestudeerden in onze maatschappij kunnen verdragen, zonder dat er van overproductie sprake hoefde te zijn. Uiteraard zullen niet allen, die zich aanbieden, voor een academische opleiding geschikte candidaaten zijn, maar in het geheel zal het peil van den gemiddelden student de basis moeten zijn voor de eischen, die wij aan hem mogen stellen. (Our society needs university graduates and in spite of all pessimistic predictions I would like to contend that in our

112

5 A New Start

society we (supposing an adequate review of certain social habits) can easily afford to double the number of graduates, without there being talk of overproduction. Of course, not all candidates who present themselves will be suitable for an academic education, but overall the level of the average student will have to be the basis for the requirements we are allowed to set.) [118]

The ‘overproduction’ Freudenthal mentioned was a spectre of the 1930s. After the Second World War the fear that the Netherlands would have to survive at a permanently lower standard of living was still predominant. However, Freudenthal deemed society’s need for academics to be so great that he argued for adjusting the requirements, within certain limits, to the supply. He foresaw in particular a large shortage of engineers. If Europe wanted to continue to compete with the United States in the scientific field a great deal had to change, to start with by establishing a fairer admittance policy, a ‘humane selection procedure’. At the initiative of Minnaert the VWO section in Utrecht established a working group in 1948 engaged in the ‘Recruitment and education of Dutch academics’ [119]. On 9 and 10 December 1949 Freudenthal lectured on behalf of this working group at the VWO-conference on ‘De toelating tot het Hoger Onderwijs’ (Admission to Higher Education). The recommendation of the State Committee was already known by that time: to give as many as possible of those suitable for an academic education the chance to go to university. According to Freudenthal this was not enough: it was not a case of allowing ‘as many as possible of those suitable’ to go to university, but “of doing complete justice to intellects suitable for Higher Education” [120]. To determine who was suitable he stated that a good ‘psychological-technical’ selection method had to be developed. This was a first step in the right direction that would, however, take a lot of time and effort. Next, an identical system of student wage for all students had to be established. That was the only way, according to Freudenthal, to achieve an admission system in the best possible democratic way, in which all students had the same budget and no longer depended on their parents. This also directly and positively influenced the students’ general education and development which in its turn matched Freudenthal’s vision in the framework of Studum Generale. The ‘benefit’ of Freudenthal’s proposal was according to him large: Een rustige en, voor zover dit met menselijke middelen mogelijk is, onbezorgde studie voor iedereen; een zekere onafhankelijkheid van de student van zijn ouders als financiële machthebbers; de mogelijkheid voor iedere student, om deel te nemen aan het studentenleven (men zou dit dan zelfs verplicht kunnen stellen) en zich behalve wetenschappelijk ook cultureel te ontwikkelen; verder, het contact van studenten uit een gegoed milieu met leeftijdsgenoten uit lagere volksklassen; en meer algemeen de opheffing der klasse-tegenstellingen, die in de tegenwoordige studenten-maatschappij duidelijk merkbaar zijn. (A quiet and, as far as humanly possible, a care-free study for everyone; a certain independence of the student from his parents as financial power; the possibility for every student to participate in university life (it could even be made obligatory) and develop himself culturally as well as scientifically; next, contact between better-off students with those of the same age from lower classes; and more generally, the removal of the class differences that are explicitly noticeable in today’s student society.) [121]

Here too Freudenthal emphasized the importance he attached to participating in university life and cultural development as he already had explained in ‘Oude en


113

nieuwe universiteiten’. Arguments and objections against such a plan were, according to Freudenthal, all surmountable: if only the students’ admission was seen as a real problem, the means to solve it could then be found. Minnaert and Freudenthal, in particular, would continue to act as advocates of the student wage in the 1950s. In his later articles in De Groene Amsterdammer and Wetenschap en Samenleving too Freudenthal regularly returned to this subject. Apart from his repeated plea for a student wage and ‘the university open to all’ he also kept going on about the development of better selection methods and in relation to this he kept criticizing the inadequate statistical studies into intelligence [122].

5.2.4 Higher Education: A Ramshackle Parthenon or a House in Order? In the three discussions mentioned above Freudenthal always contributed his very own explicit point of view. In every way possible he seized the opportunities provided by what he called “the high waves of passion to reform our university education” to disseminate his own ideas after the liberation [123]. Characteristic for all his ideas was his social democratic starting point (‘education as a right for all’). The unorthodox attitude in his plea for relinquishing established traditions was of course easier because of his ‘non-Dutch’ outlook on higher education. Freudenthal seemed convinced that the society could be shaped in the way one wished. Because of this he sometimes made a somewhat naïve impression: as if all problems of the world could be solved by establishing public libraries and organizing hikes. But Freudenthal was not blind to reality, more than that, he argued in favour of placing science more at the service of reality, even if this superficially seen seemed to go at the cost of ‘social appreciation’. For a nation to remain worth counting there was only one criterion and that was what society required of the graduates and the professions they chose. If (potential) scientific talent had to be preserved for the Netherlands and the impending exodus of that talent to the United States had to be prevented, there was nothing for it but to carry out drastic reforms: Onze Europeesche geestelijke beschaving is een kostbaar goed, en uit mijn bibliotheek, die voor mij een essentieel onderdeel van die beschaving representeert, zou ik geen stuk willen missen. Maar bij alle geestdrift voor boeken en architectuur zou ik met mijn schatten niet in een bouwvallig parthenon willen zitten, waar het aan alle kanten tocht. Lang zullen wij, fijn beschaafde Europeanen, ons niet meer kunnen permitteeren, af te geven op een liefst tusschen aanhalingsteekens geplaatste Amerikaansche cultuur. Een groot deel Europeesche beschaving is aan romantische cultuurtheorieën ten gronde gegaan. Wij staan voor de keuze de werkelijkheid te aanvaarden—dat zijn de groote beschavingen, die in dien strijd overwonnen hebben—of, trotsch op een groot verleden, genoegen te nemen met een museumbestaan: heel het vasteland van Europa één groot Marken en Volendam, waar belangstellende Amerikaansche touristen zich rond laten leiden (zooals de Romeinen in het verbloeide Athene deden), om de laatste uitloopers van een eerbiedwaardige “cultuurharmonie” beautiful te vinden, en om ons, eer wij het gewaar worden, onze laatste ingenieurs weg te engageeren, terwijl wij er angstvallig voor waken, dat zij met hun handen van onze Rembrandts afblijven. (Our European intellectual civilization is a precious good and I would not want to miss one book of my library, which to me is a vital part of that civilization. But even with

114

5 A New Start

all the passion for books and architecture I would not want to reside with my treasures in a ramshackle draughty Parthenon. We, subtly civilized Europeans, will soon no longer be able to afford to run down an American culture, preferably put between quotation marks. A large part of the European civilization has been ruined by romantic culture theories. We are confronted with the choice of accepting that reality—that is large civilizations which have been victors in that battle—or, proud of a great history, content ourselves with an existence in a museum: the whole of the continent of Europe will be one big Marken and Volendam, where American tourists are shown around (as the Romans were in an Athens past flowering) in order to characterize the last bits of a respectable “culture harmony” as “beautiful’’ and to lure away, before we are even aware of it, our last engineers, while we scrupulously make sure that they keep their hands off our last Rembrandts.) [124]

Freudenthal was not blind to the flaws in the educational system. He, more than anybody else, knew how class and financial factors still stood in the way of ‘education for all’. Even when his ideas and the ideas of the groups he represented sometimes seemed a bit utopian and brash, he really did take reality into account. With, by his own account, a heavy heart Freudenthal had abandoned the idea of tackling the problems in the educational system from the bottom (in primary education). In its place came what he called the ‘realistic tactics’: start with democratic reforms not in the primary education but on the contrary in the higher education. He worked out: Wie ons als utopisten doodverft, overwege eerst het volgende: in een smalle en goed te overziene sector van ons onderwijs kunnen we schoon schip maken en het beginsel “selectie volgens bekwaamheid” verwezenlijken, en wel met betrekkelijk kleine kosten. Binnen zeer korten tijd dragen de daar geïnvesteerde gelden rente: na 4–6 jaar wanneer de maatschappij kan profiteren van numeriek ongewoon sterke jaargangen van bijzonder begaafden. Het hoger onderwijs moet de eerste bres zijn; hier is de doorbraak het gemakkelijkst. Door beurzen te vragen voor alles en nog wat versplinteren we de krachten, en door nu reeds het middelbaar en ander voortgezet onderwijs in de plannen te betrekken, zullen we nìèts bereiken. (Those who paint us as utopians, have to consider the following before they do so: in a narrow and surveyable sector of our educational system we can put our house in order and realize the principle of “selection according to ability”, and we can do that at a relatively low cost. Within a very short time the invested money will yield interest: after four to six years when society can benefit from the unusually strong numbers of highly talented graduates. Higher education has to be the first breach; a breakthrough is easiest here. By asking scholarships for all sorts of things we will fragment the forces, and by now including secondary and other advanced education in the plans, we will achieve nothing.) [125]

5.3 The Wiskunde Werkgroep (Mathematics Working Group) Freudenthal sent a copy of his inaugural lecture to many of his colleagues, friends and acquaintances. Among those was Janssen, secretary of the Centrale Werkgroep of the WVO. Janssen was a teacher at the secondary school department of the Montessori school in Bussum. He held a (lower) certificate for mathematics among other subjects and was chairman of the WW. Freudenthal and Janssen first met at the Rhederoord Conference in 1945.

5.3 The Wiskunde Werkgroep (Mathematics Working Group)

115

Janssen wrote a letter to Freudenthal in which he thanked him for the inaugural lecture he had received. He wrote that he remembered with pleasure Freudenthal’s contribution to the Rhederoord conference and that this encouraged him to appeal once again to Freudenthal’s interest in the WVO. We zouden het nl. zeer op prijs stellen, wanneer U eens voor de Wiskunde Werkgroep van de w.v.o. een inleiding zoudt willen houden. Daar bespreken leerkrachten van moderne scholen de problemen van het vak, zowel didaktische als vaktechnische. […] Misschien kunt U eens uiteenzetten welke propedeuse noodzakelijk is met het oog op het volgen van universitair onderwijs, waarbij de wiskunde hoofd- of hulpvak is. Misschien heeft U een oordeel over de gebruikelijke methoden of ideeën voor didaktische vernieuwing. (We would really appreciate it if you would make an introduction for the Wiskunde Werkgroep of the WVO That is the podium where modern school teachers discuss the problems, both didactical and technical, of the discipline. […] Maybe you could explain which first-year curriculum is necessary in view of studying at a university with mathematics as a major or secondary subject. Perhaps you have an opinion about the usual methods or ideas for didactical innovation.) [126]

In ending his letter Janssen expressed the hope for a fruitful contact between Freudenthal and the teachers of the WW. Ik hoop, dat U het ook op prijs zult stellen in persoonlijk contact te komen met leraren die streven naar een moderne vorm van wiskunde-onderwijs, zodat de resultaten van een dergelijk contact niet al te eenzijdig zouden behoeven te zijn. (I hope that you will also value getting into personal contact with teachers who strive for a modern form of mathematics education, so that the results of such contacts are not necessarily too one-sided.) [127]

Freudenthal replied positively to this invitation to come and lecture and immediately asked the secretary of the WW, H.J. Jacobs, to register him as a member of the working group [128]. Freudenthal and Jacobs were already acquainted since Jacobs, apart from his job as a mathematics teacher at “Het Nieuwe Lyceum” in Bilthoven, was also finishing off his mathematics study in Utrecht. That was the way in which Freudenthal’s involvement in the Wiskunde Werkgroep started; as an authority in the field of higher education he was asked about his opinion and his ideas about mathematics education in secondary schools. Freudenthal would attend practically all monthly meetings of the working group in the years that followed and that indicated that he most certainly appreciated the “personal contact with teachers” [129]. The meetings were by turn in Amsterdam, Bussum, at Minnaert’s astronomy department the ‘Sonnenborgh’ in Utrecht or at Mrs. Ehrenfest’s home in Leyden. At the time Freudenthal became involved in the working group, he had in fact not yet published anything about mathematics education nor made it public in any other way. Only in his lecture ‘Opvoeding tot denken’ had he said anything about the teaching of arithmetic. His interest in education and schooling were of course fertile ground for more involvement in mathematics education in secondary schools. Seen in the light of his own points of view, it is self-evident that he subsequently chose the innovative and not particularly conservative circles of the WW of the WVO.

116

5 A New Start

5.3.1 Activities of the Wiskunde Werkgroep The state of affairs in mathematics education after the war involved remedying and catching up on the arrears caused by the war. The curriculum of 1937 had hardly had the time to become operational. Since many pupils had fallen behind during the war the level of the pupils within one class differed widely. This made it difficult for the teachers to determine which level they could demand of the pupils. In addition, the complaints about overburdening the programme still persisted. Mathematics teacher H. Streefkerk, for example, wrote in 1945 in Euclides about the Beth curriculum which was unworkable according to him, quantity-wise and content-wise [130]. In short, the climate of opinion favoured a critical review of the subject matter. L.N.H. Bunt, curator of the Paedagogisch Instituut der Rijksuniversiteit te Utrecht (Pedagogical Institute of the State University in Utrecht), even cautiously spoke of a favourable result of the after-effects of the war: Niet alleen in materiële zin doet zich de nawerking van de oorlog gevoelen, ook het denken over vele onderwijsvraagstukken heeft zich er niet aan kunnen ontrekken [sic]; en in dit opzicht zou men dan misschien, bij wijze van uitzondering en heel voorzichtig, van een gunstige uitwerking kunnen spreken. Zoals men zich namelijk bij veel handelwijzen, die men vroeger vanzelfsprekend vond, thans afvraagt waartoe men aldus handelt en of een voldoend hoge waarde van het doel van de handeling wel vast staat, zo stelt men zich deze vragen ook op menig gebied van het onderwijs. (Not only in a material sense can the aftereffects of the war be noticed, the thinking on many educational problems is also affected; and as far as this is concerned, one could perhaps, exceptionally and very carefully, speak of a favourable effect. As people now wonder about many procedures, which seemed obvious in earlier times, why they are as they are, and whether their goals represent a sufficient value, they now also ask such questions in the field of education.) [131]

These kind of critical questions about the mathematics education were perhaps felt and expressed most strongly in the Wiskunde Werkgroep (Bunt was among its members). At the WW meetings invited speakers (often members of the working group themselves) gave introductory lectures about current problems in mathematics education which were discussed afterwards. Sometimes subjects for debate came up that were reserved for a more thorough treatment in one of the following meetings [132]. In 1948 the members conceived the plan to scrutinize the complete curriculum and remove its excrescences. The goal was to formulate a mathematics programme, identical for HBS-B and Gymnasium-β, in which not the formative value but the practical use of the subject matter would be emphasized. For this purpose committees consisting of members of the WW were established for the various subjects in the curriculum (algebra, goniometry, stereometry and plane geometry, analytical geometry and descriptive geometry). Freudenthal, Minnaert, Van Hiele (mathematics teacher at the Kennemerlyceum in Bloemendaal and Freudenthal’s ex-student from Amsterdam) and later Bunt too, together formed the algebra committee. Van Hiele said about this: Er waren, vanuit de wiskunde-werkgroep van de WVO, verschillende commissies. Er was er een voor Algebra, waarin Minnaert en Freudenthal zaten. Men zocht een derde lid, maar


117

niemand vond het prettig bij die twee professoren te zitten. Laat mij het maar doen, zei ik toen, ik durf wel tegen die twee op. Waarom niet, dacht ik erbij. (Several committees arose from the WW. One of these was for Algebra and Minnaert and Freudenthal were its members. A third member was wanted but nobody liked to join the two professors. Ah well, let me do it, I said, I dare to challenge these two. Why not, I thought.) [133]

As is evident from this quotation, both Freudenthal and Minnaert still remained two awe-inspiring professors to whom the other members looked up, even in the egalitarian, non-hierarchic working atmosphere characteristic for the meetings of the WW. Others recalled in particular that Mrs. Ehrenfest and Minnaert were the most authoritative individuals in the working group during the first years after the war [134]. Nevertheless, Van Hiele liked working in the algebra committee best: De algebra-commissie vond ik het leukst. Bij alles wat er aan leerstof werd genoemd, zeiden ze: dat is toch waanzin, moet je dat nu weer horen, ha, ha, ha. Ik genoot hoor, want ik was het daar grondig mee eens. Slechts bij een enkel ding merkte ik op dat het weggooien gevolgen had voor andere onderdelen van de leerstof. Als je dit wegdoet, ben je dat ook kwijt. Maar we konden goed met elkaar opschieten. (I liked the algebra committee best. Everything mentioned as subject matter was greeted with: that is complete nonsense, hear, hear, ha, ha, ha. I enjoyed it very much since I wholeheartedly agreed. Only about one single item I made the remark that throwing it away would affect other parts of the subject matter. If you throw this away, you will lose that as well. But we could get on very well together.) [135]

The two professors clearly did not lack the wish to make a clean sweep without scruples. The reports of the WW meetings also showed that Freudenthal and Minnaert were on the same wavelength with their interpretation of the mathematics education. The various committees acted energetically: many meetings and a lot of consultations between the committees followed. The activities of the WW would in 1952 eventually lead to a completely revised curriculum for HBS-b and Gymnasium-β [136]. Little remains of what was discussed during these meetings in the form of publications, correspondence or notes. But it is known that the WW must have had its own circular at that time. The only options for publication open to the WW were the journal of the WVO ( Vernieuwing van Opvoeding en Onderwijs) and Euclides. Even though the WW could better reach the desired audience via Euclides than by publishing in Vernieuwing, it was certainly not easy in those years to get WW articles into these journals. The teachers’ associations looked at the activities of the WW with due suspicion. The members of LIWENAGEL, for instance, were rather furious about the fact that the working group was engaged in “such specific Gymnasium affairs as analytical geometry” [137]. In 1949 Jacobs wrote to Freudenthal about the possibility of publishing the WW plans in Euclides: Plaatsing in Euclides zou bovendien moeilijkheden gaan veroorzaken, want zowel LIWENAGEL als WIMECOS vinden het niet goed dat de wiskunde werkgroep veel in Euclides plaatst. Uit alles blijkt dat zij een beetje bevreesd beginnen te worden over onze activiteit. (Publishing in Euclides would stir up trouble since neither LIWENAGEL nor WIMECOS approve the mathematics working group publishing a great deal in the journal. All this shows that they are beginning to fear our activity.) [138]

118

5 A New Start

Did the teachers’ associations think the working group was a threat to their discipline? Wansink, who was not only involved in the WW, but later also became member of the board of WIMECOS, characterized the discrepancy he experienced as follows: De opvattingen van die beide groepen stonden aanvankelijk diametraal tegenover elkaar. Achteraf noem ik ze “conservatief” voor wat wimecos betrof, “revolutionair” ten aanzien van de Wiskundewerkgroep. Ik heb in die strijd der meningen vrijwel steeds een tussenpositie ingenomen, wat tot gevolg had dat ik in de conservatieve hoek als te vooruitstevend, [sic] in de vooruitstrevende hoek als te conservatief werd beschouwd. (Initially, the ideas of these two groups were diametrically opposed. With hindsight, I call them “conservative” as far as WIMECOS is concerned, “revolutionary” as far as the Wiskunde Werkgroep is concerned. I have practically always taken an intermediate position in this battle of opinions, with the consequence that I was considered too progressive in the conservative corner, and too conservative in the progressive corner.) [139]

Nevertheless, Euclides allowed the WW space to announce that they had organized a conference weekend in 1948, the first of a long series of autumn conferences [140]. Organizing such annual conferences and the mediation of teachers like Wansink eventually meant that the WW gained more prestige and publicity among mathematics teachers.

5.3.2 ‘The Algebraic and Analytical view on the Number Concept in Elementary Mathematics’ At a cost of six guilders all interested parties could register for the 2-day conference in November 1948. No less than 48 participants, most of them not members of the working group, arrived at the Maarten Maartens-Huis in Doorn on 13 and 14 November (Fig. 5.8). With Freudenthal as chairman they listened to and discussed the lectures delivered by Bunt, Freudenthal and Van Hiele. A report of these lectures, the discussions and a list of all participants were published in Euclides. Among the participants, apart from the speakers, were: Mrs. D. van Hiele-Geldof, Jacobs, Janssen, Streefkerk, Timmer, Turkstra, Vredenduin and Wansink. Mrs. Ehrenfest was absent owing to illness. LIWENAGEL was specially represented by J.E. Schrek [141]. An extensive report of this conference appeared in Vernieuwing too. The WW was the first working group that answered the WVO’s call for its working groups to organize an annual weekend “in order to discuss the problems with a wider audience” [142]. This goal of the WVO had been resoundingly achieved: Veel deelnemers, in totaal hadden zich 53 personen opgegeven, verzochten ons van dergelijke conferenties een traditie te maken. Het was ook buitengewoon verheugend zo’n grote belangstelling te ondervinden uit de meest uiteenlopende kringen. Alle confessionele richtingen waren vertegenwoordigd. (Many participants, in total 53 people were registered, asked us to make a tradition of such conferences. It was also exceptionally satisfying to experience such a large interest from a diversity of circles. All denominational directions were represented.) [143]


119

Fig. 5.8 Picture of the participants at the weekend conference of the Wiskunde Werkgroep (1948), with among others: Bunt ( front, to the left), Wansink ( front, third from the left), Freudenthal ( front, fifth from the left), J.R. Janssen ( front, sixth from the left), Vredenduin ( one but hindmost row, second from the right) and Mrs. Van Hiele-Geldof ( second row, third from the right)

In his lecture, entitled ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’ (‘The algebraic and analytical view on the number concept in elementary mathematics’), Freudenthal looked from a higher standpoint at the algebra dealt with at secondary schools. Seen from the standpoint of higher mathematics the number concept that primary school pupils brought with them to the secondary school constituted a rather primitive foundation. To gain that insight, pupils should have learned, according to Freudenthal, to see and seek problems that superficially did not seem to be problems to them. In the history of mathematics too it had taken a long time before one considered the principles of arithmetic as theorems that could be proved. The principle of complete induction produced the natural numbers, the arithmetic operations and arithmetic principles. It was in fact the same kind of principle as that used as ‘study aid’ when explaining for instance the multiplication tables: the next result was calculated by adding to the last result. The essential difference between the method of the mathematician and that of the child was according to Freudenthal found in the more concise and more general mathematic notation: U zult vragen: “Wat hebben we aan deze subtiliteiten in het onderwijs?” En u kunt deze vraag uitbreiden, door alles wat u op de unversiteit als voorbereiding tot “de klas” moet leren, en bloc te verwerpen. Ik ben er van overtuigd, dat deze geesteshouding onjuist is. Ik meen, dat alle didactische problemen alleen bevredigend kunnen worden opgelost, wanneer

120

5 A New Start

men het meest elementaire telkens weer doordenkt vanuit hoger standpunt (waarbij ik dan toegeef, dat dit hoger standpunt niet altijd het wiskundige, maar vaak ook het paedagogisch-psychologische is). (You may wonder: “What use are these subtleties to education?” And you can extend this question by rejecting en bloc everything you learned at the university as a preparation for “the classroom”. I am convinced that this mental attitude is wrong. I think that all didactical problems can only then be solved in a satisfying manner if you think through the most elementary points from a higher standpoint (and I have to admit that this higher standpoint is not always the mathematical standpoint, but often also the pedagogic-psychological one)). [144]

With this Freudenthal made an important nuance to his view on the principle ‘lower mathematics seen from a higher standpoint’ in relation to the teachers’ training. This higher standpoint did not necessarily have to be a mathematically higher standpoint. Freudenthal did not share the then prevailing view that what one has learned well, one can also teach well without any problem: pedagogic-psychological considerations were also important in determining a method. He continued his lecture with a story about one of his sons to whom he tried to explain the subtraction of negative numbers. ‘Minus times minus is plus’ was not a good explanation for a child. Gelukkig bleef mijn zoontje volharden en dwong mij na te denken. En toen liet ik hem een tafel opmaken, die hem aan zijn eerste jaren lagere school moest doen denken, nl: 5–2=3 5–1=4 5–0=5 5–(–1)=6 5–(–2)=7 5–(–3)=8 Het middel was afdoende. (Fortunately my little son persisted and forced me to think. I let him draw up a table that had to make him think of what he had learned during the first years in primary school, viz: 5–2=3 5–1=4 5–0=5 5–(–1)=6 5–(–2)=7 5–(–3)=8

That was an effective remedy.) [145] Acting as a pedagogue he had, without being aware of it, constructed the mathematical proof for humanities track this could, for a − ( −b) = a + b and subsequently he had transformed it into the language and atmosphere of the first algebra lessons. By his own account he had not given up more than strictly necessary of the exactness of the higher mathematics by substituting the complete induction by a didactical one. This was according to Freudenthal a fundamental pedagogic principle in school mathematics: Het doel van elk bewijs is helderheid van inzicht in een structuur, en de verscherping van exactheids-eisen mag enkel het gevolg zijn van het doorzien van problemen, die de helderheid hebben vertroebeld. […] We moeten nu niet de historische ontwikkeling klakkeloos in


121

onze didactiek overnemen, maar de lijn vanuit de hogere wiskunde weer terugverlengen, om de juiste didactische methoden te vinden. (The aim of all proofs is to gain a clear understanding of a structure, and the sharpening of the exactness requirements may only be the result of seeing problems that have obscured that clarity. […] Now we should not adopt the historical development indiscriminately in our didactics, but we should extend the line from the higher mathematics back again to find the right didactical methods.) [146]

Thus in finding the right approach Freudenthal attached more value to the knowledge of the accompanying higher mathematics as an aid than to the historical development of that mathematics. Starting from the development of the discipline (the so-called historic-genetic principle) could according to him contribute to finding the right approach, but it was not necessarily decisive. Next in his lecture Freudenthal showed what he meant with examples from the teaching of algebra. There was for example the problem of the introduction of real numbers. By approaching square roots in for example geometric relation (in the “historically respectable method, such as the Babylonian were already familiar with”) these proved their worth to the pupils. The primitive representation of the real numbers as infinite decimal fractions was sufficient for the pupil to calculate with in a simple way. By contrast the Dedekind cut as a basis for the real numbers was too artificial a method and did not belong to the domain of the secondary school. According to Freudenthal calculating with infinite decimal fractions was entirely justified seen from a mathematical point of view. An infinite decimal fraction was after all also a real number known only by approximation, be it with any approximation you could ever require. Additions, subtractions, multiplications and divisions could without any objection be done according to the rules learned at primary school (provided that you knew for sure that a fraction was not composed of only zeros). There is one aspect that Freudenthal did not want to obscure, neither here nor in the curriculum: even though a correction of a decimal approximation could not be more than two units of the last decimal place, this did not mean that the place before that was “saved form the eraser”. After all, the approximation of a decimal fraction ending in a long row of nines could be overthrown by a small error. Since the continuum of numbers could not be factorized discretely different decimal fractions of the form a 0 , a1 … a n 999 … en a 0 , a1a 2 … (a n + 1)000 … had to be identified as being equal. Yet Freudenthal wondered whether it was necessary to undermine the naïve belief of the pupils in (calculating with) decimal fractions: Theorieën, zoals die der reële getallen, die uitgevonden zijn, om aan een critiek het hoofd te bieden, kunnen alleen dan door een leerling goed worden begrepen, wanneer van te voren [sic] dezelfde critische houding in hem wakker wordt geroepen, waaraan de theorie haar oorsprong dankt. Ook als de leerling hiervoor toegankelijk is, blijft de vraag naar het nut ervan nog open. Dat neemt niet weg, dat men wel in het onderwijs het feit mag vermelden, dat een reëel getal twee verschillende decimaalbreuk-ontwikkelingen kan bezitten. (Theories such as those of the real numbers, invented to meet a criticism, can only be understood

122

5 A New Start

well by a pupil, when that same attitude of criticism is firstly evoked in the pupil. Even when a pupil is receptive in this respect, the question whether it is useful still remains. All the same it is permissible to mention in the curriculum the fact that a real number can have two different approximations by decimal fractions.) [147]

According to Freudenthal it was not necessary to justify things at all costs to higher mathematics in the mathematics education. Such didactical ‘compromises’ with higher theory were acceptable if they had a pedagogical reason. Freudenthal continued his lecture by proposing that the methods of secondary and higher education should be in line with each other: “to serve the boy going to university and the graduate starting to teach.” [148]. He then showed that, by adapting the language used in secondary schools to that of modern mathematics, the naïve method sketched here resulted in a logical manner in higher mathematics (the definition of a real number as a class of equivalent fundamental sequences of rational numbers). Apart from this analytical view on the number concept he also presented an algebraic view. In general, the mode of operation of school mathematics was according to Freudenthal technically algebraic. That implied, the pupils considered √2 only as a symbol with which you calculated in such a way that its square was two. The fact that √2 had a value, could then be neglected. Freudenthal even suspected that the required practice with the operations that had to be taught, could on the contrary be acquired by the ability to forget the less formal elements. Whether this attitude was correct in secondary schools again had to be determined from a higher standpoint. Freudenthal showed that this naïve formal arithmetic was a good preparation for the formal methods prevailing in mathematics (compare this case to the formal conception of solutions of algebraic equations such as were used in the theory of number fields) and, also when seen from a higher standpoint of exactness, it was completely justified. In this technical way you could also already introduce the complex numbers in secondary school. However, Freudenthal was against this: Wil de leerling er de zin van vatten, dan zou men eerst zijn naïeve vertrouwen in de onfeilbaarheid van de formele methoden grondig moeten doorzeven—met misschien als gevolg, dat hij het geloof niet alleen verliest in i maar ook nog in √2 verliest. Zonder deze critische houding is de waarde van al die finesses op zijn minst problematisch. Waarom zou men ook in het schoolonderwijs hoger eisen van exactheid stellen, wanneer men de imaginaire getallen invoert, dan bij het invoeren der gehele getallen, der gebroken getallen, der worteluitdrukkingen en der reële getallen? (For a pupil to understand this, one should first thoroughly sift his naïve trust in the infallibility of formal methods—perhaps resulting in the pupil losing faith not only in i but also in √2. Without this critical attitude the value of all these finer points is at the very least problematic. Why would one make higher demands on education at school when introducing the imaginary numbers compared to introducing integers, fractions, roots or real numbers?) [149]

To conclude Freudenthal once again effectively recapitulated his message to the mathematics teachers: Ik zeg u dit niet, om uw wetenschappelijk geweten in slaap te sussen. Ik wil het veeleer wakker schudden. Echter niet, om aan u te vragen, dat u naast een slordige schoolwiskunde ook nog een boven elke blaam verheven hogere wiskunde zoudt beoefenen. Het doel van uw bekendheid met hogere wiskunde moet veeleer zijn, een vruchtbare synthese te kunnen zoeken—vruchtbaar zowel ten bate van het onderwijs als ook ten bate van de wetenschap.


123

(I do not tell you this to salve your scientific conscience. I rather want to shake it awake. But not in order to ask you that you would practice apart from a sloppy school mathematics also a higher mathematics beyond all blame. The goal of you knowing higher mathematics should rather be to be able to look for a fruitful synthesis—fruitful both for the benefit of education and for the benefit of science.) [150]

Streefkerk started the discussion after Freudenthal’s lecture with the question to what extent teachers should in this way revert to increasingly modern developments in higher mathematics? Where did it stop? Freudenthal immediately seized this opportunity to put forward his interpretation of the teachers’ training at the university. With so many teachers in the company this resulted in a lively discussion about the teachers’ training [151]. Freudenthal’s lecture was the ‘Higher education’ component of the three lectures at the world conference. In his approach to mathematics education he aimed at mathematics for those pupils who would encounter mathematics again in higher education. Bunt’s lecture, entitled ‘De keuze van de leerstof bij het onderwijs in de wiskunde’, (‘The choice of subject matter when teaching mathematics’) was the ‘research’ component at the conference. He emphasized the wide gap in the Netherlands in the field of (sharing) the practical experience of teachers. According to him too little was known about experience with subject matter in the class room to arrive at a definite choice for and interpretation of the curriculum. Only by sharing and publishing their experience to develop criteria by, could teachers, using the knowledge thereby acquired, arrive at a well thought-out interpretation of mathematics education [152]. Van Hieles lecture, entitled ‘Een poging om de richtlijnen op te stellen voor een didactiek der wiskunde’, (‘An attempt to compose guidelines for a didactics of mathematics’) could be considered the ‘practical’ component of the weekend. He outlined guidelines, clues and questions which could help in establishing a practical and well-founded didactic of mathematics. He departed from the intended goal of education and the best and easiest way to reach that goal with pupils [153]. The various discussions after the lectures clearly showed how the teachers differed in their opinions. There was, for example, the almost classic discussion between supporters and opponents of the axiomatic structure of geometry education in the lower classes and a discussion as to whether it was good or not to have pupils learn something against their will [154]. Freudenthal, supporter of abolishing descriptive geometry at the HBS, and Wansink, passionate supporter of maintaining this subject, got caught up in a public discussion on this subject [155]. Irrespective of the diversity and mix of the participating public, the good atmosphere was not affected by these discussions. This atmoshpere was illustrated by the following phrase from the report of the weekend in Vernieuwing. ’s Avonds speelden de heren Leujes en Reckendorf een duo voor viool en altviool van Mozart. Later samen met Dr. van Tol een trio van Telemann voor piano, viool en altviool. Het was een bewijs van de hechte band onder de deelnemers, dat deze heren, die elkaar nooit eerder hadden ontmoet, aan deze avond een gezellig muzikaal tintje gaven. (In the evening messrs. Leujes and Reckendorf played a duet for violin and alt violin by Mozart. Later they played a trio for piano, violin and alt violin by Telemann with Dr.van Tol. It proved the close connections between the participants that these gentlemen, who had never met before, gave an enjoyable musical touch to this evening.) [156]

124

5 A New Start

Words of gratitude to the organizers of the “exceptionally successful conference” concluded the weekend [157].

5.3.3 ‘Mathematics for Non-Mathematical Studies’ Following the great success of 1948 one year later the working group again organized a weekend conference chaired by Freudenthal on 5 and 6 November. The general subject chosen for the conference this time was ‘Mathematics for non-mathematical studies’. Apart from being announced in Vernieuwing the conference was also announced in Euclides: Dit onderwerp zal belicht worden van algemene, van paedagogische en van niet-mathematische zijde, door de heren Dr. F. van der Blij, leraar aan het Gem. Gymnasium te Breda, P.J. van Albada, leraar aan het Lyceum voor Montessori-onderwijs te Rotterdam en H.J. Struik, zenuwarts te Deventer, terwijl Prof.Dr. H. Freudenthal, evenals verleden jaar, bereid is gevonden de algemene leiding op zich te nemen. (The general, pedagogic and non-mathematical aspects of this subject will be elucidated by Dr. F. van der Blij, teacher at the Gemeentelijk Gymnasium in Breda, P.J. van Albada, teacher at the lyceum for Montessori education in Rotterdam, and H.J. Struik, neurologist in Deventer respectively, while Prof. Dr. H. Freudenthal has been found willing to take the general chair.) [158]

Euclides’ editors neither wanted to publish the lectures that were delivered nor a report of the conference. They only incorporated a short reference to the report in Vernieuwing [159]. In Vernieuwing only Van Albabda’s complete lecture was printed in addition to this report [160]. This conference too was a success: 50 participants came to the Maarten-Maartenshuis in Doorn. Unlike at the previous weekend the lectures and the discussions did not stress mathematics education as much. Van der Blij in his lecture ‘Wiskunde in de cultuur ’ (‘Mathematics in culture’), for example, spoke about the coherence and the battle between mathematics and religion in cosmology. Struik discussed in his lecture ‘De waarde van de wiskunde voor de niet-mathematicus’ (‘The value of mathematics for the non-mathematician’) the method he adopted to explain insights in another field with mathematical figures. Van Albada (1905–1997), WW-member from the very beginning, in his lecture ‘De wiskunde voor niet-mathematische richtingen’ (‘Mathematics for non-mathematical studies’) was the only one to address the situation in education. He criticized the drilling for the final examination which every year added to the content of the curriculum a new load of useless puzzle-problems that the next senior class had to practice. He argued that in the lower classes the accent in choosing the curriculum should be on the needs of the pupils. In doing so it should be realized what was psychologically best for which age of the pupil. In the higher classes the accent could be more on the needs of society. For the α- or humanities track this could, for example, mean the introduction of statistics into the programme. Next Van Albada spoke of his positive experiences with mathematics education at a Montessori lyceum [161]. During the discussion after his lecture Freudenthal proposed to distinguish three categories of pupils: those who actually would need a complete mathematics

References

125

education, those for whom mathematics certainly had some form of practical value and for whom algebra was the most important aspect, and those who only studied mathematics for its cultural value. The difficulty was in establishing the direction that was best for the pupil in time [162]. He no longer mentioned a secondary school ‘without mathematics’ as he had proposed it in his lecture ‘Opvoeding tot denken’, here. The 3 years of WW meetings he had attended had perhaps convinced him of the possibility of ‘mathematics for all’.

5.3.4 Freudenthal’s Mathematical Working Group In 1987 Freudenthal wrote that he could not remember anything of his first years as a member of the mathematics working group: Mogelijk was de eerste bijeenkomst die ik bijwoonde die op 11 januari 1947 in het Montessorilyceum in de Lairessestraat in Amsterdam (ik woonde toen nog in die stad), maar ik herinner me er niets van, evenmin als van de bijeenkomsten daarna. (Possibly the first meeting I attended was that of 11 January 1947 at the Montessori lyceum in the Lairessestraat in Amsterdam (I lived in the city at that time), but I remember nothing of this meeting, nor of the later ones.) [163]

Nevertheless, he must have played an active role in the working group. The reports of the weekend conferences showed that he took it upon himself generally to chair the discussions. Furthermore, he did not hesitate to intervene in the discussions: his name being mentioned in practically every report of the discussions. He was asked as a professor, during the first conference, to throw light on the curriculum from a ‘higher standpoint’. The fact that he enjoyed respect and prestige in WW circles became clear from Van Hiele’s memories of the committee work. The combination of his enthusiastic devotion to the right cause, his personality geared to modernization and the respect he enjoyed as a scientist, will have been the reason that in February 1950 Freudenthal took over Janssen’s role as chairman [164]. This quiet transfer marked for Freudenthal the beginning of a period in which mathematics education in secondary schools would be taking up a prominent place in his interests and his work.165 The era in which the WW would become known as ‘Freudenthal’s Mathematics Working Group’ had started [166].

References 1. Freudenthal, Schrijf dat op, Hans 323. 2. In composing this and following paragraphs information has been taken from the following studies: Freudenthal, Schrijf dat op, Hans; Alberts, Jaren van berekening; G. Alberts, F. van der Blij, J. Nuis, ed., Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen (Amsterdam 1987); G. Alberts, H.J.M. Bos, J. Nuis, ed., Om de wiskunde. Stimulansen voor toepassingsgerichte wiskunde rond 1946 (Amsterdam 1989); Bertin, Bos, Grootendorst, ed., Two decades of mathematics in the Netherlands; Van Dalen, L.E.J. Brouwer; H.W. von der Dunk, W.P. Heere, A.W. Reinink, ed., Tussen ivoren toren & grootbedrijf; De Utrechtse

126

5 A New Start

Universiteit 1936–1986 (Utrecht 1986); Knegtmans, Een kwetsbaar centrum van de geest; L. Molenaar, De rok van het universum. Marcel Minnaert. Astrofysicus 1893–1970 (Amsterdam 2003); Molenaar, ‘Wij kunnen het niet langer aan de politici overlaten.’. Unfortunately it has not been possible to locate an archive of the Wiskunde Werkgroep 3. Alberts et al., ed., Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen 116. Van Heemert also heard from his parents that Freudenthal rode on a tank through Amsterdam. Van Heemert to Freudenthal, 5 June 1945, RANH Hans Freudenthal Papers, inv.nr. 36. 4. See also: A.W. de Groot, De universiteit van Amsterdam (Amsterdam 1946) 86. 5. Freudenthal to Mayor and Aldermen of Amsterdam, 11 June 1945; RANH, Hans Freudenthal Papers, inv.nr. 323. For that matter, Bruins had studied mathematics, but defended his thesis later based on experimental and theoretical work supervised by the physicist Clay; J.A. van Maanen, ‘In memoriam prof.dr. E.M. Bruins (1909–1990)’, Tijdschrift voor de Geschiedenis der Geneeskunde, Natuurwetenschappen, Wiskunde en Techniek 14 (1991) 116–118, q.v. 116. 6. Freudenthal, Schrijf dat op, Hans 117. 7. Hans Freudenthal, 5000 jaren internationale wetenschap (Groningen 1946) 22. 8. E.H. Kossmann, ‘Nederland in de eerste na-oorlogse jaren’, in: Alberts et al., ed., Om de wiskunde 7–16, q.v. 15. 9. P.C. Baayen, ‘Beleid om de wiskunde’, in: Alberts et al., ed., Om de wiskunde 61–74, q.v. 63–64. 10. Alberts, Jaren van berekening 198. 11. Molenaar, ‘Wij kunnen het niet langer aan de politici overlaten’ 9. 12. Hans Freudenthal, ‘Het Trippenhuis’, in: De Groene Amsterdammer, from October 1951: RANH, Hans Freudenthal Papers, inv.nr. 1450 et seq. Freudenthal became a member of the KNAW (Koninklijke Nederlandse Academie van Wetenschappen, Royal Dutch Academy of Science) on 28 April 1951. 13. Alberts et.al., ed., Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen 8. 14. Alberts, Jaren van berekening 153–154. 15. Even though he appreciated Freudenthal's mathematical qualities, Brouwer, with his eccentric character, could not have borne the “restless activity” of a mathematician like Freudenthal in his direct environment. Nevertheless, even after these disappointing developments Freudenthal continued to hold Brouwer in great esteem. For example, in the 1970s he took care of his collected works; Van Dalen, L.E.J. Brouwer 446–447; Freudenthal, ed., L.E.J. Brouwer Collected Works 2. Geometry, analysis, topology and mechanics (Amsterdam 1976). 16. F. van der Blij e.a., Een kwart eeuw wiskunde 1946–1975 (Amsterdam 1973) 81–83; syllabus accompanying the course ‘Numerieke en graphische methoden’(‘Numerical and graphical methods’): RANH, Hans Freudenthal Papers, inv.nr. 474; Freudenthal to the Board of management of the MC on 19 April 1947, Alberts et al., ed., Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen 139–148. 17. Alberts et al., ed., Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen 119. 18. Correspondence between Freudenthal and J. Sittig of the Adviesbureau voor Toegepaste Statistiek (Advisory Bureau of Applied Statistics), 1948: RANH, Hans Freudenthal Papers, inv.nr. 2; J. Sittig and H. Freudenthal, De juiste maat. Lichaamsafmetingen van Nederlandse vrouwen als basis van een nieuw maatsysteem voor damesconfectie (Leiden 1951) (The correct size. Bodily dimensions of Dutch women as a basis for a new size system for ladies clothing). 19. R.M. Nepveu et.al., Universitaire flitsen: van Illustre School tot Rijksuniversiteit (Utrecht 1986) 80; Von der Dunk et.al., ed., Tussen ivoren toren & grootbedrijf 399. 20. Conversation with Van der Blij, 20 May 1999; Nepveu et.al., Universitaire flitsen 79–80. 21. Van Est defended his thesis A generalization of a theorem of J. Nielsen concerning hyperbolic groups on 3 April 1950 cum laude; I. Moerdijk and J.P. Murre, ‘In memoriam Willem Titus van Est’, Nieuw archief voor wiskunde 5/4 (2003) 281–283. 22. Being disappointed in the mathematics field during the Second World War is said to have been an extra motivation for his didactical interest. Freudenthal is said to not have been able to publish an important mathematical discovery found by him in a roundabout way, someone

References

127

else got ahead of him. However, there is no evidence found in his personal archive for this motivation. After the war too Freudenthal's reputation as a great mathematician endured; conversation with Mirjam Freudenthal, 24 April 1999. 23. Freudenthal, Schrijf dat op, Hans 345. 24. The New Era in Home and School was the monthly journal of the New Education Fellowship. 25. Personal correspondence (among which correspondence between Freudenthal and Jan Gunning, headmaster of ‘Het Nieuwe Lyceum’ in Bilthoven): RANH, Hans Freudenthal Papers, inv.nr. 175; S.J.C. Freudenthal-Lutter, ‘Die Entwicklung der Jenaplan-bewegung in den Niederlanden’ (ongepubliceerd 1985): RANH, Papers of Susanna Freudenthal-Lutter, inv.nrs. 37, 87. 26. Correspondence between Freudenthal and Jan Gunning, headmaster of ‘Het Nieuwe Lyceum’ in Bilthoven, 1949–1953: RANH, Hans Freudenthal Papers, inv.nr. 175. 27. RANH, Susanna Freudenthal-Lutter Papers, inv.nr. 37. 28. In 1885 educator Ligthart became headmaster of the ‘Openbare School voor Onvermogenden’ (‘Public school for the Poor’) in The Hague where he developed his pedagogic ideas in practice. Ligthart's pedagogy was characterized by putting the child in the central role, the trust he had in children and the opportunity he wanted to give them to move freely to learn and to take their own responsibility. In his ‘zaakonderwijs' (teaching of practical subjects) real life was taken as a lead for didactics; S.J.C. Freudenthal-Lutter, Naar de school van morgen (Alphen aan de Rijn 1971) 161; B.C. de Jong, Jan Ligthart (1859–1916). Een schoolmeester-pedagoog uit de Schilderswijk (Groningen 1996). 29. Diaries and notebooks of Freudenthal; RANH, Hans Freudenthal Papers, inv.nrs. 275, 441, 440. 30. Pocket diaries 1946–1949: RANH, Hans Freudenthal Papers, inv.nrs. 96–99. 31. Interview with Hans Freudenthal, ‘Hans Freudenthal, tussen de bergen en de zee’, in: G.H. Jansen, S. Vess, Ballonnen en brood. Utrechtse portretten (Utrecht 1986) 101–107, q.v. 105. 32. Freudenthal to H. Guttfeld, 30 November 1947: RANH, Hans Freudenthal Papers, inv.nr. 175. 33. Freudenthal to J.R. Janssen, 31 July 1945; RANH, Hans Freudenthal Papers, inv.nr. 42. 34. J.R. Janssen to Freudenthal, 1 August 1945; RANH, Hans Freudenthal Papers, inv.nr. 42. 35. Pocket diaries 1946–1949: RANH, Hans Freudenthal Papers, inv.nrs. 96–99; Freudenthal, Schrijf dat op, Hans 343. 36. Freudenthal-Lutter, Naar de school van morgen 149. 37. This led to the establishment of the ‘Vernieuwingsraad’ (Modernization council). Eventually this initiative failed because the various modernization initiatives and the (accompanying) ideologies in the Netherlands could not be integrated. See also: Freudenthal-Lutter, Naar de school van morgen 150; C.J.J.A. Morsch, Met de moed van de hoop. Studies over de vernieuwing van opvoeding, onderwijs en maatschappij in Nederland in de periode tussen ± 1930 en 1984 (Eindhoven 1984) 224–228. 38. J.D. Imelman, W.A.J. Meijer, De nieuwe school gisteren en vandaag (Amsterdam 1986); Freudenthal-Lutter, Naar de school van morgen; Morsch, Met de moed van de hoop. 39. The text of this lecture was not published; RANH, Hans Freudenthal Papers, inv.nr. 466. 40. In this period Freudenthal also wrote the unpublished manuscript ‘Denken en logica’ (Thinking and logic) about logic forms and thinking, in which he reached the same type of conclusions as in ‘Opvoeding tot denken’ (Education for thinking); RANH, Hans Freudenthal Papers, inv.nr. 458. 41. Kohnstamm, Keur uit het didactisch werk (Groningen 1948) 91. The article originally dated back to 1934 and was published as ‘Mededeling 2’ (‘Announcement 2) of the Nutsseminarium in Paedagogische Studiën. 42. Hans Paul Roloff, ‘Vergleichend-psychologische Untersuchungen über kindliche Definitionsleistungen’, Beiheft zur Zeitschrift für Angewandte Psychologie 27 (1922), quoted in: Kohnstamm, Keur uit het didactisch werk 81–82. 43. RANH, Hans Freudenthal Papers, inv.nr. 440. In his notebook he accompanied the article of Kohnstamm about Roloff's research with: ‘This is haunted by genus proximum and differentia specifica’.

128

5 A New Start

44. RANH, Hans Freudenthal Papers, inv.nr. 466. Thijs contributed this definition. Freudenthal wrote this conversation down on 18 June 1943 in his notebook with notes about ‘arithmetic didactics’; RANH, Hans Freudenthal Papers, inv.nr. 440; Hans Freudenthal, ‘Opvoeding tot denken’ (unpublished 1945) 15 pages, 2. 45. The American Carleton W. Washburne (1889–1968) was teacher and educator, well-known because of the experimental programmes he developed as chief inspector of the ‘Winnetka Public Schools' (Illinois) from 1919 onwards. Education in these schools was individual and had a differentiated structure with a lot of attention for the differences in learning capacities of the pupils and emphasis on working independently. With Washburne as leader a ‘Department of Educational Research’ was also established, where, among other things, research was done into the time pupils needed to master certain subject matter at a certain age; J.D. Pulliam, History of education in America (Columbus 1976) 160, 179; E.L. Bakker, ‘The technology of instructional development’, in: R.M.W. Travers e.a., ed., Second handbook of research on teaching (Chicago 1973) 245–285, q.v. 248; R.H. Bremner e.a., ed., Children & youth in America. A documentary history. Volume ii 1866–1932 (Cambridge 1971). 46. Freudenthal, ‘Opvoeding tot denken’ 4; RANH, Hans Freudenthal Papers, inv.nr. 466. 47. Ibid., 6. 48. Ibid., 7. 49. Ibid., 10–11. 50. Ibid., 11. 51. As an example of the non-analytical character Freudenthal contributed the following riddle: Two Arabs are on the road. When they see the oasis they stop to rest a while. While lying lazily in the sand, one of them proposes a bet. 100 piastres for whose camel arrives last in the oasis. The other accepts the bet and the consequence is that neither shows the intention of getting up and mounting his camel, since they both do not want to risk 100 piastres. They remained horizontal, and although both are bored, they probably would still be there had a Dervish whom they could ask for advice not happened to pass. They explained their case to the Dervish. The Dervish thought for a while, called out a short sentence to them and… all of a sudden they jumped to their feet, got on their camels and headed head over heels for the oasis,—the question is: What did the Dervish say? Freudenthal considered this riddle too artificial as an arithmetic word problem, but characteristic of the synthetic type: one breakthrough, one prompting is enough to solve the riddle. 52. Ibid., 13. 53. Ibid., 15. 54. Freudenthal, Schrijf dat op, Hans 224. 55. Freudenthal, Schrijf dat op, Hans 119; Hans Freudenthal, ‘Struggle for life’ (unpublished 1945); RANH, Hans Freudenthal Papers, inv.nr. 468. 56. Freudenthal, ‘Struggle for life’ (unpublished 1945) 3; RANH, Hans Freudenthal Papers, inv. nr. 468. 57. Ibid., 4. 58. Morsch, Met de moed van de hoop 224 et seq. 59. Correspondence between Freudenthal and the editors of De Groene Amsterdammer (1946); RANH, Hans Freudenthal Papers, inv.nr. 34. 60. Ibid. 61. Alberts, Jaren van berekening 37. 62. Hans Freudenthal, ‘Rekenmachines winnen den oorlog’, De Groene Amsterdammer (16 maart 1946); RANH, Hans Freudenthal Papers, inv.nr. 1405. 63. Hans Freudenthal, ‘Redeneren en doen’, De Groene Amsterdammer (8 maart 1947), Hans Freudenthal, ‘Examen in vergiftigde toestand’, De Groene Amsterdammer (8 oktober 1949); RANH, Hans Freudenthal Papers, inv.nrs. 1407 and 1423. 64. Freudenthal, 5000 jaren internationale wetenschap (Groningen 1946). 65. Hans Freudenthal, ‘Driehoeken en spijkerschriftcijfers’, in: Hans Freudenthal, Van sterren tot inlegzolen (Arnhem 1954) 119–122, q.v. 119–120; published in De Groene Amsterdammer (8 April 1950); RANH, Hans Freudenthal Papers, inv.nr. 1431.

References

129

66. Hans Freudenthal, ‘Eikenhout is Eikenhout’, De Groene Amsterdammer (13 March 1948); RANH, Hans Freudenthal Papers, inv.nr. 1414. 67. See also paragraphs 5.2 and 5.3. From 1974 he would find a similar outlet—and get an even greater public name—in his articles for NRC/Handelsblad; RANH, Hans Freudenthal Papers, inv.nr. 1627–1700. 68. Kees Boeke, ‘Van onderwijsvernieuwing naar mensvorming’, Vernieuwing van Opvoeding en Onderwijs 7 (1949) 298–299, q.v. 299. 69. ‘Verslag van de Varende Conferentie’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 16–21, q.v. 20. 70. Ibid., 20. 71. Hans Freudenthal, ‘De medewerkende taak van de opvoeder bij de mensvorming’ (unpublished 1949) 13 pages, 10; RANH, Hans Freudenthal Papers, inv.nr. 485. 72. Ibid., 12–13. 73. De Vrije Katheder (The Free Lectern) originated in 1940 as a resistance paper of communist students at the University of Amsterdam and functioned between 1945 and 1950 as a platform for leftist intellectuals and artists. More about De Vrije Katheder in: F. van den Burg, De Vrije Katheder 1945–1950: een platform van communisten en niet-communisten (Amsterdam 1983). 74. Hans Freudenthal, ‘Onderwijsvernieuwing, maatschappelijk gezien’, De Vrije Katheder 8 (1949) 6–8, q.v. 8. 75. For this paragraph a lot of information is taken from the following studies: Alberts, Jaren van berekening; Alberts e.a., red., Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen; Alberts e.a., red., Om de wiskunde; Molenaar, ‘Wij kunnen het niet langer aan de politici overlaten’; J.C.C. Rupp, Van oude en nieuwe universiteiten (Den Haag 1997). 76. Freudenthal to H. Hopf, 7 Nov 1945; RANH, Hans Freudenthal Papers, inv.nr. 88. 77. Hans Freudenthal, ‘Oude en nieuwe universiteiten’ (unpublished 1945) 75 pages, 1; RANH, Hans Freudenthal Papers, inv.nr. 467. 78. Freudenthal, Schrijf dat op, Hans 337. 79. The fact that Freudenthal watched his step in his accompanying letter is most probably due to the important role that Van der Corput played in this period in the distribution of mathematics chairs and the dependent position Freudenthal occupied. 80. Freudenthal to Van der Corput, 20 Oct 1945; RANH, Hans Freudenthal Papers, inv.nr. 18. 81. Van der Corput to Freudenthal, 24 October 1945; RANH, Hans Freudenthal Papers, inv.nr. 18. 82. Erven P. Noordhoff Uitgevers gladly wanted to publish Freudenthal's article “subject to some minor comments”, but were confronted with a scarcity of printing paper. Freudenthal himself would have to provide printing paper for about 500 copies so that the article, while it still was topical, could be published as soon as possible; correspondence between Erven P. Noordhoff Uitgevers Groningen and Freudenthal, December 1945; RANH Hans Freudenthal Papers, inv.nr. 29. 83. RANH, Hans Freudenthal Papers, inv.nr. 1426. Freudenthal also writes in De Groene Amsterdammer about the shortcomings of Studium Generale as a means in the battle against specialization: ‘Reorganisatie van het Hoger Onderwijs ii’ of 26 November 1949, published with reference to that Report of the Reinink committee II. In the article he brought up the same arguments as in ‘Oude en nieuwe universiteiten’. 84. Freudenthal, ‘Oude en nieuwe universiteiten’ 24; RANH, Hans Freudenthal Papers, inv. nr. 467. 85. Ibid., 37. 86. Ibid., 37–38. 87. Ibid., 38. Here the theme ‘education by reading’ recurred that Freudenthal also explained in his lecture ‘Education for thinking’. 88. Freudenthal, Studienerinnerungen 4. 89. Freudenthal, ‘Oude en nieuwe universiteiten’ 66; RANH, Hans Freudenthal Papers, inv.nr. 467. 90. Ibid., 68.

130

5 A New Start

91. Freudenthal, 5000 jaren internationale wetenschap 22. 92. Freudenthal, ‘De medewerkende taak van de opvoeder bij de mensvorming’; RANH, Hans Freudenthal Papers, inv.nr. 485. In that lecture too he mentioned the anti-intellectualism as one of the dogmas of National Socialism. 93. For the discussion about the teachers' training during the interbellum period see paragraph 2.2.4. 94. See also: Alberts, Jaren van berekening. 95. Molenaar, De rok van het universum; Molenaar, Wij kunnen het niet langer aan de politici overlaten 124. 96. More about ‘student wage’ in paragraph 5.2.3. 97. Freudenthal, 5000 jaren internationale wetenschap 20–21. 98. Minnaert to Freudenthal, 7 January 1947; RANH, Hans Freudenthal Papers, inv.nr. 53. 99. Freudenthal, ‘Oude en nieuwe universiteiten’ 24; RANH, Hans Freudenthal Papers, inv.nr. 467. 100. Ibid., 41–45. 101. He gave an example of this in 1948 in his WW lecture ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides 24 (1948) 106–121. 102. Freudenthal, ‘Oude en nieuwe universiteiten’ 51–52; RANH, Hans Freudenthal Papers, inv. nr. 467. 103. Ibid., 57–58. 104. He had used this method immediately after the war in Amsterdam and later on also in Utrecht in his own lectures. He mentioned this in many letters to colleagues mathematicians such as Hopf, Van der Corput and Rosenthal (1945); RANH, Hans Freudenthal Papers inv. nrs. 18, 66, 88. 105. Hans Freudenthal, ‘Voorstellen tot reorganisatie van het Hoger Onderwijs in de Wiskunde’, 4 december 1947, 4 pages; RANH, Hans Freudenthal Papers, inv.nr. 475. 106. Ibid., 1–2. 107. This was a real problem as shown from information Wansink described in 1950: 22 % of the teachers who taught mathematics were not qualified (sufficiently) to do so in 1950: Joh. H. Wansink, ‘Mathematical teaching in Dutch secondary schools’, Euclides 26 (1950) 99–114, q.v. 111. 108. Freudenthal, ‘Voorstellen tot reorganisatie van het Hoger Onderwijs in de Wiskunde’ 2; RANH, Hans Freudenthal Papers, inv.nr. 475. 109. Freudenthal proposed the possibility to let the periodic increases in salary depend on whether these courses were attended or not; Freudenthal, ‘Voorstellen tot reorganisatie van het Hoger Onderwijs in de Wiskunde’ 4; RANH, Hans Freudenthal Papers, inv.nr. 475. 110. Addition to the discussion paper; RANH, Hans Freudenthal Papers, inv.nr. 475. 111. Hans Freudenthal, ‘Tekort aan leraren. Gebreken aan de opleiding’, De Groene Amsterdammer (3 July 1948); RANH, Hans Freudenthal Papers, inv.nr. 1419. 112. ‘Report of section M’, in: Rapport van de Staatscommissie tot Reorganisatie van het Hoger Onderwijs (’s-Gravenhage 1949) 335–353, q.v. 341–344. 113. Hans Freudenthal, ‘Tekort aan leraren. Gebreken aan de opleiding’, De Groene Amsterdammer (3 July 1948); RANH, Hans Freudenthal Papers, inv.nr. 1419. 114. ‘Joh.H. Wansink’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 42. 115. Wansink, Didactische oriëntatie voor wiskundeleraren. Part I, II, III. 116. Interview with Wansink in: Goffree, Ik was wiskundeleraar 90. 117. Freudenthal, ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides 24 (1948) 106–121, q.v. 120–121. 118. Freudenthal, ‘Oude en nieuwe universiteiten’ 26; RANH, Hans Freudenthal Papers, inv. nr. 467. 119. Molenaar, ‘Wij kunnen het niet langer aan de politici overlaten’ 90–91. 120. Hans Freudenthal, ‘De recrutering der studenten. Het plan-studieloon der Afdeling Utrecht van het Verbond van Wetenschappelijke Onderzoekers’, 24 pages, q.v. 6; Also published in De toelating tot het Hoger Onderwijs Verslag van het congres te Amsterdam 9–10

References

131

December 1949, 51–74; RANH, Hans Freudenthal Papers, inv.nr. 743. A brief summary of this text is in the article: Hans Freudenthal, ‘De universiteit—voor wie?’, De Groene Amsterdammer (5 november 1949); RANH, Hans Freudenthal Papers, inv.nr. 1425. 121. Freudenthal, ‘De recrutering der studenten’ 16; RANH, Hans Freudenthal Papers, inv. nr. 743. 122. Hans Freudenthal, ‘Zwaardere eindexamens?’, De Groene Amsterdammer (23 sep 1950); RANH, Hans Freudenthal Papers, inv.nr. 1435; Hans Freudenthal, ‘De rol der traditie in de wiskundige wetenschappen’, De Nieuwe Stem (1950) 574–589; Hans Freudenthal, ‘De dwang der denkgewoonten’, Wetenschap en Samenleving (1950) (deel I) 177–180, (1951) (deel ii) 12–16. 123. Freudenthal, ‘Tekort aan leraren. Gebreken aan de opleiding’, De Groene Amsterdammer (3 July 1948); RANH, Hans Freudenthal Papers, inv.nr. 1419. 124. Freudenthal, ‘Oude en nieuwe universiteiten’ 71; RANH, Hans Freudenthal Papers, inv. nr. 467. 125. Freudenthal, ‘De recrutering der studenten’ 22; RANH, Hans Freudenthal Papers, inv.nr. 743. Less than a year earlier he still wrote in De Vrije Katheder that one would have to start modernizing education with the most fundamental education, the primary education. 126. Janssen to Freudenthal, 10 Dec 1946; RANH, Hans Freudenthal Papers, inv.nr. 42. 127. Ibid. 128. Jacobs to Freudenthal, 22 Dec 1946; RANH, Hans Freudenthal Papers, inv.nr. 42. 129. Pocket diaries 1947–1950; RANH, Hans Freudenthal Papers, inv.nrs. 97–100. 130. H. Streefkerk, ‘Het leerplan voor de wiskunde op de h.b.s.’, Euclides 21 (1945) 30–39. He gains support in this article from among others M. Eilander: M. Eilander, ‘Caveant consules’, Euclides 22 (1947) 212–221. 131. L.N.H. Bunt, De leerstof van ons wiskundeonderwijs. Een onderzoek naar opvattingen en gebruiken dienaangaande (Groningen 1949) 7. This book contains the results of an inquiry among mathematics teachers that Bunt conducted on behalf of the Paedagogisch Instituut in Utrecht. In it the overloading and the desire for various parts of the programme were investigated. See also paragraph 6.3. 132. More about the Wiskunde Werkgroep in: E. de Moor, ‘Didactische pioniers’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 193–206. 133. Interview with Van Hiele in: Goffree, Ik was wiskundeleraar 122. 134. J.K. Timmer, ‘De wiskunde werkgroep na 1945’, Vernieuwing van Opvoeding en Onderwijs 183 (1961) 185–186, q.v. 186. 135. Interview with Van Hiele in: Goffree, Ik was wiskundeleraar 122–123. 136. About the origin and the content of this curriculum see also paragraph 6.3. 137. Jacobs to Freudenthal, 3 April 1949; RANH, Hans Freudenthal Papers, inv.nr. 42. 138. Ibid. 139. Interview with Wansink in: Goffree, Ik was wiskundeleraar 90. 140. ‘Wiskunde werkgroep der W.V.O.’, Euclides 24 (1948) 56. The same announcement was also published in Vernieuwing van Opvoeding en Onderwijs (1948) 63. 141. ‘Het conferentie-weekend op het Maarten-Maartens-huis te Doorn georganiseerd door de Wiskunde Werkgroep der W.V.O.’, Euclides 24 (1948) 81–133. 142. ‘Weekend-conferentie uitgaande van de Wiskunde Werkgroep der wvo’, Vernieuwing van Opvoeding en Onderwijs 7 (1948) 134–136, q.v. 134. 143. Ibid., 134. 144. Freudenthal, ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides 24 (1948) 106–121, q.v. 108. 145. Ibid., 109. 146. Ibid., 110. 147. Ibid., 114. 148. Ibid., 114. 149. Ibid., 119. 150. Ibid., 119.

132

5 A New Start

151. For a description of this discussion, see also paragraph 5.2.2. 152. L.N.H. Bunt, ‘De keuze van de leerstof bij het onderwijs in de wiskunde’, Euclides 24 (1948) 83–105. 153. P.M. van Hiele, ‘Een poging om de richtlijnen op te stellen voor een didactiek der wiskunde’, Euclides 24 (1948) 122–133. 154. Ibid., 131. 155. Freudenthal, ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides 24 (1948) 106–121, q.v. 121. 156. ‘Weekend-conferentie uitgaande van de Wiskunde Werkgroep der wvo’, Vernieuwing van Opvoeding en Onderwijs 7 (1948) 134–136, q.v. 135. 157. Ibid., 136. 158. Announcement in: Euclides 25 (1949) 36. Van der Blij would come to Utrecht in the 1950s to replace Bockwinkel. 159. ‘Tweede conferentie-weekend uitgaande van de wiskunde werkgroep der wvo’, Euclides 25 (1950) 172. Streefkerk, editor of Euclides, at first did not want the WW to publish in Euclides. He later reconsidered his decision and informed Jacobs that he would speak with Wijdenes about publishing a short report if such should be requested. In the end Van Albada's lecture was the only one that was published three years later in Euclides. P.J. van Albada, ‘De wiskunde voor de niet-mathematische richtingen’, Euclides 28 (1952) 1–11. See also the correspondence between Freudenthal and Jacobs, 11 November 1949; RANH, Hans Freudenthal Papers, inv.nr. 42. 160. ‘Tweede conferentie-weekend, georganiseerd door de wiskunde werkgroep’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 174–176; P.J. van Albada, ‘De wiskunde voor nietmathematische richtingen’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 154–159. 161. P.J. van Albada, ‘De wiskunde voor niet-mathematische richtingen’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 154–159. 162. ‘Tweede conferentie-weekend, georganiseerd door de wiskunde werkgroep’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 174–176. 163. Freudenthal, Schrijf dat op, Hans 343. 164. Jacobs to Freudenthal, 27 Feb 1950; RANH, Hans Freudenthal Papers, inv.nr. 42; Wansink, ‘De Wiskunde-Werkgroep van de w.v.o.’, Euclides 28 (1953) 197–205, q.v. 199. 165. From 1950 (criticizing) mathematics education became a repeated subject in Freudenthal’s articles in De Groene Amsterdammer. Examples are: ‘Zwaardere eindexamens?’ (23 september 1950); ‘Practische rekenkunde’ (24 October 1950); ‘En nu…de wiskunde’ (2 December 1950); ‘De impopulariteit der wiskunde’ (unknown date); RANH, Hans Freudenthal Papers, inv.nrs. 1435, 1436, 1439, 491. 166. Morsch, Met de moed van de hoop 222.

Chapter 6

From Critical Outsider to True Authority

Ik heb het altijd leuk gevonden om tot een gehoor te spreken; een college, een voordracht, een tafelspeech, leuke dingen. Ik kom als een acteur over, hè?(I have always enjoyed speaking to an audience: a lecture, a discourse, an after-dinner speech, it is always fun. I come across like an actor, don’t I?) Hans Freudenthal in ‘Hans Freudenthal, tussen de bergen en de zee’ (Hans Freudenthal, ‘Between the mountains and the sea’), 1986 [1]

Public speaking came naturally to Freudenthal (Fig. 6.1) [2]. There had to be a very legitimate reason for him not to accept an invitation to offer his opinion and ideas for some association or organization. In the mid 1950s these numerous lectures, discourses and accompanying publications not only turned Freudenthal into a wellknown mathematician with a well above average interest in educational affairs, but also resulted in him gradually coming to be considered an authority in this field. This did not happen overnight. It was not only Freudenthal’s own activities that caused this: the times were on his side. Immediately after the war the innovative spirit reached its peak in the Netherlands. By the early 1950s everything had become more or less familiar again. This was also the case in mathematics: the establishment of the Mathematisch Centrum (Mathematical Centre) was an example. In mathematics education things went on a bit more cautiously. Five years after the liberation the understanding sank in, in mathematics too that changes were inevitable and modernizing ideas gained ground. Freudenthal’s hopes were running high about these modernizations: Ik heb de indruk, dat er in geen vak zoveel aan paedagogiek en didactiek wordt gedaan als in het onze. Wanneer ik b.v. zie, hoe in de werkgroep-wiskunde van de Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs maand aan maand soms hele zondagen over de problemen van het wiskundeonderwijs wordt gediscussieerd, door een groot aantal leden, die daar lange reizen voor over hebben, dan voel ik me elke keer gesterkt in de overtuiging, dat we de goede kant op gaan. Want het besef, dat er veel te verbeteren valt, is er, en de voorstellen dienomtrent zijn zeer concreet. De tirades over de heerlijke wiskunde, waarmee je zo prachtig het verstand kunt scholen, zijn verstomd, sinds de wiskunde eenvoudig een bittere noodzaak is gebleken, en juist die onderdelen van het programma, die lange tijd beschouwd werden als tuchtmiddel, om jeugdige hersenen in te plooien, worden nu het scherpst gecritiseerd. (I am under the impression that no other discipline goes as much for pedagogy and didactics as ours. When I see for example how in the mathematics © Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_6

133

134

6 From Critical Outsider to True Authority

Fig. 6.1 Freudenthal’s speech at the opening of the ‘Boothstraat 1c’ (1958) working group of the Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs (WVO, Cooperative for Modernization of Upbringing and Education), month after month sometimes, complete Sundays are spent by a large number of members, who have taken the trouble to travel far to attend, in discussions about the problems of mathematics education, then I feel steeled each time in the conviction that we are heading in the right direction. Since the awareness that a lot can change for the better is present and the propositions about this are very concrete. The ranting about the lovely mathematics, which can so wonderfully be used to train the mind, has died down since mathematics has appeared to be a dire necessity, and in particular those parts of the programme that have long been considered as disciplinary measures to yield youthful brains, are now being criticized most severely.) [3]

From Critical Outsider to True Authority

135

The WW’s activities were taken seriously by an increasing number of teachers, something that could be observed in the large number of teachers and educational representatives that frequented the weekend conferences. In the course of the 1950s, members of the working group regularly published in Euclides and an increasing number of members of the formerly somewhat conservative associations of teachers WIMECOS and LIWENAGEL became members of the working group. In 1946 the WW started with just 20 members, but in 1953 there were already 128 members [4]. That changed the Wiskunde Werkgroep in the 1950s from a revolutionary, and therefore somewhat elite, group into a respected and valued party in the discussions about modernizations in mathematics education even having its own monthly bulletin from 1952 onwards. Furthermore, the members of the working group no longer considered themselves to be lonely fighters for innovation. Freudenthal wrote in a report entitled ‘Onderwijsvernieuwers bijeen’ (‘Education innovators assembled’) about a weekend on educational innovations organized by a number of working groups of the WVO: Met de “onderwijsvernieuwers” in het opschrift heb ik nu niet alleen de W.V.O.-leden bedoeld—het zou aanmatigend zijn, deze titel voor hen te reserveren. Overal in den lande gist het in het onderwijs. Nieuwe wegen worden gezocht, verkend en begaan. Een grote lijn begint zich af te tekenen. Vaak ontbreekt nog het contact tussen voortrekkers, die het in dezelfde richting zoeken. De Amersfoortse conferentie leek dan ook soms op een jaarbeurs voor onderwijsvernieuwing, druk bezocht door inspecteurs, vertegenwoordigers van lerarenorganisaties, rectoren, directeuren en leraren. (The “education innovators” of the title are in my view not solely the members of the WVO—it would be presumptuous to reserve that title for them. Everywhere in the country education is in a state of ferment. New roads are being searched for, explored and walked on. A gist is starting to stand out. What is often lacking is the contact between pioneers looking for it along the same lines. Consequently, the Amersfoort conference sometimes looked like a trade fair for educational innovators, busily visited by inspectors, representatives of teachers’ organizations, headmasters, directors and teachers.) [5]

The crowning glory of the innovative initiatives in the 1950s was the realization of the new mathematics curriculum [6]. Even when the collaboration between the WW and the teachers’ associations did not always work out as smoothly as desired, in the end a curriculum would be realized with which eventually all parties agreed. In the 1950s Freudenthal published with ever increasing regularity on mathematics education. In the 1940s only a single article in that field written by him was published, in 1956 alone there were five, whereas the actual number is higher since his articles in newspapers and weeklies should have been added. Incidentally, this number was not so high at the cost of his mathematical publications. The educational sector increasingly considered Freudenthal an expert in the field of mathematics education, as evidenced among other things by his invitation in June 1956 from the editors of Euclides to become a member of the permanent staff of writers. He took to this “very delightedly” [7]. Freudenthal attached a lot of value to the good contact he had recently established with the teachers’ associations, declining for example an invitation to lecture for the Nederlandse Onderwijzers

136


Vereniging (Dutch (primary) School Teachers’ Association) at the time. He wrote that he preferred to cooperate with the teachers’ associations, since these represented the majority of the teachers [8]. That ‘majority aspect’ was in keeping with his conviction that it was sometimes more effective if you cooperated (even when you did not always agree with the operational method and the ideas), compared to being ‘against’ and standing aloof. His membership (and in 1954 chairmanship) of the ‘Nederlandse Onderwijscommissie voor Wiskunde’ (NOCW, Subcommittee for the Netherlands of the International Committee on Mathematical Instruction) can be seen in this light and is a good example of the recognition of his expertise. Of course, Freudenthal would not have been Freudenthal if he had not also regularly attacked some sacred cows and in doing so brought the usual criticism of the teachers on himself. In 1952 for example his lecture on mechanics education created quite a sensation and he managed to touch a sore point with the teachers when he expressed his opinion on the teachers’ training. Even so the first half of the 1950s showed Freudenthal’s development from critical outsider to authority in the field of mathematics education. In the early 1950s he was given a good opportunity to clearly formulate his standpoints on mathematics education in the pamphlet Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? ( Can mathematics education contribute to the education of the intellectual capacity?)

6.1 Mathematics Education and the Education of the Intellectual Capacity One of the most well-known publications by the Wiskunde Werkgroep was perhaps this pamphlet entitled Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? ( Can mathematics education contribute to the education of the intellectual capacity?) This contained a discussion between Mrs. Ehrenfest and Freudenthal about the influence of geometry education on the intellectual capacity. Mrs. Ehrenfest and Freudenthal primarily knew each other from the meetings of the Wiskunde Werkgroep which were regularly held at her house in Leyden. A letter written by Mrs. Ehrenfest to Freudenthal much later showed how the pamphlet was produced. Mrs. Ehrenfest had initially written the first article, upon which Freudenthal, who totally disagreed with her statements, proposed to publish her article, his reaction and both their further answers in one volume [9]. Mrs. Ehrenfest started the discussion with an explanation of how mathematics education, and in particular geometry education, could contribute to educating the intellectual capacity and how the curriculum should be organized to achieve that effect. Thinking logically was according to her more than “drawing conclusions from given premises”, and also included thinking qualities such as being able to abstract and formulate: De hierbij behorende denkgewoonten zijn: naar het essentiële in een gegeven situatie zoeken (abstractievermogen)[;] zich daarvan rekenschap te geven (wat niet hetzelfde is); te trachten dit scherp te formuleren; een zich aanbiedend antwoord op een vraag met het gehele complex van gegevens te confronteren en nooit de totale situatie uit het oog te

6.1 Mathematics Education and the Education of the Intellectual Capacity

137

verliezen (kritisch vermogen). (The thinking habits that belong to this are: seeking the essentials in a given situation (power to abstract)[;] realizing this (which is not the same); trying to formulate this sharply; confronting an obvious answer to a question with the total range of data and never losing sight of the total situation (ability to criticize).) [10]

Most of the mathematics education however, consisted of subject matter offered as a theory which the pupil simply had to accept, and exercises for the pupils were deriving the more difficult statements and applications from this. According to Mrs. Ehrenfest, in this way pupils not only explicitly failed to practice good learning and thinking manners, but practiced ‘non-thinking’ by uncritically accepting the instructions of others. Good thinking manners and real understanding were only taught in mathematics education if the pupil was allowed to experience the total build-up of the subject matter. According to her, contact with practice offered an opportunity to practice independently discovering essential elements, an important part of learning to think logically. In those circumstances there was certainly a chance of enlarging the thinking capacity of the pupils by mathematics education. Prerequisite for a really high level of mathematics education according to Mrs. Ehrenfest was that the scope of the programme was to be reduced most drastically and that pupils were allowed to set the working pace they themselves deemed necessary to be able to use their own intellect at every step. Freudenthal presented himself in his answer to Mrs. Ehrenfest as her opponent, even when situations could be thought of where he would defend her point of view “as decidedly as I attack it here” [11]. According to Freudenthal there were few reasons to sing the praises of the formal mathematical methods as an example of the art of thinking as actually practiced. Other ‘good thinking habits’ such as thinking in analogies were not done justice in mathematics according to him. Men begrijpe mij goed: ik acht het niet onmogelijk, dat men, een speciaal vak beoefenende, ook opvoedt tot denken in de meer algemene zin—goed begrepen moedertaalonderwijs is hiervoor vooral aangewezen, omdat denken voor een groot gedeelte taalbeheersing is; ik sluit ook de mogelijkheid van bijdragen in denkoefening van de kant van de wiskunde geenszins uit. Maar ik vrees, dat men op drijfzand bouwt, wanneer men de uren, die het één of ander schoolvak opeist, wil rechtvaardigen met een beroep op de denkoefeningen, waaraan die tijd zou worden besteed. (Please do not misunderstand me: I do not deem it impossible that someone, practising a certain subject, also learns to think in a more general sense—well understood education in the mother tongue is the first obvious means to achieve this, since thinking is for a large part mastering a language; I do not at all exclude the possibility that mathematics contributes to practicing thinking. But I fear that one is building on quicksand when one wants to justify the periods that some school subject requires by making an appeal to the thinking exercises that those periods would be devoted to.) [12]

Even when he did not exclude the fact that ‘learning to think by mathematics education’ was possible to a certain extent, Freudenthal was still of the opinion that the very first—and also the very last—task of mathematics education was teaching mathematics, in which the technical and cultural meaning of mathematics were the criteria. ‘Learning to think’ was not to be used as a justification for mathematics education. Apart from that Freudenthal very much admired the kind of (ideal) education that Mrs. Ehrenfest had in mind for secondary education: a somewhat rigid,

138


logically transparent structure of geometry in which a lot of the dead weight of the time could be thrown overboard. Replying to Freudenthal’s reaction Mrs. Ehrenfest once again emphasized why only mathematics education offered the possibility to practice good thinking manners: precisely because of the simplicity of the subject matter it was possible that the effort to understand that matter could succeed. She reproached Freudenthal (correctly for that matter) for surpassing the true topic of the discussion when he spoke of the value of mathematics as a subject in its own right, something that Mrs. Ehrenfest had certainly not missed, but which she had not touched upon in this discussion. Freudenthal ended the discussion by admitting that Mrs. Ehrenfest was right: he had indeed read “has to” instead of “can” in the title of the discussion. But he was absolutely justified in this in his own opinion. After all: Wat Mw. Ehrenfest doet, is een methodiek van het wiskundeonderwijs te schetsen, die wellicht tot het gestelde doel [het ontwikkelen van het algemene denkvermogen] zou leiden, indien de vraag of dit doel wel bereikbaar is, van tevoren op grond van algemenere overwegingen positief beantwoord zou zijn. Beantwoordt men die vraag ontkennend, dan vervallen Mw. Ehrenfest’s argumenten. (What Mrs. Ehrenfest does is sketching a methodology of mathematics education which perhaps can lead to the specified goal [developing the general power of thinking], provided that first the question whether this goal is reachable at all has been answered positively based on more general considerations. If this question is answered negatively, Mrs. Ehrenfest’s arguments are no longer valid.) [13]

As much as he appreciated and respected Mrs. Ehrenfest’s practical advice for the organization of mathematics education, in his eyes the goal of mathematics education remained mathematics itself. And the question—“should mathematics education contribute to the development of intellectual capacity?”—should indeed be answered negatively according to Freudenthal: “Het trachten om iets te begrijpen in plaats van een oordeel of een regel op vreemd gezag te slikken” kan nergens zo goed als in de wiskunde worden beoefend—aldus Mw. Ehrenfest. Absoluut juist—tot op ’t ogenblik, dat de kwestie van overdracht aan de orde komt. De wiskunde kweekt deze goede gewoonte in hoge mate aan—ten bate van de wiskunde. (“Trying to understand something instead of accepting a judgement or a rule on some authority” can be practiced nowhere better than in mathematics—said Mrs. Ehrenfest. Absolutely correct—until the moment that the question of transfer comes up. Mathematics greatly cultivates that good habit—for the benefit of mathematics.) [14]

The transfer of (thinking) abilities acquired when learning mathematics made or broke the formative value of mathematics. And according to Freudenthal that transfer was, in particular because of the (overly) simple structure and abstract nature of mathematics at the most only present for the benefit of another mathematical field. He did not mention it here, but he probably based this conviction on the results of the American study into transfer he spoke of earlier [15]. For that matter, the argument that mathematics would be too abstract to transfer to thinking outside the field of mathematics, had already been put forward in the 1930s by Kohnstamm [16]. The rejection of the formative value terminated all of Mrs. Ehrenfest’s arguments for a mathematics education organized for ‘learning to think’. Balancing contradictions, the pros and cons of certain actions was not practiced in mathematics and thus Freudenthal stressed unnecessarily:


139

Ik ben nog een keer zo uitvoerig op principiële kwesties ingegaan, om te kunnen zeggen, waaròm ik het aankweken van goede denkgewoonten door middel van de wiskunde zo secundair acht: omdat de wiskunde hiervoor het minst aangewezen terrein is. (I once again went deeply into fundamental questions to be able to say why I consider cultivating good thinking habits by way of mathematics of so minor importance: because mathematics is the least proper field for this.) [17]

And this ended the discussion. More than 20 years later Van Hiele identified the pamphlet as a good example of how two scientists had been talking at cross purposes because they failed to sort out their field of terms in such a way that the word symbols they used had the same meaning [18]. The two scientists were indeed talking at cross purposes. The main reason was that they did not debate about the same statement, something that only became obvious from the last two contributions in the pamphlet. In the end the misunderstanding seemed to originate from the word “can” in the title. Mrs. Ehrenfest argued: ‘Yes, mathematics education (and in particular geometry education) can contribute to the education of the intellectual capacity’, and subsequently schematically indicated what that education should look like according to her. In both his answers Freudenthal continually skipped the question or the word can and maintained instead ‘no, mathematics education should not contribute to educating the intellectual capacity’ (or the can only could be ascertained experimentally and was not very probable). To him the goal of mathematics education was mathematics itself. Goed wiskundeonderwijs lijkt me op zichzelf al een zo geweldig probleem, dat ik Mw. Ehrenfest’s vraag of dat onderwijs tot de opvoeding van het denkvermogen kan bijdragen gevaarlijk acht, wanneer zij gesteld wordt met de bedoeling, vanuit het antwoord het onderwijs te beïnvloeden. (Good mathematics education already seems in itself such a tremendous problem that I consider Mrs. Ehrenfest’s question whether that education can contribute to developing the intellectual capacity, when it is posed with the intention of influencing the education starting from the answer, a dangerous one.) [19]

So in the end there was no such thing as a real discussion. But both had seized the opportunity to elucidate their ideas about those parts of mathematics education that were the most important to themselves: Mrs. Ehrenfest about the organization of geometry education, Freudenthal about the goal and use of mathematics education in general. In the discussion Freudenthal touched upon a deeper layer by posing the question whether the formative value indeed existed and—even more importantly—what that formative value of mathematics education meant in practice. In the discussion about a possible introduction of geometry education with visual aids of 1924 Mrs. Ehrenfest and Dijksterhuis and co were opponents, but they both always took the formative value of mathematics education for granted. Freudenthal, who on the contrary agreed with Mrs. Ehrenfest’s organization of mathematics education, denounced the fact that the formative value was obvious and took up a position opposite to both other ‘camps’. In this respect Van Dantzig, who also rejected the argument of the formative value of mathematics, was his closest ally. But Van Dantzig stressed the argument that mathematics had a social use whereas Freudenthal nowhere elaborated on this point in his discussion with Mrs. Ehrenfest. But he did touch on the ‘technical meaning’ of mathematics as an argument for teaching the subject.

140


Freudenthal’s point of view in this discussion in which he put the emphasis on the lack of a scientific foundation for the alleged formative value mostly resembled Kohnstamm’s. But compared to Freudenthal Kohnstamm was less emphatic in expressing his doubt about the existence of formative value in the case of mathematics. More so than Freudenthal in this discussion he refused to a certain extent to commit himself when speaking about the formative value and deemed mathematics education, in any case for learning to think logically, pre-eminently appropriate [20]. H. Streefkerk, mathematics teacher and editor of Euclides, wrote a favourable review of the pamphlet in his journal, in which he joined Freudenthal’s side but also encountered a lot of useful things in the texts of Mrs. Ehrenfest. He warmly recommended the pamphlet to his readers and hoped that “many mathematics teachers would put up with the mere cost of acquisition; the interest of our pupils is at stake” [21]. In a letter to Freudenthal he wrote about the pamphlet, even before the review was published: De laatste 3 bladzijden van het boekje zijn de beste en de aardigste [dit is de slotreactie van Freudenthal]. Mevrouw Ehrenfest heeft geen idee van de koeien, uilen, schapen, ganzen en ezels die onze klassen bevolken. Het door haar bedoelde—inderdaad ideale—onderwijs kàn, bij de huidige samenstelling van de schoolbevolking en de klassen, zeker in de eerste 3 à 4 klassen niet gegeven worden. Dat zou hoogstens in klas 5 gaan, als klas 5 geen examenklas was. (The last three pages of the little book are the best and the nicest [this is Freudenthal’s concluding reaction]. Mrs. Ehrenfest has no idea of the cows, owls, sheep, geese and donkeys that populate our class rooms. The education she intends—and indeed that would be ideal—cannot be provided in the present composition of the school and class population, certainly not in the first three or four years. That would at the most be possible in the fifth year, were it not for the fact that this was the final year.) [22]

This support from someone on the Euclides teachers’ side was not really unexpected. As early as 1945 Streefkerk had written an article in Euclides about the overload and the impracticability of the HBS curriculum and had already corresponded regularly with Freudenthal for a few years about various subjects from the didactics of mathematics in general, and in particular in the capacity of editor of Euclides about the content of two short articles on HBS exercises Freudenthal wrote for Euclides [23]. They also knew each other from the meeting of the Wiskunde Werkgroep. Yet this review in Euclides was in fact a cautious beginning of the reverberation and recognition of Freudenthal’s ideas among teachers. There were of course still voices to be heard in favour of the ‘formative value’ argument when the curriculum was being selected. Wansink, for example, who with his often moderate attitude was perhaps a better representative of the average teacher, wrote in a letter to Freudenthal about the pamphlet: Ik ben het met U eens dat geen leerstof alleen terwille van de opvoeding van het denkvermogen een plaats op de roosters verdient. Daartegenover zou ik leerstof (onverschillig voor welk theoretisch vak ook) die op eigen gebied, in een zekere omgeving van de behandelde leerstof, niet tot beter denken zou voeren, van de roosters willen verwijderen. (I agree with you that including subject matter in the programme solely for the sake of the development of the intellect is not desired. But then again, I would want to remove from the programme subject matter (for any theoretical subject) that in its own field, in a certain environment of the subject matter to be treated, would not lead to a better intellect.) [24]


141

Three years later Freudenthal returned to the discussion with Mrs. Ehrenfest anew. In the German Archimedes. Anregungen und Aufgaben für Lehrer, Schüler und Freunde der Mathematik ( Archimedes. Stimuli and exercises for teachers, students and friends of mathematics) he once again explained his point of view in a reaction to the article of Mrs. Ehrenfest printed in the same issue: Mein Kriterium für guten Mathematikunterricht ist, ob er als Mathematikunterricht etwas tauge, ob sie [de leerlingen] den Kulturwert begreifen, ob die Mathematik ihnen spaß macht. (My criterion for good mathematics education is, whether there is any use in it, whether they [the pupils] understand the cultural value, whether they have fun with mathematics.) [25]

To conclude he once again emphasized the misconception that mathematics through its simplicity would constitute a good practice ground for thinking, since: Denken lernt man nicht im Einklang, sondern im Widerspruch. (Harmony does not teach you to think, contradiction does.) [26]

To develop the intellect it was necessary to learn to weigh arguments in a pool of ‘pros’ and ‘cons’. “Widerspruch”, which according to Freudenthal was a characteristic of a democratic structure. Mathematics, with its straightforward provability and order, did not contribute anything to such thinking methods. Freudenthal’s ideas about educating the intellect, and in particular the role mathematics would play, does not seem to change much in those years. As before in ‘Rekendidactiek’ (1944) and ‘Opvoeding tot denken’ (1945) he held on to the conviction, that exactly because of its simple, transparent structure mathematics was not a good subject to learn to think and most probably a transfer or formative value of that education to other fields could not be expected [27]. To possibly reach this he also saw more good in (mother tongue) language education, provided that this was indeed language education, and not language analysis or linguistics in disguise [28]. This last ‘provided that’ was something Freudenthal most probably added after he had personally experienced the practice of language education via his children. He wrote, for example, regularly about education in De Groene Amsterdammer. In the article ‘Mevrouw ze groeien ook zo hard’ (‘Madam, they grow so quickly’) he criticized the tendency to choose subject matter in school ‘allowing for growth’. To teach grammar in primary school for the sake of foreign languages in secondary school was according to Freudenthal putting the cart before the horse: first learning the discipline (grammar) and only then learning the language itself, whereas children of that age are not at all interested in or have a notion of the meaning of grammar [29]. Three weeks later he criticized the word drilling method by which one of his sons had to learn his mother tongue [30]. He trode on dangerous ground with this opinion as evidenced by the pile of reactions that De Groene Amsterdammer received, mostly coming from teachers who felt challenged in their expertise by a ‘layman’. He did not take such criticism seriously as shown by his letter to the physicist R.A.J. Bosschart, in which Freudenthal wrote about the education in primary school:

142


Ik weet niet, of men het kind op de lagere school beslist in contact moet brengen met enige wetenschap. Mocht het zo zijn, dan zou ik onder de natuurwetenschappen de biologie beslist prefereren. Maar de gekkenhuis-wetenschap van het lijdend voorwerp hoort in het lager onderwijs zeker niet thuis. Ieder [sic] psycholoog weet, dat een kind op die leeftijd niet vatbaar is voor taalanalyse […] Het is een oude hobby van mij, en ik heb er herhaaldelijk over geschreven: dat taalwetenschap niet hetzelfde is als taal, jaartallen niet hetzelfde als geschiedwetenschap, encyclopedie-materiaal niet hetzelfde als aardrijkskunde. Maar helaas elke keer, dat ik het lijdend voorwerp of het jaartal 1296 of de rivierklei aanraak, gaan de mensen in hun heilige huisjes voor verontwaardiging door het dak. De werels [sic] zou ten gronde kunnen gaan, als hun kinderen iets verstandigers leerden dan zijzelf. (I am not sure whether or not children in primary school should be brought into contact with any science. If so, I would definitely prefer biology among the sciences. But the madhouse science of the direct object certainly does not belong in the domain of primary education. Every psychologist knows that a child of that age is not susceptible to language analysis […] It is one of my old hobbies and I have written repeatedly about it: that linguistics is not the same as language, dates are not the same as history, the contents of encyclopaedia are not the same as geography. But alas every time I touch upon the direct object, or the year 1296 or the river clay, the people whose ground I tread upon blow their tops with indignation. It might be the end of the world if their children were to learn something more sensible than they themselves had learned.) [31]

In his articles he mentioned language education as a part of the education that possibly had even more problems than mathematics education [32]. But all the same he would mainly continue to concentrate on the problems of the mathematics education.

6.2 A Body Under the Floorboards: The Mechanics Education “The last word about mechanics as a subject in secondary schools has apparently not yet been spoken,” that was the end of Dijksterhuis’ article in Euclides in 1930 [33]. With this he concluded a few years of discussions about the subject of mechanics between the mathematics and physics camps in the 1920s. These discussions were between advocates of a logically-deductively structured mechanics (most mathematics teachers were in favour of this), and the advocates of treating a more experimentally based mechanics, which would be in harmony with the methods used in the physics education [34]. Even when in 1929 mechanics became an independent subject, an agreement about the desired approach (empirical or deductive?) of the mechanics education had by no means been reached. In the following years the mechanics education had hardly been subject of discussion, until in November 1952 Freudenthal, in his own direct way, resurrected the subject again. He did it in his lecture during the weekend conference of the Werkgroepen Wiskunde en Natuurkunde (The Physics and Mathematics Working Groups) of the WVO in De Tempel in Rotterdam. The reason was simple and concrete: his eldest son got around to doing mechanics at his school. Freudenthal, who had not been in contact with the mechanics education before, was shocked by what he saw, and quoted Dante to describe his feelings:

6.2 A Body Under the Floorboards: The Mechanics Education

143

Toen ik de levensweg had half voltooid, hervond ik me in een donker woud verdwaald. (Midway on my journey of life, I found myself astray in a dark wood.) [35]

He blamed himself, as someone interested in education and teaching, for not having been involved in this subject and its problems before and he felt “like someone who lives for years in the same room and suddenly discovers that there is a body buried under the floorboards” [36]. As could be expected Freudenthal had consequently thrown himself into a thorough dissection of this ‘body’ and treated his audience to his findings. In a footnote of the printed version of this lecture he firstly accounted for the method he had followed. In such a study into a school subject examination, questions and textbooks were the only scientifically reliable sources according to Freudenthal. The information that was acquired in that way was statistical information and consequently the conclusions drawn from it were statistical conclusions. In that way Freudenthal safeguarded himself from the very start against possible critics who might accuse him of ‘generalizing’, since he generalized just as much as “someone who, for instance, derives from statistical data that the average adult Dutchman is 1.64 m in height, and who would then be reproached for neglecting all those who are shorter or taller” [37]. In doing so he indirectly showed that he thought it inevitable that his conclusions would be criticized. Which was understandable in itself, since Freudenthal’s conclusions were very outspoken. His first objection against the mechanics education was that it had nothing to do with the original goal (teaching applied mathematics). Met ontsteltenis leest men tegenwoordig ontboezemingen van wiskundigen, waarin de wiskunde gezuiverd wordt van de smaad, haar aangedaan door natuurkundigen en technici, die de euvele moed bezaten haar toe te passen. Het lijkt, of men in het gevlij wil komen bij de beoefenaren der geesteswetenschappen, zo spant men zich in om aan te tonen, dat ook de wiskunde een wetenschap van de geest is en niet van de verachtelijke stof. (With dismay one can, at present, read outpourings from mathematicians purging mathematics of the slander caused by physicists and technicians having the nerve to apply it. The great pains they take to show that mathematics is also a science of the mind and not of the despicable matter, makes it look as if they want to butter up the science of humanities.) [38]

In the period between the two World Wars, the ambition, also expressed in the BethDijksterhuis curriculum, was to teach mechanics as a science based on axioms with the goal of “language analysis and introduction to a rationalistic natural philosophy and history of science” [39]. In practice, textbooks, however, showed something different. Next Freudenthal discussed two textbooks, Beginselen der theoretische mechanica ( Principles of theoretical mechanics) by Schogt (Freudenthal did not know whether it was ever used in the secondary education) and Mechanica voor het M.O. (Mechanics for secondary education) by H.J.E. Beth and P.J. van Loo. He was not happy about either of these. Both books deluded both teacher and pupil with an exactness that was not around. Nothing was left of the empirical background of the subject. In addition, Freudenthal found Beth’s book in particular incomprehensible (despite “the otherwise excellent production by Beth”) [40].

144


He criticized not so much the didactics that were followed but rather the incompleteness, the sloppiness and the difficult language used by the authors. Consequently his conclusion was: Ik zou beide boeken niet ‘slecht’, maar ‘zeer slecht’, noemen, indien ik de auteurs kon verwijten, dat ze verzuimd hadden, betere voorbeelden na te volgen. (I would call both books not ‘bad’, but ‘very bad’, if I could blame the authors to have omitted following better examples.) [41]

Since there were no really better examples. Other textbooks were even worse if possible. One of the worst problems for Freudenthal was the love of definitions that the mechanics education cultivated. He illustrated this ‘mechanics style’ with numerous examples of various textbooks: Men kan de eenvoudigste begrippen nu eenmaal niet tot nog eenvoudigere herleiden, maar wel als het moet, tot ingewikkeldere. Van deze mogelijkheid wordt gretig gebruik gemaakt: Wanneer twee lichamen zich zo ten opzichte van elkaar bewegen, dat zij op een gegeven ogenblik met elkaar in aanraking komen, dan zegt men, dat zij op dat ogenblik tegen elkaar botsen. (There is nothing to be done about it but it is not possible to convert the most simple concepts to even more simple concepts, but you can reduce them, if you have to, to more complex concepts. This opportunity is used eagerly: When two bodies move with respect to each other in such a way that they touch each other at some moment in time, then it is said that they collide at that moment.) [42]

Freudenthal saw only one solution if one wanted to silence the complaints about the poor ability to express themselves of the pupils and that solution was to dispense with the mechanics education, “the perfect school for sloppiness” [43]. The mechanics education as a whole was still nothing more than an incoherent theory with the ultimate goal to “solve silly examination problems raised by people who have no idea of what has changed the last 90 years in the relationship between mathematics and its applications” [44]. There was one good book. Wansink’s book, Mechanica voor het M.O. ( Mechanics for secondary education), was according to Freudenthal perhaps the best book ever to be written in that vein: well-organized, clear, sober, concise, with a good style and didactics, business-like, logically clear, with problems that had been deliberately chosen, containing no examination problems, and sometimes there even was some contact with the physical reality. The fact that this book existed could be a reason not to lose heart and hold on to the subject. However, the subject itself was also no good at all, not even when taught in the best possible way. Freudenthal advanced and underpinned two propositions: 1. Het is onjuist, de mechanica op de H.B.S. volgens de deductieve methode te onderwijzen; 2. Het is onjuist, de omvang van het mechanica-programma te laten bepalen door de draagwijdte van de H.B.S.-wiskunde (d.w.z. van dat gedeelte der wiskunde, dat op de H.B.S. wordt en kan worden onderwezen). (1. It is not correct to teach mechanics at the H.B.S. by deduction; 2. It is not correct when the volume of the mechanics curriculum is determined by the scope of the mathematics of the HBS (that is to say of that part of mathematics which is and can be taught at the HBS).) [45]


145

The first argument was, of course, also applicable to geometry education. Freudenthal considered starting to teach science a priori in a deductive manner a didactically improper method. The pupil should learn geometrical quantities such as force, mass, velocity and acceleration by measuring them and not by representing them by characters such as K, m and a in a set of mathematical formulas. Je mag in de mechanica (evenals in de wiskunde) alles, waar je zin in hebt, met elkaar vermenigvuldigen en er een nieuwe naam voor bedenken—zo ongeveer moet het er in de mechanica-hersens van onze H.B.S.-leerlingen uit zien. (In mechanics (just as in mathematics) you are allowed to multiply everything you fancy with anything else you like and think up a new name for it—that is what it probably looks like in the mechanics brains of our HBS pupils.) [46]

As far as the second argument is concerned: the criterion to limit the subject matter to that which could be handled with HBS mathematics and which could serve to generate problems was the wrong criterion. Freudenthal mentioned a long list with subjects and questions the pupils missed in the education of the time, containing things ranging from the stability of a bicycle to the dynamic action of the lever [47]. Laat de lijst, die ik hier heb opgemaakt, goed op u inwerken! Al deze dingen missen de H.B.S.-leerlingen. Is het niet een schrikwekkende lijst? Wat leer je op de H.B.S. in plaats van deze mooie en nuttige dingen? Uitrekenen, hoe een balletje in een verticale cirkelbuis met aansluitend hellend vlak beweegt en hoe een kegel zonder wrijving op een vlak kan staan, terwijl een staaf, opgehangen aan een katrol, met wrijving tegen die kegel leunt. Dat is H.B.S.-mechanica! Schaf die prulwetenschap af! (Let the list I made here sink in well! All of these things are missed by HBS pupils. Isn’t it a terrifying list? What is it you learn at the HBS instead of these beautiful and useful things? You calculate how a ball moves in a vertical circular tube connected to an inclined plane and how a cone without friction can stand on a plane, while a bar suspended from a pulley leans with friction to that cone. That is the HBS mechanics! Scrap that trash science!) [48]

The lessons and the subject matter made available by that action could easily be divided between the disciplines where they belonged: between physics and mathematics, so that better use can be made of a more physics-oriented sense of mechanics. If the HBS curriculum did not change quickly, the HBS would be ousted as a school type by a less rigid school type, just as had happened—at least in Freudenthal’s point of view—with the Gymnasium 90 years ago: “Veritably a counterproductive progressiveness!” [49]. It did not take long before Freudenthal’s plea was noticed. It was not only published in a brochure along with the other lectures, a report of the meeting was also printed in De Groene [50]. During the annual meeting 6 weeks after the lecture G.A. Janssen, WIMECOS’ chairman, took stock of the modernizing voices and initiatives in mathematics education in 1952 [51]. Apart from a lot of valuable things (that for him included also the “passionate, youthful enthusiasm” of the Wiskunde Werkgroep) he first and foremost wanted to warn the teachers and evoke watchfulness, so that the danger that “non-mathematicians could with some pretence refer to statements of acknowledged mathematicians in order to harm science education” could be averted [52]. Besides Mrs. Ehrenfest (whose good intentions were by no

146


means questioned by Janssen) it was in particular Freudenthal whose mechanics lecture caused such a danger: Als een spreker, die zelf verklaart, dat zijn belangstelling voor en verdieping in de desbetref fende didactische materie nog maar enige maanden oud is, het resultaat van zijn onderzoek laat culmineren in een: “Dat is h.b.s.-mechanica. Schaf die prulwetenschap af!”, zonder een constructief idee aan de hand te doen voor wat er na die afschaffing moet gebeuren, en in zijn betoog en passant veler pennevruchten o.a. van iemand van zo grote verdienste als dr. Beth Sr meent te mogen “kraken”, dan zal de spreker zich wel niet kunnen verwonderen over het feit, dat wij ten aanzien van de hechtheid van het hierbij behorend betoog argwanend worden, en ten aanzien van geest en strekking van dat betoog verontrust en verontwaardigd. We waarderen uiteraard, dat van universitaire zijde voor de didactische problemen van het V.H.M.O. belangstelling aan de dag wordt gelegd, maar er zal aan enige voorwaarden moeten hebben voldaan, wil het gebodene voor ons onderwijs positieve waarde hebben. Mede op grond van wat pers en syllabus ons leerden, achten we het betoog van prof. Freudenthal […] beneden de door ons bedoelde maat. Toch willen we hier ook gaarne verklaren, dat we zijn kennis en belangstelling node zouden missen; zijn beweringen zijn echter, naar het ons toeschijnt vaak te impulsief, de gebezigde woorden te “dik”. (If a lecturer who in his own words says that he has been interested in it and has gone deeply into the relevant didactical matter for only a few months, lets the result of his research reach its climax in: “That is the HBS mechanics! Scrap that trash science!”, without giving a constructive idea about what is to be done after the abolishment, and incidentally in his argumentation thinks he can “scathe” the writings of many among whom a man of merit such as dr. Beth senior, then that lecturer should not be surprised by the fact that we, with respect to the strength of the accompanying argumentation, become suspicious and with respect to spirit and thrust of that argumentation become alarmed and indignant. Of course we appreciate university interest in the didactical problems of the VHMO, but some conditions will have to be met before what is offered will have a positive value for our education. Partly based on what we learn from the press and the workbook, we consider prof. Freudenthal’s plea […] not up to our mark. Yet we will gladly mention here, that we will be sorry if we were to miss his knowledge and interest; his statements, however, it would seem to us, are often too impulsive, the words he uses too “heavy”.) [53]

The greatest danger lay according to Janssen in the negative effect that Freudenthal’s statements, which in his opinion were not sufficiently well motivated, could have on the public at large. In a letter to Freudenthal E.W. Beth (son of H.J.E. Beth) mentioned Janssens’ comments—as expressed during the WIMECOS meeting. Freudenthal replied to Beth that he had sent Janssen the printed version of his lecture so that he “after having written the criticism could also read what he [Janssen] has criticized” [54]. The fact is that the text printed in Nieuwe wegen was much longer than Freudenthal’s lecture, in which he for example mentioned the book by Beth senior only briefly. In his letter to Beth junior Freudenthal wrote that he had got used to such criticism a long time ago. As far as his point of view with respect to mechanics education was concerned he explained to Beth: Dat ik als wiskundige dit standpunt inneem, is het gevolg ervan, dat ik nogal veel met toepassingen te maken heb, en dus weet, hoe wiskunde wordt toegepast, en dat de toegepaste wiskunde van ons M.O. een caricatuur is. Ik heb opgemerkt, dat zelfs de natuurkunde en scheikunde door de zinloze vermathematisering zijn besmet. (That I, as mathematician take this point of view, is the consequence of the fact that I am rather involved in applications and therefore know how mathematics is applied, and that applied mathematics in our secondary education is a caricature. I have noticed that even physics and chemistry have been affected by this pointless mathematizing.) [55]


147

He observed this by the great difference in education of his two sons (‘A’ and ‘B’) at two different schools: one of them received good, experience-based education and got the hang of it, the other (according to Freudenthal “by far the more intelligent of the two”) received bad education and therefore ended up with “the characteristic babble of someone who has never been in touch with scientific reasoning”: Bij hem ontbreekt totaal het begrip ervoor, dat natuurwetenschap geen wiskunde is, dat je in de natuur- en scheikunde niet iets bewijst, maar dat je een systeem zoekt, om de verschijnselen te verklaren en te ordenen. A zal natuurkunde gaan studeren, B, die er op zichzelf vermoedelijk even geschikt voor is, beslist niet. De zinloze vermathematisering van de natuurwetenschappelijke vakken gaat mij dus zeer ter harte. (He completely lacks the understanding that natural science is not mathematics, that in physics and chemistry you do not proof something, but that you are looking for a system to explain and order the phenomena. A will study physics, B, who taken on his own is probably equally suited to do so, will certainly not. I am therefore greatly concerned about the useless mathematizing of the science subjects.) [56]

An immediate reaction by Janssen failed to come. Freudenthal did not give in and more than 6 months later sent a manuscript for publication to the editors of Euclides in which he once again explained the course of events. It was not his intention, Freudenthal wrote, to invalidate the arguments of the criticism (that is to say his inexpertness and the great merits of H.J.E. Beth) since these were purely personal. Nor did Freudenthal challenge the WIMECOS’ chairman to mention some objectively founded arguments, since he had understood that there was no reaction to be expected: Het enige, dat mij rest, is een beroep op de leden van WIMECOS, om zichzelf de zakelijke voorlichting over mijn rede te verschaffen, die hun van de kant van hun Bestuur wordt geweigerd. Ik nodig hen uit, kennis te nemen van mijn rede. (The only thing left for me to do, is to appeal to the members of WIMECOS to avail themselves of the objective information of my lecture, which their committee withholds them. I invite them to take note of my lecture.) [57]

This article (entitled ‘Pro domo’) was not published, but Freudenthal did receive Janssen’s late reaction in which he thanked him for sending the lecture. But Janssen saw no reason to change his earlier reaction: Van onkunde, in welke richting ook, bent u niet beticht. ’t Zou te dwaas zijn. Wat ik hiervoor in het midden bracht sloeg op door u zelf gesproken woorden. Waar het om gaat, naar de mening van ons bestuur, is: “zijn beweringen zijn echter, naar het ons toeschijnt, vaak te impulsief, de gebezigde woorden te “dik”.” Bij alle waardering die we voor uw werk hebben, was dit ons bezwaar en meenden we goed te doen U daar op, langs de gebezigde weg, te wijzen. (You have not been accused of ignorance, in whatever direction. That would be too foolish. What I put forward in this respect referred to words you spoke yourselves. The point is, according our committee: “his statements, however, it would seem to us, are often too impulsive, the words he uses too “heavy”.” In addition to the appreciation we have for your work this was our objection and we meant well in indicating this to you in the manner we used.) [58]

The chairman thus avoided an actual discussion about the content of the mechanics education. Not surprisingly, the WIMECOS committee had already taken their point of view a long time previously: mechanics education had to remain an independent subject [59]. According to Janssen Freudenthal had put himself out of action by his

148


perhaps overly impulsive language and he ought to be reprimanded for it. But, Janssen concluded in his letter, “for the rest I personally hope that you will unflaggingly want to continue devoting your interest and knowledge to our education” [60]. Freudenthal did not seem to be much affected by what happened: he received support and understanding in academic quarters from Beth junior who apparently did not have a problem with Freudenthal attacking his father’s book. This support was perhaps a little surprising since Freudenthal and Beth were advocates of opposite convictions, in particular where the formative value of the mathematics education was concerned. After the HBS Willem Evert Beth (1908–1964) went to university and studied consecutively mathematics, psychology, philosophy and law [61]. In 1946, after having taught mathematics and science for years, he was appointed extraordinary professor of logic and philosophy of science and its history. The formative value of the mathematics education was an important theme to Beth—just as it was to his father. As late as 1937, in a lecture for the WW, he stated that forming the intellect was one of the main reasons for geometry education. As Janssen did not discuss the content of his lecture any more deeply, Freudenthal could not take his criticism seriously. Before that time Freudenthal had already written in his letter to Beth: “Anyhow, thanks for your interest, this sweet talk is never boring” [62]. In the end the net result was that both Freudenthal’s views (and with those indirectly also those of the Wiskunde Werkgroep) received the necessary—be it negative—attention and that Janssen and the teachers associations had, in Freudenthal’s eyes, made fools of themselves since they did not want to be engaged in the discussion. It did not seem to bother Freudenthal that, due to the reckless words of his lecture and the less than tactful approach of the teachers, he had made an open discussion in Euclides almost impossible and in the process got his fingers burned. That Freudenthal was ‘right’ would eventually become clear when mechanics was abolished as an independent school subject in 1962. The lessons reserved for mechanics were divided between mathematics and physics, as had already been decided by the Beth and Fokker committees in the 1920s [63].

6.3 Preparations for a New Curriculum As mentioned in paragraph 5.3.1, in 1948 the Wiskunde Werkgroep set out to compose a completely new mathematics curriculum. Freudenthal, Minnaert, Van Hiele and Bunt made up the algebra committee. With great enthusiasm the committee pruned the traditional subject matter considerably. What the exact role of Freudenthal in this pruning process was, is difficult to assess in retrospect, but from Van Hiele’s recollections it appeared that he was not particularly conservative [64]. In fact, Freudenthal did not want to stop there. In a letter to the working group he brought forward a working plan for future actions: Met het programma-werk kan onze werkzaamheid in die richting niet zijn afgelopen. Ik vrees zelfs, dat wij het door ons gestelde doel niet bereiken, indien wij onze eisen niet kracht

6.3 Preparations for a New Curriculum

149

bijzetten. Ik meen, dat wij dit het beste kunnen doen, door gezamenlijk als Werkgroep een Wiskunde-leergang voor M.O. en V.H.O. te schrijven. (When we finish the curriculum our activities in that direction have not yet reached an end. I even fear that we cannot reach our goal if we do not enforce our arguments. My idea is that we, as a working group, can do that best by writing a teaching method for mathematics in MO and VHO.) [65]

Next Freudenthal sketched a working plan for such a project and he proposed some points which could be discussed when the general trend of such a teaching method was to be determined: 1. Wiskunde-onderwijs in de twee eerste klassen geheel en al propaedeutisch, dus constructief en experimenteel, zonder bewijstechniek. 2. Verschuiving van enige wiskunde-uren naar de hogere klassen; de in de lagere klassen vrijkomende uren gebruiken, door reeds in de eerste klas met natuurkunde-, mechanica- en sterrekunde-onderwijs te beginnen. (Deze vraag eventueel bespreken in samenwerking met de werkgroep van deze vakken). (1. Mathematics education in the first two years should be completely introductory, that means constructive and experimental, without proving-skills. 2. Shift some of the mathematics hours to higher classes; use the hours thus available by starting with physics, mechanics and astronomy education in the lower classes right away. (Discuss this question, if needed in cooperation with the working groups for these subjects).) [66]

Jacobs, secretary of the working group, discussed Freudenthal’s proposition with chairman Janssen and the Van Hieles. The former rather preferred to postpone the discussion of such a plan till once the new curriculum had been discussed. Van Hiele took the view, Jacobs wrote, that the working group would be too heterogeneous to compose such a curriculum. Hoewel hij werken in de door U aangegeven richting zou toejuichen zou hij vooraf willen nagaan in hoeverre spreiding der meningen optreedt en hoeverre het mogelijk is, deze spreiding aanvaardende, tot verbetering van methoden te komen. Op deze wijze, aldus de heer en Mevr. van Hiele, kunnen we de constructieve krachten binden i.p.v. afstoten. (Even though he would applaud working in the direction you indicated he would still want to check how much the opinions disperse and, accepting this dispersion, whether it would be possible to improve the teaching methods. In that way, according to the Van Hieles, we can attract instead of reject constructive forces.) [67]

That was how Freudenthal’s proposition disappeared into a deep desk drawer and the working group concentrated on composing the new curriculum. Early in January 1950 the first concept of the algebra curriculum was finished and Freudenthal had written a general introduction for the total curriculum concept [68]. All the same, Freudenthal did not think that composing a new curriculum would be the ultimate way to solve the problems in mathematics education. He addressed this subject in De Groene Amsterdammer. The Dutch mathematics education, for many a stumbling block, was up a blind alley, he wrote. The subject matter was outof-date and was at least 50 years behind the living mathematics. The subject matter with which pupils were trained was worthless and every year it was complemented with new useless lumber: the examination problems of the last year. Met een verbetering van het wiskundeprogramma zijn we er nog lang niet, al zijn heel wat leraren overtuigd, dat hun onderwijs vruchtbaarder zou zijn, als de dril op examenacrobatiek verdween. (Improving the mathematics curriculum will by no means be enough, even

150


though quite a number of teachers are convinced that their education would bear more fruits if the drill in examination acrobatics were to disappear.) [69]

A more important question was left untouched according to Freudenthal: how could it be that intelligent people had difficulties with mathematics? Dat uit te zoeken is een urgente taak voor paedagogen en psychologen. En zolang we hierop geen antwoord krijgen, kunnen we niets anders doen dan door blijven te borduren op een oude traditie, die eist dat een middelbare scholier wiskunde leert. Het blijft een sukkelen, zeer ten spijt van allen, die met het wiskunde-onderwijs te maken hebben. (To figure that out is an urgent task for pedagogues and psychologists. And for as long as we do not have an answer to this question we cannot do anything else but continue elaborating on an old tradition which requires that a secondary-school pupil learns mathematics. We have to put up with it, much to the regret of all who are involved in mathematics education.) [70]

Nevertheless Freudenthal took part in this ‘putting up with’ and worked diligently at designing the curriculum. It did suit his attitude: even when he did not agree he kept involved to be able to at least adjust matters and in any case have an influence on the results. In 1953 Vernieuwing published the article ‘De programma’s wiskunde voor Gymnasium β en HBS-b’ (The mathematics curricula for Gymnasium-β and HBSb). This was an extended and in some small points adapted version of the introduction Freudenthal wrote in 1950 as a general introduction to the formulated curriculum [71]. The same introduction, followed by the concept of the curriculum, was published in Euclides shortly afterwards. Wansink added an introduction on the activities of the working group. This was necessary he noticed, since: Telkens weer blijkt me in gesprekken, dat er omtrent die werkgroep misverstanden bestaan, die soms hardnekkig van aard zijn. Te gemakkelijk wordt de groep soms vereenzelvigd met één of met enkele harer leden. De invloed van enkele buiten het V.H.M.O. staande medewerkers wordt dikwijls verkeerd aangeslagen; van de ideeën die in de groep leven, wordt soms, door emotionele remmingen, niet of ternauwernood kennis genomen. (In conversations I notice, over and over again, that misunderstandings, which sometimes are of a persistent nature, exist about the working group. The group is sometimes too easily identified with one or some of its members. The influence some of its co-workers, who do not originate from the VHMO have, is often wrongly assessed. Owing to emotional inhibitions, the ideas the group cherishes are sometimes hardly or not at all noticed.) [72]

In view of the effect Freudenthal had with his statements (such as those about mechanics education) it seemed obvious that Wansink was referring to Freudenthal’s influence. Wansink’s introduction to the history and the activities of the working group was followed by the introduction (‘Algemene inleiding’) to the formulated mathematics curriculum written by Freudenthal. Here the working method and the starting point of the working group were explained. The most important criticism of the curriculum in force was that certain areas were overrun by inferior subject matter. Apart from that another important motive for revision was the overburdening of the pupils as had become clear, amongst other things from a research project of the Paedagogisch Instituut (Pedagogical Institute) in Utrecht [73]. Freudenthal’s complaint as he stated in De Groene Amsterdammer could be heard in the following formulation:


151

We weten wel, dat op het ogenblik nog vrijwel alle gegevens ontbreken voor een wetenschappelijk-paedagogisch gefundeerd programma voor het onderwijs in de Wiskunde en we hebben dit gebrek dan ook tijdens onze besprekingen sterk gevoeld. Aan de andere kant is de behoefte aan herziening zo urgent, dat doortastende maatregelen niet mogen worden uitgesteld. (We do know that at the moment we lack practically all information for a scientifically-pedagogically founded mathematics curriculum, and we did experience that deficiency strongly in our discussions. But on the other hand the need for a revision is so urgent that thorough measures should not be postponed.) [74]

The most important criterion in composing the curriculum was the practical use of mathematics as an auxiliary subject for numerous school subjects, studies and professions. Het criterium van het practisch nut nu heeft geleid tot gevoelige schrappingen vooral in de Algebra en tot een drastische schrapping in de thans gedoceerde trigonometrie. Deze twee gebieden zijn immers overwoekerd door leerstof, die geen enkel verband meer houdt met welke praktijk dan ook en die geheel is geabsorbeerd door een op zichzelf wenselijke, maar in feite verkeerd gerichte routine. (The criterion of practical use have now led to the leaving out of some crucially important items in algebra and some equally vital areas in the trigonometry currently taught. For these two areas have been overgrown by subject matter that has no bearing whatsoever on whatever practice and is completely absorbed by a routine that in itself is desirable, but in fact is wrongly focused.) [75]

It was the first time in mathematics education that the arguments of ‘practical use’ and ‘connection to the practice’ were preferred to those of the formative value. Since this criterion was so explicitly mentioned in the general introduction, there was enough support apparently—at least in the working group of 128 members— for this starting point, a situation completely different to that in which Van Dantzig had put forward this point of view 30 years earlier. Differential and integral calculus, probability theory and statistics were added to the curriculum based on that same practical use. The working group had made an exception for geometry: Ook zonder diep in te gaan op de vraag, in hoeverre het meetkundig-onderwijs vormende waarde (b.v. voor de ontwikkeling van het denken of van het ruimte-inzicht) bezit, hebben we, met het oog op de culturele betekenis van dit vak, gemeend, dat een scherpe breuk met de traditie niet verantwoord was. Het is de leidende bedoeling van het voorgestelde meetkundeprogramma, om de leerling inzicht te geven in een logisch opgebouwd systeem, dat zekere afgerondheid bezit; omwegen en afdwalingen worden dus zoveel mogelijk vermeden. (Without pursuing further the question to what extent geometry education has formative value (for example for developing the intellect or the three-dimensional understanding), we have, with an eye on the cultural meaning of this subject, thought that a clean break with tradition was not justified. It is the guiding principle of the proposed geometry curriculum to give the pupil insight into a logically founded system, that has a certain completeness; detours and wanderings are avoided as much as possible.) [76]

Another controversial point was the descriptive geometry to which ultimately some time was devoted in the HBS curriculum. Wansink was one of the strong advocates of upholding the descriptive geometry. Freudenthal was one of the opponents. In his eyes, the descriptive geometry was “a subject, isolated from all other parts of mathematics, from science and from engineering” [77]. The only reason to uphold the subject at the HBS was “because stereometry has expanded to such an extent

152


that a lot of pupils would no longer pass the geometry exam, if this very easy part of the curriculum, ‘descriptive geometry’, were to disappear” [78]. The final point mentioned in the introduction was the question of the examination problems. The working group disapproved of the overburdening of the pupils by continually introducing a new kind of examination problems and argued in favour of a larger number of singular, simple and mutually independent ones. Here too Freudenthal’s opinion could be heard. In De Groene Amsterdammer, as well as in the Vernieuwing, he regularly complained about absurd examination problems and the useless ‘drilling’ of the pupils in learning methods to solve more and more complicated types of problems every year, with the consequence that the examinations determined the curriculum instead of the other way around [79]. The working group could not find any reason why the mathematics curriculum for the HBS-b and Gymnasium-β should have to differ from each other, but: Onze pogingen, om tot een uniform programma te geraken zijn gestrand, toen tenslotte bleek, dat de verwezenlijking ervan te ingrijpende wijzigingen in de programma’s veronderstelde. We hebben derhalve getracht, de programma’s enigszins naar elkaar toe te buigen, in de hoop, hierdoor de algehele gelijkschakeling op een later tijdstip voor te bereiden. (Our attempts to reach a uniform curriculum failed when it eventually transpired that its realization would be predicated on overly drastic changes in the curricula. We have therefore tried to flex the curricula towards each other a bit, hoping to prepare in that way a complete bringing into line at a later stage.) [80]

The working group therefore proposed introducing analytical geometry at the HBS and constructive geometry at the Gymnasium. In the event of the curriculum presented here leading to a less demanding curriculum, the time made available should be spent on an improvement of the methodology of the mathematics education. It was certainly not the intention of the working group to diminish the number of lessons. The reports of the committees for algebra, analytical geometry, goniometry and descriptive geometry with further details of the proposed material per subject followed this general introduction. A direct reaction to the working group’s proposal in Euclides failed to materialise. Indirectly there might have been a reaction: on 2 January 1954 at the annual meeting of WIMECOS the WIMECOS board with the cooperation of LIWENAGEL decided to appoint a committee that “would have to report on the total of subject matter and examination curriculum” [81]. This committee consisted of Messrs. Alders, Holwerda, Bunt, Vredenduin and Wansink (chairman), the last three had also been active in four out of the five working group committees (viz the committees for algebra, geometry, analytical geometry and descriptive geometry). Less than a year later this committee presented their curricula. The committee emphasized in their report that a lot of preparatory work had already been done by the Wiskunde Werkgroep, amongst others, and that the curriculum they proposed here “should not be considered separately from the amount of work that has been done elsewhere” [82]. Consequently, the broad outlines of the two curricula resembled each other. In the WIMECOS curriculum too the emphasis was on cutting back the subject matter, introducing such new items as statistics, differential calculus and integral calculus in view of their ‘practical use’ and the ‘formative value’ starting point was also pushed into the background (the committee


153

proposed for example an empirical and intuitive introduction for education in plane geometry). For the rest the committee criticized in their report the tendency for examination problems to become more complicated in a technical sense every year and it supported the WW standpoint in this respect by preferring “more singular and simple, mutually independent examination problems […] rather than a smaller number of complicated, more difficult problems, sometimes even built up in such a way that the next question cannot be answered if you had failed to solve the preceding ones” [83]. It is remarkable that, whereas the WW proposed maintaining the descriptive geometry (be it only partly and under the name of constructive geometry), the WIMECOS committee proposed fitting a drastically reduced version of it in stereometry. In composing the proposal the WIMECOS committee had deliberately avoided cooperation with the WW, as revealed in the words of WIMECOS’ chairman Wansink. De WVO, die zo doortastend voor vernieuwingen had gepleit, was buiten ons commissiewerk gehouden, omdat samenwerking er maar op grote weerstanden zou zijn gestuit. Toch heeft het me grote voldoening gegeven dat enige belangrijke ideeën van de WVO in het commissierapport ingang hebben kunnen vinden. (The WVO, that had argued so boldly for modernization, was kept out of our committee work, since cooperation with them would only have resulted in strong resistance. Yet, to my great pleasure some of the important ideas of the WVO have been adopted in the committee report.) [84]

That did not go down well with some of the committee members. Van Hiele, for example, said the following about the course of events: En het [WW-voorstel] was nog maar net uitgekomen, of Wansink en Vredenduin gingen in een commissie zitten van WIMECOS en LIWENAGEL. Ze kwamen later met een eigen programma, en er werd over de Wiskunde Werkgroep niet meer gepraat. Wel werd gesteld dat we dat ‘WIMECOS-programma’ maar moesten slikken, omdat het niet veel van het andere verschilde. Maar die link werd niet eerlijk, niet openlijk gelegd. Ik heb dat nooit erg fijn gevonden, maar goed. (And the [WW] proposal had only just been published before Wansink and Vredenduin sat on a WIMECOS and LIWENAGEL committee. They later composed a curriculum of their own and after that not a word was heard about the Wiskunde Werkgroep. It was supposed, however, that we had to swallow the ‘WIMECOS curriculum’ since that did not differ much from the other one. But the two were not linked in a fair way, not openly. Anyway, I have never liked that very much.) [85]

Nevertheless the working group agreed to the proposals. After those present at a WIMECOS’ extraordinary general meeting had accepted the curriculum, Freudenthal and Jacobs as representatives of the WW wrote a petition to the Minister of OK&W (Onderwijs, Kunsten and Wetenschappen, Education, Arts and Science) and the inspectors of the VHMO: Zij [de WW] is verheugd over de moed van de voorbereidende commissie en van de leden van WIMECOS, die niet teruggeschrikt zijn voor fundamentele wijzigingen van het thans vigerende programma. Van harte betuigt zij haar instemming met het ontwerpprogramma van WIMECOS. Zij geeft er haar medewerking aan door een didactisch onderzoek in te stellen naar de middelen om het te verwezenlijken. Op de najaarsconferentie van de wiskunde werkgroep zullen fundamentele problemen van dit programma worden behandeld en detailvragen zullen op de maandelijkse bijeenkomsten ter sprake komen. (They [the WW] are pleased about the courage of the preliminary committee and the members of

154


WIMECOS, who were not afraid of fundamental changes in the curriculum that is in force now. They wholeheartedly expressed their approval of the WIMECOS curriculum draft. They will assist by conducting a didactical study into the means to realize this. Fundamental problems of this curriculum will be treated at the autumn conference of the Wiskunde Werkgroep and detailed questions will be discussed at the monthly meetings.) [86]

To the greater annoyance of Freudenthal—caused by the coincidence of the summer courses of the Mathematisch Centrum (for which also the new curriculum was chosen as one of the subjects) with the planned weekend of the working group the Wiskunde Werkgroep would eventually abandon discussing this topic [87]. Freudenthal seemed to take the fact that the working group was more or less put out of action, well and gave WIMECOS credit for their success with the curriculum. He did not mind that the WW curriculum had been forgotten: Wat zou het lot zijn van dat programma, waaraan vier jaren moeite was besteed? Welnu, het is begraven, en het mag begraven blijven, al staan er lezenswaardige dingen in van betrekkelijke eeuwigheidswaarde. En dat is goed zo, want er zijn nu eenmaal zaken, die gedaan worden met het doel zichzelf overbodig te maken. (What would be the fate of that curriculum into which four years of effort has been put? Well, it is buried, and it can remain so, even when it contains things of a relatively eternal value worth reading. And that is good, since there happen to be things that are done with the aim to make themselves superfluous.) [88]

The most important thing to him was that, even if that was only implicit, there would indeed be some results from the work and the ideas of the working group. And that was the case now that WIMECOS had presented proposals for a new curriculum. Een programma, niet minder revolutionair dan dat, waar de wiskunde werkgroep van de “vernieuwing” trots op was. Ook vernieuwing is een relatief begrip. De vernieuwers van nu zijn gedoemd de behoudenden van straks te zijn. Vernieuwers zijn die mensen, die zich ex officio overbodig maken. (A curriculum, no less revolutionary than one the mathematics working group for “modernization” could be proud of. Modernization is also a relative notion. The modernizers of today are destined to be the conservatives of the future. Modernizers are those people who make themselves superfluous in their official capacity.) [89]

In this article Freudenthal one by one praised the changes in the curriculum as presented by WIMECOS and once again emphasized the courage of the association in choosing this curriculum despite the many discussions its proposal had caused. Freudenthal was particularly impressed by the decision to (re-)introduce statistics, differential and integral calculus, and analytical geometry. Men kan gemakkelijk zeggen, dat alles wat in dit programma wordt voorgesteld, dringend noodzakelijk en niet meer te vermijden is. Maar hoeveel sectoren zijn er niet van onze samenleving, waar de problemen even urgent zijn en de menselijke traagheid schier onoverwinnelijk is? (It can easily be said that everything that is proposed in this curriculum was urgently needed and not to be avoided any longer. But how many sectors of our society are there with equally urgent problems where the human slowness is almost invincible?) [90]

A “brave decision” he continued to rejoice, inspired by “courage and self-confidence and the serious wish to take a giant step to a better mathematics education” that breaks with a 90 year-long old established tradition. Full of hope for the future he concluded his plea. If this curriculum was accepted the work started to give meaning to

6.4 Probability Theory and Statistics: A Text Book

155

… aan een programma, dat niet, zoals veelal, vijftig jaar achter de feiten aanholt. Niet voor de school, maar voor het leven—mogen we dan zeggen en met “leven” dan niet dat van onze grootouders bedoelen, maar de wereld van vandaag. (… a curriculum that is not, as is usually the case, fifty years behind the facts. Not for the school, but for life itself—we may say and with “life” we mean not that of our grandparents but today’s world.) [91]

Once the teacher’s association LIWENAGEL had also approved the curriculum there was hardly anything that prevented its introduction. With a few alterations (for example, statistics and descriptive geometry were eventually cancelled for fear of overburdening the pupils) the new curriculum for HBS and Gymnasium would become operative starting from 1 September 1958 [92]. Thus for the first time Freudenthal had indirectly made headway in the mathematics education with his ideas, without his name or even that of the WW being associated with it.

6.4 Probability Theory and Statistics: A Text Book Now that the practical applicability of the mathematics to be taught at the VHMO had become an ever more important starting point, the new subjects of probability theory and statistics also came up for discussion. Early in the 1950s under the leadership of Bunt a test with statistics education for Gymnasium-β had been conducted with positive results [93]. Now that its use in the mathematics education was proven, statistics was subsequently introduced in the changes to the curriculum proposed by WIMECOS and LIWENAGEL. Freudenthal was also involved in this development. In the preface to the report on Bunt’s test, for example, he was thanked for reading the manuscript and giving advice [94]. In the beginning of January 1955 Freudenthal wrote a letter to the publisher Erven F. Bohn NV in which he proposed writing a volume on probability theory and statistics for the 'Volksuniversiteits-Bibliotheek’ (‘Library of adult education’) series. The editors enthusiastically accepted his proposal. Before the year ended Freudenthal had finished the first version. The book would be published in 1957 as the 57th issue of the series, in between Latijnse letterkunde ( Latin literature) by H.H. Janssen and Het ontstaan van het zonnestelsel ( The origin of the solar system) by H.P. Berlage [95]. Statistics and probability theory were far from being new subjects to Freudenthal. At the end of the 1940s, for example, he had been commissioned by ‘De Bijenkorf’ along with J. Sittig to work on a new sizing system for ready to wear clothing [96]. A few years later Freudenthal was asked by Dobbelman NV to cooperate on the statistical processing of the results of market research on soap (Fig. 6.2) [97]. From 1952 to 1954 Freudenthal advised the company on various items of market research. In 1952 he assisted the Rechter-Commissaris (examining judge) of the Utrecht district in answering the question “whether the slot machine confiscated at the suspect’s premises is of such a nature that playing with it is a game in which, generally speaking, the possibility of winning depends on chance, and whether that chance increases when the player is more practiced or more dexterous” [98].

156


Fig. 6.2 Request to Freudenthal for statistical advice by the Dobbelman firm (1952)

What was new for Freudenthal was writing a book with this outline. He was, of course, accustomed to writing synopses of the lectures he delivered at the university, but never before had he written a textbook for a more general audience [99].


157

Freudenthal was convinced that statistics had become a vital aspect of society and that its use would certainly not be limited to the insurance business and games of chance: Maar heden ten dage zijn statistieken en kansrekeningen niet meer een specialiteit van verzekeraars. Wil men tot een besluit komen, om de hoogte van onze dijken te bepalen, dan kan men niet wachten tot theorie en praktijk ver genoeg zijn gevorderd, om mathe matisch streng de hoogst mogelijke waterstand uit te rekenen, maar men moet trachten uit de beschikbare gegevens conclusies te trekken, die met een zekere—zeer kleine—kans ook foutief zouden kunnen zijn. Medici kunnen niet wachten met het toedienen van nieuwe middelen en het toepassen van nieuwe operatiemethoden, tot er geen twijfel meer bestaat over hun uitwerking, maar ze moeten kansen kunnen nemen en bij die becijfering van die kansen zich kunnen laten helpen door de kansrekenaars of statistici. (But these days, statistics and calculations of probability are no longer a specialty of insurers. If we have to decide on the height of our dykes, we cannot wait until theory and practice have advanced far enough to be able to calculate the highest possible water levels with mathematical strictness, but we have to try to reach a conclusion from the available data, which could be wrong with a certain—very small—chance. Medical doctors cannot delay in administering new drugs and applying new surgery methods until no doubts remain about their effects: they have to be able to take chances and in calculating these chances they could enlist the help of calculators of probability and statisticians.) [100]

So there were reasons enough for Freudenthal to inform a more general audience about the mathematical side of statistics. The audience that Freudenthal had in mind when writing Waarschijnlijkheidsrekening en statistiek had at least a secondaryschool training with sciences, but from time to time he demanded a bit more from his readers: Ik vraag de lezer daarom dezelfde clementie, als die hij beoefent, wanneer hij een roman leest: af en toe, als het moet, wat bladzijden over te slaan. (I therefore ask the reader to show the same mercy he shows in reading a novel: from time to time, if necessary to skip a few pages.) [101]

The book was certainly neither intended as teaching material for the VHMO, nor for a university course, but for the somewhat mathematically educated interested ‘layman’. In practice it would turn out to be mainly of use to mathematics teachers. In 1966, for example, it was used in the reorientation courses for teachers in mathematics education organized by the Commissie Modernisering Leerplan Wiskunde (Committee for Modernizing the Mathematics Curriculum). It is interesting to examine the book further in the light of Freudenthal’s didactical ideas. In the first place Freudenthal wanted to write a book made up of a collection of short stories, subjects brought together from probability theory and statistics that he thought were “fundamental and interesting”. He did not avoid proofs, because: Met het begrip waarschijnlijkheid wordt tegenwoordig op zoveel gebieden gewerkt, dat menigeen er iets voor voelt, het fijne ervan te weten te komen. Ja, maar voor de wiskundige, die het hem moet uitleggen, is het fijne juist niet het resultaat en het recept, waar men iets mee kan beginnen, maar de wijze, waarop men aan resultaten en recepten komt. De wiskundige vraagt voor elke uitspraak een bewijs, niet omdat hij zijn collega’s niet vertrouwt, maar omdat veelal het bewijs het enige middel is, om te weten te komen, wat met de uitspraak is bedoeld. De bewijzen en redeneringen worden u daarom ook in dit boekje niet bespaard. Ook de formules niet. Er zijn er trouwens niet zo bijster veel, want er wordt veel

158


geredeneerd in dit boekje, en dat geschiedt meestal zonder formules. (Nowadays the notion of probability is used in so many fields that many people feel like finding out the details of it. Yes, but to the mathematician who has to explain it the crucial point is neither the final outcome nor the formula which can be of use, but the way in which these results and formulas are obtained. The mathematician demands a proof for every statement, not because he does not trust his colleagues, but mainly because the proof is the only means of finding out what the meaning of the statement is. Consequently, you are neither spared the proofs nor the argumentation or the formulas in this book. There are none too many anyway, as the book contains a lot of argumentation which is usually done without formulas.) [102]

The book consisted of nine chapters with a length varying from 4 to 37 pages. In Chap. 1, entitled ‘Het vaasmodel’ (literally the ‘vase model’), Freudenthal, starting from a merely intuitive notion of ‘chance’ his readers might have, built a framework of mathematical concepts with the accompanying notation system from definitions and practical examples. Subsequently he introduced concepts such as chance, the stochastic variable x and the model of the vase, after which he took the reader by the hand and derived various rules for calculating with probabilities. Every definition or rule was followed by a series of examples, often closely related to practical situations and presented with a dash of humour. This is what he wrote after explaining the vase model: De kans, om bij een sprong van de Domtoren te Utrecht de nek te breken, is practisch 100 . —Een statistiek hieromtrent is mij niet bekend, maar ik kan er een maken. Ik gooi een honderdtal mensen van de toren (natuurlijk in gedachten, want na de eerste zou ik niet veel gelegenheid meer krijgen, de proef voort te zetten) en ga na of ze hun nek breken (in gedachten). Voor elk, die de nek breekt, werp ik een zwart balletje in de vaas, voor elk die het overleeft, een wit. (The chance of breaking your neck when jumping from the Utrecht Domtoren (at the time the highest church tower in the Netherlands) is practically 100 .n I do not know a statistic about this, but I can make one. I will throw about hundred people from the tower (theoretically, of course, since after the first I would not have much opportunity to continue with this test) and assess whether they break their necks (again theoretically). For everyone who does break his neck I put a black ball into the vase, for everyone who doesn’t a white one.) [103]

Following the examples are a few exercises with answers to be found in the appendix (Fig. 6.3). Freudenthal kept addressing his reader. Every other paragraph of the text and the exercises contain statements such as “But take care!”, “Calculate this but do it in a clever way!” or “Figure this out!” [104] Not all of the exercises required calculations or mathematical notations. Particularly in the first few chapters these were often ‘deep ones’ meant to make the reader aware of the difference between the intuitive notion of chance and the concept of chance in probability theory. A good example was the following exercise at the end of the paragraph on conditional probability. Twee personen in een trein praten met elkaar. Ze komen bij een overweg. De één zegt: “De mensen, die op deze weg zijn aangewezen, zijn niet te benijden. De spoorbomen hier zijn praktisch altijd dicht. Ik kom hier sinds jaren dagelijks twee keer met de trein voorbij en heb ze nog nooit open gezien”. Wat heeft deze anekdote met het onderwerp te maken? (Two persons in the train are talking to each other. They arrive at a level crossing. One says: “The people who have to resort to this road are not to be envied. The level-crossing barriers are practically always closed. For a couple of years I have passed this crossing twice daily and have never seen them open.” What has this anecdote to do with the subject?) [105]


159

Fig. 6.3 Freudenthal during his lecture ‘Orde and wanorde’ (‘Order and chaos’) during the ‘Statistische dag 1960’ (‘Statistical day’) of the Vereniging voor Statistiek (1960)

Freudenthal did not furnish an answer to this type of exercise. He did give the solutions to the more difficult exercises, albeit without explanation or calculation. The exercises often resembled the ones that preceded them and which were already detailed by Freudenthal, so much so that the reader ought to be able to solve them easily, as becomes clear from the following exercise in the paragraph about estimating expectations: 4.8.5. In een steekproef van 5001 Nederlandse vrouwen was het gemiddelde gewicht 66.75 kg en de steekproefstrooiing hierop 10,91 kg. Welke uitspraak zal ik omtrent de gemiddelde Nederlandse vrouw doen? x is het gewicht in kg, x´ is het gewicht (in kg) in de steekproef.

ε x ′ = 66, 75, Dx ′ = 10.91, 1 Dx ′ = 0,16. n −1 Dat is dus de maatstaf, waarmee ik moet rekenen. Ik kan bijvoorbeeld de uitspraak

ε x ≥ 66 kg doen met dezelfde zekerheid als de uitspraak

ε z ≥ z − 0, 75 / 0,16,

160


d.w.z. z ≤ 4, 5. ( z = de-Moivre-stochastiek). Met welke waarschijnlijkheid op juistheid kan ik de uitspraak ε x ≥ 66,5 doen? Welke uitspraken kan ik met een zekerheid van 95 % doen? Beantwoord dit zelf! (4.8.5. In a random sample of 5001 Dutch women the average weight was 66.75 kg and the accompanying variance was 10.91 kg. What will I say about the average Dutch woman? x is the weight in kg, xʹ is the weight (in kg) in the random sample.

ε x ′ = 66.75, Dx ′ = 10.91, 1 n −1

Dx ′ = 0.16.

That is therefore the standard, with which I have to calculate. I can for example say

ε x ≥ 66 kg with the same certainty with which I can say

ε z ≥ z − 0.75 / 0.16, that is z ≤ 4.5 . (z = de-Moivre-stochastic). What is the probability that it is correct when I say What can I say with a certainty of 95 %? Find the answer yourself!) [106]

ε x ≥ 66.5 ?

The first four chapters, which dealt with the vase model, permutations and combinations, expectation and variance, and the limit laws respectively, contained the most exercises. By the way, these chapters were difficult to read independently of each other (or as ‘separate stories’). If only because of the adopted notation they built on each other and formed a unity. The following chapters (‘De kunst van het gissen’ (‘The art of guessing’), ‘Spel, strijd en strategie’ (‘Game, competition and strategy’), ‘Stochastische processen’ (‘Stochastic processes’), ‘Waarschijnlijkheid in de natuurwetenschap’ (‘Probability in the natural sciences’) and ‘Filosofie der waarschijnlijkheid’ (‘Philosophy of probability’)) better served the purpose Freudenthal mentioned in his preface; he wanted to write a bundle of short stories. Every small chapter mainly consisted of detailed examples from practice, in which Freudenthal took the reader by the hand in order to find, together and by argumentation, the accompanying solution. In that sense it was a book bearing an informative rather than a didactical load, even though Freudenthal did his utmost to keep the reader actively alert. But Freudenthal had chosen the right tone and structure in his book, as became clear from the favourable review that was published in Euclides. Apart from the use of statistics in society, the reviewer H.W. Lenstra also emphasized in particular the good ‘publicity’ that Freudenthal provided for mathematics as a subject with this book.

6.5 Paedagogums, Paeda Magicians and Scientists

161

Zo lijkt het mij van groot belang, dat een boekje, dat naar wij hopen in de handen van vele niet-wiskundigen zal komen, duidelijk demonstreert, dat de wiskunde geen “droog” vak is, dat de wiskundige zich niet alleen met abstracte redeneringen ophoudt, maar dat wel degelijk ook de humor om de hoek komt kijken en dat bij wiskundige beschouwingen en redeneringen ook zeker de aesthetica in het geding is. Naar mijn mening is prof. Freudenthal er bij uitstek in geslaagd deze elementen naar voren te doen komen. (For example, it seems to me of great importance that a booklet, which as we hope will end up in the hands of many non-mathematicians, clearly shows that mathematics is not a “dry” subject, that the mathematician is not only engaged in abstract reasoning, but that indeed humour is involved and that in mathematical considerations and reasoning aesthetics are certainly also an issue. In my view Professor Freudenthal has succeeded pre-eminently in putting these elements forward.) [107]

According to Lenstra, the book was exactly in time for the mathematics teachers who might have to teach statistics in the VHMO—something that most probably provided extra motivation for Freudenthal to write the book. The reviewer warmly recommended the book to the readers of Euclides, “all the more because the whole set-up shows the didactical qualities of the author and that can be instructive” [108]. The sale of Waarschijnlijkheidsrekening en statistiek went well. Over the years the book would be translated into several languages and several editions were published.

6.5 Paedagogums, Paeda Magicians and Scientists: The Teacher Training Apart from the attempts to modernize mathematics education by designing a new curriculum the discussion on how to structure the teacher training also played an important role. In the period between the two worlds wars such discussions had also been running in parallel [109]. In the meantime not much had changed. In fact, many of the complaints of that time were still extant. Freudenthal had already formulated a first point of view about the teacher training at the university in his unpublished ‘Oude en nieuwe universiteiten’ [110]. Apart from his untiring efforts in the 1950s to introduce student wages Freudenthal also developed an outspoken opinion on the set-up of a university-based teacher training that in fact was still missing. The proposals in the report of the Reinink committee to introduce a “master degree for teachers” (at the time also supported by Freudenthal) were met with such protest by the teachers that these had never been accepted. On 28 August 1952 it was decided by Royal Decree that in order to obtain teaching credentials a master graduate should have attended lectures in pedagogy, psychology of puberty and general didactics for a year and also have done his teaching practice for a period of 3–6 months. Not a new idea since this had already been proposed by the Beth committee in the interwar period. That was the situation when Freudenthal, the day before the implementation of this Royal Decree, delivered his lecture ‘Noodzakelijkheid en mogelijkheid van een leraarsopleiding’ (‘Necessity and possibility of a teacher training’) at the convention

162


of the ‘Genootschap van leraren aan Nederlandse gymnasia en lycea’ (‘Association of teachers at Dutch Gymnasia and Lycea’) (Fig. 6.4). He wrote later about the impression that he must have made: Bominslag? Knuppel in het hoenderhok? Nee—dat werd het pas in de verslagen van de grote pers. Ik denk veeleer dat ze me aanzagen voor een wildeman, een clown, een op effectbejag verzotte demagoog […] De secretaris schreef me later mijn lezing toch maar liever niet te publiceren, omdat ‘niemand van de leden erom gevraagd had’. (Direct hit? No—it only became that in the press reports. I rather think that they took me for a madman, a clown, a demagogue playing to the gallery […] The secretary later wrote to me that he would rather not publish my lecture since ‘none of the members had asked for it’.) [111]

Almost a year later the lecture finally was published in the weekly Christelijk Schoolblad. Onze vacatures ( Christian School paper. Our vacancies) with the title ‘Vakgeleerde of paedagoog?’ (Fig. 6.4). In this article Freudenthal emphasized the absurdity of the existing system. The new graduate was qualified to teach: not the subject matter he had learned at university, but: … hetgeen de jongeman, voor hij de Alma Mater omhelsde, heeft opgedaan, en wat vijf à zes jaar in zijn hersens heeft gesluimerd en nu ontwaakt. Hij heeft ondertussen geleerd “boven de schoolstof te staan”. Op Eiffeltorenhoogte, vanwaar alles beneden bespottelijk klein lijkt, en nu gaat hij met de lift naar de begane grond, om in het vervolg op te kijken tegen de Eiffeltoren van de wetenschap. Wèl, mocht hij tijd en geld hebben, om eens een entréé-kaart [sic] te kopen, dan mag hij ook weer eens een keer naar het hoogste platform reizen. (… what the young man has learned before he embraced the Alma Mater that has slumbered in his brain for five or six years and now awakens. In the meantime he has learned “to view the school material from above.” From the height of the Eiffel Tower, where everything down there looks ridiculously small, and now he takes the elevator down to look up at the Eiffel Tower of science. But, should he have time and money, to buy an entry ticket, he is once again allowed to travel up to the highest platform.) [112]

Such wrongs had been observed earlier, for example by Dijksterhuis in 1926 [113]. But Freudenthal took it one step further. The cause according to Freudenthal was the dominant fear of “paedagogums and paeda magicians” (Freudenthal refers to the play with the word pedagogue: paedagoog  paedagogum, paedagoochelaar, which associates the pedagogue with goochem, i.e. shrewd, and goochelaar, i.e. magician.) that could contaminate the discipline. The universities did not train graduates to teach and that was wrong: a teacher should first and foremost be trained to become a teacher. Yet, Freudenthal was not all that pessimistic about the future. We beschouwen het thans als vanzelfsprekend, dat een toekomstige arts voor arts wordt opgeleid, en over enkele tientallen jaren zal de opleiding voor leraar even vanzelfsprekend zijn. De hoogmoedige grapjes over paedagogums en paedagoochelaars zullen we dan even weinig begrijpen als die der 18e eeuwse artsen over de chirurgijnen en die der 19e eeuwse artsen over hun collega’s-gynaekologen, die als een raar soort mannelijke vroedvrouwen met de nek worden aangezien. (Today we consider it obvious that a prospective medical doctor is trained as a medical doctor, and in a few decades training to be a teacher will be as matter of course. We will then understand equally less of the arrogant jokes about paedogogums and paeda magicians as we understand of the jokes of the medical doctors of the 18th century about the barber surgeons or the jokes of the 19th century doctors about their colleague gynaecologists, who were looked down upon as a strange sort of male midwives.) [114]


163

Fig. 6.4 Freudenthal’s lecture on the teacher training was eventually published as ‘Vakgeleerde of paedagoog?’ (‘Specialist or pedagogue?’) in Christelijk Schoolblad (1953)

164


Next Freudenthal refuted the two most oft heard arguments that until then had wrecked the attempts to fit a teacher training into the university curriculum. First of all, the argument was that lowering the requirements for the discipline would harm the standing of the teaching profession. According to Freudenthal, that was nonsense since: Een maatschappelijke waardering voor de leraar, die men meent te moeten redden, bestaat in ’t geheel niet. In elk geval is de leraar de minstgewaardeerde onder alle academisch gevormden. En dàt in een tijd, dat warme paedagogische belangstelling overal te bespeuren is. Daar klopt iets niet—zou ik zeggen. (The standing of the teacher in society, that some think needs to be saved, does not exist at all. In any case the teacher is the least esteemed profession among the academics. And this occurs at a time when fervent pedagogical interest is manifest everywhere. There is something wrong here—I would say.) [115]

According to Freudenthal the majority of the teachers were less familiar with what went on in the pedagogical world than any parents interested in upbringing. En nog erger: een kleine minderheid weet er net genoeg van af, om de hele wetenschappelijke paedagogiek als poespas en de paedagogisch geïnteresseerden onder de ouders als paedagogums te kwalificeren. Mogen we er ons over verbazen, dat de leraarstand zo laag gewaardeerd wordt? (And even worse: a small minority knows just enough of it to qualify the complete scientific pedagogy as hoo-ha and the parents interested in pedagogy as pae dagogums. Can we be surprised that the profession of teacher is not so highly esteemed?) [116]

The poor textbooks (“trash” Freudenthal called them), some of which had already reached their twentieth or thirtieth print editions, also contributed to the negative image of the teachers. As an example he mentioned a textbook “for a modern language” that had been analysed by his wife some time before, in which you could easily find, even without studying the language for 6 years, countless blunders [117]. The second oft heard argument was the didactical value of the disciplinary education of the prospective teacher at the university. According to Freudenthal this formative value was zero. The ever increasing specialization in science made it necessary to choose. Within the scope of the mathematics and science programmes educating students to become ‘all-round’ scientists, even if only in one discipline, was no longer possible. U ziet: van het befaamde “boven de stof te kunnen staan, dank zij de universitaire opleiding” is hier geen sprake meer; er is niets dan een technische opleiding in een zeer bepaalde richting. Ik beweer, dat ook waar dit nog niet het geval is, de Eiffeltorenhoogte van de universitaire opleiding geen uitkijkpost is, om de schoolstof te overzien. Voor wie iets van psychologie afweet, spreekt dit haast vanzelf. Transfer of training tussen verwijderde gebieden is er niet zonder meer. Wetenschappelijk critisch besef op het ene gebied verworven is misschien voor overdracht vatbaar; maar wil dat zo zijn, dan moet die overdracht zelf worden geoefend. (You see: the famous “being able to gain a better view of the school material thanks to a university education” is absolutely out of the question; it is nothing more than a technical education in a very specialized direction. I claim, that even where it is not yet the case, the Eiffel Tower height of the university education is not a look-out to overview the school material. To people who know something of psychology this is almost self-evident. Transfer of training between fields far away from each other does not exist just like that. Scientific critical understanding acquired in one field is perhaps open to transfer, but even if it is, the transfer itself has to be practised.) [118]


165

Exactly what the teacher training should look like, Freudenthal could not tell either. But he did see how a change had to be brought about. The attitude of wait and see had to be broken up, initiated by the teachers themselves. The universities should, for so long as they could not yet educate teachers, evoke interest for the didactical, pedagogical, psychological and sociological problems of the teaching profession and pass on knowledge about these subjects. This would give rise to a new generation of teachers who could, by “searching in their work as a teacher for a starting point for scientific activities”, contribute to developing being and becoming a teacher into a discipline. This ‘working on a discipline of mathematics education’ in and through practice was a starting point that would come to the fore more expressly in Freudenthal’s later work. The first cautious beginnings of it can be found here [119]. The development of a teaching methodology—an expression Freudenthal did not yet explicitly use, but only circumscribed here—as a discipline was very urgent according to him. Ik zou het normaal vinden als iedere leraar promoveerde, en wel over die problemen, die hem van het meest nabij raken: die van het leraarschap. Ze zouden voorlopig niet uitgeput raken. We weten van geen onderdeel van de schoolstof óf, wanneer en hoe het zou moeten onderwezen worden. We weten niet wat voor allen en wat voor enkelen geschikt is, we kennen van ons onderwijs geen enkele der psychologische en sociale drijfveren en consequenties, en dat nog niet alles op losse schroeven staat, vindt zijn oorzaak alleen hierin, dat al die schroeven vastgeroest zijn. (I would consider it normal if every teacher were to obtain a PhD, and then one in those problems that are closest to his daily work: those of being a teacher. For the time being there would be no shortage of problems. We do not know whether, when and how any of the parts of the school material should be taught. We do not know what is suitable to all and what only to a few, we know none of the psychological and social drivers and consequences of our education, and the fact that not everything is falling apart yet is only because the screws holding it down are rusted.) [120]

In the end Freudenthal was glad that the division of scientific education into a research and teaching version as proposed by the Reinink committee had not come off: in his eyes it would not have been a real teacher training after all and it would only have cut off future developments prematurely [121]. Freudenthal saw as the best solution, combining the teaching qualification with only those Master’s degrees in which one of the subsidiaries was “pedagogy and didactics”, so that the attention of the future teacher was brought to the problems of the teaching profession. Apart from that, “pedagogy” needed to be created as a main subject in mathematics and physics, so that candidates would be in a position to more intensively devote themselves to educational affairs without missing out on the teaching qualification in a mathematics or physics subsidiary. The main point was that the student should be informed about the existence of a scientific pedagogy so that he would become interested in it—‘in the best case’, because Freudenthal did not consider it a real preparation for the teaching profession. Apart from that, students should be brought into contact with children, both in and out of school, as early as possible and preferably in the form of pedagogical research. For as long that had not yet been realized it would result in an ever increasing separation between disciplinary and pedagogical functions in schools.

166


Inmiddels kan men er zeker van zijn, dat de lerarenstand elk jaar weer een jaar achterraakt bij een maatschappij, waarin de belangstelling voor alle paedagogische problemen groeiende is. (In the meantime one can be sure that year after year the teaching profession will fall behind a society in which interest in all pedagogical problems is growing.) [122]

That is how Freudenthal concluded this article. He was not impressed by the measures that were set out the day after in the Royal Decree, he informed his readers in De Groene Amsterdammer. Nevertheless it was a step (albeit a small one) in the right direction, but not yet in the way Freudenthal would have wanted it: Is dit K.B. nu iets om over te juichen? Ja, voorzover eindelijk de noodzakelijkheid van een leraarsopleiding officieel is erkend. (Is this R.D. now something to be exulted? Yes, but only because of the fact that finally the necessity of a teacher training has been recognized officially.) [123]

In this article too he stressed once again the danger of a division between scientists and pedagogues, something that according to him would inevitably happen. He was afraid that the effect of the Royal Decree would appear to be only negative in the end: Ik vrees, dat er niet veel meer aan te doen is. Zolang er in ’t geheel geen leraarsopleiding was, stonden we sterk met onze eisen, er één in te stellen. We hebben nu een noodoplossing gekregen. Aan noodoplossingen pleegt een lang en taai leven beschoren te zijn. (I fear, that there is nothing to do about it anymore. So long as there was no teacher training at all we were in a strong position with our requirement to establish one. And now we are confronted with a make-shift solution. And make-shift solutions are usually granted a long and tough life.) [124]

In 1955 it was decided by Royal Decree that didactical training was to be made compulsory for all disciplines for which a teaching qualification was required (including the secondary-school certificates). The period in which teaching practice had to be done was specified to a minimum of 3–6 months. In Utrecht the first lectures on general didactics were given on the authority of Professor M.J. Langeveld. In November of the same year Freudenthal once again devoted a lecture to the subject during a weekend conference of the WW. Het verhaal wordt eentonig—zult u zeggen, en terecht. Ik heb over de leraarsopleiding herhaaldelijk geschreven en gesproken, en het zou mezelf vervelen, als ik er niet op kan rekenen dat, alle monotonie ten spijt, in de discussie toch steeds weer nieuw leven zal sprankelen […] En zelfs zoals ik het vandaag zal formuleren, zal het ook nog niet definitief zijn. Was het anders, dan was het voor mij een veeg teken, dat ik al op het punt stond, er niets meer bij te leren. (The story becomes monotonous—you will say, and rightly so. I have time and again spoken and written about the teacher training and I would be bored myself if I could not rely on the fact that, in spite of all monotony, the discussions would not reveal again and again new sparkling life […] And even my way of formulating it today will also not yet be definite. If it was different it would be a bad omen to me, I would be on the verge of learning nothing new.) [125]

The function of the education was twofold: develop knowledge, abilities and standards on the one hand, and selection on the other. To effect changes in the educational system was a very difficult and tiresome process Freudenthal observed. A new teacher should primarily have learned to criticize the system and point the way


167

to new roads, instead of resuming the thread of his own secondary-school experiences. Er zal altijd moeten worden geselecteerd—dit is buiten kijf, maar we wensen, dat dit geschiede terwille, maar niet door middel van onderwijs. De aard van de opleiding zou niet mogen worden bepaald door percentages, die aangeven, hoeveel er moeten mislukken. (Selection will always be needed—that is beyond doubt, but we wish that it would happen for the sake of and not by means of education. The nature of the education should not be determined by the percentages indicating how many pupils should fail.) [126]

Good criterions for determining the proper subject matter were still lacking. The most important guidelines were tradition and the perfectionism of the teachers who wanted to send the children out into the world with as much intellectual substance as possible. Studieprogramma’s worden bepaald door mensen, die het de jongeren kwalijk nemen, dat ze niet in drie à vier jaren dezelfde schatten van wetenschap en beschaving verzamelen, waarvoor zìj [sic] er dertig à veertig nodig hadden; door mensen, die de voornaamste deugd van de opvoeder missen: het geduld. Ze zouden moeten worden ontworpen door hen, die kunnen geloven in de jeugd en in de ernstige voornemens van een mens, niet op de lauweren van een verworven bevoegdheid te gaan rusten, maar met zijn pond te blijven woekeren, ook als hij voor geen tentamen meer behoeft te blokken. (Curriculums are determined by people who blame the youngsters for not gathering in three or four years the same treasures of science and civilization as they themselves did in thirty of forty years; by people who miss the most important virtue of the educator: patience. They should be designed by those who can believe in youth and in the serious intentions of a person not to rest on the laurels of an acquired qualification, but to make the most of his pound, even when he no longer needs to study for an exam.) [127]

Freudenthal held the view that a person who could not be trusted to continue his education with the degree certificate in his pocket, did not deserve that certificate either “even though he had studied 20 years to get it” [128]. Social esteem should not be measured by the time that a person needed to complete his studies. He therefore proposed lightening the burden of the education and next brought forward concrete proposals for the organization of the teacher training at the university. For the prospective teacher he proposed a short training after the bachelors’ degree exam (a bachelorship of 2 or 3 years) with the time divided between lecturers in the discipline and those in pedagogy and didactics. Apart from that the teacher should also be allowed to continue his studies at the university after taking his qualification—an ideal that he shared with his colleague Minnaert [129]. Regarding the objectives of such an education Freudenthal wrote: Van de wiskundige opleiding eis ik: 1. dat zij de aanstaande leraar brengt tot dieper inzicht in de stof, die hij straks moetonderwijzen, 2. dat zij hem een idee geeft van de wijze, waarop de wiskunde in andere wetenschappenen in de techniek wordt toegepast, 3. dat zij hem in contact brengt met de verworvenheden der wiskunde, die van algemeneculturele betekenis zijn, 4. dat zij hem tot een enigszins zelfstandige wetenschappelijke activiteit kan leiden. (My requirements for the mathematical education are: 1. that it brings the prospective teacher deeper insight in the subject matter he will subsequently have to teach,

168


2. that it gives him an idea of the manner in which mathematics is applied in science and technology, 3. that it brings him into contact with those achievements of mathematics that have a general cultural meaning, 4. that it can guide him to a slightly independent scientific activity.) [130]

The principle that a pedagogical-didactical education was essential had by now been generally accepted. Yet Freudenthal did not argue in favour of an extended pedagogical-didactical education for the simple reason that academically trained didactics lecturers did not yet exist, and thus prudence was called for: Een jonge wetenschap is een onbeschermd jachtterrein, toegankelijk voor iedereen, serieuzen en fortuinzoekers, naieven en geschoolden, nuchter denkenden en fantasten […] Mijn wetenschappelijk hart zou bloeden, wanneer mijn studenten behoorlijke wiskunde zouden moeten ruilen voor het quasi-wetenschappelijke jargon van (op hun terrein mis schien verdienstelijke) pedagogen, die zich met een aantal mooi klinkende maar onbegrepen mathematische termen op dikdoenerij toeleggen—om van de volslagen charlatans maar geheel te zwijgen. (A new discipline is an unprotected hunting ground, accessible to everybody, the serious and the adventurers, the naïve and the trained, the sober-minded and the dreamers […] My scientific heart would bleed, if my students were obliged to trade in decent mathematics for the quasi-scientific slang of pedagogues (who may have earned credits in their field), who concentrate on bragging by using a number of euphonious but uncomprehended mathematical terms—let alone the complete charlatans.) [131]

Under the motto “the realism of tomorrow will be more progressive than that of today” he therefore proposed extending the pedagogical-didactical education as much as possible, but only to the extent that it did have the same level of the disciplinary education. De vakdidactiek moet zich opwerken tot akademisch peil, en voorzover zij het reeds bereikt heeft, moet zij het handhaven. De leraarsopleiding moet groeien, tegen de verdrukking in, niet op de weg van de minste weerstand. (The teaching methodology has to climb the ladder to the academic level, and should it already have reached it, then it has to maintain that level. The teacher training has to grow, in spite of opposition, not along the line of least resistance.) [132]

To his regret Freudenthal could not corroborate his opinion and proposals as stated above with more facts. But: Dat die er niet of nauwelijks zijn op het gebied van mijn lezing, is niet mijn schuld—als ik me dan toch op zulk een gebied begeef, moet ik de consequenties aanvaarden. De consequentie van een lezing is een discussie—op staande voet of met enig uitstel van executie. (That these are not or hardly to be found within the scope of my lecture is not my fault—if I nevertheless make my way to such a field, I will have to accept the consequences. The consequence of a lecture is a discussion—on the spot or with some stay of execution.) [133]

A few months before Freudenthal made the proposal for a condensed education in this lecture, he had explained it in an article in Universiteit en Hogeschool [134]. He reproached the universities for their overly-cautious wait and see attitude, in which laws and regulations took the responsibility the university should have. There was for example the threat that the university would lose the influence it had on the teacher training and that would be a fatal and dangerous development in a time, in which the shortage of teachers only increased.


169

In 1955 Freudenthal delivered a lecture with the title ‘Relations entre l’enseignement secondaire et l’enseignement universitaire en Hollande’ (‘Relations between secondary education and university education in the Netherlands’) at an international symposium in Genève. It was published a year later in L’Enseignement Mathématique ( Mathematics education) [135]. He used this opportunity to expose, one by one, all the things that were wrong in the Dutch secondary education: the backward curriculums (that fortunately were altered), the bad examination problems and the enormous gap between the caricature of mathematics that was presented to pupils in a secondary school and what students learned mathematics really was in their academic studies. Here too he mentioned the lack of a real teacher training and the difficult developments in establishing such an education. Nevertheless, Freudenthal managed to end in a positive tone (albeit with a slightly cynical undertone) about the Dutch education situation: Ce sont là nos problèmes en Hollande. Je ne dis pas que nous n’en ayons pas d’autres, que nous ne discutions pas la meilleure méthode d’enseigner les lieux géométriques et les proportions. Au contraire, de toutes ces questions nous faisons des études aussi profondes que nos collègues d’autres pays et que l’autruche le fait du sable où elle cache sa tête. L’opposition de ceux qui, malgré l’évolution de la structure sociale, veulent conserver un système d’enseignement suranné, est opiniâtre. Tout de même, ma confiance est grande. La réforme est en marche et elle viendra, comme tout chose dans l’enseignement, trente ans trop tard. (These are our problems in the Netherlands. I do not say that we do not have others, that we do not discuss the best method to teach geometrical loci or proportions. On the contrary, we study all these questions as deeply as our colleagues from other countries do and as the ostrich studies the sand in which it buries its head. The opposition of those that despite the evolution of the structure want to conserve an outdated educational system is unyielding. Nevertheless, I am very confident. The modernization is on its way and it will, like everything else in education, come thirty years too late.) [136]

The fact that practically no problems had been solved in didactics could also be used to pep up the image of the profession of teacher, as Freudenthal argued in his article ‘Traditie and opvoeding’ (‘Tradition and education’) in 1957. The belief that being a teacher required nothing else than rattling off traditional curriculums had harmed the reputation of the teacher. In onze maatschappij, waar elke nieuwe techniek met bekwame spoed in de praktijk wordt gebracht, is opvoeding en onderwijs een ouderwets beroep. Ik ben er van overtuigd dat kinderen dat aanvoelen en dat jonge mensen om deze redenen zich tot andere beroepen voelen aangetrokken. Willen we meer leraren en onderwijzers hebben—en de behoefte is er—dan is de weg, die we moeten inslaan: opvoeding en onderwijs te verheffen tot het peil van andere technieken in onze maatschappij. (In our society, in which every new technique is put into practice with due speed, educating and teaching are old-fashioned professions. I am convinced that children feel this and that young people are drawn to other professions for that reason. If we want more teachers and schoolmasters—and there is a need for them—we have to take this road: to raise education and teaching to the same level as other professions in our society.) [137]

In the end it would still take a couple of decades before Freudenthal would see the ideals of the teacher training as described here implemented—and what is even more important—an academic didactical discipline realized.

170


6.6 Freudenthal Internationally The lack of a serious teacher training was inherent in the more or less contemptuous attitude that the majority of the mathematics professors displayed with respect to teaching methodology and mathematics education and that certainly prevailed well into the 1950s. When a mathematician decided to engage in teaching methodology it was often described as ‘second-choice’ work. His colleagues considered Freudenthal’s increasing activities in the field of teaching methodology ‘a waste of that man’ [138]. In the course of the 1950s this view cautiously started to change. Recognition of the need for teaching methodology and taking it seriously as a discipline gradually gained ground, in an international context too. Since he had made his reputation as a mathematician (who also laboured for all pedagogic, didactic and educational matters in relation to the discipline), Freudenthal increasingly appeared in that role on the international stage. On 17 November 1953, for example, he received a letter from Koksma, chairman of the Wiskundig Genootschap, with an invitation to take a seat in the ‘Nederlandse Onderwijscommissie voor Wiskunde’ that was being established. This committee, established by the Wiskundig Genootschap, was the national subcommittee of the ICMI and had as its objective studying mathematics education in both a national and an international context. The committee’s assignment was: • in Nederland belangstelling voor de wiskunde te bevorderen onder jongeren, in het bijzonder leerlingen van het Algemeen Voortgezet en Voorbereidend Wetenschappelijk Onderwijs. Het gaat hierbij niet om de wiskunde als een specialisme, maar om de bete kenis die het beoefenen van de wiskunde heeft voor mens en maatschappij. • in het buitenland contacten te leggen en onderhouden met gelijksoortige organisaties die zich bezig houden met wiskundeonderwijs en die thans verenigd zijn in de International Commission on Mathematical Instruction (ICMI), die onder meer eens in de vier jaar een International Congress on Mathematical Education organiseert. • to promote interest for mathematics in the Netherlands among young people, in particular among pupils of the Algemeen Voortgezet en Voorbereidend Wetenschappelijk Onderwijs. The point is not mathematics as a discipline, but the sifnificance practicing mathematics has for man and society. • to make and maintain contact outside the Netherlands with similar organizations that engage in mathematics education and that form at the moment the International Commission on Mathematical Instruction (ICMI), which among other things organizes a four yearly International Congress on Mathematical Education.) [139]

In 1953 all countries involved in the ICMI received such a request to set up a national subcommittee for mathematics education [140]. E.W. Beth became chairman of the NOCW and with it the Dutch representative of this committee. Freudenthal along with Bunt, J.C.H. Gerretsen, D.N. van der Neut, Wansink and Van Dantzig were the other members of the committee. Less than a year later Beth resigned his membership and during the meeting of 25 November Freudenthal was appointed as the new chairman, a function he fulfilled for the next 20 years [141]. Referring to the difference of opinion they had on mathematics education he wrote in Schrijf dat op, Hans [142]:

6.6 Freudenthal Internationally

171

Het lijkt ironie dat ik […] bij de Internationale Commissie voor Wiskunde Onderwijs (ICMI) als Nederlandse vertegenwoordiger E.W. Beths opvolger werd. Ironie, maar ook niet veel meer dan dat. Ik was, ook later als president, een vreemde eend in die bijt—maar dat is alweer vooruitlopen op de geschiedenis. (It seems ironical that I […] became E.W. Beth’s successor in the Internationale Commissie voor Wiskunde Onderwijs (ICMI). Ironical, but not much more than that. I was, later too as president, the odd one out—but then again that is running ahead of history.) [143]

Even though it seems as if Freudenthal is proud of the fact that he as “the odd one out” would play such an important role in the committee, him being chosen as chairman was really not that exceptional. Koksma had intended to form a committee with members who were “didacticians by profession” and there were not many of those to choose from in the mathematical world [144]. Freudenthal, with his reputation as a great mathematician who in addition had shown exceptional interest in didactical affairs, was a logical choice. Freudenthal himself was not really convinced of the use of such a committee. He preferred to be engaged in what he called “the interior of the education”, the things he did in the Wiskunde Werkgroep: Wat doet een commissie voor het wiskunde-onderwijs? Zich met de buitenkant van het onderwijs bezighouden. En een werkgroep? Met de binnenkant. De buitenkant, dat zijn de soorten onderwijs met hun doelstellingen, programma’s, examens, bevoegdheden, alles waarvoor je de vereiste kennis met enquêtes vanuit en studies achter het bureau kunt verwerven. Heel nuttig soms, maar niet mijn cup of tea. (What is it that a committee does for the mathematics education? It is occupied with the exterior of the education. And a working group? With the interior. The exterior, that is the sort of education with its objectives, curriculums, exams, qualifications, everything for which you can gather the necessary information from inquiries and studies at your desk. Very useful, sometimes, but not my cup of tea.) [145]

Being the Dutch representative of the NOCW, Freudenthal was present at the symposium in Geneva in 1955. The lecture he gave there was one of the first times that he spoke about educational affairs outside of the Netherlands. A meeting of the Executive Committee of the CIEM was organized immediately after the symposium. At that meeting he scored what he called “his first success” when the plan of action for the preparation of the Education section of the next Internationaal Mathematisch Congres in Edinburgh in 1958 was decided on [146]. Freudenthal proposed putting a comparative study into the methods for initial geometry education on the agenda as one of the three subjects to be dealt with this. This was agreed. It was of course no coincidence that Freudenthal brought in exactly this subject which for some years had already received much attention at the meetings of the Wiskunde Werkgroep. The other subjects were ‘the mathematics education for pupils up to and including the age of 15’ and ‘the scientific base of the mathematics education in secondary education’, characteristic examples of what Freudenthal called ‘the exterior of education’. In this way Freudenthal managed to get one subject from the ‘interior of education’ onto the agenda after all. ‘Initiation into geometry’ was also the subject of the lecture he delivered on behalf of the ICMI at the Tata Institute of Fundamental Research in Bombay during the ‘Mathematical Education in South Asia Conference’ in February 1956 [147].

172


This conference, jointly organized by UNESCO, among others, was focussed on discussing the problems of the mathematics education in South Asia. In a trip report in De Groene Amsterdammer Freudenthal concluded that these were enormous problems. The many organizational problems in those countries were exactly the reason that there was not much interest in the pedagogical-didactical side of the education. In that field the conference did not offer much that was useful according to Freudenthal. After 2 weeks in Asia he went home with een flauw besef van de geweldige problemen, waarmee Aziatische landen worstelen. Of ik van mijn kant hun iets heb kunnen geven, moet ik betwijfelen, en wel niet uit overdreven bescheidenheid. Twee weken Azië is te weinig. (a faint idea of the enormous problems Asian countries were struggling with. Whether I could contribute to solving them, I have to doubt, and this not because I am excessively modest. Two weeks in Asia is too short.) [148]

The only ray of hope during his trip was a visit to one of the few progressive schools in the region. Dit was de enige plaats waar ik echt kon leven in het gevoel, dat de diepere problemen van onderwijs en opvoeding overal ter wereld dezelfde zijn. Hier werd niet strijdbaar geworsteld met een verouderd systeem, maar hier was er de bewogen rust van de zoekende, die weet dat hij op de goede weg is. (This was the only place where I felt that the more serious problems of education and teaching are the same everywhere in the world. It was not a militant struggle with an outdated system, but the eventful rest of someone searching who knows that he is on the right track.) [149]

The work for the NOCW and afterwards for ICMI would occupy more and more of Freudenthal’s time at the end of the 1950s. That this work was not always closest to his heart was evident. Nevertheless he seized this opportunity with both hands and on the principle that ‘cooperating is better than doing nothing’ he managed to establish a reputation as a mathematics educator, in an international field as well.

References 1. Jansen, Vess, ‘Hans Freudenthal, tussen de bergen en de zee’, Ballonnen en brood. Utrechtse portretten 101–107, q.v. 106. 2. The following studies were used to derive data for this paragraph: Van Berkel, Dijksterhuis; Freudenthal, Schrijf dat op, Hans; De Moor, Van vormleer naar realistische meetkunde; Wansink, Didactische oriëntatie voor wiskundeleraren. Delen I, II, III and the volumes Euclides 26–32 (1950–1957). 3. Hans Freudenthal, ‘En nu… de wiskunde’, De Groene Amsterdammer (2 december 1950); RANH, Papers of Hans Freudenthal, inv.nr. 1439. Freudenthal probably meant the logicaldeductive structure of Euclidean geometry when he spoke of “disciplinary measures”. 4. Wansink, ‘De wiskunde werkgroep van de W.V.O.’, Euclides 28 (1953) 197–205, q.v. 198. 5. Hans Freudenthal, ‘Onderwijsvernieuwers bijeen’, De Groene Amsterdammer (24 November 1951); RANH, Hans Freudenthal Papers, inv.nr. 1453. 6. For a description of the realization of this curriculum see paragraph 6.3 7. Correspondence between the editors of Euclides and Freudenthal, June 1956; RANH, Hans Freudenthal Papers inv. nr. 29. 8. Correspondence between the Nederlandse Onderwijzers Vereniging (NOV) and Freudenthal; RANH, Hans Freudenthal Papers, inv.nr. 56. According to Freudenthal the NOV would be in opposition to the teachers' associations.

References

173

9. Mrs. Ehrenfest to Freudenthal, 3 March 1964; RANH, Hans Freudenthal Papers, inv.nr. 26. 10. T. Ehrenfest-Afanassjewa and H. Freudenthal, Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? (Purmerend 1951) 6. 11. Ibid., 12. 12. Ibid., 16. 13. Ibid., 24. 14. Ibid., 25. 15. See the discussion of the lecture ‘Opvoeding tot denken’ in paragraph 5.1.3. 16. Ph.A. Kohnstamm, ‘De formele logica en het kinderlijke denken’, published in 1934 as ‘Mededeling 2 van het Nutsseminarium’, in Paedagogische Studiën; Kohnstamm, Keur uit het didactisch werk 67–92; Ph. A. Kohnstamm, ‘Over de “vormende waarde” der leervakken’, published in 1929–1930 as ‘Mededeling 16 van het Nutsseminarium’, in Paedagogische Studiën; Kohnstamm, Keur uit het didactisch werk 1–31. For that matter, Kohnstamm was convinced that practicing mathematics and classical languages was pre-eminently suited to learning to think logically. 17. Ehrenfest and Freudenthal, Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? 26. 18. P.M. van Hiele, ‘Freudenthal en de didaktiek der wiskunde’, Euclides 51 (1975) 8–10. 19. Ehrenfest and Freudenthal, Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? 24. 20. The parallels between Freudenthal's and Kohnstamm’s ideas clearly emerge in Kohnstamm’s article written on the occasion of a lecture in 1938; ‘De verhouding der anorganische natuurwetenschappen onderling en tot de wiskunde uit didaktisch oogpunt’, in: Kohnstamm, Keur uit het didactisch werk, q.v. 376–390. 21. H. Streefkerk, ‘Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen?’, Euclides 27 (1951) 57–59, q.v. 59. 22. Streefkerk to Freudenthal 27 April 1951; RANH, Hans Freudenthal Papers, inv.nr. 73. 23. Correspondence between Streefkerk and Freudenthal, 1948–1951. In these articles, entitled ‘De dwarskijker’, Freudenthal briefly treated an exam problem and the kinds of mistakes that pupils made in solving it; RANH, Hans Freudenthal Papers, inv.nr. 73; Hans Freudenthal, ‘De dwarskijker I’, Euclides 26 (1951) 245–251; Hans Freudenthal, ‘De dwarskijker II’, Euclides 28 (1952) 58–62. 24. Wansink to Freudenthal, 29 April 1951; RANH, Hans Freudenthal Papers, inv.nr. 82. 25. Hans Freudenthal, ‘Erziehung des Denkvermögens’, Archimedes 6 (1954) 87–89, q.v. 88; RANH, Hans Freudenthal Papers, inv.nr. 787. 26. Ibid. 27. See also: Freudenthal, ‘Rekendidaktiek’ 101–102; RANH, Hans Freudenthal Papers, inv.nr. 465. 28. Hans Freudenthal, ‘Opvoeding van het denkvermogen? Answer to Mrs.. Ehrenfest’, in: Ehrenfest and Freudenthal, Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? 12–18, q.v. 15–16. 29. Hans Freudenthal, ‘Mevrouw, ze groeien ook zo hard!’, De Groene Amsterdammer (4 Aug 1951); RANH, Hans Freudenthal Papers, inv.nr. 1447. 30. Hans Freudenthal, ‘Weet u wat een escalator is?’, De Groene Amsterdammer (25 August 1951); RANH, Hans Freudenthal Papers, inv.nr. 1448. 31. Freudenthal to R.A.J. Bosschart, 25 August 1955; RANH, Hans Freudenthal Papers, inv.nr. 12. 32. See also for example: Hans Freudenthal, ‘Traditie en opvoeding’, Rekenschap 3 (1957) 95– 103, q.v. 101. 33. E.J. Dijksterhuis, ‘Opmerkingen over het onderwijs in mechanica als onderdeel der physica’, Euclides 7 (1930/31) 76–94, q.v. 94. 34. See for the discussion on the subject mechanics in the period between the two World Wars paragraph 2.2.3. 35. Hans Freudenthal, ‘Het mechanica-vraagstuk’, in: Nieuwe wegen bij het onderwijs in de wiskunde en de natuurwetenschappen (Purmerend 1952) 22–43, q.v. 22. The quote is taken

174


from the epic ‘Divine Comedy’ by Dante Alighieri, 1300. The “dark wood” in the work is a symbol for a troubled phase of life. 36. Ibid., 22. 37. Ibid., 23. 38. Ibid., 26. 39. Ibid., 26. 40. Ibid., 27. 41. Ibid., 27. 42. Ibid., 31. 43. Ibid., 33. 44. Ibid., 34. 45. Ibid., 34–35. 46. Ibid., 39. 47. Kohnstamm too mentioned in his article in 1938 the example of the dynamics of cycling as counterpart of the ideal of a logically-deductively structured mechanics education as supported by the mathematicians; Ph. A. Kohnstamm, ‘De verhouding der anorganische natuurwetenschappen onderling en tot de wiskunde uit didaktisch oogpunt’, in: Kohnstamm, Keur uit het didactisch werk 376–390, q.v. 378. 48. Freudenthal, ‘Het mechanica-vraagstuk’, Nieuwe wegen 42. 49. Ibid., 42. 50. Van Berkel, Dijksterhuis 588. 51. This chairman of WIMECOS should not be confused with J.R. Janssen, the secretary of the Centrale Werkgroep of the WVO and chairman of the WW in the 1940s. 52. G.A. Janssen, ‘Over onze tijd met een heenwijzing naar onze taak’, Euclides 28 (1953) 116– 122, q.v. 121. 53. Ibid., 119. 54. Freudenthal to E.W. Beth, 4 Aug 1953; RANH, Hans Freudenthal Papers, inv.nr. 8. For that matter, the lecture Freudenthal sent to Janssen was accompanied by an extremely decent and friendly note; Freudenthal to G.A. Janssen, August 1953; RANH, Hans Freudenthal Papers, inv.nr.42. 55. Freudenthal to E.W. Beth, 4 Aug 1953; RANH, Hans Freudenthal Papers, inv.nr. 8. 56. Ibid. 57. RANH, Hans Freudenthal Papers, inv.nr. 508. 58. Janssen to Freudenthal, 4 July 1954; RANH, Hans Freudenthal Papers, inv.nr. 42. 59. See also: ‘Adres inzake het mechanica-onderwijs op de H.B.S.-b’, Euclides 28 (1952) 123–128; ‘Adres van WIMECOS aan de minister van onderwijs, kunsten en wetenschappen inzake het mechanica onderwijs’, Euclides 29 (1953) 213–217. 60. Janssen to Freudenthal, 4 July 1954; RANH, Hans Freudenthal Papers, inv.nr. 42. 61. A.G.M. van Melsen, ‘Herdenking van Evert Willem Beth (7 juli 1908–12 april 1964)’, Jaarboek der Koninklijke Nederlandse Akademie van Wetenschappen 1964–1965 (Amsterdam 1965). 62. Freudenthal to E.W. Beth, 4 Aug 1953; RANH, Hans Freudenthal Papers, inv.nr. 8. 63. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel I 99–100. See also paragraph 2.2.3. 64. See Van Hiele's remark on the way of working in the algebra committee as described in paragraph 5.3.1. 65. Correspondence between Freudenthal and the WW, undated, probably written in the spring of 1949; RANH, Hans Freudenthal Papers, inv.nr. 84. 66. Ibid. 67. Jacobs to Freudenthal, 3 April 1949; RANH, Hans Freudenthal Papers, inv.nr. 42. 68. Jacobs to Freudenthal, 22 January 1950; RANH, Hans Freudenthal Papers, inv.nr. 42. See also the drafts of the introductions in Freudenthal's notebooks; RANH, Hans Freudenthal Papers, inv.nr. 755 and 499. 69. Hans Freudenthal, ‘En nu… de wiskunde’, De Groene Amsterdammer (2 december 1950); RANH, Papers of Hans Freudenthal, inv.nr. 1439. 70. Ibid.

References

175

71. ‘De programma’s wiskunde voor Gymnasium-β en HBS-b’, Vernieuwing van Opvoeding en Onderwijs 11 (1953) 216–218. See also the drafts of the introductions in Freudenthal’s notebooks; RANH, Hans Freudenthal Papers, inv.nr. 755 and 499. 72. Wansink, ‘De Wiskunde-Werkgroep van de W.V.O.’, Euclides 28 (1953) 197–205. 73. Here Bunt's study among mathematics teachers of the VHMO, which concluded that the curriculum was seriously overloaded, was meant; L.N.H. Bunt, De leerstof van ons wiskundeonderwijs. Een onderzoek naar opvattingen en gebruiken dienaangaande (Groningen 1949). 74. ‘Het wiskunde-programma voor het V.H.M.O. Een ontwerp van de Wiskunde-Werkgroep van de W.V.O.’, Euclides 28 (1953) 206–226, q.v. 208. This phrasing is therefore almost literally equal to the draft of this introduction in one of Freudenthal's notebooks; RANH, Hans Freudenthal Papers, inv.nr. 755. 75. ‘Het wiskunde-programma voor het V.H.M.O.’, Euclides 28 (1953) 208. 76. Ibid., 209. 77. Freudenthal, ‘Het mechanica-vraagstuk’, Nieuwe wegen 24. 78. Hans Freudenthal, ‘Examens en leerstof’, Vernieuwing van Opvoeding en Onderwijs 10 (1952) 193–198; RANH, Hans Freudenthal Papers, inv.nr. 765. 79. Hans Freudenthal, ‘Zwaardere eindexamens?’, De Groene Amsterdammer (23 September 1950); Hans Freudenthal, ‘Examens en leerstof’, Vernieuwing van Opvoeding en Onderwijs 10 (1952) 193–198; Hans Freudenthal, ‘De kinderexamens’, De Groene Amsterdammer (16 August 1952); Hans Freudenthal, ‘Hoe oud is Maria?’, De Groene Amsterdammer (23 May 1953); RANH, Hans Freudenthal Papers, inv.nrs. 1435, 765, 1464, 1478. 80. ‘Het wiskunde-programma voor het V.H.M.O.’, Euclides 28 (1953) 206–226, q.v. 210. 81. ‘Rapport van de leerplan-commissie-1954 van WIMECOS inzake het opstellen van een ontwerp-leerplan en een ontwerp-eindexamenprogramma voor wiskunde voor de H.B.S.-b’, Euclides 30 (1955) 149–176, q.v. 150. 82. Ibid., 151. 83. Ibid., 157. 84. Interview with Wansink in: Goffree, Ik was wiskundeleraar 93. 85. Interview with P.M. van Hiele in: ibid., 123. 86. ‘Adres van de Wiskunde-werkgroep van de W.V.O. inzake het ontwerp-leerplan van WIMECOS’, Euclides 30 (1955) 202–203, q.v. 202–203. 87. In the end only one third of the usual number of participants will therefore show up at the weekend conference of the working group; Hans Freudenthal, ‘Ons weekend’, Mededelingenblad van de Wiskunde Werkgroep der W.V.O. (1955); Alberts e.a., red., Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen 119. 88. Hans Freudenthal, ‘Naar een nieuw wiskunde-onderwijs’, De Groene Amsterdammer (2 april 1955); RANH, Hans Freudenthal Papers, inv.nr. 1513. 89. Ibid. 90. Ibid. 91. Ibid. 92. Something of the subject statistics remained: ‘principles of statistics’ became an optional subject for Gymnasium-α, completed with an oral examination; Wansink, Didactische oriëntatie voor wiskundeleraren. Deel III 270. 93. L.N.H. Bunt, Statistiek als onderwerp voor het gymnasium α. An investigation into the possibilities of teaching descriptive and elementary mathematical statistics in secondary schools (Groningen 1956); L.N.H. Bunt, Statistiek als onderwerp voor het gymnasium α. Verslag van een proefneming. A Inrichting en resultaten van het onderzoek (Groningen 1957). 94. Ibid. 95. Correspondence between Erven F. Bohn and Freudenthal, 1955; RANH, Hans Freudenthal Papers, inv.nr. 29. 96. Correspondence between Freudenthal and J. Sittig of the Adviesbureau voor Toegepaste Statistiek, 1948; RANH, Hans Freudenthal Papers, inv.nr. 2; J. Sittig and H. Freudenthal, De juiste maat. Lichaamsafmetingen van Nederlandse vrouwen als basis van een nieuw maatsysteem voor damesconfectie (Leiden 1951). 97. Correspondence between Freudenthal and the Chem. Dept. of Dobbelman N.V., 1952–1954; RANH, Hans Freudenthal Papers, inv.nr. 24.

176


98. Correspondence between Freudenthal and the Rechter-Commissaris (examining judge) responsible for the criminal cases in the district Utrecht, 1952; RANH, Hans Freudenthal Papers, inv.nr. 64. 99. Before he lectured on (the history of) statistics (such as the more popular ‘De opkomst der statistische denkwijze in de moderne wetenschap’ for Maatschappij Diligentia in 1951) and he wrote on statistics and probability theory in De Groene Amsterdammer; Hans Freudenthal, ‘De opkomst der statistische denkwijze in de moderne wetenschap’, Natuurkundige Voordrachten 29 (1951) 57–65; Hans Freudenthal, ‘Op het telraam van het toeval’, De Groene Amsterdammer (11 juli 1953); Hans Freudenthal, ‘Jagerslatijn’, De Groene Amsterdammer (17 July 1954); RANH, Hans Freudenthal Papers, inv.nrs. 757, 1481, 1502. 100. Freudenthal, ‘Jagerslatijn’, De Groene Amsterdammer (17 July 1954); RANH, Hans Freudenthal Papers, inv.nr. 1502. 101. Hans Freudenthal, Waarschijnlijkheidsrekening en statistiek (Haarlem 1957) v. 102. Ibid., v. 103. Ibid., 5. 104. Ibid., 7, 7, 13. 105. Ibid., 21. 106. Ibid., 86–87. In composing his problems Freudenthal regularly used numerical data from the study De juiste maat. 107. H.W. Lenstra, ‘Boekbespreking. Prof. Dr. Hans Freudenthal Waarschijnlijkheidsrekening en statistiek’, Euclides 32 (1957) 248–249, q.v. 249. 108. Ibid. 109. See paragraphs 2.2.2 and 2.2.4. 110. See also paragraph 5.2.2. 111. Freudenthal, Schrijf dat op, Hans 361–362. 112. Hans Freudenthal, ‘Vakgeleerde of paedagoog?’, Christelijk Schoolblad. Onze vacatures (7 May 1953, 15 May 1953, 21 May 1953); see also: RANH, Hans Freudenthal Papers, inv. nrs. 779, 780. 113. E.J. Dijksterhuis, ‘Beschouwingen over de universitaire opleiding tot leeraar in wis-en natuurkunde’, Bijvoegsel 2 (1925–1926) 81–95. 114. Freudenthal, ‘Vakgeleerde of paedagoog?’; RANH, Hans Freudenthal Papers, inv.nrs. 779, 780. 115. Ibid. 116. Ibid. 117. As a result of the fact that in 1949 and 1950 criticism from Freudenthal and his wife of the English textbook their eldest son had to use, the book was replaced by another; correspondence between the Freudenthals and the headmaster of the Nieuwe Lyceum in Bilthoven, 1949–1950; RANH, Hans Freudenthal Papers, inv.nr. 175. 118. Freudenthal, ‘Vakgeleerde of paedagoog?’; RANH, Hans Freudenthal Papers, inv.nrs. 779, 780. 119. Ibid. In Weeding and sowing Freudenthal further explained his line of reasoning on the development of a discipline of mathematics education. The work of the IOWO in the 1970s was more or less a concrete interpretation of it. 120. Ibid. 121. Ibid. He considered this at the end of the 1940s as a somewhat acceptable alternative for the existing situation: see for example: Hans Freudenthal, ‘Voorstellen tot reorganisatie van het Hoger Onderwijs in de Wiskunde’, 4 December 1947, 4 pages; RANH, Hans Freudenthal Papers, inv.nr. 475. Hans Freudenthal, ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides 24 (1948) 106–121, q.v. 120–121. 122. Ibid. 123. Hans Freudenthal, ‘De leraarsopleiding’, De Groene Amsterdammer (25 October 1952); RANH, Hans Freudenthal Papers, inv.nr. 1468. 124. Ibid.

References

177

125. Freudenthal, ‘De leraarsopleiding’, Vernieuwing van Opvoeding en Onderwijs 109 (1956) 173–180, q.v. 173. 126. Ibid., 175. 127. Ibid., 176. 128. Ibid., 176. 129. Molenaar, De rok van het universum 414. 130. Freudenthal, ‘De leraarsopleiding’, Vernieuwing van Opvoeding en Onderwijs 109 (1956) 173–180, q.v. 178. 131. Ibid., 179. 132. Ibid., 179. 133. Ibid., 180. 134. Hans Freudenthal, ‘Vrijheid en verantwoordelijkheid’, Universiteit en Hogeschool 2 (1956) 295–301. 135. Hans Freudenthal, ‘Relations entre l’enseignement secondaire et l’enseignement universitaire en Hollande’, L’Enseignement Mathématique 11 (1956) 238–249. This was a symposium of the ICMI, more information can be found in paragraph 6.6. Freudenthal delivered his lecture on 2 July 1955. 136. Ibid., 249. 137. Hans Freudenthal, ‘Traditie en opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 98–99: This article, entitled ‘Tradition and education’, was first published in English in: The New Era in Home and School 37 (1956) 127–132. 138. “Didactiek was tweede garnituur onder hoogleraren”, according to Prof.dr. A.W. Grootendorst; conversation with Grootendorst, 23 April 1999. 139. Taken from article 1 of the regulations of the NOCW dd 25 November 1954, quoted in: W. Kleijne, ‘NOCW vijftig jaar’, Nieuw Archief voor Wiskunde 5 (2004) 308–313, q.v. 309. 140. Other names for the ICMI are: Internationale Mathematik-Unterrichts Kommission (IMUK) or: Commission Internationale pour l’Enseignement Mathématique (CIEM); RANH, Wiskundig Genootschap, inv.nr. 240; RANH, Hans Freudenthal Papers, inv.nr. 1747. 141. Freudenthal’s successor in 1975 was H.J.A. Duparc; Kleijne, ‘NOCW vijftig jaar’, Nieuw Archief voor Wiskunde 5 (2004) 308–313, q.v 313; NOCW-correspondence; RANH, Hans Freudenthal Papers, inv.nr. 1770. 142. Freudenthal specifically meant Beth jr’s remark in Piaget’s L’Enseignement Mathématique (1955): “Le rôle de la formation mathématique dans l’enseignement secondaire consiste presque exclusivement, me paraît-il, Ã familiariser les élèves avec la méthode déductive” quoted in: Freudenthal, Schrijf dat op, Hans 341. 143. Freudenthal, Schrijf dat op, Hans 342. 144. W. Kleijne, ‘NOCW vijftig jaar’, Nieuw Archief voor Wiskunde 5 (2004) 308–313, q.v. 309. 145. Freudenthal, Schrijf dat op, Hans 346. 146. Ibid., 347. 147. Correspondence between the New Education Fellowship and Freudenthal, 1955–1956; RANH, Hans Freudenthal Papers, inv.nrs. 57, 1770. 148. Hans Freudenthal, ‘Opvoeding tot hele mensen’, De Groene Amsterdammer (21 april 1956); RANH, Hans Freudenthal Papers, inv.nr. 1525. 149. Ibid.

Chapter 7

Freudenthal and the Van Hieles’ Level Theory A Learning Process

Ik heb geboft. Alwaar ik tegen een muur aankeek ging ten langen leste een deur open. Een van eigen inzicht, maar meestal andermans (zoals Van Hieles niveaus). (I have been lucky. Everywhere where I ended up looking at a wall a door opened at long last. A door of my own insight, but most of the time one of another person’s insight (such as Van Hiele’s levels).) Hans Freudenthal in Schrijf dat op, Hans, 1987 [1]

7.1 Introduction: A Special PhD Project Freudenthal supervised dozens of PhD students. Most of them wrote their theses on subjects stemming from pure mathematics. On 4 July 1957 Pierre van Hiele (1909– 2010) was the first student to obtain his doctorate on a subject from the didactics of mathematics [2]. His wife Dina (Dieke) van Hiele-Geldof (1911–1958) got her doctorate on the same day, on a subject from the didactics of mathematics as well; her project was done under the supervision of Professor of Pedagogy Langeveld (Fig. 7.1). Freudenthal wrote about these events at the time: Promoties op didactische onderwerpen zijn nog steeds uiterst zeldzaam. Wetenschappelijk onderzoek op mathematisch-didactisch terrein van een formaat, zoals thans door de Van Hiele’s verricht, mag zelfs in het internationale vlak uniek heten. In hun toespraken gaven de promotoren als hun mening te kennen, dat hoe men ook over twintig of dertig jaar over de resultaten van de Van Hiele’s zal oordelen, het vaststaat dat zij door hun aanpak baanbrekend werk hebben verricht; elk toekomstig mathematisch-didactisch onderzoek zal ermee rekening moeten houden als uitgangspunt en voorbeeld. (Obtaining doctorates on didactical subjects is still extremely rare. Scientific research in a mathematical-didactical field of a stature as now shown by the Van Hieles can, even in an international context, be called unique. In their speeches the supervisors put forward the opinion that, whatever the judgement on the results of the Van Hieles might be in twenty to thirty years, one thing is certain they have broken new ground; every future mathematical-didactical research project will have to take it into account as a starting point and an example.) [3]

This fact on its own makes it worthwhile devoting our attention to the work of the Van Hieles in this study. A further reason is that the results of both PhD projects not only had a strong influence on (research in the field of) didactics of mathematics in © Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_7

179

180

7 Freudenthal and the Van Hieles’ Level Theory

Fig. 7.1 A newspaper cutting from the Nieuw Utrechts Nieuwsblad about the double doctorate of the Van Hieles that Freudenthal had kept (1957)

7.2 Freudenthal as Supervisor

181

general, but certainly also on Freudenthal’s work in particular. Freudenthal himself often expressed his great admiration for the work of the Van Hieles: I should like to draw the reader’s special attention to the work of Mr. and Mrs. Van Hiele, not only because it is the most revolutionary attempt, but also because of the underlying pedagogical and didactical theory. In my opinion the Van Hieles’ theory of thinking levels as well as their experimental work are achievements of educational research which should be recommended to all people interested in mathematical education for closer inspection. [4]

Wherever he saw his chance he pushed the theory and very soon integrated it in his own ideas. In several places he described the work of the Van Hieles as crucial to the development of his own mathematical-didactical insights. The quote opening this chapter is one example. Here is another. Het mathematiseringsproces waar de Van Hieles zich vooral mee bezighielden, was dat van de meetkunde, preciezer gezegd: ze waren de eersten die het meetkundig leerproces als proces van mathematiseren interpreteerden (al gebruikten ze de term niet, evenmin als de term heruitvinding). Zodoende ontdekte Pierre in het onderwijs, zoals Dieke het beschreef, de niveaus waarvan ik eerder sprak. Ik pakte die ontdekking op—wellicht het belangrijkste element in mijn eigen wiskunde-onderwijskundig leerproces. (The mathematizing process the Van Hieles were engaged in was that of the geometry, more precisely: they were the first who interpreted the geometrical learning process as a process of mathematization (even though they neither used that term, nor the term re-invention). In doing so Pierre, as Dieke described it, discovered in the education the levels I spoke about earlier. I incorporated that discovery in my work—perhaps the most important element in my own mathematical-educational learning process.) [5]

The PhD research of the Van Hieles was indirectly a concrete result of the ideal supported by Freudenthal that mathematics teachers should continue to do academic research after their graduation, as he phrased it in 1953 in his article ‘Vakgeleerde of paedagoog?’ (‘Specialist or pedagogue’) [6]. Consequently it seems very worthwhile to find answers to the following questions: who were the Van Hieles? What was their theory and why was it so special? Why was their work so important to Freudenthal and what influence did it have on Freudenthal’s mental legacy? [7].

7.2 Freudenthal as Supervisor Didactics of mathematics was the leitmotiv in the lives of Mrs. and Mr. van Hiele. They knew Freudenthal from when they studied mathematics in Amsterdam. Van Hiele said about this: Tijdens mijn studie moesten we af en toe voordrachten houden. Daarin probeerde ik soms al wat didactiek naar voren te brengen, door leerlingen en hun problemen met de leerstof in de beschouwingen op te nemen. Vooral Freudenthal placht dit aan te moedigen. (During my studies we had to give lectures from time to time. I sometimes tried to bring up some didactics by including pupils and their problems in my observations. Freudenthal. in particular used to encourage this.) [8]

After their studies the Van Hieles started working as mathematics teachers. They got married in 1940. After the war they co-wrote several exercise books for the geometry

182


education which were well received particularly in those educational circles geared to modernization [9]. In their work they strongly emphasized self-motivation and letting pupils discover geometry for themselves using material from everyday life. Freudenthal recognized them as pioneers in the field of intuitive initial geometry education (besides Van Albada who worked on the same basis) [10]. From the beginning the Van Hieles were also involved in the work of the Wiskunde Werkgroep, where the geometry education was regularly discussed [11]. On the whole Freudenthal and the Van Hieles could be counted as belonging to the same group of education reformers. Freudenthal wrote in 1957: De wiskunde-boekjes, de publicaties en de dissertaties van de Van Hiele’s dragen dan ook duidelijk de stempel van wat men vernieuwing van het onderwijs pleegt te noemen: critisch te staan tegenover al hetgeen in het onderwijs door de traditie is gegeven, en zich telkens opnieuw te bezinnen niet alleen op de didactische methoden, maar evenzeer op het doel van het onderwijs als zodanig. (The mathematics books, the publications and the theses of the Van Hieles clearly bear the stamp of what is usually called modernization of education: being critical of all the tradition in the education, and time and again reflecting not only on didactical methods but also on the objective of education as such.) [12]

Writing a thesis in mathematics on a didactical subject was far from being common in the 1940s. The thesis of mathematics teacher Hendrik Mooij from 1948 was the only one in this field for a long time. And this thesis was not even completely didactical because it had a purely mathematical appendix [13]. It was Van Hiele’s initiative to write a thesis on a subject from the didactics of mathematics. Freudenthal first suggested—as Van Hiele recalled– a mathematical subject for his thesis, whereupon Van Hiele asked whether it would be possible to acquire a doctorate in didactics [14]. Freudenthal agreed and next came up with a substantive idea for the scheme of such a study. In 1944 in ‘Rekendidaktiek’ he had expressed the wish to be able to observe children who were learning “collectively” at some time. Now he considered doing a very detailed study of one pupil, from the moment this pupil arrived at the school up to and including the final exam, to be more useful. With this Freudenthal appeared to be anticipating the role he would start to attach to ‘observing the learning process’ within the scope of didactical research at the end of the 1960s. Years later Van Hiele said about this: Van die leerling zou dan alles genoteerd moeten worden, wat hij deed, wat hij zei, hoe hij zijn proefwerken deed enz. enz. Dat kan natuurlijk nooit, dacht ik. Als je één leerling neemt is de kans groot dat je die onderweg kwijtraakt. Dus dan moet je er een stuk of zes nemen. (Everything related to this pupil would have to be noted down, what he did, what he said, how he did his tests etc. etc. That can never be done, I thought. If you take one pupil, chances are that you will lose him along the way. So you should take six or so.) [15]

However this idea was simply not feasible. So Van Hiele first detached himself from Freudenthal as his supervisor and started to cooperate with pedagogue Langeveld. After leaving the HBS Martinus Jan Langeveld (1905–1989) went to the Gemeentelijke Universiteit van Amsterdam (University of Amsterdam) and studied Dutch and history with philosophy and pedagogy as subsidiary subjects. In 1934 he defended his thesis in linguistics, Taal en denken: een theoretiese en didaktiese bijdrage tot het voortgezet onderwijs in de moedertaal, inzonderheid tot dat der

7.2 Freudenthal as Supervisor

183

grammatika (Language and thinking: a theoretical and didactical contribution to secondary education in the native language, especially to that in grammar), with Kohnstamm [16]. Through the intercession of Kohnstamm he was appointed professor in pedagogy, didactics and development psychology in Utrecht 5 years later. Langeveld was the founding father of what would become known as the Utrecht direction in pedagogy and was, by his own account, responsible for the introduction of phenomenology of the humanities in Utrecht. In his personalistic pedagogy he took the anthropology of the child and the (pedagogical) situation of the child as his starting point. In his pedagogy he drew attention to the strictly personal, unique nature of every child with his or her perception of the environment [17]. Maturity, autonomy, forming of conscience and responsibility were to him central issues and with this he joined the personalistic pedagogy of his tutor Kohnstamm [18]. After the war Langeveld was for many years the dominant person in pedagogy in the Netherlands. He played a leading role in conquering a full position for pedagogy as an independent discipline (in particular at the state universities). He was responsible for the fact that pedagogy was recognized as an independent academic discipline alongside psychology in the Netherlands [19]. In his farewell speech in 1972 he said: Samenwerking met de meest diversen en zeker met psychologen is geen probleem, waar wij onze opvoedingswetenschappelijke identiteit niet hoeven te vervagen of te verliezen. De daartoe noodzakelijke wetenschappelijke bewustwording, de onontkoombaar nodige vorming van die identiteit van de pedagoog en andragoog moeten thans volop hun kans krijgen, wil uit samenwerking niet een identiteitsverlies resulteren. Een verlies, dat noch wetenschappelijk vruchtbaar zou zijn noch voor de betrokken kinderen, volwassenen, opvoeders, enzovoort een eerlijke dienst mogelijk zou maken. (Cooperation with the most diverse people and certainly with psychologists is not a problem in which we need to diminish or lose our educational academic identity. The academic awakening and the forming of the identity of the pedagogue and the andragogue that is inevitably needed today should be fostered and strengthened so as to prevent an identity loss as a result of that cooperation. A loss that would be neither academically productive nor make an honest service possible for the children, adults, and the educators etc. involved.) [20]

Langeveld was part of the ‘Utrechtse School’, a group of Christian and humanistoriented scholars who advocated a phenomenon-oriented psychology in the years between 1945 and 1960 [21]. This psychology was chiefly characterized by an aversion (in any individual present to a greater or lesser extent) to the psychology modelled on the methods of the natural sciences, and by a great emphasis on personal responsibility and personal development. In this psychology the situation to be investigated was considered without a preconceived theoretical scheme of thought. For that matter, Langeveld applied the phenomenological psychology mainly in the preparatory phase of his research and finally followed it up with empirical research. Freudenthal was not involved in the Utrechtse School as such. His contacts with Langeveld were primarily those between colleagues. After Van Hiele had approached Langeveld, in December 1948 he also strengthened the ties with Freudenthal by way of a letter. Van Hiele wrote: Ik heb de mogelijkheid om te gaan werken voor een promotie in de didactiek der wiskunde nader overwogen en ik zou het zeer op prijs stellen, als ik daarbij op Uw steun mocht reke nen. Het schijnt mij nl., dat de problemen, die mij het meest liggen meer in Uw lijn liggen

184


dan van die van Prof. Langeveld. (I have reconsidered the possibility of starting the work for a doctorate in the didactics of mathematics and I would appreciate it greatly if I could count on your support in this matter. It seems to me that the problems which mostly appeal to me are more in your line than in that of Professor Langeveld.) [22]

According to Van Hiele it was Freudenthal who stimulated Mrs. van Hiele to obtain her doctorate as well [23]. The final subject of the research project, “the issue of insight studied in the didactical context of geometry education”, was evidently inspired by the daily educational practice of the Van Hieles. In the end Van Hiele would pursue his doctorate with Freudenthal and Mrs. van Hiele with Langeveld. Langeveld, however, kept playing a part in supervising Van Hiele’s thesis. His influence on the end result was substantial. For example, he proposed the title among other things. He also taught Van Hiele a lot about psychology and pedagogy, lent him the necessary books and from the beginning had the whole framework of the thesis in his mind [24]. The cooperation between Freudenthal and Van Hiele was not always the easiest as becomes clear from Van Hiele’s recollections of that period: Nu moet je niet denken dat Freudenthal en ik het altijd eens waren. Soms stonden we met woedende koppen tegenover elkaar. Op een keer is het gebeurd dat hij Langeveld opbelde. Die Van Hiele is zo eigenwijs, had hij toen gezegd. Later hoorde ik dat van Langeveld, die hem had geantwoord: dat kan wel waar zijn, Freudenthal, maar wat zeg je van jezelf? En van mij? (You must not think that Freudenthal and I always agreed with each other. We sometimes confronted each other furiously. Once he happened to call Langeveld on the telephone saying that Van Hiele is such a know-all. Langeveld later told me that he had responded by saying: that could easily be true, Freudenthal, but what do you think of yourself on that score? Or about me?) [25]

One of the differences of opinion was related to Van Hiele’s interpretation of Freudenthal’s discussion with Mrs. Ehrenfest in Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? [26] He proposed that the difference of opinion between Freudenthal and Mrs. Ehrenfest was based on the different implications of the concept of mathematics. Freudenthal would highlight the capacity to substitute in mathematics, and afterwards assess the limited implications of mathematical thinking habits. According to Van Hiele, Mrs. Ehrenfest also included in mathematics the thinking process that precedes mathematics, something that substantially increased the implications of mathematical thinking. According to Van Hiele this would to a large extent explain their difference of opinion on the formative value of the mathematics education. Freudenthal also raised objections against the classification in types of pupils that Van Hiele made: the so-called “algorithm-type” (more inclined to master algorithms) and the “structuring type” (more inclined to structure the observed reality). What Freudenthal’s objections precisely were can no longer be assessed exactly. It can be concluded from Van Hiele’s defence of this classification that Freudenthal saw the danger of classifying pupils from the very start, into these two categories, since this would be stigmatizing and teachers would therefore be too passive in this respect. Van Hiele emphasized that in his classification of the types of pupils he was only aiming at the result of mathematics education that was organized in an algorithmic

7.3 ‘Problems of insight’: Van Hiele’s Level Theory

185

or a structuralizing manner and not at a classification of the pupils as such. He deemed it necessary for the phenomenological method he supported to give such a description of pupils—a statement that showed Langeveld’s influence on Van Hiele’s study. Van Hiele wrote to Freudenthal: Op minstens vier plaatsen in de dissertatie staat aangegeven dat het hier gaat om een kweekprodukt van het onderwijs. Hiermee wordt op duidelijke wijze de leraar ervoor behoed te vervallen tot een afwachtend fatalisme. En het beschrijven van typen behoeft U ook niet te verontrusten: het is immers kenmerkend voor de fenomenologische metode [sic] (die ik voor een studie als deze noodzakelijk acht), dat men zich klaarheid over een situatie verschaft door deze voor enkele speciale gevallen zeer nauwkeurig te beschrijven. Ik zou geen ander doeltreffend middel weten. (In at least four places in the thesis it is indicated that a cultivated product of the education is meant. With this the teacher is clearly kept from lapsing into an expectant fatalism. There is also no need to be distressed about the description into types: after all, it is characteristic for a phenomenological method (which I deem necessary for a study such as this), that light is shed on a situation by describing it very accurately for some special cases. I would not know of any other effective means.) [27]

In the end these differences of opinion between Van Hiele and Freudenthal were not fundamental. There was nothing that prevented Freudenthal accepting Van Hiele’s work completely and admiring it with an open mind. The level theory as described by Van Hiele would become well known internationally, even outside of the didactics of mathematics. The Utrecht educational psychologist C.F. van Parreren (1920–1991) for example referred to the theory in his Psychologie van het leren (Psychology of learning), where he discussed learning to solve problems with insight [28]. In Freudenthal’s correspondence, letters are to be found in which he was asked to provide information about the theses and the methods of the Van Hieles until long after the date of the doctorate [29].

7.3 ‘Problems of insight’: Van Hiele’s Level Theory In her thesis Van Hiele’s wife described an analysis of the practice of the initial geometry education, whereas he was more involved in the theoretical side [30]. Freudenthal considered both theses to be inextricably bound up with each other.other. Het is moeilijk te zeggen, welk van de twee proefschriften belangrijker is: het meer bespiegelende van Dr. van Hiele of het meer proefondervindelijke van zijn echtgenote. In elk geval vullen zij elkander op bijzonder fraaie wijze aan. Men mag zich erover verheugen, dat de twee doctores naast hun dagelijk werk als leraar (en Mw. van Hiele bovendien naast haar werk als huisvrouw en moeder van twee kinderen) de tijd hebben gevonden, om deze waardevolle proefschriften te schrijven. (It is difficult to say which of the two theses is the more important: the more reflective one of Dr. van Hiele or the more experimental one of his wife. In any case they complement each other in a particularly splendid manner. One may rejoice at the fact that both doctors in addition to their daily work as teachers (and for Mrs. van Hiele there is also her role as housewife and mother of two children) have found the time to write these valuable theses.) [31]

In spite of that Van Hiele himself had the idea that Freudenthal appreciated Mrs. van Hiele’s work the most.

186


Eigenlijk beweerde hij ook altijd dat Dieke veel beter was dan ik. En als je nu beide dissertaties ziet, na 27 jaar, dan vind ik dat haar proefschrift absoluut nu nog waardevol is. De theorie van de denkniveaus heb ik gebracht met Dieke’s naam erbij, maar hij is helemaal door mij gevonden. (In fact he always maintained that Dieke was much better than I. And if you see both theses now, after 27 years, I think that her thesis is absolutely valuable, even now. I presented the theory of the levels of thinking with Dieke’s name attached to it, but it was entirely originated by me.) [32]

Freudenthal’s preference for Mrs. van Hiele’s work also emerged from the words he spoke at her funeral. Originally there would have been only one thesis according to Freudenthal, that is Van Hiele’s, and the observations made by his wife would be included in it, maar wat in jullie werk precies van de één was en wat van de ander, zullen we niet uit elkaar kunnen rafelen, ook waar jullie je publicaties apart hebben getekend. Laten wij zeggen, dat de helft van haar was, en dan natuurlijk de betere helft. Want zoals de mens als man én vrouw is geschapen, zo is jullie werk menselijk, juist omdat het mannelijk en vrouwelijk tegelijk is. (but what in your work exactly belonged to the one and what to the other, we will not be able to unravel, even if you have signed your publications separately. Let us say that half of it was hers and then, of course, the better half. Because as man is created as man and woman, in the same way your work is human, exactly because it is male and female at the same time.) [33]

It is not surprising that Freudenthal was very enthusiastic about Mrs. van Hiele’s work [34]. At long last there was a didactical study that was primarily based on the practice of teaching. A study, conducted by doing observations, noting down protocols and analysing the practice in the class room of a teacher, and above all: published and so made accessible to everyone. In Freudenthal’s eyes this way of working turned Mrs. Van Hiele’s thesis into a unique and valuable didactical document. The intuitive approach to the initial geometry education in which pupils worked with material from everyday life that she proposed, appealed to him. Her approach in the very first geometry lessons in her class with 12-year olds bore a remarkable resemblance to the way of working Freudenthal described for a similar group in his article ‘Traditie en Opvoeding’ [35]. Whereas Mrs. Van Hiele in one of her first lessons for example showed how she let her pupils construct a ‘right angle’ by folding a sheet of paper twice, Freudenthal argued in ‘Traditie en Opvoeding’ for something similar [36]. Het was heilzamer geweest, het kind zelf te laten zoeken en ontdekken, hoe men rechte lijnen en rechte hoeken maakt, door papier te vouwen, en hoe men cirkels trekt met een punaise en een touwtje. Uitvindingen zijn waardevoller dan schema’s en patronen. (It had been more beneficial to let the child search and discover by themselves by folding paper how straight lines and right angles are made, and how circles are drawn with a drawing pin and a piece of string. Inventions are more valuable than schemes and patterns.) [37]

There was, however, one difference between both methods: Freudenthal thought it dangerous to give children a compass, a ruler and a setsquare before they had found out themselves how to construct a circle, a straight line or a right angle without using these aids [38]. It would cut off “the road to adventure” for children if they were presented with ready-made schemes to solve problems. But Mrs. van Hiele let her pupils make paper cubes and they were allowed to use the tools that

7.3 ‘Problems of insight’: Van Hiele’s Level Theory

187

Freudenthal deemed so dangerous, right away in the first lesson. The advantage of starting geometry education in this way, was that the children could begin in the three-dimensional space that was more tangible to them, an idea that very much stirred Freudenthal’s enthusiasm [39]. Probably also because of this Freudenthal did not object to this kind of introduction and he saw no harmful consequences of the practice, since he never ever returned to this difference in their views. Despite his enthusiasm for Mrs. van Hiele’s work, the results of Van Hiele’s study, and in particular his so-called level theory, have had a directly provable, more concrete and above all a permanent influence on Freudenthal’s ideas [40]. In his thesis Van Hiele analysed the role that insight played in the mathematics education. He confined himself to geometry education, because he saw no fundamental differences between insight in geometry and insight in mathematics in general. ‘Insight’ performed a key role in the development of didactics according to Van Hiele: In de didactiek der wiskunde blijkt de betekenis van het “inzicht” zo fundamenteel te zijn, dat door bestudering van dit begrip de struktuur van die didaktiek voor een groot deel ontwikkeld kan worden. (In the didactics of mathematics the meaning of “insight” appears to be so fundamental that by studying this concept the structure of didactics can for a large part be developed.) [41]

The first time that Van Hiele encountered the gap between understanding and not understanding, a phenomenon that he recognized from his own teaching practice, was in the work of developmental psychologist Jean Piaget. Piaget had studied extensively how knowledge was effected in children [42]. According to Piaget, the development of children followed a logical order, in which among other things the child’s age determined whether or not it was ready for a new insight. Contrary to Piaget, who considered the gap between understanding and not understanding bound to age, Van Hiele thought that this gap could actually be bridged by a learning process [43]. This view was not only inspired by Piaget’s work, but also by Koning’s thesis. Jan Koning (1907–1989) defended his thesis with the title Enige problemen uit de didactiek der natuurwetenschappen, in het bijzonder van de scheikunde (Some problems from the didactics of science, in particular of chemistry) in 1948. In it he spoke of ‘levels’ in the subject matter and described how and where these emerged in chemistry education [44]. Koning’s study was the first doctorate study supervised by Langeveld. Langeveld spoke well of Koning’s approach in his Inleiding tot de studie der paedagogische psychologie van de middelbare schoolleeftijd (Introduction to the study of the pedagogic psychology of the secondary-school age) [45]. When Van Hiele also encountered these levels in the results of his wife’s study he had enough confidence to actually incorporate this idea in his thesis. At the time of the doctorates, Freudenthal summarized the core of the level theory as follows: De ontwikkeling van de geestelijk groeiende leerling is of dient te zijn een opeenvolging van elkaar afwisselende perioden: de leerling tracht op te klimmen tot een bepaald niveau, om na het bereiken van dit niveau zich zekere vaardigheden te verwerven, die steeds meer worden geautomatiseerd; is hier de nodige routine bereikt, dan kan de sprong naar het volgende niveau worden gewaagd. Gaat men tot automatisering over voor het hiervoor vereiste niveau is bereikt, dan worden bekwaamheden ontwikkeld, die voor geen overdracht

188


op andere gebieden vatbaar zijn. Maar evenzeer wordt de toepassing bemoeilijkt, als te weinig routine wordt verworven. (The development of the mentally growing pupil is or should be a succession of alternating periods: the pupil tries to reach a certain level, and when this is reached he acquires certain abilities which become increasingly automated; once the necessary routine has been reached the jump to the next level can be ventured. If you proceed to automation before the required level has been reached, then abilities are being developed that are not susceptible to transfer to other fields. But the application is just as much hindered when too little routine has been acquired.) [46]

The Van Hieles described these levels specifically for the geometry education as follows. At the so-called zeroth level the pupil works with geometric figures that are based on experiences with physical objects and images. The properties that a pupil discovers in familiar figures means that the figures will start to function as symbols for the pupil, where a symbol is equivalent to the complete set of properties of such a figure. After that the properties characteristic for a certain figure form ‘signals’. Once these symbols and signals (of properties of the figures) become the pupil’s subject of study themselves, the pupil has gone from the zeroth level (rough structuring) to the first level. Geometric relations such as congruence, parallelism and so on, will become a means to order the signals and symbols of the zeroth level. At the second level these relations subsequently become subjects of research. For that purpose the pupil uses ‘the relations between the relations’: he learns that relations can be ordered. At the moment the pupil is able to order the relations found at the second level he has reached the third level. Concepts such as ‘implication’ can now be used. At this level it is in principle possible to develop an axiomatically constructed geometry. Finally, at the fourth level logical thinking (such as axiomatics as such) becomes a subject of study [47]. In his article from 1955 in Paedagogische Studiën entitled ‘De niveau’s in het denken, welke van belang zijn bij het onderwijs in de meetkunde in de eerste klasse van het V.H.M.O.’ (‘The levels in thinking that are important for geometry education in the first year of the VHMO’) Van Hiele described in a concrete example how the levels could be found in the subject matter: Wanneer bijv. een leerling weet, dat in een ruit de diagonalen elkaar loodrecht door midden delen, dan kan hij na het bereiken van het eerste niveau daaruit de konklusie [sic] trekken, dat indien twee gelijke cirkels elkaar snijden, het lijnstuk dat de snijpunten verbindt en het lijnstuk dat de middelpunten verbindt elkaar loodrecht door midden delen. Het kan dan misschien nog zijn, dat hij de ruit niet direkt ziet, maar hij is in ieder geval klaar, als hij daarop opmerkzaam is geworden. De leerling daarentegen, die het niveau nog niet bereikt heeft, ziet het belang niet in van de wetenschap, dat de figuur een ruit bevat. (For example, when a pupil knows that in a rhombus the diagonals orthogonally divide each other in halves, he can, after having reached the first level, conclude from it that if two equal circles cross each other the line connecting the two intersections and the line connecting the two centres divide each other orthogonally in halves. It may be that he does not directly recognize the rhombus, but he is in any case ready for it as soon as his attention is drawn to it. But the pupil who has not yet reached the level, does not see the importance of the knowledge that the figure contains a rhombus.) [48]

Just like Freudenthal, Van Hiele greatly admired Mrs. Ehrenfest’s work, in particular her Uebungensammlung from 1924. Its influence can be identified immediately, especially in Mrs. van Hiele’s work. Both describe an initial geometry education with the use of visual aids and exercises in keeping with practice.

7.4 Freudenthal and the Theory of the Van Hieles

189

In his thesis Van Hiele also referred to Freudenthal’s work several times. He analysed for example the discussion between Freudenthal and Mrs. Ehrenfest on the formative value of the mathematics in Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? and quoted from Freudenthal’s article on the mechanics education [49]. Van Hiele concluded that neither Mrs. Ehrenfest nor Freudenthal had a lot of faith in the formative value of mathematical algorithms. Van Hiele himself had more confidence in it, provided that the education would meet specific conditions: Zodra men in het wiskunde-onderwijs de moed zal hebben gevonden te breken met de gewoonte de wiskunde uit zichzelf te ontwikkelen in de vorm van een deduktief logisch systeem, zodra men in de plaats daarvan de wiskunde gaat ontdekken door er van buiten af in door te dringen, zal men mogen verwachten, dat er een behoorlijke transfer zal zijn naar vrijwel alle transferkringen. (As soon as people in mathematics education have found the courage to break with the habit of developing mathematics from itself in the form of a deductive logical system, as soon as instead of that people start to discover mathematics by penetrating it from the outside, one may expect that there will be a reasonable transfer to practically all transfer circles.) [50]

7.4 Freudenthal and the Theory of the Van Hieles: From ‘level theory’ to ‘guided re-invention’ From several quotes and from Freudenthal’s retrospective on his own work in Schrijf dat op, Hans from 1987 it is clear how fundamental Freudenthal thought the work of the Van Hieles was for the development of his own ideas [51]. Not surprisingly, since the studies by the Van Hieles satisfied exactly the demand that Freudenthal again and again raised for good scientific-pedagogic research based on the practice of the mathematics education [52]. Of course he had often shown some admiration for other people’s ideas in the field of didactics and education of mathematics before, though it was often only between the lines, for example for the work of Mrs. Ehrenfest and that of Van Albada in the field of initial geometry education. But never before had there been a theory that Freudenthal had accepted so totally and with open arms, had praised and subsequently incorporated in his own ideas. In the 1940s and 1950s Freudenthal already had clearly in mind how things should not go in mathematics education: the logical order of mathematics should not be decisive for the psychological and educational order. His aversion against purely deductively constructed (geometry) education, his enthusiasm for a more intuitive, concrete approach and his scepticism about basing the legitimation of the mathematics education on the alleged formative value followed directly from this. In his ‘Rekendidactiek’ he was already an opponent of teaching arithmetic in order to learn to think [53]. In his lecture ‘Opvoeding tot denken’, and even more obviously in Kan het wiskundeonderwijs tot de opvoeding het denkvermogen bijdragen? he argued in favour of concrete arithmetic word problems taken from real life and he explicitly rejected the formative value of arithmetic and mathematics education (for things other than ‘arithmetical thinking’) [54]. He described the stimulus for

190


finding the right solution to well-selected problems, amongst other things, in his lecture ‘Opvoeding tot denken’ and in ‘Oude en nieuwe universiteiten’ [55]. The question was where and how that psychological and educational order of the mathematics education could be found: Uit het onderwijs dat er slecht aan voldeed? Uit proefnemingen die ver boven mijn mogelijkheden uitstegen? Hoe kon ik mij en anderen overtuigen van wat die orde behelsde? Niet dat die vraag mij kwelde, maar zolang ik die vraag niet kon beantwoorden, bleef het bij algemeenheden. Het antwoord—een theoretisch antwoord—kwam toen de Van Hieles aan hun proefschrift werkten, dus ongeveer in 1955/56: de niveaus van de Van Hieles. […] Wat ik van de Van Hieles leerde heb ik op mijn eigen wijze verwerkt—zo gaat dat nu eenmaal. (From the education itself that served it so badly? From experiments that far exceeded my possibilities? How could I convince myself and others what that order amounted to? It was not so that that question troubled me, but as long as I could not answer it, it never got beyond generalities. The answer—a theoretical answer—followed when the Van Hieles were working on their theses, so about 1955/56: the levels of the Van Hieles, […] What I learned from the Van Hieles I processed in my own way—that is the way it goes.) [56]

In his publications too the change that the level theory implied for Freudenthal was very noticeable. Until that moment his articles contained little if any descriptions of concrete educational things such as micro didactics or a learning process in mathematics education. Freudenthal restricted himself primarily to pointing out abuses and criticizing what were generally known as the ‘traditional’ (Dutch) educational methods. He did not achieve more than indicating the direction in which improvements and solutions should be sought. But he was also the first to frankly admit that he too could do no more than that as long as results of good research work were lacking. His article ‘De algebraïsche en de analytische visie op het getalbegrip in de elementaire wiskunde’ from 1948 was the exception. In it he not only discussed a concrete approach for specific subject matter, he also formulated some more general requirements for that teaching. He argued for example in favour of cultivating in pupils a critical attitude towards the theory to be learned, so that the pupil would feel the need to solve the problem. He also held the opinion that the requirements set for the exactness of the theory to be learned by the pupils should be adapted to their levels and their insights [57]. This article, his first on didactics of mathematics to be published, was as far as the theme was concerned practically isolated from the articles he wrote in subsequent years. With the advent of the Van Hieles’ work—and even more specifically: with Van Hiele’s level theory—this changed. For the first time Freudenthal described the work of the Van Hieles in his two articles about initial geometry education that resulted from the lecture he held on this subject as a representative of the ICMI in 1956 [58]. In these he discussed different methods for the initial geometry education that were, however, all based on intuition. To write these he relied on the work that several teachers within the Wiskunde Werkgroep had done in this field. Hence he did not yet refer to the theses but to the educational methods for the geometry education of the Van Hieles that had been published up till then. He also used a lot of what he wrote in these articles on geometry education, in the article ‘Traditie en Opvoeding’ from 1957 [59].


191

The question ‘why teach geometry in secondary schools?’ could not yet be answered by the Wiskunde Werkgroep, and by consequence also not by Freudenthal, early in the 1950s [60]. In ‘Traditie en Opvoeding’ Freudenthal raised that question again and he himself answered it as follows: Sommigen zeggen: omdat meetkunde het verstand oefent. Anderen: omdat meetkunde de kennis is van de ruimte waarin wij leven. Of: omdat meetkunde de eerste wetenschap der mensheid is geweest. Ik zei: omdat meetkunde als logisch systeem een middel is—en wellicht het meest doeltreffende middel—om kinderen te doen voelen de macht van de menselijke geest—dat is van hun eigen geest. (Some say: because geometry trains the intellect. Others: because geometry is the knowledge of the space in which we live. Or: because geometry was the first science of mankind. I said: because geometry as a logical system is a means—and perhaps the most effective means—to make children feel the power of the human intellect—that of their own mind.) [61]

This was not at all a plea for the traditional logically-deductively constructed geometry education à la Dijksterhuis. Because, Freudenthal argued, to be able to reach this goal the initial geometry education had to meet a number of specific conditions. First of all the school teacher had to be able to summon up the patience—“the greatest pedagogical virtue”—to wait until the child itself asked for the ‘why’ of phenomena [62]. ‘Maturity for logics’ could not be a starting point, but should become the goal of the initial geometry education. Vital in this was that the pupil was allowed to experiment and to make mistakes. Het is een oud gezegde dat men uit zijn fouten leert—in ’t Engels zegt men: by trial and error. Inderdaad, de fout is een onmisbare schakel in de didactische methode. […] Voor de methode van “trial and error”, van gissen en missen, zijn echte problemen vereist. Een ingewikkelde breuk en een zinsontleding zijn geen echte problemen, want een echt pro bleem bevat meteen ook een criterium, dat vertelt of de poging om het op te lossen is gelukt. Een echt probleem is een taak om iets te maken en het is opgelost als het gevraagde ding geschapen is. In het traditionele systeem is het teken voor een fout een rode streep, dus een oordeel dat door de leraar of onderwijzer wordt opgelegd. Als echte problemen moeten worden opgelost, is “fout” synoniem met “falen”. Falen is iets ernstigers dan fouten maken in de traditionele zin, maar een falen kan leiden tot een heroriëntering en tot nieuwe pogingen, terwijl een fout geen andere betekenis heeft dan bijvoorbeeld een onregelmatigheid in een patroon, die gecorrigeerd moet worden. (It is an old proverb that one learns from ones mistakes—in English you say: by trial and error. Indeed, the mistake is an indispensable link in the didactical method. […] For the method of “trial and error”, of guessing and missing, real problems are required. A complex fraction and a parsing are not real problems, since a real problem also inherently contains a criterion that decides whether or not the attempt to solve it has succeeded. A real problem is a task to create something and it is solved if that something actually has been created. In the traditional system the sign for a mistake is a red line, thus a judgement imposed by the teacher or school master. If real problems have to be solved, “mistake” is synonymous with “failure”. Failure is something more serious than making mistakes in the traditional sense, but a failure can lead to a reorientation and to new attempts, whereas a mistake can have no other meaning than for example an irregularity in a pattern that has to be corrected.) [63]

When he elaborated on the power of the phenomenon ‘fitting’ as a part of initial geometry education (something that also took up an important place in Mrs. van Hiele’s thesis) Freudenthal made, with hindsight, some striking remarks in both articles from 1956:

192


De wonderen van het passen moeten het kind doen rijpen voor het meetkundig-logische systeem. Maar tevens zal het kind in dit stadium de grondstoffen moeten vergaren èn de problemen die het op een telkens hoger peil zal moeten aanpakken. Het kind moet zijn eigen ontwikkeling kunnen zien en beoordelen, als iemand die in een fotoalbum bladert of die een berg beklimt en terugblikt over de weg, die hij is gegaan. (The miracles of fitting have to ripen the child for the geometric-logical system. But the child will also have to collect the raw materials and the problems which they have to approach in each case at a higher level. The child must be able to see and judge their own development, as someone who thumbs through a photo book or climbs a mountain and looks back on the way they came.) [64]

In the English version he spoke in the corresponding passage about “re-call and reconsider the treasure of old problems and re-examine the old solutions at each new stage” [65]. He did not elaborate further on this retrospect and the development of the pupil, nor did he make a possible relation between the two explicit. Yet it seems that here already the foundation was laid for what he later would recognize and describe as ‘reflection as a level-enhancing means’. The first time that Freudenthal mentioned the level theory of Van Hiele as such was in the preface of the geometry report with the title Report on a comparative Study of Methods of Initiation into geometry. This report constituted the Dutch contribution to the part of the International Mathematical Congress in Edinburgh in 1958 of that same name. Freudenthal was the editor of this report in which authors from the world of education explained their various educational methods for initial geometry education. None of these articles was based on the so-called axiomatic method. In his preface Freudenthal could not help drawing the reader’s attention to the Van Hieles’ contribution in particular. He not only wrote that their work was the most revolutionary, but also described both the level theory and their experimental work as unique achievements in the field of didactical research [66]. From this moment he rarely missed an opportunity to emphasize the value of the theses in an article or a lecture [67]. In his account about the report in Euclides Freudenthal goes one step beyond ‘describing’ for the first time when he mentions the Van Hieles’ level theory. Here he compared the relation between the levels described by the level theory with the logical relation between theory and meta-theory: The relation between one level and the next higher one is analogous to that between a system and a meta-system. At every level the subject matter is a certain field that will be organized on this level. The devices of organizing on a certain level will form the field, and therefore the subject matter, on the subsequent higher level. [68]

In that way Freudenthal described the level theory in a manner in which the direct link with the geometry, essentially and emphatically present in the Van Hieles’ work, had disappeared. It seemed that by this abstraction Freudenthal started to see the level theory as separate from the context (the geometry education) in which he became familiar with it: a new level had been reached … It came even more clearly to the fore in his lecture ‘Logica als methode en als Onderwerp’ (‘Logic as method and as subject’) of 1960 [69]. Here he explained the requirements that he wanted a teacher to meet with respect to his knowledge of


193

logic. Not only should the teacher be above the subject matter, he should also be above the logical form of that subject matter: Men zou nog een stap verder kunnen gaan, door van de leraar ook te eisen, dat hij boven de door hem gebezigde methode van aanbieding van de stof staat en zichzelf deze methode bewust kan maken. Ook hierbij kan de logische analyse diensten bewijzen. Niet in die triviale zin, dat de methode door de logische structuur worde bepaald, maar om, naar het voorbeeld van de van Hieles, de niveau’s [sic] van inzicht te herkennen èn hun onderlinge relatie, die volstrekt logisch van aard is. (One could go even one step further by requiring that the teacher should also be above the method of offering the subject matter that he uses and that he can make himself aware of this method. Here too the logical analysis can be of use. Not in the trivial sense, that the method is determined by the logical structure, but, according to the example of the Van Hieles, to recognize the levels of insight and the relation between them, which has a completely logical character.) [70]

Consequently it was not the knowledge of the logical order or structure of the subject matter to be taught that the teacher should use in his education, but the knowledge of the logical relations between the levels which the pupils would encounter in the learning process—a subtle but essential difference. Thus Freudenthal interpreted the level theory of the Van Hieles step by step in his own way. He abstracted the theory from the geometry education, from which it originated, to a way of working by giving insight in the (levels of the) subject matter by logical analysis. An article which at first sight seemed to be separate from this subject was ‘Trends in modern mathematics’ (1962) [71]. In it he dwelled on the (modern) mathematical way of thinking and working. In his view it was characteristic of the modern mathematics practice that mathematics itself had become subject of axiomatization: Creative mathematical thinking has always been concerned with the organisation of a subject matter, but never before this century has mathematics itself become the subject matter of this organizing activity. The local view on organization has been superseded by a global one, known under the name of axiomatization. Though Hilbert’s axiomatics of Euclidean geometry (1899) meant the organization of one mathematical system only, it started the development of axiomatics into a tool for unifying mathematics. [72]

In modern mathematics the mathematical systems that came about by organizing and arranging of the subject (the mathematical subject matter concerned) had themselves become subject to organization and axiomatization. A striking parallel between this remark and Freudenthal’s interpretation of the level theory can be drawn: in that way there was in mathematics the matter of (logically definable) jumps to a higher level, comparable to the jumps between the levels that appeared to exist in the education process. All of these articles and thoughts seemed, sometimes consciously, sometimes unconsciously, a preparation for the formulation that Freudenthal finally articulated in his articles ‘Logical analysis and critical survey’ from 1962, an article that appeared in the Report on the relations between arithmetic and algebra, published by the NOCW, and the article from 1963 ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ (‘Education of modern mathematics or modern education of mathematics?’) which appeared a bit later [73]. The theoretical part of the content of both articles overlapped to a large extent (sometimes it was even a literal translation).

194


In these articles a critical step in the development of Freudenthal’s mental legacy can be discerned. The fact is that here he uses two important pillars of his mental legacy for the first time in a publication. In these articles he introduced not only the so-called ‘anti-didactical inversion’ (or ‘inversion anti-didactique’) but also the concept of guided re-invention’ (or ‘invention dirigée’). ‘Guided re-invention’ comes up best in ‘Logical analysis’. It is believed that knowledge and ability acquired by re-invention is better understood and easier preserved than if acquired in a less active way. Whether this is true and whether re-invention has to play a part in the guided learning process, is a question which must be answered by pedagogical psychology, not by logical analysis, but I think it is a general experience among adult mathematicians that the easiest way to understand a mathematical paper, is to re-invent its results. […] In the following I will take for granted a more universal part to be played by re-invention in the learning process. I will even suppose that this part is one of first priority. Then the logical analysis which precedes the design of a teaching method has to concern itself with the process of inventing the subject matter rather than with the subject matter itself. Unfortunately we do not know too much about mathematical invention. […] Original invention has been unguided. Guided invention or rather reinvention may enjoy different opportunities. [74]

Freudenthal did not commit himself completely: he still wanted the extent to which the ‘re-invention’ in a learning process would positively contribute to understanding and sinking in of the subject matter, to depend on the research results from child psychology, if there were any. For the time being he assumed that this contribution would be positive. According to him ‘guided re-invention’ was in that case an important principle when designing a teaching method. In ‘Enseignement’ in a content-wise identical extract he did not get beyond mentioning ‘invention dirigée’ [75]. This seems peculiar, since ‘Enseignement’ was written later. However, since further on in ‘Enseignement’ where he discusses the initial geometry education, he does speak of ‘réinventer’ and ‘d’expériences réinventives’ of the pupil, it seemed as if he in this context did not make a real fundamental distinction between ‘inventing’ and ‘re-inventing’ [76]. He also mentioned the concept of re-inventing (réinvention) more than once in describing the level theory [77]. It therefore seems plausible that Freudenthal did indeed in both articles mean the same with ‘guided re-invention’ and ‘invention dirigée’. The second new concept—anti-didactical inversion (‘inversion anti-didactique’)—popped up first in ‘Enseignement’. As an example of anti-didactic inversion Freudenthal mentioned enforcing definitions on pupils even before they have had the possibility to gain experience with the subject matter concerned and feel the need to formulate such a definition. Mathematicians themselves do not proceed like that when creating mathematics! Mais la méthode traditionnelle, c’est de lui offrir des définitions dans un domaine où il n’a pas eu d’expériences, et même avant qu’il ne puisse saisir la raison de définitions formelles—méthode tout à fait contraire à celle que nous observons en créant des mathématiques Nous savons bien qu’il est impossible de faire des définitions utiles dans un domaine qui n’est pas encore exploré, mais nous demandons à l’élève de commencer l’exploration où elle devrait aboutir. Cette inversion anti-didactique est poussée à l’extrême dans les programmes récents de géométrie axiomatique. (But the traditional method, that is one in which they are confronted with definitions in a field in which they do not yet have e xperience, and


195

even before they can understand the reason of formal definitions—is a method completely opposed to the one we observe in mathematicians who create mathematics. We well know that it is impossible to formulate useful definitions in a field that has not yet been explored, but we ask the pupil to start the exploration where it should end. That anti-didactical inversion is carried out to the extreme in the recent programmes of axiomatic geometry.) [78]

The concept ‘anti-didactic inversion’ was also based on an important starting point of Freudenthal: the freedom to re-invent, the freedom for pupils to create mathematics by themselves. The fact that this did not simply mean following the historical development of the subject, was inherent in the surplus value of the additional ‘guided’ in ‘guided re-invention’. ‘Guided re-invention’ and ‘anti-didactical inversion’ did not come out of the blue. As was clear from the preceding articles both concepts were already mentioned before in more guarded terms. But it is the first time that Freudenthal mentioned and defined them this explicitly. In the past he postulated—whether explicitly or not—a preference for the educational-psychological order above the logical order of mathematics, while in the education around him he observed that the logical order was preferred. Now he had a term for this phenomenon, in which not only was the phenomenon itself effectively defined, but which also clearly reflected Freudenthal’s aversion: something called ‘anti-didactical’ could, of course, not possibly mean anything good in education. With the term ‘guided re-invention’ Freudenthal concisely summarized the consequences which according to him followed from the level theory. Concretely starting with mathematics education, giving pupils the opportunity to experiment and explore, offering the subject matter to an extent of exactness that matched the level of the pupils at that moment, it was all cloaked in the term ‘guided re-invention’. In Freudenthal’s view it was clear that the level theory of the Van Hieles was also at the basis of this term: after all, in education based on the principle of guided re-invention the analysis of the learning process of the pupils was the starting point. This ‘re-inventing’ as a didactical principle was according to Freudenthal also found in the earlier textbooks by the Van Hieles. It is not by chance that the Van Hieles seized upon this idea. To my knowledge they were the first who wrote a textbook in which the learning process is purposively initiated and kept up as a process of re-inventing. [79]

The term ‘re-invention’, just as ‘anti-didactical inversion’ did, also comprised for Freudenthal that other important property of the mathematics to be taught: mathematics should not be identified with the result, but should be considered, analysed and taught as an activity. The inspiration for Freudenthal’s intuitionistic streak can be recognized in this: mathematics is a mental creation and that principle should also be the basis of its education. The traditional approach presupposes that language or mathematics have to be taught as ready-made products. If this stand is adopted, it is quite consistent to base teaching on the logical analysis of language or mathematics, considered as funds of knowledge rather than as activities. If, however, teaching means initiating into an activity, than (sic) the teaching subjects have to be analysed as activities, not as ready-made products, before a teaching method is designed. [80]

196


The fact that such an analysis subsequently led to the application of the level theory was obvious to Freudenthal. It looks as if in this quotation he heaped the ‘traditionalists’ and the supporters of a logical-deductive approach together. Or in any case, to the supporters of a strict approach too the activity was more important than the results. Since they were led by (the logical construction of) the mathematical end result when organizing the education, and not by (re)inventing that mathematics they did not achieve ‘mathematics as an activity’ with their logical-deductively constructed education according to Freudenthal, but quite the contrary. New in ‘Logical analysis’ was also Freudenthal’s description of the application of the different levels with regard to a subject other than the initial geometry education. If he had detached them from the geometrical context before by putting their mutual relationship on a par with that between theory and meta-theory, now he even went one step further. By means of subjects such as complete induction and the theory of (ir)rational numbers he described the various levels that pupils had to go through in acquiring insight in this matter. He also deviated from the approach of the Van Hieles in further generalizing the (content of the) levels. Freudenthal’s subject of research was no longer the four steps from ‘tangible context’ to ‘logic’ as established by the Van Hieles, but he made the number of steps dependent on, amongst other things, the relevant subject. He gave a complete interpretation of how this would work in the case of complete induction: If on a certain level complete induction had been a significant activity, this activity may become a matter of conscious consideration and finally of explicit formulation on the next level. The higher level is here clearly indicated by the heavy quantification over all properties of natural numbers which occurs in the statement of complete induction.[…] Operations with integers such that multiples of a certain m are considered unessential, and operations with pairs of integers according to the equivalence relation for fractions, are unproblematic activities. On a higher level they can be promoted to subject matter and finally described by the device of abstraction through equivalence classes. On a still higher level the equivalence relation and the abstraction by equivalence classes can become subject matter. [81]

As it happened the Van Hieles themselves had also thought about applying the level theory to other subjects from mathematics education. In 1957 in an article about the phenomenology of the initial geometry education the Van Hieles already indicated that that possibility should not be excluded. De hiervoor aangeduide wijze om het meetkunde-onderwijs te beginnen heeft het voordeel, dat de leerlingen ervaren, hoe men een kennisgebied, waarvan men globale strukturen bezit, door analyse voor objectieve beschouwingen toegankelijk kan maken. De hier voor de meetkunde aangegeven weg kan nl. ook voor andere kennisgebieden gebruikt worden. Of het daar ook mogelijk zal zijn het kennisveld tenslotte te matematiseren [sic], hangt van de aard van het veld af. Noodzakelijk daarvoor is immers o.a., dat de relaties niet gedenatureerd worden, wanneer zij in logische relaties worden omgezet. (The way to start geometry education as indicated here has the advantage that the pupils experience how a field of knowledge, of which the broad outlines are known, can be made accessible to objective observations by analysis. The fact is that the route indicated here for geometry can also be used for other fields of knowledge. Whether it will be possible to mathematize the field of


197

knowledge eventually, will depend on its nature. Since for doing that it is necessary that, amongst other things, the relations are not denaturized when they are converted into logical relations.) [82]

The fact that Freudenthal interpreted the level theory so freely indicated that he had made himself literally and figuratively familiar with it in the meantime. Whereas some years earlier he had not gone further than recommending the content of the theses of the Van Hieles, here Freudenthal had integrated the level theory and all practical and theoretical consequences in his own body of ideas and had given them an important, if not dominant, place. By doing that Freudenthal could now further accentuate his old ideas. For example, he once again repeated in ‘Logical analysis’ the condition that adapting the exactness of the language used in the mathematics education was allowed, something he also mentioned in his article of 1948. However, he could now couple the degree of exactness of the language to be used by the pupil with the help of the level theory to the level of the pupil. Every level in the learning process asks for corresponding levels of rigour and formalization, but traditional teaching tends to move on one level of rigour and formalization from the start of teaching to the end. In actual teaching this means too high a level to start with and too low a one to finish. In geometry Euclidean rigour in the beginning prevents teaching, at the end it blocks modern rigour. [83]

In the French equivalent he was even clearer: … il n’y a pas d’exactitude absolue. A chaque niveau correspond une forme d’exactitude qui lui est proper. Exiger de l’élève l’exactitude d’un niveau où il ne se trouve pas, est malhonnête au nom de l’honnêteté. (There is no absolute exactness. A degree of exactness that belongs to it corresponds with every level. Requiring of a pupil the exactness of a level which is not his’, is unfair in the name of honesty.) [84]

The premature use of too strictly mathematical language in mathematics education did not work, as Mrs. Van Hiele had already written in her thesis as well [85]. The development of the mathematical understanding had to go hand in hand with developing the (degree of exactness of the) formulation of that understanding. Another new point Freudenthal touched upon in these articles was learning of axiomatics as opposed to learning to axiomatize. He saw an essential difference between the two for mathematics education: Je nie absolument que quelque axiomatique que ce soit puisse être un des buts de l’enseignement scolaire en géométrie, mais j’admettrais bien l’axiomatisation Il faut distinguer axiomatique et axiomatisation. (I absolutely deny that any axiomatics can be one of the goals of geometry education at school, but I would certainly allow axiomatization. We have to distinguish between axiomatics and axiomatization.) [86]

‘Learning axiomatics’ was for Freudenthal a case of anti-didactical inversion, but for ‘learning to axiomatize’ on the other hand—in which the pupil was assigned an active, re-inventing role—he saw a valuable role in mathematics education. In that same year he devoted a complete article (‘Was ist Axiomatik, und welchen Bildungswert kann sie haben? (‘What is axiomatics and what educational value can it have?’)) to the question of axiomatics in education [87]. Many of the ideas from

198


‘Logical analysis’ and ‘Enseignement’, as well as a complete passage of ‘Tradition and education’ from 1957, returned in this article. [88]. Yet there was an essential difference. For with this new article Freudenthal had a completely new, and at the time extremely urgent, goal in view: entering the battle with the tendency to haphazardly introduce modern mathematics in the mathematics education [89]. This shift of emphasis had more or less already been started by Freudenthal when he wrote ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’. The level theory as such disappeared into the background in Freudenthal’s publications and he primarily referred to ideas such as ‘anti-didactical inversion’ and ‘guided re-invention’ which to him were (implicitly or not) connected to this theory. He still mentioned the Van Hieles and their work in his articles, but now only incidentally. To Freudenthal the work of the Van Hieles was promoted to fundamental knowledge. He would use the concepts he derived and the vision he developed from that work in the battle that he more and more explicitly entered in the subsequent years: the battle against the movement that would rage though the 1960s ‘as a bandjir’, New Math [90].

7.5 Analysis of a Learning Process: Reflection on Reflection In Freudenthal’s book Revisiting mathematics education. China lectures, that was published in 1991, he subjected his own ideas on the level theory, and in particular those on the transfer between the different levels in the learning process, to an interesting analysis [91]. By comparing different descriptions from his own work he concluded that there was a crucial difference between the articles ‘Logical analysis and critical survey’ and ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ [92] Crucial, since it was a leap in his own learning process with respect to his insight in didactics of mathematics. In essence the difference could be reduced to the content of the following quotation from ‘Enseignement’ that could not be found, as such, in ‘Logical analysis’: Ce qui caractérise la hiérarchie des niveaux en général, c’est que la technique des opérations d’un certain niveau devient matière à réflexion à un niveau supérieur… (What characterizes the hierarchy of the levels in general is that the technique of the operations of a certain level become material to reflect on one level higher…) [93]

The essence was in the word “réflexion”. According to Freudenthal this was the source of his awareness of reflection as an activity to raise one level in the learning process of the pupil. In ‘Logical analysis’ the description that most resembled this was “conscious consideration”, observing consciously an activity, as described in the next passage: If on a certain level complete induction had been a significant activity, this activity may become a matter of conscious consideration and finally of explicit formulation on the next level. [94]

7.6 To Conclude

199

Here he did not make the connection with reflection as an activity to raise one level. At the end of the 1980s Freudenthal described the difference between both views, called “a fundamental distinction” by himself, as follows: [95] The second version [that from ‘Enseignement’] adds to the first [that from ‘Logical analysis’] the means by which operational matter on the lower level is transformed into subject matter on the higher level: being made conscious and subjected to reflection is the means of level-raising, which is applied to the—still intuitive (operational)—activities of the lower level, rather than to the subject matter. [96]

By his own account Freudenthal only discovered in ‘Enseignement’ reflection as the ‘motor’ of the mathematical discovery and at that moment recognized it as an essential didactical principle for the education based on the principle of re-invention. This conclusion is apparently not completely correct. It is going too far to give the article in ‘Enseignement’ the same key role in the development of Freudenthal’s ideas as he did himself. It is true that he used the actual term “reflection” in this context for the first time, but this description of “looking back” was not really new. After all in ‘Logical analysis’ he also mentioned ‘observing consciously’ activities of a lower level, and in ‘Het aanvankelijk meetkunde-onderwijs’ and its English version from 1956 he already spoke of reviewing and judging the personal development [97]. Apart from that, in ‘Enseignement’ Freudenthal did not connect reviewing activities at a lower level and the level-raising effect of this action for the pupil as explicitly as he formulated it at the end of the 1980s. Maybe he is right when he says that the seeds of (a part of that) insight were already present in ‘Enseignement’. But it was not yet a real breakthrough here. Just as with so many of his key notions ‘reflection’ also developed more gradually and more diffusely in other different publications than he wished to make us believe in Revisiting mathematics.

7.6 To Conclude In the following years the work of the Van Hieles, and in particular the level theory, would become well known both nationally and internationally. It would be a source of inspiration for many researchers and developers of mathematics education, in particular in the United States and the Soviet-Union [98]. The influence of their work had not yet worn off Freudenthal either. From ‘level theory’ via ‘mathematics as an activity’, ‘guided re-invention’ and ‘axiomatizing’ it seemed only a small step to ‘mathematizing locally’, a didactical concept he introduced later. Yet in the further development of his ideas he would never again refer so directly to the theory of the Van Hieles as he had done in the years described in this chapter. In the early 1960s through his introduction of notions such as ‘guided re-invention’ and ‘antididactical inversion’ he had a machinery of thought of his own at his disposal. A machinery of thought to study in depth, to build on, but especially: to disseminate.

200


References 1. Freudenthal, Schrijf dat op, Hans 359. 2. Feestboek. Prof. Dr. Hans Freudenthal—70 jaar (1975) (publication of the Mathematisch Instituut and the IOWO at the occasion of Freudenthal's 70th birthday) 40. 3. Article in Nieuw Utrechts Nieuwsblad (4 July 1957), written by Freudenthal at the occasion of the doctorates of the Van Hieles, 29 May 1957, 3 pages, q.v. 1; RANH, Hans Freudenthal Papers, inv.nrs. 37, 142 4. Hans Freudenthal, ‘Preface of the editor’, in: Hans Freudenthal, ed., Report on methods of initiation into geometry (Groningen 1958) 5–7, q.v. 6–7. 5. Freudenthal, Schrijf dat op, Hans 354. 6. Freudenthal, ‘Vakgeleerde of paedagoog?’, Christelijk Schoolblad. Onze vacatures (1953); RANH, Hans Freudenthal Papers, inv.nrs. 779, 780. See also paragraph 6.5. 7. In composing this and following paragraphs information has been taken from the following studies: P.M. van Hiele, De problematiek van het inzicht (Amsterdam 1957); D. van Hiele-Geldof, De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. (Amsterdam 1957); De Moor, Van vormleer naar realistische meetkunde. 8. Interview with Van Hiele in: Goffree, Ik was wiskundeleraar 104. 9. In a book review of this workbook in Vernieuwing there is talk of “an excellent method”; ‘Book review of: Drs. P.M. van Hiele en Dra. D. van Hiele-Geldof: Werkboek der meetkunde 1e deel’, Vernieuwing van Opvoeding en Onderwijs 7 (1948/49) 175. 10. Freudenthal writes: “The procedures of this group (I may fairly speak of a school of teachers) depend heavily on the use of self-made material of a different kind, cards bearing instructions, models and construction pieces, like Meccano parts but of a still more flexible structure. This movement has originated from two different sources. One is the work of P.M. van Hiele and his wife, Mrs. D. van Hiele-Geldof.” in: Hans Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956), 83–97, q.v. 87. 11. At the WW-meeting of Saturday 2 May 1953 Mrs. van Hiele for example gave a lecture with the subject: ‘Waar zullen we ’t zwaartepunt van het meetkunde-onderwijs in de eerste klas leggen?’, Vernieuwing van Opvoeding en Onderwijs 103 (1953) 222. 12. Article in Nieuw Utrechts Nieuwsblad 4 July 1957; RANH, Hans Freudenthal Papers, inv.nrs. 37, 142. 13. H. Mooij, Over de didactiek van de meetkunde benevens benaderingsconstructies van een hoek in gelijke delen (Amsterdam 1948). In this very small-scale project Mooij researched the effect of using learning conversations in the class room in the usually logically-deductively organized geometry education. Van der Corput was supervisor and Kohnstamm was also involved in the project. With respect to content this project was of little account to the project of the Van Hieles. More about Mooij's project in: F. Goffree, ‘Een halve eeuw onderzoek. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 233–243; De Moor, Van vormleer naar realistische meetkunde 286. 14. Interview with Van Hiele in: Goffree, Ik was wiskundeleraar 110. 15. Ibid., 110. 16. J.W. van Hulst, ‘Martinus Jan Langeveld. 30 October 1905–15 December 1989’, Koninklijke Nederlandse Akademie van Wetenschappen. Jaarboek 1991 (Amsterdam 1991), 154–162; Frederieke Schenk, De Utrechtse school. De geschiedenis van de Utrechtse psychologie tussen 1945 en 1965 (Utrecht 1982) 2. 17. I. Weijers, ‘Mondige burgers. Een cultuurhistorische plaatsbepaling van de pedagogiek van M.J. Langeveld’, Pedagogisch Tijdschrift 19 (1994) 189–206, q.v.190. 18. I. van Hilvoorde, Grenswachters van de pedagogiek. Demarcatie en disciplinevorming in de ontwikkeling van de Nederlandse academische pedagogiek (1900–1970) (Amsterdam 2002) 87; Weijers, ‘Mondige burgers’, Pedagogisch Tijdschrift 19 (1994) 189–206. 19. B. Levering, ‘De betekenis van M.J. Langeveld voor de naoorlogse pedagogiek (met het accent op de periode 1945–1960)’, Pedagogisch Tijdschrift 16 (1991) 147–160.

References

201

20. M.J. Langeveld, Opvoedingshulp als groeiende wetenschap (Amsterdam 1972) 15–16. 21. Other social scholars of the Utrechtse School were criminal lawyer W. Pompe, psychiatrists H.C. Rümke, P.H.A. Baan and physiologist-psychologist F.J.J. Buytendijk; Frederieke Schenk, De Utrechtse school. De geschiedenis van de Utrechtse psychologie tussen 1945 en 1965 (Utrecht 1982); Schuyt en Taverne, eds., 1950. Welvaart in zwart-wit 384, 387, 416; Weijers, ‘Mondige burgers’, Pedagogisch Tijdschrift 19 (1994) 189–206. 22. Van Hiele to Freudenthal, 2 December 1948; RANH, Hans Freudenthal Papers, inv.nr. 37. 23. Interview with Van Hiele in: Goffree, Ik was wiskundeleraar 110. 24. Ibid., 113. 25. Ibid., 110. 26. Van Hiele to Freudenthal, 3 February 1957 (the answer from Freudenthal is missing); RANH, Hans Freudenthal Papers, inv.nr. 37. 27. Van Hiele to Freudenthal, 3 February 1957; RANH, Hans Freudenthal Papers, inv.nr. 37. 28. C.F. van Parreren, Psychologie van het leren. Verloop en resultaten van leerprocessen (Deventer revised edition 1971) 239 et seq. 29. K.S. Gangadharan from Ceylon to Freudenthal (in which, as a result of Freudenthal's lecture in Bombay, he asks for information about the textbooks and the methods of the Van Hieles), 22 March 1956; Secondary School of Hashomer Hatsair Mishmar Haemek-Hazorea to Freudenthal (in which they ask for more information about the theses of the Van Hieles), 23 May 1957, with an answer from Freudenthal, 29 May 1957; John C. Moyer to Freudenthal (in which he, as a result of what Freudenthal writes in Mathematics as an educational task, asks for more information about the work of the Van Hieles), 16 March 1977; RANH, Hans Freudenthal Papers, inv.nrs. 32, 37, 54. 30. For an overview of all mathematical-didactical theses of the second half of the last century and the position the Van Hieles have in this overview, see also: F. Goffree, ‘Een halve eeuw onderzoek. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 233– 243 and F. Goffree, ‘De opbrengst. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 333–345. 31. Article in Nieuw Utrechts Nieuwsblad 4 July 1957; RANH, Hans Freudenthal Papers, inv.nrs. 37, 142. 32. Interview with Van Hiele in: Goffree, Ik was wiskundeleraar 113. 33. From Freudenthal's speech at the funeral of Mrs. van Hiele, July 1958; RANH, Hans Freudenthal Papers, inv.nr. 521. 34. He spoke for example about her “marvellous reports on classroom observations” and he wrote: “She took notes while teaching; and I was astonished how many more—and more important—things she observed than I was able to do.” In 1958 Mrs. van Hiele died. Freudenthal spoke in his speech at her funeral of “a teacher by the grace of God”, in: Hans Freudenthal, Revisiting mathematics education. China lectures (Dordrecht 1991) 93; Freudenthal speeched at the funeral of Mrs. van Hiele, July 1958; RANH, Hans Freudenthal Papers, inv. nr. 521. 35. Hans Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 3 (1957) 95–103. 36. D. van Hiele-Geldof, De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. 16. 37. Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 100. 38. Ibid., 100. 39. He made an exception for the (also used by among others the Van Hieles) two-dimensional approach of the project ‘paving stones’, in which the concept ‘fitting’ was central. See for example his descriptions of courses for initial geometry education in: Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 17–18; Hans Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956), 83–97, q.v. 91–93. 40. Since the themes of both theses were closely connected, Freudenthal often spoke about ‘the work of the Van Hieles' in general, without distinguishing them specifically. 41. Van Hiele, De problematiek van het inzicht 8.

202


42. A more elaborate description of Piaget's work and ideas and (Freudenthal's) critique will be treated in paragraph 9.6.3. 43. Langeveld too, regularly published critiques on Piaget. His critiques were mainly based on— according to Langeveld—Piaget's poor insight in the social world of the child; B. Levering, ‘De betekenis of M.J. Langeveld voor de naoorlogse pedagogiek (met het accent op de periode 1945–1960)’, Pedagogisch Tijdschrift 16 (1991) 147–160. 44. In this thesis too Freudenthal played a (small) role: in his preface Koning thanks Freudenthal for his help with the statistical part of his thesis; J. Koning, Enige problemen uit de didactiek der natuurwetenschappen in het bijzonder van de scheikunde (Dordrecht 1948) 7, 218. 45. M.J. Langeveld, Inleiding tot de studie der paedagogische psychologie van de middelbare schoolleeftijd (Groningen, fifth revised edition 1954, first edition 1937) 459. 46. Article written by Freudenthal in Nieuw Utrechts Nieuwsblad 4 July 1957; RANH, Hans Freudenthal Papers, inv.nrs. 37, 142. 47. Van Hiele, De problematiek van het inzicht; Van Hiele-Geldof, De didaktiek van de meetkunde in de eerste klas van het V.H.M.O.; the article by the Van Hieles in: Freudenthal, ed., Report on Methods of Initiation into Geometry 67–80. 48. From: P.M. van Hiele, ‘De niveau's in het denken, welke van belang zijn bij het onderwijs in de meetkunde in de eerste klasse van het V.H.M.O.’, Paedagogische Studiën 32 (1955) 289–297, as quoted in: P.M. van Hiele, De problematiek van het inzicht 127. 49. Van Hiele, De problematiek van het inzicht 91–98. 50. Ibid., 100. 51. Freudenthal, Schrijf dat op, Hans 352, 359. 52. This is a wish Freudenthal expressed at several occasions in lectures and articles, for example in: ‘Rekendidaktiek’ (unpublished 1944); RANH, Hans Freudenthal Papers, inv.nr. 465; ‘En nu… de wiskunde’, De Groene Amsterdammer (2 December 1950); RANH, Hans Freudenthal Papers, inv.nr. 1439; also in ‘De leraarsopleiding’, De Groene Amsterdammer (25 October 1952); RANH, Hans Freudenthal Papers, inv.nr. 1469: about “raising education and teaching to the same level other techniques in our society have” in Hans Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 98–99; finally about the urgency of a scientific-pedagogic study into mathematics education in the introduction to: ‘Het wiskunde-programma voor het V.H.M.O. Een ontwerp van de Wiskunde- Werkgroep van de W.V.O.’, Euclides 28 (1953) 206–226, q.v. 208. 53. Hans Freudenthal, ‘Rekendidactiek’ (unpublished 1944) 101–102; RANH, Hans Freudenthal Papers, inv. Nr. 465. 54. Freudenthal, ‘Opvoeding tot denken’ (unpublished 1945); RANH, Hans Freudenthal Papers, inv.nr. 466 Ehrenfest and Freudenthal, Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? 55. Freudenthal, ‘Opvoeding tot denken’ (unpublished 1945); RANH, Hans Freudenthal Papers, inv.nr. 466; Freudenthal, ‘Oude en nieuwe universiteiten’ (unpublished 1945); RANH, Hans Freudenthal Papers, inv.nr. 467. 56. Freudenthal, Schrijf dat op, Hans 352. 57. Hans Freudenthal, ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides 24 (1948) 106–121, q.v. 110, 114. 58. Hans Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956), 83–97; and the Dutch abridged version: Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18. 59. Hans Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103. 60. ‘Het wiskunde-programma voor het V.H.M.O. Een ontwerp van de Wiskunde- Werkgroep van de W.V.O.’, Euclides 28 (1953) 206–226, q.v. 208. 61. Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 97–98. This phrase is also found in ‘Het aanvankelijk meetkundeonderwijs' (1956) and in ‘Initiation into geometry’ (1956). 62. About ‘patience’ as the most important virtue of the educator he also wrote in: Freudenthal, ‘De leraarsopleiding’, Vernieuwing van Opvoeding en Onderwijs 109 (1956) 173–180, q.v. 176; Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, 16.

References 63. 64. 65. 66.

67.

68. 69.

70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84.

203

Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 100–101. Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 16. Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956), 83–97, q.v. 92. Hans Freudenthal, ‘Preface of the editor’, in: Hans Freudenthal, ed., Report on methods of initiation into geometry 5–7, q.v. 6–7. In 1987 he says about this report: “The report derives its great historical meaning from the contribution of the Van Hieles—in fact composed by me from their several publications. The contribution was the first internationally accessible explanation of their level theory …” In: Freudenthal, Schrijf dat op, Hans 347–348. In subsequent years he mentioned the theory of the Van Hieles in, among other texts, the following ones: Hans Freudenthal, ‘Preface of the editor’, in: Freudenthal, ed., Report on methods of initiation into geometry 5–7; Hans Freudenthal, ‘Report on a comparative Study of Methods of Initiation into Geometry’, Euclides 34 (1959) 289–306, Hans Freudenthal, ‘A Comparative Study of Methods of Initiation into Geometry’, l’Enseignement Mathématique 5 (1959) 119–139; Hans Freudenthal, ‘Logica als Methode en als Onderwerp’, Euclides 35 (1960) 241–255; Hans Freudenthal, ‘Logical analysis and critical survey’, in: Freudenthal, ed., Report of the relations between arithmetic and algebra (Groningen 1962) 20–41; Hans Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’, L’Enseignement Mathématique 9 (1963) 28–44; Hans Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29; Hans Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22; see also: the lecture ‘Integratie achteraf of vooraf’ from 1961 and the lecture Freudenthal gave for the MC: ‘Axiomatiek in het wiskunde-onderwijs bij het VHMO’ in 1962; RANH, Hans Freudenthal Papers, inv.nrs. 536, 537. Freudenthal, ‘Report on a comparative Study of Methods of Initiation into Geometry’, Euclides 34 (1957) 289–306, q.v. 306. He gave this lecture at the weekend conference of the WW: Freudenthal, ‘Logica als Methode en als Onderwerp’, Euclides 35 (1960), 241–255. This is identical to the lecture, ‘Logik als Gegenstand und als Methode’, he gave at the ICMI seminar in Aarhus in 1960, also published as article: Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22. Freudenthal, ‘Logica als Methode en als Onderwerp’, Euclides 35 (1960), 241–255, q.v. 255. Hans Freudenthal, ‘Trends in Modern Mathematics’, ICSU Review 4 (1962) 54–61. This article was also published in German: ‘Tendenzen in der modernen Mathematik’, Der Mathematische und Naturwissenschaftliche Unterricht 16 (1963) 301–306. Freudenthal, ‘Trends in Modern Mathematics’, ICSU Review 4 (1962) 54–61, q.v. 55. Hans Freudenthal, ‘Logical analysis and critical survey’, in: Freudenthal, ed., Report of the relations between arithmetic and algebra; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 28–44. Freudenthal, ‘Logical analysis and critical survey’ 23–24; italics mine. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 32. Ibid., 39. Ibid., 34. Ibid., 41; italics mine. This same passage can also be found in: Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 34; Freudenthal, ‘Logical analysis and critical survey’ 27; see also: ibid., 23, 26. Freudenthal, ‘Logical analysis and critical survey’ 23. Ibid., 27. Dr. P.M. van Hiele and Dr. D. van Hiele-Geldof, ‘Een fenomenologische inleiding tot de meetkunde’, Euclides 33 (1957) 33–47, q.v. 45; NB.: the Van Hieles consistently used “matematiseren” instead of “mathematiseren”. Freudenthal, ‘Logical analysis and critical survey’ 32. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 43.

204


85. She wrote: “If the mathematical language is used too early and the teacher has not started from everyday speech, the mathematical language will be learned without the corresponding mathematical understanding”, in: D. van Hiele-Geldof, De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. 44. 86. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 41. 87. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29. 88. More about this article in paragraph 8.6.3. 89. The background to this tendency and Freudenthal's role in these developments will be treated extensively in chapter 8. 90. Freudenthal, Schrijf dat op, Hans 348. 91. Freudenthal, Revisiting mathematics education. China lectures 96–102. 92. By his own account he had written ‘Logical analysis' at the end of the 1950s, ‘Enseignement' dated according to him from 1961. 93. Freudenthal, ‘Enseignement des mathétiques modernes ou enseignement moderne des mathématiques?’ 33. 94. Freudenthal, ‘Logical analysis and critical survey’ 27. 95. Freudenthal, Revisiting mathematics education. China lectures 98. 96. Ibid., 99. 97. Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’ 16; Freudenthal, ‘Initiation into Geometry’, 92. 98. Goffree, Ik was wiskundeleraar 101; John C. Moyer to Freudenthal, 16 March 1977, and answer; RANH, Hans Freudenthal Papers, inv.nr. 54.

Chapter 8

Method Versus Content New Math and the Modernization of Mathematics Education Ich bin kein ausgeklügelt Buch, ich bin ein Mensch mit seinem Widerspruch. (I am not a sophisticated book, I am a man with my own inconsistencies.) Hans Freudenthal in Schrijf dat op, Hans, 1987 [1]

8.1 Introduction: Time for Modernization After the realization of the 1958 curriculum in the mid 1950s a period of peace and quiet seemed to have begun in mathematics education [2]. After all, this curriculum was unique since it was supported by both the establishment in the mathematics education—represented by the members of the teachers’ associations WIMECOS and LIWENAGEL—and by the faction of the members of the Wiskunde Werkgroep who were known as persistent reformers. And what is more, the opinions of the advocates of a ‘more strict’ (logically-deductively structured) mathematics education were hardly heard any more. A curriculum so broadly based simply had to lead to deep satisfaction and solidarity. The opposite was true: years of great turmoil in the world of mathematics education had come, fed by the wish for a drastic change of course, based on various causes, both inside and outside of the Netherlands. It was as if the whole world of mathematics education had been struck by a fever blown over from the United States: the ‘New Math’ fever, most tersely characterized by the notion ‘modernization’. Modernization, not so much of the didactics, but of the content of the mathematics education, which had lost the connection with the modern mathematics that originated in the twentieth century. What exactly was this so-called New Math movement? Where and how was the influence on the Dutch mathematics education noticeable? And what was Freudenthal’s attitude towards it? Now that he was respected everywhere as an authority in the field of education and didactics of mathematics—with his involvement in the ICMI both inside and outside the Netherlands—he had to take up a position in this field: was he an advocate or an opponent? This is the kind of question, which can only be posed in this way in retrospect. The matter was not so black-and-white, particularly in the first years after the advent of New Math—the mid 1950s in the United States. It took a while for New Math to © Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_8

205

206

8 Method Versus Content

penetrate the world of education in Europe and even longer before the consequences for the mathematics education in the Netherlands became apparent. It is tempting to put Freudenthal’s activities in the field of mathematics education in either the ‘pro’ or ‘contra’ camps. But initially these camps were not at all that clearly distinguishable, particularly in the early years of New Math. In addition, but also as a consequence, Freudenthal’s activities at that time were not easily heaped together. In Schrijf dat op, Hans he described himself as a passionate opponent of New Math and characterized his actions as: …een dam opwerpen tegen de internationale bandjir van New Math. […] De geïsoleerde, van de realiteit hermetisch afgesloten wiskunde van toen—denk aan mijn citaten—, daar ging ik tegenin, en des te harder naarmate de isolering, de afsluiting van de realiteit in New Math tot de top werd gedreven. (…erecting a dam against the international flood of New Math. […] The isolated mathematics, at the time hermetically disconnected from reality— remember my quotations—, that was what I opposed, and the more strongly as the isolation, the disconnection from the reality in New Math was pushed to the end of its tether.) [3]

With hindsight the following picture was the most plausible: Freudenthal as a passionate opponent of everything New Math represented. This picture was also in agreement with the state of affairs in the early 1970s. But Freudenthal’s conviction that he was an ‘opponent’ had certainly not taken shape clearly in the 1950s. At first sight, his activities and involvement in various initiatives and symposia in the field of (the modernization of) mathematics education did not paint an image of a lone opponent—in the world of mathematics education he described himself as a “drummer for deaf ears”—of New Math [4]. Initially, his reaction to the suddenly omnipresent drive to innovate by modernizing was enthusiastic and optimistic. Wrapped up and collaborating in this rising development—like many others—he only gradually seemed to see the potential ‘dangers’ of New Math, to process them in his ideas and to disseminate them in publications and lectures. With hindsight Freudenthal himself also thought that he should have seen the ‘danger’ of New Math earlier and should have stopped it: Gezaghebbende wiskundigen demonstreerden het voor onderwijskundigen die er zich aan vergaapten[,] al had geen van hen een flauw idee waar deze begrippen in de wiskunde toe dienden. Het was een absurde bevlieging wiskunde vanuit dit perspectief op school te onderwijzen—ik heb deze dwaze geestesgesteldheid toen niet au sérieux genomen en dat was misschien de grootste taxatiefout die ik in mijn leven heb gemaakt. (Leading mathematicians demonstrated it in front of the educators who gaped at it [,] even though none of them had the faintest idea what good these notions in mathematics were. It was an absurd whim to teach mathematics from that perspective at schools—I have not taken this foolish mentality seriously at the time and that was maybe the greatest misappraisal I have made in my life.) [5]

Not taken seriously? Or not recognized in time as being a “foolish mentality”? Yet he would emerge from the turmoil of the 1960s as the person who managed to erect a dam against New Math. What exactly had happened?

8.2 New Math

207

8.2 New Math The leitmotiv in the history of mathematics education in the 1950s can best be described using the term ‘modernization’. Modernization of the mathematics education, or even more precisely: modernization of the mathematics taught. Influenced by initiatives from abroad, originating in the United States, but later also coming from some other (West) European countries, in the Netherlands too the idea that more modern mathematics had to be taught at the secondary schools, to start with at the HBS and the Gymnasium, began to gain ground. Where did that idea come from?

8.2.1 The Gap between Modern Mathematics and Mathematics Education The idea in itself was not new. Such opinions had already been voiced from the end of the nineteenth century. In Germany, for example, the most important objective of the ‘Reform Bewegung’ inspired by Felix Klein’s ideas was to reform mathematics education with respect to content and adapt it to the modern mathematics that had developed in previous decades. Putting the notion of function in the centre and introducing differential and integral calculus were the most important consequences [6]. In 1964 Freudenthal wrote about this: Half a century ago German educators spoke about the “doppelter Bruch”, the twofold rupture in the education of the mathematics student who first has to forget about school mathematics when he passes from the secondary school to the university, and a few years later about university mathematics when, returning to school as a teacher, he had to tie on the thread where it was cut off a few years ago. This double rupture, though early recognized as an evil, was not bridged when it was relatively mild. It has grown sharper and sharper, and finally we have been left with no other choice than drastic overhauling mathematical education at the secondary level. [7]

The New Math movement too was a result of the dissatisfaction with the gap between mathematics at the secondary school and the academic mathematics. This gap was made obvious as a result of, among other things, the work of Bourbaki [8]. ‘Nicolas Bourbaki’ was not a person but the name of a group set up by young mathematicians of the École Normale Supérieure in Paris in 1935 [9]. Dissatisfied with what they saw as the outdated content of the education they received at that school they resolved to think out anew the mathematics from its base and reformulate it as a logical unity. This resulted in the organization of dozens of seminars and the publication of a series of collectively written books. The Bourbaki group was in principle neither in direct contact with nor involved in secondary education. But one of its members, Jean Dieudonné (1906–1992), as advocate of New Math did play an important role in its dissemination in Europe and in particular in France. The impressive work of the Bourbaki group inspired the necessary awe, also in the world of the mathematics teachers, and so again underlined the gap between the academic mathematics and the school mathematics [10].

208


At the end of the 1980s Freudenthal saw it as follows: In onze eeuw heeft de wiskunde—net als vele andere wetenschappen—een geweldige groei in breedte, diepte en hoogte te zien gegeven. Maar er was meer: een aloude neiging om de wiskunde telkens weer te reorganiseren, en dat ook weer met wiskundige middelen, verstevigde zich tot een bewust beleefde stormachtige trend. Een Franse groep—Bourbaki—gaf er gestalte aan: een systeem van de wiskunde, vanaf de meest algemene begrippen, afdalende naar de meer en meer gespecialiseerde—een lange weg die in feite nooit werd afgelegd, maar toch als systeem indrukwekkend van architectuur. Indrukwekkend, maar onvoltooid, want onder het reorganiseren groeit de wiskunde tussen je vingers door. Niettemin, qua structuur een schoonheid, die vooral die mensen fascineerde die alleen maar die structuur zagen en nauwelijks wisten waartoe die structuur diende. In de jaren vijftig was het nog maar een kleine schare die erdoor gegrepen werd: volgens dit fraaie systeem moest wiskunde op school worden onderwezen. (In our century mathematics—like many other sciences—has shown an enormous growth in width, depth and height. But there was more: an ancient tendency to reorganize mathematics again and again, and this then also with mathematical means, was deepened to become a consciously experienced tempestuous trend. A French group—Bourbaki—gave shape to it: a system of mathematics, set up from the most general notions, descending to the increasingly specialized notions—a long path which in fact was never travelled, but as a system it was of an impressive architecture. Impressive, but unfinished, because when you are reorganizing mathematics it grows on between your fingers. Nevertheless, as far as structure is concerned a beauty, which mainly fascinated those people who saw only structure and hardly knew what its function was. In the 1950s it was still only a small crowd that was affected by it: in accordance with this splendid system mathematics had to be taught at schools.) [11]

Even in the Netherlands over the years complaints had been heard about the gap between mathematics at the university and at the secondary schools. These had indeed led to adaptations of the curriculum, albeit with a delay and less clearly recognizable as such. An example was the struggle that lasted for decades to introduce differential and integral calculus into the curriculum, something that only really came through with the introduction of the 1958 curriculum. Bridging the gap between the ‘higher mathematics’ of the university and the school mathematics did not even play the most important role in this discussion [12]. In the end attempts to introduce new subjects such as statistics failed at the last moment [13]. Prior to this Freudenthal had also complained about the fact that the school mathematics lagged behind modern mathematics. In 1950 he wrote in De Groene Amsterdammer about the outdated school mathematics which was “at a dead end” and lagged at least 50 years behind the ‘living’ mathematics [14]. But yet he too seemed satisfied with the change in the curriculum in 1958. In any case at the time he applauded the fact that the 1958 curriculum now no longer “had been overtaken by events” [15].

8.2.2 Modernization of the Mathematics Education in the United States The New Math movement was the result of the general dissatisfaction with the mathematics education in the United States after the Second World War [16].

8.2 New Math

209

In 1972 the mathematician Morris Kline (1908–1992) made a list of the inadequacies of that education: the algebra teaching relied too much on rote learning of an infinite number of arithmetical methods and tricks (‘drilling’) [17]. The deductive aspect of Euclidian geometry did not live up to its promise in the traditional American high-school curriculum. For many pupils it boiled down to learning proofs by heart. In addition, some parts of mathematics that had no longer been essential, were maintained in the curriculum for traditional reasons only, and some of those simply contained faulty mathematical statements. With such an outdated curriculum the mathematics education at the secondary school of course prepared pupils badly for the practical, technological and economic role that mathematics played in society. On the one hand there was an ever more obvious and increasing role for the application of mathematics. On the other hand different, more modern parts of mathematics (such as statistics and logic), which were not yet part of school mathematics, played an increasingly more important role. Above all the connection between mathematics education and the science studies at the university was bad, because mathematics education lagged decades behind the state of affairs in science. The solution, that ought to bring a didactical improvement as well, was sought in a modern and transparent structure of the content of the mathematics education by presenting mathematics as one logical entity. Within that mathematics the notion of structure was pivotal. Sets, relations and the emphasis on the accompanying exact mathematical usage of language were the most important spearheads. Transformation geometry and the usage of vectors ought to replace the Euclidian school geometry. In that sense it was a movement in which, perhaps even more than in the traditional education, the mathematics to be educated was the central point. The logicaldeductive structure of the mathematics was determinative for the structure of the subject matter and thus for the didactics to be used. This was based on the implicit idea that in mathematics education the mathematical structure could be compared to the psychological structure, or in any case that the psychological structure would not be a hindrance to the mathematical structure. The argumentation was that a logical structure from the beginning, in particular, would keep mathematics transparent for the pupils and this would increase the attraction of mathematics. Debating about the content of the didactical aspects of the education as advocated by New Math played (notably in the early years of the movement) a minor role. The origin of this approach could be found in a project from 1951. Led by mathematics teacher and educator Max Beberman (1925–1971), who worked at the University of Illinois, a project was drawn up to provide recommendations to raise the mathematical level of the prospective ‘Engineering’ students (which was later extended to comprise other exact sciences). These recommendations mainly consisted of a list of (more) modern subjects that would have to be raised up to a greater or lesser extent in the secondary mathematics education for these prospective students. This project created the ‘University of Illinois Committee on School Mathematics’ (UICSM), which in the early 1950s experimented with modern mathematics curricula for senior high school.

210


In 1955 the ‘College Entrance Examination Board’ created the ‘Commission on Mathematics’, chaired by professor A.W. Tucker, with the aim of investigating the “mathematical needs of today’s American youth” [18]. A large number of the terms and modern subjects from the work of the UICSM (University of Illinois Committee on School Mathematics) ended up in the curriculum designed by this committee. Originally also meant for senior high-school pupils who wanted to continue their studies, it later strongly influenced curriculum designs for junior high-school and primary education. The catalyst for the great diffusion of the New Math movement, however, was the launching of the Russian Sputnik in 1957. The fear that the United States might lag behind the Soviet-Union caused by this event, put modernization and improvement of the American education in mathematics and other exact subjects high on the Federal agenda. To stimulate this modernization substantial subsidies became available and various funds and committees were established. Fear and a feeling of inferiority in the scientific field were the reason that the public at large supported these developments. As a result the New Math supporters in the United States initially had an open field. The largest and most well-known project for experiments with modern subject matter was that by Yale professor Edward G. Begle, the ‘School Mathematics Study Group’ (SMSG), conceived in 1958 [19]. In the project a New Math curriculum for the complete primary and secondary school paths was designed and disseminated. It was remarkable that in the design of the curricula comparatively many mathematicians and only a few teachers were involved. Professional development courses at the institutes established with that purpose, on the other hand, were massively attended. Publishers broke into the developments by issuing textbook schemes based on New Math, even before these were adequately tested in practice. Yet these initiatives were actually criticized in the United States. In 1962, for example, on the initiative of Morris Kline, The Mathematics Teacher published a protest letter in which great concern was expressed about the effect of the new curricula which would present the pupils overly quickly with too much and too isolated mathematics, and in doing so would pass over the essence of mathematics: its connection with the reality of every day life and the link with other fields of study [20]. Dozens of leading American mathematicians signed this letter. In 1972 Morris Kline would elaborate on his objection to New Math in his book Why Johnny can’t add. In his book he described an amusing caricature of teaching of arithmetic based on New Math. One parent asked his eight-year-old child, “How much is 5 + 3?” The answer he received was that 5 + 3 = 3 + 5 by the commutative law. Flabbergasted, he rephrased the question: “But how many apples are 5 apples and 3 apples?” The child didn’t quite understand that “and” means “plus” and so he asked, “Do you mean 5 apples plus 3 apples?” The parent hastened to say yes and waited expectantly. “Oh,” said the child, “it doesn’t matter whether you are talking about apples, pears or books; 5 + 3 = 3 + 5 in every case”. [21]

In the course of the 1960s, ‘New Math’ with an abstractness that sometimes had been enforced to an absurd degree would gradually lose the sympathy of the public at large (and in particular of the pupils’ parents). Increasingly forced by critics the New Math movement dropped out of the picture in the early 1970s.

8.2 New Math

211

In a nutshell, the most important considerations at the introduction of New Math originated in society and science. A more ideologically biased motivation or ‘the interest of the child’ did not play a role. The psychological aspect too only played a marginal and implicit role: there were indications from psychology that pupils were, earlier than formerly thought, ready for ‘logic’ and ‘structure’. Support for this view was found in the work of psychologists such as Piaget and Bruner [22]. This motivated the New Math advocates to teach (mathematical) structure earlier and more emphatically. The need to underpin the new subject matter psychologically and if necessary pedagogically was hardly felt. The most important psychological foundation was this: ‘it is possible to teach this to pupils, and the transparent logical structure will make mathematics education easier and more attractive to the pupils’. Such an implicit projection of psychological findings on the practice of education was a thorn in Freudenthal’s flesh. The fact that certain subject matter could be taught in his view hardly influenced the decision to proceed to actually teaching it, let alone being a decisive reason. More than that: the fact that something could be taught in itself had little value, since with the right attention and effort everything could be declared ‘teachable’. Such a line of thought—it can be taught, so there is no reason not to do it—was according to Freudenthal simply abuse of the statistical results of psychology [23]. In the 1970s he would increasingly often give his opinion on the doubtful role that some of the psychological theories play in the development of mathematics education. Twenty years earlier he had not realized the threat to this extent. In his publications of the late 1950s and early 1960s he never spoke of the role of psychology in the development of the ideas based on New Math, nor did he mention Piaget or his work in this respect [24]. In Schrijf dat op, Hans he looks back at the course of events when New Math developed and at the role that psychologist Piaget would have played in it: Het begon dus niet pas met de Spoetnik-shock. Het begon al vroeg in de jaren vijftig. Zelfs Piaget, die er niets van begreep behalve dat het woord ‘structuur’ hem aansprak, hadden ze ertoe weten over te halen. Met Piagets naam op het reclamebord meenden ze zich verzekerd van de steun van de psychologie. Wat psychologie! Wiskunde kent een logische orde en voor wie wiskunde onderwijst is de verleiding groot de psychologische, de onderwijskundige orde aan de logische op te offeren. Ik heb mijn best gedaan dit te vermijden en in mijn ‘fragment Rekendidactiek’ van 1942, zo niet al vroeger, heb ik er anderen voor willen waarschuwen. Maar wat zag ik nu gebeuren? Een ten top gedreven logische orde, een systematiek van de wiskunde als geheel—zo zou de wiskunde moeten worden onderwezen. (It thus did not begin with the Sputnik shock. It had already begun in the early 1950s. They had even managed to convince Piaget, who did not understand anything of it except for the fact that the word ‘structure’ appealed to him. With Piaget’s name on the billboard they felt confident of the support of psychology. What now, psychology! Mathematics is ruled by a logical order and he who teaches mathematics is easily seduced to sacrifice the psychological, the educational order to the logical order. I have done my utmost to avoid this and in my ‘fragment Rekendidactiek’ of 1942, if not earlier, I wanted to warn others. But what was now happening before my very eyes? A logical order brought to ecstasy, a systematic of mathematics as a whole—that is how mathematics should be taught.) [25]

Nevertheless, the American developments of the late 1950s generally inspired great respect in Western Europe and led to various initiatives. The courses on modern mathematics for mathematics teachers in Belgium and the teaching material developed

212


by the ‘Scottish Mathematics Group’ in Great Britain (which was later known as the ‘Scottish method’ in the Netherlands) were only a few of the examples of the activities developed and were perhaps the most influential on the Dutch mathematics education. Within the ICMI the modernization of the mathematics education was also put on the agenda. The German mathematician (and chairman of the ICMI) Heinrich Behnke (1898–1979), for example, spoke at the ICMI congress in Edinburgh about the discrepancy between modern mathematics practiced at the university and mathematics taught at secondary schools. ‘Modernization of mathematical teaching’ was chosen as one of the subjects for the educational part of the ICMI congress, for which the NOCW composed a report entitled Modernization of mathematical teaching in the Netherlands [26]. This is only a small selection of the various initiatives towards a more modern mathematics education that were taken in Western Europe [27]. With hindsight the starting signal for these developments seemed to have been given at a seminar in the French city of Royaumont in 1959, and after that the New Math movement definitely spread over Western Europe.

8.3 Royaumont: A Bridge Club with Unforeseen Consequences At the end of 1959 the ‘Organisation for European Economic Co-operation’ (OEEC) organized a seminar about the modernization of mathematics education [28]. The OEEC was established in 1948 with the aim of coordinating the American financial aid for the reconstruction of (Western) Europe after the war [29]. At the end of the 1950s this goal had more or less been reached and in 1961 the organization was transformed into the ‘Organisation for Economic Co-operation and Development’ (OECD), which emphasized in its objectives the promotion of economic growth, employment and prosperity of its member states [30]. At the end of the 1980s Freudenthal saw the whys and wherefores of this seminar crystal-clear: Dankzij de Spoetnik-shock zagen ze hun kans schoon. De schoolwiskunde was zo ver achtergebleven bij de wiskundige wetenschap dat er schoon schip moest worden gemaakt. De OESO (toen nog OEEO), de organisatie voor economische samenwerking, pakte de zaak op. Wat had de OESO met onderwijs te maken? Het was het Bureau voor wetenschappelijk en technisch personeel waaronder de vraag naar nieuw wiskundeonderwijs ressorteerde. (Thanks to the Sputnik shock they saw their chance. The school mathematics had lagged behind the mathematical science so far that a clean sweep would have to be made. The OECD (at the time still called the OEEC), the organisation for economic cooperation, took the initiative. What business did the OECD have with education? It was the Bureau for scientific and technical employees where the demand for novel mathematics education belonged.) [31]

Freudenthal, by his own account, was “fed up with this kind of conference” and refused to be sent there as a delegate [32]. In Schrijf dat op, Hans he called this staying away “a cardinal mistake” [33].

8.3 Royaumont: A Bridge Club with Unforeseen Consequences

213

Royaumont was niet zo maar een conferentie, regeringen hadden er vertegenwoordigers naartoe gestuurd—een conferentie met gezag bekleed. Had ik, als ik er geweest was, de besluiten van Royaumont tegen kunnen houden? Ik had in elk geval nog harder kunnen schreeuwen dan mijn vermaarde vriend Dieudonné. Achteraf bekeken was het niet alleen een fout dat ik niet naar Royaumont ging, maar het ging dwars tegen mijn gewoonte in mij niet afzijdig te houden maar mee te werken om bij te sturen, een gewoonte die ik bijvoorbeeld later in de CMLW getrouw naleefde. (Royaumont was more than just a conference, governments had sent representatives—it was an authoritative conference. Could I, if I had been there, have stopped the decisions taken in Royaumont? In any case I could have shouted even harder than my illustrious friend Dieudonné. With hindsight it was not only a mistake that I did not attend the conference at Royaumont, but I went right against my habit of not standing aloof but rather cooperating to adjust the plans, a habit which I faithfully complied with in, for example, the CMLW.) [34]

But, contrary to what Freudenthal described in Schrijf dat op, Hans, it had not been such a distinct refusal of the invitation to attend the congress. Initially he was not at all negatively inclined towards the plans put forward by the OEEC and he did his utmost to influence the organization of the congress.

8.3.1 Freudenthal in ‘the group of experts’ It all started with a letter written by J.R. Gass, ‘Head of Division of the Office for Scientific and Technical Personnel’ (OSTP) that Freudenthal received from the OEEC in October 1958 [35]. The OSTP was established in 1958 with, amongst other things, the aim of upgrading the quality of scientists and engineers. The description of the aims in mind also included “increasing the effectiveness of sciences and mathematics teaching” [36]. In his letter to Freudenthal Gass made it clear that the OEEC had decided to support the attempts to improve mathematics education at secondary schools in the various member states. The Belgian mathematician W. Servais had pointed out Freudenthal as “a leading pioneer in the field of mathematics teaching” and as such he was asked to support this initiative. Freudenthal complied with this request by supplying a list with all Dutch initiatives for improvement of mathematics education. He mentioned among other things the work done by the ICMI, Euclides and the theses of the Van Hieles. Subsequently he joined the ‘OEEC group of experts’ which consisted of the professors Choquet, Servais, G. Walusinski (France), M.H. Stone (Chicago), Tucker and Howard F. Fehr (New York). The last three mentioned constituted a substantial representation of the New Math movement from the other side of the Atlantic. Freudenthal was invited as an ‘expert’ to the meetings of the OEEC. The most important aim of the OEEC was the (financial) support of existing activities. Apart from that new initiatives were also discussed. For the meeting in Paris on 13 and 14 March 1959, for example, the OEEC sent the experts a discussion paper about two projects. The first project was the plan to organize “a survey of the ‘Current practices and trends in school mathematics’”, followed by the second project that would consist of a seminar about “New thinking in school mathematics and possibilities of implementation”. The aim of the seminar was described as follows:

214


The seminar is to bring together a number of key people from the OEEC countries for a fruitful exchange of views on the basis of the results of this survey. From the experience of similar activities the Secretariat hopes that this project will also influence national action and will result in the more effective channelling of new ideas from other countries to those responsible for mathematical education in the OEEC countries. [37]

Freudenthal attended this meeting and a week later received a letter from R.C. Kwantes of the Ministry of Education. Kwantes, the Dutch member of the ‘Governing Committee’ of the OEEC, asked on behalf of the minister of Education for advice with respect to Dutch participation in a course in Darmstadt entitled ‘Mathematics, programming and introduction into mechanics of electronic computing’, which was financially supported by the OEEC. By return Freudenthal seized this opportunity to properly explain what he thought of the state of affairs in the OEEC and in particular what he thought of the meeting in Paris: Het lijkt me nuttig, U mijn algemene indruk uiteen te zetten. Deze was in hoge mate ontgoochelend. Met enige collega’s heb ik me afgevraagd, waarvoor men ons eigenlijk nog naar Parijs laat komen. Immers, het Governing Committee heeft al alles besloten en wij als deskundigen behoeven alleen nog maar Ja en Amen te zeggen. […] We waren niet in staat essentiële wijzigingen in het voor ons liggende plan aan te brengen, omdat het immers door het Governing Committee was goedgekeurd. (It seems useful, to give you my general impression. It was disillusion to a high extent. With some colleagues I wondered why we had been asked to go to Paris in the first place. After all, the Governing Committee has already taken all the decisions and we, as experts, only have to agree. […] We could not introduce essential changes in the plan put before us since the Governing Committee had already approved it.) [38]

With respect to the content Freudenthal saw no use at all in the OEEC plans. After all, he wrote, since time immemorial the ICMI produced the kinds of reports as envisioned by the OEEC and these never had any positive influence whatsoever. This report would also fail to contribute to improving mathematics education according to Freudenthal. Ik heb ter vergadering bij herhaling de vraag gesteld, naar de doelstelling van het rapport en geen antwoord gekregen […] Men had volstrekt de indruk, dat het alleen maar er om te doen was, een rapport te fabriceren, waarmee O.E.E.S. voor de openbaarheid kan treden, om zijn bestaansrecht te bewijzen. (During the meeting, I have repeatedly asked for the objective of the report and not received an answer […] One was completely convinced that the only reason was to produce a report which the OEEC could use for publicity to prove its right to exist.) [39]

The plan for the seminar was also evidence that the OEEC did not think beyond the standard terms according to Freudenthal: Bijeenkomsten zoals hier bedoeld zijn er te kust en te keur. De in het wiskunde-onderwijs belangstellenden ontmoeten elkaar geregeld—een der commissie-leden sprak van de bridgeclub—we kennen elkaar en elkaars ideeën en we kunnen op die manier nog enkele tientallen jaren door blijven sukkelen. Als de O.E.E.S. er niets nieuws aan kan toevoegen, hoeft ze er niet aan te beginnen. Ik vind het jammer voor de ton, die O.E.E.S. voor het wiskunde-onderwijs uitgeeft en die bijzonder nuttig had kunnen worden besteed, en ik vind het zonde voor de tijd, die voor ons in deze dilettantische projecten gaat zitten. Ik heb ter vergadering positieve voorstellen gedaan, die algemene instemming vonden, maar ze konden niet eens echt aan de orde worden gesteld, omdat het Governing Commitee nu eenmaal


215

alles had beslist. Mr. Gass lijkt me een uitstekende figuur, maar hij is ook aan handen en voeten gebonden. (There are meetings galore of the type such as intended here. The people interested in mathematics education meet regularly—one of the committee members spoke of the bridge club—we know each others’ ideas and can plod along in that way for many years to come. If the OEEC cannot add something new there is no need to start it. It is a shame about the hundred thousand guilders that the OEEC spends on mathematics education and that could have been spent particularly useful and it is a shame about the time we have spent on these amateurish projects. I have during the meeting made positive propositions that met with general approval but these could not even be really discussed, since the Governing Committee had already taken all the decisions. Mr. Gass looks a first-rate person to me, but he is also bound hand and foot.) [40]

Subsequently Freudenthal mentioned briefly that he was not the only one expressing this criticism and that the same sort of objections were also true for organizations such as UNESCO. As an example of such mismanagement Freudenthal mentioned the UNESCO congress in Bombay in 1956, which he attended, where hundreds of thousands of guilders were thrown away since not a single mathematics teacher had attended [41]. Yet he also contributed a positive proposal towards a possible task of the OEEC: establishing an international research institute for mathematics education. In such an institute the few people who really did scientific work in the field of mathematics education would be able to cooperate sustainably. Even when Freudenthal had expressed the wish for thorough scientific-pedagogic research before, this was the first time that he spoke of a research institute. Ik ben er van overtuigd, dat het wiskunde-onderwijs doeltreffend alleen maar kan worden geholpen door wetenschappelijk onderzoek, zoals wij uit onze instituten en laboratoria zijn gewend. Zodoende kunt U misschien begrijpen dat ik denk in termen van instituten en laboratoria, waar men in de administratieve sector aan enquêtes en congressen denkt. (I am convinced that mathematics education can only progress effectively by scientific research such as we are used to in our institutes and laboratories. Perhaps you can thus understand that I think in terms of institutes and laboratories, where people in the administrative sector think of inquiries and congresses.) [42]

In mid April Freudenthal received a letter and the accompanying documents from H. Metzger, manager of the ‘Office for Scientific and Technical Personnel’ of the OEEC in Paris, and was confronted with a completely detailed programme for the coming seminar. Three large sessions were planned: ‘New thinking in mathematics’, ‘New mathematics in education’ and ‘New materials and teacher training’ [43]. Investigating the need for research in the field of mathematics education was mentioned as one of the sub-goals, but the emphasis in the description of the goals was mainly on the modernization of the subject matter to be taught and the accompanying professional development courses of the teachers. Later it would become clear that this plan was in essence designed by the seminar chairman Stone—an outspoken advocate of a New Math curriculum [44]. All this was of course exclusively seen in the light of the enormous need of the society for technically-mathematically trained employees: the so-called ‘scientific manpower’ [45]. In the part ‘New thinking in mathematics education’ for example, the emphasis was not on educational methods and didactics, but on the algebra that

216


would serve as the basis of all mathematics education, and besides that “to bring to the fore, the relative importance of structure-concepts and -skills”, one of the most striking characteristics of the New Math movement [46]. The audience the organizers had in view for the seminar would consist of people with the following qualifications: The persons invited to the Seminar wil [sic] have the following qualifications (1) is recognised nationally (2) has contributed to the literature in the field (3) is a leader in his respective area of work (4) will give time and study wholeheartedly to the work of the seminar. [47]

Moreover the selected persons together would also have to represent the following factions: (a) mathematicians (university professors) (b) applied mathematicians (in industry, economics, physical sciences, social sciences) (3) educators and ministers of education, inspectors (concerned with teacher education and the curriculum) and (d) classroom teachers of mathematics. [48]

The speakers, to conclude, would be selected from those “who have done thinking, study, or publication of the topic involved” in such a way that all countries would be represented. Freudenthal’s objections against this scheme, of course, were substantial. In his answer to Metzger he blamed the seminar’s organization for neglecting all activities already existing and all discussions that had previously been held in the field of mathematics education (Fig. 8.1). It is true that O.E.E.S. can stimulate new research but they cannot create it by stamping on the ground. Addresses to be delivered at such a seminar should be the result of long years research, and not a collection of opinions. I would propose you first to ask serious investigators what they can tell, and afterwards to settle the programme. [49]

According to Freudenthal there was also an exaggerated emphasis on new subjects as the basis for a new curriculum. But the actual discussion in mathematical education runs on the question, whether new topics are really a panacea. There are people who believe that new topics will be worse than the old ones if they are taught with the old methods. [50]

This remark was the core of Freudenthal’s critical attitude on the consequences of ‘New Math’. He next proposed selecting ‘Methods versus topics in new mathematical education’ as a subject for the seminar [51]. According to him not only could greater unity be achieved in the programme, it would also imply that experts with different opinions could meaningfully discuss what Freudenthal called “this fundamental point of the modern educational controversy in mathematical education” [52]. In the answer the organization sent to Freudenthal they ‘thanked’ him kindly but decidedly for his suggestions and hints. The only essential and for the OEEC most attainable goal of the seminar according to Gass was to bring the activities in this field to the attention of the governments. In this way the desired support could be acquired for those actions that were seen as the most meaningful by the most


217

Fig. 8.1 a, b, c, d, e and f Letters written by Freudenthal to Metzger, Kwantes, De Vos van Steenwijk and Piekaar in which he lodged a complaint concerning the organization of the OEEC seminar in Royaumont (1959)

218

Fig. 8.1 (continued)




219

220





221

222




223

competent persons active in that field: mathematicians, education administrators and teachers. This is essentially the purpose of the seminar we are proposing. This is the reason why it should be rather wide-ranging. It is not intended that it should contribute particularly to the latest developments in professional discussions, but rather that it should produce the ‘bilan’. [53]

The subject for the seminar Freudenthal had suggested would come up for discussion in practice as a matter of course. There would also be a contribution about the state of affairs in research in the field of mathematics education. Following the written response of Freudenthal, the meeting of the group of experts on 25 May 1959 was no longer deemed necessary. Freudenthal would not settle for that. By way of Mrs. De Vos van Steenwijk of the Dutch delegation to the OEEC in Paris he once again tried to clarify his objections and standpoints: Indertijd was besloten op 25 mei bijeen te komen voor de voorbereiding van het “Seminar” over wiskunde-onderwijs. Kort geleden kregen we van Mr. Gass een uitvoerig ontwerp van het programma van het “Seminar”. Dit programma was een verschrikkelijke hutspot, waar werkelijk geen verstandig woord uit op te maken was. Ik heb dit ook zeer beleefd in mijn schriftelijke critiek gezegd. Ik ontving thans een brief van Gass, waarin met zoveel woorden staat, dat de organisatie van het “Seminar” al zo ver gevorderd is, dat een bespreking van het programma op 25 mei niet nodig meer is. (At the time it had been decided to meet on 25 May to prepare the “Seminar” about mathematics education. Not long ago Mr. Gass sent us a complete design of the programme of the “Seminar”. This was a terrible hotchpotch which really doesn’t convey a sensible word. I have mentioned this very politely in my written critique. Today I received a letter from Gass which in so many words states that the organization of the “Seminar” has proceeded thus far that a discussion of the programme on 25 May is no longer necessary.) [54]

Freudenthal in particular felt he had been put out of action as an advisor: the programme seemed to be definite even before any advice could influence it. Ik heb Mr. Gass hieromtrent niet geschreven. Ik vind het vervelend, om aldoor maar in de contramine te zijn. Zodra definitief komt vast te staan dat de bemoeienissen van OEES met het wiskunde-onderwijs niet in het serieuze vlak liggen, kan ik me altijd nog terugtrekken, maar sneu vind ik het hele zaakje toch wel. (I did not write to Mr. Gass about this. I feel uncomfortable to be continually uncooperative. As soon as is assessed definitely that the activities of the OEEC with the mathematics education are not seriously intended, I can always withdraw, but nevertheless I find the whole business a shame.) [55]

De Vos van Steenwijk explained in her answer that a few things were due, among other things, to the organizational structure and a lack of time. Nevertheless she pointed out that in particular this congress, “despite the fact that you disapprove of the programme”, could offer a valuable forum for what experts like Freudenthal saw as the most important problems, such as the educational methods. She thus stated implicitly that Freudenthal had better cooperate in the congress in order to be heard as an education expert [56]. Freudenthal also lodged a complaint in similar words about the course of events and about the “rather vague letter” he had received from Gass with Kwantes of the Ministry of Education [57]. He suspected the OEEC of not being willing to treat the

224


matter seriously, but only “to do something more or less spectacular, irrespective of what that would be” [58]. He concluded with the statement that he would leave it aside completely, whether something had to be done about it or not. In July of that same year Freudenthal was asked by dr. A.J. Piekaar, head of the department for university education of the Ministry of Education to recommend some people who would be eligible to participate in this congress, that is to say: a representative of the higher education (Freudenthal recommended Bunt or himself), an inspector of the VHMO and a representative of the teachers’ associations (Freudenthal recommended Wansink or Vredenduin). But, Freudenthal could not help adding, als u deze praktijk niet wilt volgen, zou u bepaalde mensen sturen wegens hun wetenschappelijke prestaties of belangstelling op mathematisch-didactisch terrein. Maar ik weet echt niet of u hun er een plezier mee doet. Het gehele programma is op het ogenblik zo vaag en verward, dat men van niemand kan verwachten, dat hij op zo iets toehapt. (if you do not want to pursue this practice, you should send certain people because of their scientific achievements or their interest in the mathematical-didactical field. I am not sure, however, if you do them a favour in doing so. The complete programme is at this moment so vague and confused that you cannot expect anybody to jump at it.) [59]

In the event of Piekaar feeling inclined towards this last option, Van Hiele, Chr. Boermeester and W.P. Thijssen ought to be considered, according to Freudenthal, noting that the first is the “only one who has done original and significant work in the mathematical-didactical field and has a world-wide reputation” [60]. The two others belonged to those people “with an intensive interest and a substantial passive scientific level”. Strikingly Freudenthal did not count himself as being in either of these categories. Or perhaps he did, but wanted to prevent, no matter how, being sent to the seminar after all [61].

8.3.2 Royaumont without Freudenthal: The Launch of New Math Freudenthal no longer wanted to be involved in the organization and ignored the seminar. Bunt, Vredenduin and Leeman did go to the seminar in Royaumont and reported on it in Euclides [62]. In this report, most attention was paid to the speech about the requirements that had to be set for a new mathematics curriculum by the Bourbakist Dieudonné. This was the speech in which Dieudonné spoke the famous words “Away with Euclid!” (“À bas Euclide!”) and in which he argued in favour of a modern system of axioms as a basis for the geometry education [63]. The description of Dieudonné’s speech in Euclides did not start without a previous warning to the readers: We waarschuwen de lezer er echter voor, dat hij met voorzichtige kritiek het betoog moet volgen. Leest men te snel, dan lijkt het geheel vaak fantastisch en verliest men de belangrijkste aanwijzingen voor verbetering, die de spreker gegeven heeft, uit het oog. De algemene mening was, als we ons niet vergissen, dat opvattingen van Dieudonné enerzijds zeer waardevol zijn, echter anderzijds met een korreltje zout genomen moeten worden. (We warn the


225

reader, however, that he should with mild criticism follow the line of reasoning. If you read too fast, the whole often seems fantastic and you lose from sight the most important clues for improvement the speaker gave. If we are not mistaken, the general feeling was that Dieudonné’s views are on the one hand very valuable, but on the other hand should be taken with a pinch of salt.) [64]

Despite this warning, the oversimplified, superficially interpreted version of Dieudonné’s ‘Away with Euclid!’ nonetheless became the slogan for the imminent modernization of the 1960s. In 1967 Freudenthal returned to this slogan in a review he wrote about Dieudonné’s study Algèbre linéaire et géométrie élémentaire (Linear algebra and elementary geometry), in which he spied the spirit of Euclid: The inventor of the slogan “A bas Euclide” a follower of Euclid? It looks odd, but it does so simply because few slogans have been misunderstood as badly as this one. Partly, it was Dieudonné’s own fault. When he cried “A bas Euclide”, he actually meant “A bas ‘Euclide’”, (viz. the Euclid of French lycée textbooks), but in oral discussion it is a hard thing to pronounce quotation marks. [65]

At the end of the Royaumont Seminar in 1959 several resolutions were adopted. Among those was a resolution about the necessity of teaching probability calculus and statistics in secondary schools and one about the necessity to prepare new textbooks in several countries. There was also a resolution in which the OEEC was asked “to inform the governments about the importance of this work and urge them to demonstrate the required flexibility with respect to the curricula and exams to enable the execution of experiments” [66]. Despite this careful formulation in the reporting in Euclides about the seminar, the Royaumont seminar and the New Math movement had made a serious impression on the Dutch participants. To Piet Vredenduin (1909–1996), text book author and mathematics teacher at a Gymnasium and one of the Dutch representatives in Royaumont, it was a revelation to become acquainted with modern mathematics. In 1985 he described his attendance at ‘Royaumont’: Laat ik beginnen met te vertellen dat ik een van de minst belangrijke deelnemers was. Maar, misschien daarom wel, heb ik er ontzettend veel geleerd. Ik had sinds mijn doctoraal veel aan logica gedaan en aan de grondslagen van de wiskunde, maar de wiskunde zelf had ik verwaarloosd. Ik merkte nu dat er sinds 1930 wel een heleboel veranderd was. Van structuren wist ik maar weinig en, je zult wel schrikken, hier kwam ik voor het eerst in aanraking met de set-builder voor het noteren van verzamelingen. Ik liep dus wel achter! (Let me start by saying that I was one of the least important participants. But maybe that is why I have learned an awful lot there. After my master’s degree I had done a lot of logic and foundations of mathematics, but I had neglected mathematics itself. I now noticed that since 1930 indeed a lot had changed. I did not know a lot about structures and, you will be shocked, here I firstly got into contact with the set builder for the notation of sets. I surely was behind!) [67]

The transparency, the clarity of the structure of the modern mathematics and the strictness of the logic that had to be applied to it appealed strongly to many teachers, particularly when it was compared with the patchwork that the content of mathematics education in the Netherlands had become after decades of tradition. The publications dealing with modern mathematics in Euclides made the teachers aware of the gap between mathematics at school and academic mathematics. A breeding

226


ground came into being for the wish to close this gap in order for teachers to keep up with science academically. Even though the influence of New Math in the Netherlands, particularly compared to that in neighbouring countries, was to remain relatively limited, with hindsight Freudenthal blamed himself for having underestimated the whole New Math movement. At the end of the 1980s he seriously thought that he could have done something at the Royaumont seminar to positively influence and limit the consequences and the effects of New Math (Freudenthal spoke of the “devastating trail all over the world through mathematics education”) [68]. But what exactly happened in the Netherlands? And what part did Freudenthal play?

8.4 Freudenthal on Modern Mathematics and Its Meaning for Mathematics Education 8.4.1 The Nature of Modern Mathematics ‘Structures’ and ‘relations’ were the primary ideas of the so-called modern mathematics that received so much attention and was so popular in mathematics education. The school mathematics of that moment was composed of barely interconnected parts such as trigonometry, algebra and geometry. Apart from acquiring and extending knowledge and techniques within the various sectors of mathematics, the emphasis in modern mathematics was increasingly put on investigating the logical structure and connections between the parts (the theorems and axioms) of such a sector. These had some form of hierarchy with respect to each other and constituted a logical framework of relations. By abstracting this framework and loosening it from the concrete mathematical context, the structures thus acquired could, in their turn, also be analysed at a higher logical level—the so-called axiomatizing. To Freudenthal, axiomatizing was the essence of modern mathematics. Using a concrete example (‘the Pieterse family’) Freudenthal had previously clarified the principle of the axiomatic method in his lecture at the Mathematisch Centrum’s summer course for teachers in 1948. First of all, he took the reader by the hand and started with an axiomatic construction of the geometry. Een exacte opbouw van synthetische meetkunde draagt heden de volgende kenmerken. Men begint met de dingen en relaties op te sommen, waarvan sprake zal zijn, dus b.v. “punten”, “rechten”, “ligt op”, “gaat door”, “snijden”, “verbinden”, enz. Wil men het zuiniger doen, dan beperkt men zich tot één enkele relatie tussen punten en rechten, “incidentie” genaamd, en zegt b.v. in plaats van “twee rechten a en b snijden elkaar in punt C”: C incideert met a en b. […] Het is trouwens niet nodig, dat op de opsomming der fundamentele begrippen meteen definities volgen, maar in elk geval komen nu de “axioma’s”. Dat zijn zekere uitspraken, waarin de ongedefiniëerde begrippen of reeds gedefiniëerde begrippen optreden. B.v. op elke rechte liggen minstens drie punten; door elk punt gaan minstens drie rechten; door elk tweetal punten gaat één rechte; elk tweetal rechten heeft één punt gemeen (aan het laatste axioma ziet u, dat het de projectieve meetkunde is, waar wij over spreken). Dergelijke axioma’s worden als het ware verordend, en alleen hierin schuilt hun waarheids-

8.4 Freudenthal on Modern Mathematics and Its Meaning for Mathematics Education

227

gehalte. (An exact construction of synthetic geometry currently has the following characteristics. You start with listing the things and relations that will be involved, so for example “points”, “straight lines”, “pass through”, “intersect”, “connect”, etc. If you want to be more economical, then you can limit yourself to one single relation between points and straight lines, called “incidence” and say for example instead of “two straight lines a and b intersect in point C”: C is incident with a and b […] It is, as matter of fact, not necessary that after listing the fundamental notions for these to be followed immediately by definitions, but in any case now the “axioms” come. These are certain statements in which undefined notions or already defined notions appear. For example, on a straight line there are at least three points; at least three straight lines pass through every point; one straight line passes through every pair of points; every pair of lines has one point in common (from this last axiom you can derive that it is projective geometry which we are talking about). Such axioms are, as it were, prescribed and only in that hides their veracity.) [69]

Next he showed by means of a parallel with ‘the holiday of the Pieterse family’ (an example well-chosen in the “holiday course”) the essence and the power of axiomatizing: Er zijn [sic] een heel stel meetkundige axioma’s nodig, om aan de ongedefiniëerde begrippen “punt”, “rechte”, “incidentie” een zin te geven. Dat zijn de spelregels, waarvan ik mij uitsluitend mag bedienen. Punt, rechte, incidentie zijn niets anders dan dingen, die aan deze spelregels voldoen (en wel bij wijze van een verordening), en het heeft absoluut geen zin, nu te vragen, wat die punten, rechten en incidentie “werkelijk” zijn. Wel mag men deze begrippen achteraf interpreteren. B.v. kunnen wij afspreken, met “punt” nu te bedoelen een “lid van het gezin Pieterse” en met “rechte” een “vacantiedag” en voor de uitspraak “het punt A ligt op de rechte a” nu te lezen: “A neemt op de dag a deel aan een ritje met de auto van het gezin Pieterse”. Onze axioma’s van daarstraks zeggen dan, dat op elke dag minstens drie personen uitrijden, dat elk lid van het gezin op minstens drie dagen mee mag, dat ieder met ieder precies één keer op stap gaat, en dat bij telkens twee uitstapjes precies één gezinslid is te vinden, dat aan beide uitstapjes deelneemt. Hoe groot het gezin Pieterse is, hoeveel mensen in die auto gaan en hoe lang die vacantie duurt, is door deze axioma’s nog niet bepaald. Maar als u nog als nieuw axioma toevoegt ‘het gezin Pieterse is niet leeg, maar overigens zo klein mogelijk”, dan kùnt u uitrekenen, dat het uit zeven personen bestaat, dat aan elke rit drie deelnemen, en dat de vacantie zeven dagen duurt. U hebt een projectieve meetkunde gekregen, die uit zeven punten en zeven rechten bestaat, waarvan elke rechte drie punten bevat en elk punt op drie rechten ligt. (Quite a few geometrical axioms are needed to give a meaning to the undefined notions “point”, “straight line” and “incidence”. Those are the only rules that I can use. Point, straight line, incidence are nothing else than things that meet these rules (and they do so by being prescribed), and there is absolutely no sense in asking here what these points, straight lines and incidences are in reality. What is allowed is that these notions are interpreted later. We can for example agree that “point” here means “a member of the Pieterse family”, and “straight line” is “a day off” and then the equivalent of the statement “point A is on straight line a” is “A joins on day a the Pieterse family for a drive intheir car”. The axioms we defined previously tell us, that on every day at least three persons go for a drive, that every family member can go for a drive at least on three days, that every family member joins every other family member for exactly one drive, and that in every two drives exactly one family member can be found who participates in these two drives. How many persons the Pieterse family numbers, how many persons fit in the car and how long the holiday lasts is not yet determined by these axioms. But if you add as a new axiom “the set of members of the Pieterse family is not empty, but as small as possible” then you can calculate that it consists of seven family members, that three of them participate in every drive and that the holiday lasts seven days. You ended up with a projective geometry consisting of seven points and seven straight lines, and of those every straight line contains three points and every point is part of three straight lines.) [70]

228


According to Freudenthal the possibility of this interpretation (in fact ‘mapping isomorphously’) between two (albeit mathematical or not mathematical) systems was the power of axiomatizing: Door niet expliciet te zeggen, wat punten en rechten zijn, maar slechts zekere, broodnodige, eigenschappen van hen te eisen, houdt men een ruimer armslag. Men wordt in staat gesteld, dezelfde conclusies overal meteen te trekken, waar men met dingen te maken heeft, die dezelfde eigenschappen vertonen als hetgeen men van de punten en rechten heeft verondersteld. Zo gaat de ontwikkeling in de moderne wiskunde telkens weer in de richting van het axiomatiseren. (By not stating explicitly what points and straight lines stand for, but only require that they have certain, much needed, properties, you have more elbowroom at your disposal. You are able to immediately draw the same conclusions everywhere, where things are involved with the same properties as those supposed of the points and straight lines. In that way mathematics again and again goes in the direction of axiomatizing.) [71]

Language and notation system play an important role in modern mathematics. Precisely because of a well-chosen, often efficient, concise notation the abstracting steps seemed to proceed ‘automatically’. This resulted in beautiful mathematical structures, which could easily be handled and be used for many more concrete (mathematical) interpretations by retranslating them into a specific case. Consciously organizing, manipulating and adapting the (mathematical) language, the so-called formalizing, was the latest trend in the modern mathematics of the preceding years. Because of the rise of this trend Freudenthal foresaw tremendous opportunities for the (applications of) mathematics. Formalization means the conscious organization of mathematical language, or more generally of language as such; and because language is more universal than mathematics, the impact of formalization is tremendous, and I am pretty sure that in a few decades formalizing will be the major activity of people who have to apply mathematics in other fields. […] Adapting the vernacular to match the increasing complexity of mathematical structures is in my opinion the most conspicuous trend in present mathematics, not only because of its proper value for mathematics, but still more as a paradigm which shows how mathematics can be applied in chaotic working fields like human language, namely by remodelling the field rather than simply by describing it. [72]

Thus considered, modern mathematics seemed only to have become more transparent and more organized compared to the ‘previous’ mathematics. In the early 1960s Freudenthal wrote about this in De Groene Amsterdammer: Het is niet gemakkelijk, om aan de buitenstaander het karakter van de moderne wiskunde duidelijk te maken. Als ik hem vertel, dat in de loop der eeuwen de wiskunde steeds gemakkelijker is geworden, zal hij mij niet geloven. Toch is het een onbetwistbaar feit. De wiskunde heeft zich fantastisch uitgebreid, en is desniettemin een menselijke bezigheid gebleven, beoefend door mensen, die niet meer hersencellen hebben dan hun voorouders, door veel meer mensen, dan men ooit voor mogelijk had geacht, door mensen, die zonder Archimedes’ evenknie te zijn, ontdekkingen kunnen doen, die niet voor die van Archimedes onderdoen. De wiskundigen zelf zorgen ervoor, dat de wiskunde hun niet boven het hoofd groeit. Met de uitbreiding van de wiskunde loopt parallel een proces van vereenvoudiging. Van grotere hoogte en uit een ruimer perspectief kan men de oude stof beter organiseren. (It is not easy to explain the character of modern mathematics to an outsider. If I tell him that in the course of centuries mathematics has become easier and easier, he will not believe me. Yet it is an indisputable fact. The mathematics has extended itself fantastically and has just the same remained a human activity, practiced by people who had no more brain cells


229

than their ancestors, by many more people than one thought would ever have been possible, by people who, without being equal to Archimedes, can make discoveries that measure up to those of Archimedes. Mathematicians themselves take care that mathematics does not become too much for them. Parallel to the extension of the mathematics a process of simplification is running. Seen from greater heights and with a broader perspective one can better organize the old subject matter.) [73]

All in all these developments in mathematics implied a great revolution in practicing it. The gap between the way mathematics was presented in mathematics education at secondary schools and the way in which mathematics was approached by mathematicians, had become even wider. It therefore seemed a logical step to bend the modernization of mathematics education in the direction of the most modern developments in mathematics itself and in the process use the umbrella tools such as ‘structures’ and ‘relations’ that had exhibited such tremendous use and power.

8.4.2 Modern Mathematics for the Public at Large Initially Freudenthal was positively inclined towards the attempts to incorporate more modern mathematics in mathematics education. This was in line with his ambition to make science more accessible to a larger audience, as he had devoted much attention to this ambition in his articles and lectures. His book Van sterren tot inlegzolen ( From stars to insoles) from 1954, a collection of short essays that for the greater part had been published previously in De Groene Amsterdammer, was illustrative of this ambition [74]. But also when he addressed an audience with more expertise he liked to speak and write about the most modern developments in science, as is evidenced by his MC lecture in 1948. A more popularizing book, which was also very comprehensible for the interested layman, was the book he wrote with the title Wiskunde in wetenschap en dagelijks leven ( Mathematics in science and daily life), and that was published in 1967 in several languages by the Wereldakademie [75]. Freudenthal had been invited to write this book by the initiators of this series [76]. This book too consisted of several essays about the role and function of mathematics, ranging in subjects from topology to the use and functioning of computers. In the introduction Freudenthal wrote to his readers: De wiskunde is niet alleen in de loop van haar ontwikkeling buiten haar oorspronkelijke grenzen gegroeid, maar ook zijn de grenzen tussen de verschillende gebieden der wiskunde vervaagd […] Het is één van de bedoelingen van dit boek, dat te laten zien, en de lezer moet niet schrikken als er in één en het zelfde hoofdstuk sprake is van ver uiteenliggende dingen als een koffiemolen en werelden ver weg in het heelal. De lezer moge het mij vergeven als ik het hem af en toe te moeilijk gemaakt mocht hebben; daar staat immers tegenover, dat ik het hem nooit te makkelijk gemaakt heb. (In the course of its development, mathematics has not only grown beyond its original boundaries, the boundaries between the different fields of mathematics have also become blurred […] It is one of the intentions of this book to show just this and the reader should not be frightened if in one and the same chapter very extreme subjects such as a coffee grinder and the worlds far away in the universe are discussed. I hope the reader will forgive me if I now and again made it too difficult for him; after all, on the other hand, I have never made it too easy for him.) [77]

230


From such publications it became clear that, to a certain extent, Freudenthal was confident that he could make most modern mathematics accessible for the public at large—in any case for an audience with an HBS level education. This confidence was also based on his old belief that people are not only educable, but they also want to be educated. In his lecture ‘Opvoeding tot denken’ this had already come strongly to the fore, as did the (independent) role he wanted to reserve for teachers to decide about their (refresher) training [78]. It was not only the possibility to make mathematics accessible to the public at large that Freudenthal saw, from his tirelessly enthusiastic initiatives in that direction it also became clear that he thought it desirable and necessary that this was done, if only to remove the existing misconceptions about mathematics. This was shown by statements such as the following, in which Freudenthal spoke about the wrong image the public at large had of mathematics, caused by the negligence of the mathematicians themselves: I admit that if anybody is to blame for these misjudgements it is the mathematicians themselves, who have failed to introduce students at the secondary level of instruction to true mathematics. [79]

More than that, if mathematicians did not want to recognize that they were on the wrong track, according to Freudenthal they ran the risk of completely losing their grip on mathematics education. Wenn uns die heutige Bedeutung der Mathematik in den Kopf steigt, so ist das Hochmut vor dem Fall. Der “Fall” wird sein, daß man den Mathematikern verbietet, Mathematik zu unterrichten, und diese lebenswichtige Aufgabe denen überträgt, die wissen, wie die Mathematik in die Erziehung und ins Leben paßt. (If the present meaning of mathematics has gone to our heads, then that is pride before the fall. The “fall” will be that mathematicians will be forbidden to teach mathematics and that this vital task will be given to those who know how mathematics fits in the education and in life.) [80]

“True mathematics” had to be taught, but in such a way that it fitted “in die Erziehung und ins Leben”. That was what Freudenthal saw as the assignment of mathematics education. Yet Freudenthal in his lectures and articles hardly ever gave his opinion about a concrete interpretation of the education, let alone declaring a curriculum built from modern mathematics. For the time being he stuck to a few hints for the direction in which improvements should be sought and the warning he repeated in several places that the introduction of new, modern subjects on their own would not imply an improvement of mathematics education [81].

8.4.3 The Mathematician “in der Unterhose auf der Straße” (“In his Underpants on the Street”) To Freudenthal, modernizing the subject matter was not by definition an improvement. According to him active changes in the didactical methods were needed to improve mathematics education. But modern mathematics could come in extremely


231

handy in the process. For, according to Freudenthal, a correct presentation and structure of school mathematics followed the principle of the level theory of the Van Hieles, in which the subject matter of a certain level of a pupil in the learning process became the subject of reflection at a higher level. And that was precisely one of the characteristics of axiomatizing. Apart from that the pupil should get, according to Freudenthal, the opportunity to (re)discover the mathematics himself, guided by the teacher—the so-called ‘Wiederentdeckung unter Fürhung’ or ‘guided re-invention’. In his article ‘Logical analysis and critical survey’ of 1962, Freudenthal admitted that clear pedagogic-psychological proof which should support this approach, had not yet been furnished. In this article he postulated that pedagogic-psychological research should decide whether the knowledge a pupil had acquired while ‘rediscovering’ was indeed better understood and would sink in better. Logical analysis could not provide an answer to this [82]. Guided re-invention, in fact the working method of a mathematician practicing mathematics, could hardly be found in text books. Freudenthal blamed the tradition of the contemporary publications in mathematics, in which only the end result, polished afterwards and stripped of all ancillary thoughts, was presented and the route towards it was dropped. According to him a kind of false modesty of the textbooks’ authors played a role. They had the misplaced fear that they would not come across sufficiently scientifically, sufficiently mathematically—in particular in the opinion of the professors. He said of the mathematical textbook authors: Wenn er von den Überlegungen, die ihn zum Ziele führten, etwas veröffentlichte, käme er sich vor, als stände er in der Unterhose auf der Straße. Dieser objektivierenden Gewohnheit bleibt er in didaktischen Publikationen treu. Er publiziert das Resultat und schweigt über den Weg zum Resultat. (When he would publish some of the considerations that led him to his goal, he would feel as if he was on the street in his underpants. He sticks to this objectifying habit in didactical publications. He publishes the result and keeps silent about the path to reach it.) [83]

And it was precisely this ‘path to reach the result’ that counted. There lay according to Freudenthal the key to developing the kind of mathematics education in which the pupil was allowed to learn to practice mathematics in a way that matched his level of understanding—the ultimate goal Freudenthal had in view. In his article ‘De algebraïsche en de analytische visie op het getalbegrip in de elementaire wiskunde’ of 1948 Freudenthal already mentioned briefly the benefit of knowledge of the so-called higher mathematics for teachers and he did it: “to be able to search for a fruitful synthesis—fruitful not only for the benefit of education but also for the benefit of science” [84]. Fifteen years later he wrote again about the usefulness of knowledge of ‘the higher mathematics’ for the teacher. But now he stressed the usefulness of the knowledge of modern mathematics: Konnte man früher zufrieden sein, wenn der Lehrer den deduktiven Stufenbau der Mathematik kannte, so müssen wir heute fordern, daß er im didaktischen Gedankenexperiment den Stufenbau des Wiedererfindens erkennt und solche Gedankenexperimente am realen Experiment überprüft Moderne Mathematik, die auch andere als deduktive Strukture zeigt, kann ihm dabei helfen Rein stofflich kann man von ihr aber nicht erwarten, daß sie die Schäden heilt, die die alte Mathematik im Unterricht angerichtet hat. (In the past we could

232


be satisfied when the teacher knew about the deductive stepwise structure of mathematics, but today we have to require that he recognizes in a didactical thought experiment the stepwise structure of re-invention and that he tests such thought experiments against reality. Modern mathematics that shows structures other than deductive structures can also be of help to him. Purely materially, however, we cannot expect it to remedy the damage that the old mathematics has done to education.) [85]

This statement concisely summarized the role Freudenthal saw for the modern mathematics in the mathematics education. Knowledge of modern mathematics would supply the teacher with the necessary insight in (deductive) structures. With that insight the teacher could subsequently present the subject matter to the pupil by thinking through the so-called thought experiment and analysing the path that the pupil was taking according to the principle of ‘guided re-invention’ [86]. The aim he had in view in mathematics education was to make the pupils familiar with mathematics in such a way that they could apply what they had learned—on problems outside the scope of mathematics as well [87]. Ich habe erfahren, daß die Mathematik mehr zu bedeuten hat, d.h. etwas für mehr Menschen als nur die, denen ein mathematisch bestimmter Schultyp angemessen ist; und diese Bedeutung kann dann sowohl praktisch wie theoretisch sein. (I have noticed that mathematics has more meaning, that is something for more people than just those who fit into a mathematical type of school; and this meaning can both be practical and theoretical.) [88]

The solution to this problem—structuring the mathematics education in such a way that (more) pupils could apply it—was hardly given attention, “since no one has thought it worth his while to really think about it” [89]. To Freudenthal, on the other hand, it became ever clearer that herein would lie the core for a sensible improvement of mathematics education. An improvement that was urgently needed, since: …eerlijkheidshalve moeten we bekennen, dat de meerderheid der leerlingen niet in staat is van de wiskunde die zij geleerd hebben, ook maar iets toe te passen—hetzij op andere schoolvakken, hetzij in beroepsopleiding of beroep. (…in all fairness we have to admit that the majority of the pupils is not capable of applying even the smallest part of the mathematics they have learned—albeit in other school subjects, in a vocational training or a profession.) [90]

8.4.4 Fairy Tales and Dead Ends Freudenthal also investigated the use of modern mathematics as a means to make mathematics more understandable and more attractive to a larger audience. His lecture on the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht from 19 to 23 December 1964, made this clear. He talked about two kinds of mathematics that had been created: the noble science reserved for scientists and the school mathematics, a diluted extract of the former, full of adaptations, which had assumed a complete life of its own: Sometimes in education we need adaptations and mother stork tales, and things would not have been so bad, if the adaptation, the stork tale had not developed its own life. After a century of separate grow [sic] our school mathematics looks like a dead end which leads nowhere, neither to higher mathematics nor to life. To be more definite, I estimate that in our Dutch secondary school mathematics, two thirds of the subject matter and all of the method is an end in itself instead of a starting point as it should be. [91]


233

The origin of this separation would lie in the conviction that mathematics as a science would be too abstract and too exact for the pupils. These prejudices could according to Freudenthal be eliminated by letting the pupils (re)invent the abstractness themselves and by adapting the exactness requirements to the level on which the pupils controlled the subject matter. An example that Freudenthal used in various lectures that illustrated what happened if these levels were not respected, consisted of the anecdotes of the service station attendant and the counter clerk [92]: De eerste [anekdote] speelt bij een P.T.T.-loket. Ik moest een buitenlands telegram opgeven. Het tarief was 17 ct. per woord, en het aantal woorden bleek na telling 23 te zijn. De man achter het loket loste het vermenigvuldigingsprobleem feilloos op. Na veiligheidshalve het sommetje nog eens na te hebben gerekend, vroeg hij mij ƒ3,91. Ik gaf hem vier gulden en hij gaf mij 9 ct. terug. De volgende klant was al aan de beurt toen hij mij terugriep. Hij controleerde nog eens de tekst van het telegram, en het bleek, dat hij volgens de geldende regels twee woorden als één had moeten rekenen. Toen klom hij opnieuw in de rekenkunst, vermenigvuldigde 22 × 17 = 374 , trok af 391 − 374 = 17 en vergoedde mij 17 ct, die ik teveel had betaald. Het andere verhaal speelde zich bij een benzinepomp af. Ik had van mijn wagen de olie laten vernieuwen, d.w.z. ik had 6 quarters olie à 43 dollarcents gekocht. De man bij de pomp schreef op: 0.43 0.43 0.43 0.43 0.43 0.43 2.58 (The first anecdote happens at a post office counter. I had to dictate a telegram to a foreign country. The rate was 17 cents per word and after counting them, the number of words appeared to be 23. The counter clerk solved the multiplication problem faultlessly. After he had recalculated the problem once again to be certain he asked me for ƒ 3.91. I gave him four guilders and he returned 9 cents. He was already helping the next customer when he called me back. He once again checked the text of the telegram and it appeared that he had to count, according to the statutory regulations, two words as one. Then he once again called on the art of arithmetic, multiplied 22 by 17 which gave 374, deducted 391 − 374 = 17 en gave me back the 17 cents I had paid too much. The other story happened at a service station. I had the oil refreshed in my car, which meant that I had bought six quarters of 43 dollar cents each. The man at the station wrote down: 0.43 0.43 0.43 0.43 0.43 0.43 2.58)

234


Freudenthal’s sympathy was undisputedly for the latter man, who “couldn’t care less about clever algorithms such as those used in multiplying” and indeed mastered a less sophisticated technique, but could apply this in practice in the relevant manner. The counter clerk on the other hand knew “thanks to acquired algorithms how to operate at a level, which was not really his” [93]. To this man arithmetic education (in any case as far as multiplying was concerned) was eventually a ‘dead end’. The danger that the innovated, modern mathematics subject matter would become a ‘dead end’ in the education ought, according to Freudenthal, to be overcome by professional development courses for teachers. Not only should the teacher become acquainted with modern mathematics, he also ought to be able to oversee it from a higher point of view and analyse it: He must be able to look upon the subject matter from a higher point of view. This was just a drawback of old school mathematics that it was a narrow dead end with no view from above (or at most so high a view that it could not be illuminating). [94]

Freudenthal had no doubts about the very real danger that mathematics in the curriculum could go in the wrong direction. Teachers could lapse into drilling techniques and tricks, without actually teaching the pupils mathematics that would be useful to them. More than that, he had seen already some signs of it in the application ofset theory in education [95]. The use of set theory was one of the most clearly apparent influences of New Math in arithmetic and mathematics education [96]. According to Freudenthal set theory was presented wrongfully as an independent mathematical subject: If set theory, with due problems, is taught at an early stage as a self-contained subject without being applied anyhow during many years or even at all, something is wrong. Set theory is not meaningful mathematics, but rather a linguistic feature of mathematics, and it has to be taught by the fact that meaningful things can be expressed in this language. A teacher who does not know much more about modern mathematics than he has to implement himself, will teach those subjects as an end in itself, as a dead end. [97]

In this way replacing old subject matter by modern mathematics obviously was not an improvement. The key to successfully introducing modern mathematics in mathematics education was according to Freudenthal to be found in the professional development courses for teachers.

8.4.5 Modern Mathematics as the Solution? Freudenthal seemed to have a deep-rooted, almost naïve trust in the effects of the professional development for teachers. If the professional development courses in the field of modern mathematics were adequate the education in mathematics would follow suit automatically. After participating in such a course the teachers would be capable of making their own choices from the new subjects and determining in what way they would have to teach these. It was not necessary to impose this from above. In particular the Dutch system, in which (apart from a legally regulated curriculum


235

and examination) teachers had great freedom in how to organize their lessons, was according to Freudenthal a valuable construction, which could also be maintained in a modern mathematics curriculum: In the past it were fundamentally the teachers who settled the programmes. […] In our country textbooks for secondary schools are not being written by university professors but by teachers, and there are no officially prescribed or even approved textbooks. Nobody wants to change this situation. However, I am pretty sure at the time when the new programme will become effective, our teachers will be able to write textbooks, to choose the subject matter to be taught and the didactic method to be applied in their class rooms. In any case this is a goal worthwhile to work for. [98]

Freudenthal was not immediately convinced whether this would also actually work out well. In his article ‘Het staartje van de mammoet’ (‘The little tail of the mammoth’; Sect. 8.5.5 explains the allusion) in De Groene Amsterdammer of 28 July 1962 he kept his options open as far as the positive influence that modern mathematics could have on the curriculum: Na enkele jaren is de “moderne” wiskunde net zo gedenatureerd als de traditionele en ze is bovendien in plaats van gemakkelijker een stuk moeilijker geworden. Zal dit weer het lot van de vernieuwing zijn? Of is er met moderne wiskunde iets anders te bereiken? Ja en neen. Neen, als men zich ertoe beperkt, nieuwe leerstof te introduceren, en driewerf neen, als het nieuwe leerstof is, die de leraren zelf niet uit een hoger perspectief kennen. Aan de andere kant is het een feit: men kan de ervaring, dat wiskunde denkactiviteit is en niet sommetjes oplossen, nergens beter opdoen dan in de modernste wiskunde zelfs bij een bescheiden hoeveelheid stof. En hoe vroeger de leerling deze ervaring opdoet, des te beter. Maar wil dit lukken, dan moet één voorwaarde vervuld zijn: dat de leraar zelf ervan doordrongen is, dat moderne wiskunde gemakkelijker is, en dat het erop aankomt, door wiskunde de denkacti viteit te stimuleren. De schoolwiskunde is in een vicieuze cirkel gevangen. Zal de moderne wiskunde ons helpen, hem te doorbreken? Ik hoop van wel. Maar dan niet in de vorm van nieuwe leerstof, die binnenkort even diep gedegradeerd zal zijn als de oude. Wel wanneer de kennis van moderne wiskunde leidt tot een nieuwe houding tegenover de wiskunde. Allereerst bij de leraren, dan bij de leerling, en tenslotte wellicht ook bij examinatoren. (After a few years the “modern” mathematics has been denatured as much as the traditional mathematics and on top of that it has become more difficult instead of easier. Will this again be the fate of the modernization? Or can we attain another goal with modern mathematics? Yes and no. No, if we limit ourselves to introducing new subject matter, and thrice no if it is new subject matter that the teachers do not know from their own higher perspective. On the other hand it is a fact that you can best experience mathematics as a thinking activity, and not just solving problems, in modern mathematics rather than anywhere else, even when the subject matter is limited in size. And the earlier pupils get this experience the better. But for this to work one condition must be met: the teacher himself must be convinced that modern mathematics is easier and that the crucial point is to stimulate thinking by doing mathematics. School mathematics has been caught in a vicious circle. Will modern mathematics help us to break it? I hope so. But preferably not in the form of new subject matter which will soon be degraded just as far as the old one. It will when the knowledge of modern mathematics results in a new attitude towards mathematics. First of all with the teachers, than with the pupils and finally with the examiners.) [99]

Once Freudenthal was convinced of the success of the professional development courses organized by the Commissie Modernisering Leerplan Wiskunde (CMLW, Committee for modernization of the Mathematics Curriculum), he did not doubt for one moment that this would actually work out.

236


8.5 Modernization of Mathematics Education in the Netherlands A ‘New Math movement’, such as the one that took shape in the United States and after that played a big role in the mathematics education in various WesternEuropean countries, never really happened in the Dutch mathematics education. But the same urge to modernize the mathematics education that New Math was based on was also clearly felt in the Netherlands. The mathematics curriculum of 1958, which was perhaps the first curriculum broadly supported both by the conservative and the progressive side, had hardly been adopted when the first calls for more and more drastic changes—‘real modernization’—resounded again. Gerretsen wrote in Euclides in 1958: Wij staan thans voor de dwingende opgaaf ons grondig te bezinnen op een volkomen nieuwe didactiek, waarbij men niet moet terugschrikken voor zeer progressieve veranderingen in het programma. Deze progressiviteit komt in het WIMECOS-programma nog weinig naar voren. Te veel wordt aan het bestaande vastgehouden, te weinig wordt rekening gehouden met de nieuwere ontwikkeling van de wiskunde die ook de school wat te bieden heeft. Zijn de leraren in staat een nieuw leerplan te ontwerpen, dat voldoende is afgestemd op de eisen, die de maatschappij stelt, rekening houdt met opvoedkundige idealen en voorts nauwer dan tot dusverre het geval is verband houdt met de levende wetenschap? Waarschijnlijk zal een dergelijke opgaaf slechts in internationaal verband tot een goed einde kunnen worden gebracht, want de problemen zijn voor alle landen van West-Europa analoog. (We are now confronted with the imperative task to profoundly consider a completely new didactics and in performing that task we should not be afraid of very progressive changes in the curriculum. This progressiveness is not yet very clear in the WIMECOS curriculum. The existing practice is held on to too much and the newer developments of the mathematics that also have something in store for schools are not sufficiently taken into account. Are the teachers capable of designing a new curriculum that is sufficiently tuned to the requirements of the society, takes into account educational ideals and is more closely linked to the living science than has been the case until now? Such a task will probably only be executed successfully at an international level, because the problems are the same for all countries in Western Europe.) [100]

This article gained immediate support. In his ‘Didactische revue’ (‘Didactical review’) in the same issue Wansink, for example, set himself the target of “showing that the call for a new curriculum with a progressive spirit can already be heard repeatedly both inside and outside of the Netherlands” [101]. After all, the 1958 curriculum was more an adaptation of the law to the educational practice, a careful step in the direction of less pretentious and more practically organized mathematics education, than a real modernization of that education. This was also the reason that it was immediately generally acceptable. In 1961 the WIMECOS chairman put it in the following way: Het programma van 1958 streefde geen modernisering na maar consolidering en uniformisering. Consolidering: voorkomen werd dat onderwerpen als differentiaal- en integraalrekening die reeds sinds 1921 op het gymnasium en sinds 1937 op de HBS op het programma stonden in de onderwijspraktijk een dode letter zouden kunnen blijven; uniformisering: de programma’s voor de B-leerlingen van hbs en gymnasium werden gelijk getrokken. […] Maar door de algemeen aanvaarde herziening, tot stand gekomen in de

8.5 Modernization of Mathematics Education in the Netherlands

237

vijftiger jaren, wordt de weg geëffend voor een effectieve modernisering in de zestiger jaren. Een modernisering, waardoor Nederland internationaal gezien niet te ver bij andere landen zal behoeven achter te blijven. (The 1958 curriculum did not aim for modernization but for consolidation and uniformization. Consolidation: such subjects as differential and integral calculus, which were part of the curriculum at the Gymnasium since 1921 and at the HBS since 1937, were prevented from remaining a dead letter in the educational practice; uniformization: the curricula for the HBS-B and Gymnasium-β pupils were levelled. […] But this generally accepted revision, brought about in the 1950s, paved the way for an effective modernization in the 1960s. A modernization, which means that the Netherlands will not have to lag too far behind other countries.) [102]

Yet it was remarkable that a curriculum, which Freudenthal euphorically described in 1955 as a “clear-cut” decision, inspired by “courage and self-confidence and the serious wish to take a substantial step towards better mathematics education” and breaking with a “90-year old, familiar tradition”, was considered outdated so soon [103].

8.5.1 Initiatives Inside and Outside of the Netherlands The pressure from abroad and the resulting additional fear it brought that Dutch mathematics education would lag behind kept increasing. This effect was even enhanced by the fact that international cooperation in the field of didactics and education of mathematics started to play a larger role as a result of, among other things, the work of such organizations as the ICMI, UNESCO and OEEC [104]. These international developments caused the dormant discontent with the size and the content of the Dutch mathematics education and the lack of boldness in the curricula reviews of both 1939 and 1958 to re-surface. In these years Euclides was filled with articles about modernization initiatives and the accompanying curricula and experiments in other countries. In Belgium for example the modernization of mathematics education was well under way. Besides Servais Professor George Papy and his wife, teacher Frédérique Papy, were the prominent individuals within the Belgian modernization movement of mathematics education. All three of them were advocates of modernization of mathematics education in the direction of ‘New Math’ and as such were active in the OECD. The ‘Commission Internationale pour L’Etude et l’Amélioration de l’Enseignement des Mathématiques’ (CIEAEM) (International Committee for the Study and Improvement of Mathematics Education), in which Papy was active, had a strong influence in Belgium [105]. In 1950 this international group had been established by among others the Bourbakist Dieudonné, Caleb Gattegno, Piaget, Papy and E.W. Beth. Freudenthal would officially also belong to the founding fathers of the CIEAEM, but by his own account he was never aware of it and would have kept himself apart from the group until the 1970s [106]. As a result the CIEAEM initially had little influence on the Dutch situation [107]. As a result of a CIEAEM congress in 1958 a group of Belgian participants framed a curriculum ‘modern mathematics’ and after the (practically immediate)

238


commitment of the minister of education Van Hemelrijck it was experimented with [108]. Papy was pioneering these experiments, which would continue until a curriculum ‘modern mathematics’ was introduced on a national scale in 1968 [109]. The experiments involved mathematics curricula for kindergarten (Venn diagrams and simple topological structures) up to and including the final year in secondary school. They included completely renewed subject matter in which room was reserved for sets, relations and vector calculus in the lower classes (12–15 years) and for linear algebra, analysis, statistics and probability theory in the upper classes. In the Netherlands the modernization of the Belgian mathematics education was followed attentively but it was not directly copied in the Dutch mathematics education. Krooshof characterized the Belgian modernization plans in 1966 to be “new houses” whereas his proposal for the Dutch mathematics education was only “renovation” [110]. Vredenduin too felt hesitant about the extreme revisions that Papy particularly supported [111]. But at the same time he was also very impressed by the Belgian efforts. In particular the experiments with modern mathematics in Belgian schools fuelled the feeling that the Netherlands was ‘lagging behind’. In Brussels in 1959 Papy established the ‘Centre Belge de Pédagogie de la Mathématique’ (‘Belgian Centre for Mathematics Pedagogy’), that among other things offered a course on subjects in modern mathematics for mathematics teachers every year [112]. In 1984 Vredenduin spoke about refreshing his knowledge of modern mathematics at the end of the 1950s: Verder heb ik veel geleerd in België. Daar waren ze ons voor. Papy hield elk jaar een cursus van een week over moderne onderwerpen in Arlon. Die waren uitstekend. (For the rest I have learned a lot in Belgium. They were ahead of us. Every year Papy organized a course of a week on modern subjects in Arlon. These were excellent.) [113]

In addition, Vredenduin regularly wrote admiring reviews of newly published volumes of Papy’s Mathématique moderne ( Modern mathematics) [114]. In this period, apart from descriptions of foreign New Math initiatives many articles about subjects of the modern mathematics were also included in Euclides. The (micro)didactical articles which were less often included, retreated into the background because of this. Modernization of mathematics education with respect to content became a goal in itself. The ICMI, and because of it the NOCW too, were soon also engaged in the question of the modernization of mathematics education. In a NOCW report from 1962 about the modernization of mathematics education in the Netherlands, Loonstra and Vredenduin wrote in the introduction: The teaching of mathematics at the secondary level has become old-fashioned in many countries, the Netherlands not excluded. One may even say that it has not accounted for the progress of mathematics in the last two centuries. The rapid development of society and science in the last few decades and, as a consequence of this, the tremendously increased need for mathematically skilled people, compels us to look for the drawback [sic] in our education and to remedy them if we can. [115]

On an international level, mathematicians (including Freudenthal) worked in an ICMI context, on the realization of several volumes of the works Grundzüge der Mathematik (Foundations of Mathematics), intended as a handbook for mathematicians and in particular for mathematics teachers since 1955 [116].


239

The Wiskunde Werkgroep too found it hard to avoid this tendency of modernization. As before, the WW successfully organized the weekend conference every autumn; with the texts of the lectures being published in Euclides. The subject of the weekend conference of 1959, for example, was ‘Vernieuwing van het wiskundeonderwijs door nieuwe programma’s en door nieuwe didactische methoden’ (‘Renewal of mathematics education by new curricula and new didactical methods’) [117]. However, with the realization of the 1958 curriculum the modernizing role of the Wiskunde Werkgroep appeared to be played out. The most stimulating, innovating impulses came from outside the Netherlands. The battlefield, in which until well into the 1950s the conservative side (represented by the teachers associations WIMECOS and LIWENAGEL) and the progressive side (represented by the WVO and the WW) were clearly separated, dissolved. In 1962, for example, Euclides became the official medium of the WW and the WW activities were more or less absorbed in the flow of innovating initiatives of that time.

8.5.2 Freudenthal: From WW to ‘Cooperate with a view to Adjust’ Even though, apart from 1960 when he was in the United States because of his sabbatical year, Freudenthal remained chairman of the WW weekend conferences, his involvement in the working group was dwindling. Once his children at the end of the 1950s had left their schooldays behind them, Freudenthal’s direct stimuli in that field disappeared. Mirjam, the youngest, was the only one still attending the secondary school (the Amersfoort’s Lyceum) at the end of the 1950s. Mrs. Freudenthal’s activities in the educational field—and in particular those resulting from the WVO activities—increased considerably in this period. From 1950 onwards she was secretary of the Centrale Werkgroep of the WVO and a member of the ‘International Council’ of the New Education Fellowship. As such she played an important part in maintaining the international contacts between both groups [118]. In addition she was from 1956 onwards an editor of the WVO journal called Vernieuwing van Opvoeding en Onderwijs, in which she very regularly published articles herself. But she spent most of her energy on the ensuing work that the ‘Werkgroep Jenaplan’, established in 1959, and on writing publications in the field [119]. She was the one who introduced the Jenaplan Education in the Netherlands [120]. In 1969 as a result of the increased interest in the concept of the Jenaplan schools the ‘Stichting Jenaplan’ (‘Jenaplan Foundation’) was established along with its journal Pedomorfose: een internationaal oriënterend tijdschrift voor en over scholen die op pedagogische en didactische gronden, flexibele, verticale en horizontale groeperingen nastreven (an internationally oriented journal for and about schools which based on pedagogic and didactical grounds strive for flexible, vertical and horizontal groupings) [121]. Mrs. Freudenthal was secretary of the foundation and editor of Pedomorfose [122].

240


Despite the similarities and the joint origin of their interest fields and professions, Freudenthal and his wife were mainly active in education alongside each other. Occasionally Freudenthal published in Pedomorfose, but he never gave his opinion on Jenaplan Education in public. Mrs. Freudenthal would at the end of the 1960s be involved in teaching of arithmetic for a short while. At that time she was, along with J.J. de Iongh, a former student of Freudenthal and from 1960 onwards mathematics professor in Nijmegen, involved in the project Denken en Rekenen ( Thinking and Arithmetic). Following international tendencies this initiative from publisher Malmberg had as its aim the modernization of teaching of arithmetic [123]. But Freudenthal did not interfere with this initiative with respect to content [124]. It seems as if, since Freudenthal left the WVO and the WW behind at the end of the 1950s, Mrs. Freudenthal on the contrary became very close with these organizations. Freudenthal had in the early 1960s apart from educational affairs other things that also claimed a significant part of his attention. First of all there was the academic year 1960–1961 at Yale University when he lived with his wife and his daughter in New Haven. Preceding his stay there, many letters had to be written, from October 1959 onwards. At first Freudenthal wrote to several American universities and one of the important requirements was always that there had to be a good school for Mirjam in the neighbourhood [125]. Mrs. Freudenthal used the year in the United States well by visiting schools and institutes that were important to her work. Freudenthal enjoyed his time there tremendously. At his own request he lectured on ‘Lie groups’ and on ‘Foundations of geometry’ and visited several other universities to lecture. Many of these lectures were about his book Lincos. Design of a language for cosmic intercourse. Part I [126]. In this book Freudenthal described a ‘language’ built from signals and based on logic, Lincos ( Lingua Cosmica), which would enable us to communicate with extraterrestrial intelligence. Starting from the standpoint that elementary mathematics “was the most universal thing there is in the universe” Freudenthal formulated a programme for extra-terrestrial communication that started with communicating the natural numbers in code. Then examples of basic arithmetic techniques followed from which the receiver might learn the language [127]. It was—in spite of the aim of the language that fired the imagination—a rather logical-theoretical discourse. The book was published in the series ‘Studies in logic and the foundations of mathematics’ (of which Brouwer, E.W. Beth and A. Heyting were the editors) and the artificial language was a popular subject for lectures and reviews [128]. The Freudenthals concluded their stay overseas with a seven-week trip through the United States. Back in the Netherlands Freudenthal wrote his colleagues in New Haven a letter, in which he indicated how much he missed everything and how difficult he found re-acclimatizing in the Netherlands. Even though he had been kept up to date of the ins and outs of the Mathematisch Instituut by his secretary, Mrs. Breughel, and by Van der Blij during his absence, in a practical sense he had been ‘absent’ for a whole year [129]. Two years later Freudenthal was again more or less absent for an academic year, since on 16 September 1963 W.C. van Unnik, professor in theology, turned the rec-


241

Fig. 8.2 Van Unnik turns the rectorship of the Utrecht University over to Freudenthal (1963)

torship of the Utrecht University over to him (Fig. 8.2). This job would completely occupy Freudenthal for a whole year [130]. In particular such things as the ‘student quota’ resulting from the scarcity of lecturers and the ragging (Freudenthal was opposed to this by the way) that were discussed widely in the press kept him busy that year. On 21 September 1964 he turned the rectorship over to Professor L.J. Hijmans van den Bergh.

242


Apart from the sabbatical year and the rectorship there was another reason for Freudenthal’s diminishing interest in the WW. It seemed his conscious choice to put his energy into other education committees, such as on a national level the NOCW and internationally the ICMI. ‘Cooperate with a view to adjust’ was Freudenthal’s motto in such committee work even when at first he did not agree to its objectives. Seen in that light the WW no longer had much to offer him. The real game had long been played out in other committees installed by the government and Freudenthal’s reputation almost automatically ensured that he was fully involved.

8.5.3 The Commissie Modernisering Leerplan Wiskunde One of those committees was the Commissie Modernisering Leerplan Wiskunde (CMLW) that was appointed in June 1961 by the then State Secretary OK&W, G.C. Stubenrouch. The objective of this committee was “to study the modernization of mathematics education in the secondary education” [131]. The committee usually met in Utrecht. Its task was twofold: to decrease the gap between school mathematics and academic mathematics and to increase the social relevance of that school mathematics. In the first place the committee mainly had an advisory function on all aspects of the modernization of mathematics education: ranging from problems in choosing the modern subjects, if there were any, its consequences for the curriculum and the final examinations to the professional development for teachers. It was the first time that such an initiative to revise the mathematics curriculum was taken by the government and not by the field itself in the Netherlands. This tendency had already become apparent in the realization of the OEEC seminar in Royaumont as well. It was among other things connected to an increasing involvement of the Dutch authorities in the organization and modernization of the education at the end of the 1950s and the accompanying legislation in that field [132]. In the 1960s considerably more government money also became available for education. The enormous increase in the number of pupils (more pupils, in addition originating from a broader spectrum of social classes) made this necessary [133]. There was an increasing need for education and training of the complete (working) population. The result was an increase in scale of the secondary education with to a certain extent a professionalization of the education modernization. The establishment of the CMLW befitted this tendency. The committee with chairman H.T.M. Leeman, professor in mathematics, mainly consisted of professors and only a few mathematics teachers. Apart from Freudenthal among others Van der Blij, the Utrecht lecturer in didactics Bunt and A.F. Monna (1905–1995) were appointed. Monna was at the time both mathematician and deputy head of the department for secondary education of the Ministry of OK&W, and in addition one of the initiators to establish the CMLW [134]. The main results of the CMLW’s activities were twofold: first of all the reorientation courses for mathematics teachers and secondly the development of a new mathematics curriculum.


243

In particular the success of the first result made Freudenthal very happy. Nevertheless, as appeared from the picture he sketched at the end of the 1980s, he had the usual doubt when he started his work as a CMLW member. Mijn eerste reactie op de CMLW-plannen was negatief (25 augustus 1961). Toch heb ik— voor zover men het me toestond—loyaal meegewerkt. Trouwens, ik had niets anders kunnen doen—de tijd was nog niet rijp voor mijn denkbeelden. Ik was nog steeds het enfant terrible. Ik mocht er wel voor spek en bonen meedoen. (My first reaction to the CMLW plans was negative (25 August 1961). Nevertheless I have—as far as I was allowed—cooperated loyally. I could not have acted otherwise—the time was not yet ripe for my ideas. I still was the enfant terrible. I was allowed to take part for decoration.) [135]

The ‘negative reaction’ Freudenthal spoke about, could not be found in a letter of 25 August, but was present in a letter to Monna dated 23 August 1961. In that letter Freudenthal did not agree with the emphasis on the modernization of the curriculum in the committee’s assignment: Ik heb diverse malen, naar bekend zal zijn, betoogd, dat ik de modernisering van het leerplan zoals deze op het ogenblik door velen wordt gepropageerd, geen urgent probleem acht, en wel niet omdat ik aan moderne wiskunde een hekel zou hebben, maar omdat in de diverse voorstellen de introductie van moderne leerstof als principieel doel wordt gezien. Dientegenover zie ik als eerste en enige urgentie een verbetering van het wiskunde-onderwijs. (I have, as will be known, argued several times that I do not deem the modernization of the curriculum, as propagated by many, an urgent problem, and I do so, not because I would hate modern mathematics, but because the introduction of modern subject matter is considered a principal goal in the various proposals. On the other hand I consider as the first and only urgency an improvement of mathematics education.) [136]

Freudenthal considered the danger of a worsening as a result of the modernization not purely hypothetical if the subject matter “could be denatured as easily or even more easily than the old subject matter and when it could not be expected that the teachers had mastered the new subject matter sufficiently” [137]. It was not the practical use of the subject matter itself that had to decide the choice, but the subject matter of the secondary school should be of a fundamental nature, “which does not alter the fact that applications of the subject matter can be directed at practical use” [138]. He wanted the committee to design an alternative curriculum for both lower and upper secondary school, and to offer this for a certain period alongside the current curriculum in order to abolish the old curriculum at a suitable time later. Freudenthal repeated these points in the meeting in October 1961. He argued in favour of starting with the first changes in the first few years. His plea indeed resulted in the installation of a committee, under his leadership, for revision of the curriculum for the lower secondary level, besides some committees for the modernization of that for upper secondary level. He was also a member of the committee concerning the ‘professional development for teachers’. For the outside world the findings and the results of the CMLW remained a secret at first. In 1962 WIMECOS’ chairman B. Groeneveld appealed implicitly to the committee to speed up its work and not to waste too much time on such surmountable problems as ‘didactics’—a typical attitude in this period filled with desire for innovation:

244


Wel zal het, volgens de voorzitter Prof. Leeman, lange tijd duren voor de commissie met haar werk gereed is. De zeer dringende noodzaak van de modernisering—we zijn toch al zover achter t.o.v. het buitenland—maakt het noodzakelijk, dat er in vlotter tempo wordt gewerkt. Over de doceerbaarheid van nieuwe onderwerpen behoeft de commissie zich niet al te veel zorgen te maken. Dikwijls lijken de onderwerpen didactisch veel moeilijker dan ze blijken te zijn, omdat juist de didactiek zich meestal weet aan te passen. (It will, according to the chairman Professor Leeman, take a while before the committee finishes its work. The very urgent need for modernization—we are already so far behind with respect to other countries—makes it necessary to work at a more rapid pace. Whether the new subjects can be taught or not, the committee does not need to worry too much. The subjects often seem didactically more difficult that they eventually are, since it is in particular the didactics that knows how to adapt.) [139]

Even the WW—which in particular gave the modernization of the didactics the greatest attention—started to consider the modernization of the subject matter a goal in itself. In 1964, for example, within the WW a plan was realized by appointing several committees to investigate in which way modern mathematical methods could be incorporated in the curriculum. Adaptation of the current curriculum was the main thing and not—as the CMLW had as an objective—a complete renewal of that curriculum [140]. A call for cooperation directed at the readers of Euclides was signed by Freudenthal as chairman of the WW. Six months later the call was repeated [141]. But the initiative only got airborne after a lot of effort and had little influence on the rest of the proceedings. It made little difference to Freudenthal who was involved in both the curriculum activities of the WW and the CMLW. The question is to what extent he still believed in such an initiative of the WW. Perhaps he gradually realized at the time that the days of such small-scale working groups were over.

8.5.4 A Professional Development Programme for Teachers It took until 1965 before a first official announcement of the CMLW appeared in Euclides [142]. But before that the committee had become known through the courses on modern mathematics it had organized for teachers since 1963. At the start of these courses within the CMLW there was the usual scepticism with respect to the results of such courses. At the end of the 1960s Freudenthal said about this: Geen onzer had verwacht, dat de overgrote meerderheid der deelnemers de denken werkmethoden der moderne wiskunde en haar begripsstructuren zó intens zou assimileren als in feite is geschied. Wiskunde van welk soort dan ook is blijkbaar een solide basis voor andere soorten wiskunde, hoe sterk de onderlinge verschillen ook mogen zijn. (None of us had expected that the vast majority of the participants would assimilate the thinking and working methods of modern mathematics and its structural concepts so intensively as has actually happened. Mathematics of any kind apparently is a solid base for other kinds of mathematics, however great the differences between them may be.) [143]

The approach, a combination of lectures and working groups (the way Freudenthal also lectured in Utrecht) unexpectedly appeared to be very effective. The participants—almost half the number of active mathematics teachers at the HBS and the


245

Gymnasia, a turnout that exceeded all expectations—were not taught a ready-made part of modern mathematics to pass that on to the pupils. By contrast an attempt was made to convey the spirit of modern mathematics to them, so that they would understand the necessary ideas and theory behind a modern mathematics curriculum and would be able to develop a well-founded approach themselves. In a lecture at the international colloquium ‘Modern curricula in secondary mathematical education’ in 1964, Freudenthal discussed the unexpected success of these courses: The success was overwhelming, above any expectance. Our sceptical prejudice was put to deadly shame. […] The main thing: the vast majority of the attendants were able to assimilate the taught matter, a fact that was proved by their ability to solve the set problems. Still more, all things happened in a climate of sweeping enthusiasm. […] Attempts are made to transform this refreshing work into a standing institution. [144]

Since it became clear that the reorientation of the teachers would be a durable phenomenon the CMLW filed a request for a professional institute with the State Secretary in 1964 [145]. The huge success of the CMLW courses was according to Freudenthal the result of the active participation in the working groups: Listening courses on new subjects like modern mathematics are a source of frustration rather than of diffusion of knowledge. The listener mainly learns that modern mathematics is a good thing for professors or perhaps for geniuses, but inaccessible for simple teachers. It has been a revelation for many people that modern mathematics is a beautiful thing, that it is something you can do yourself and you can learn by doing—the only thing you need is to get the opportunity to do it. [146]

Freudenthal was convinced that after the successes of the courses the consequences of the modernization of mathematics education would prove to be better than expected. Now that they were well-prepared for that modernization the teachers could consciously design their lessons. Any bad influence of the textbooks that aimed too much at the content instead of at the form would be limited because of it. In The New Era he wrote in that same year: Mathematics has to be taught not by textbooks, but by teachers. Modern textbooks in the hands of teachers who do not know or have not understood modern mathematics, are not a dangerous weapon, they are simply useless. The first thing we have to do is […] to retrain teachers so that they are able to judge modern mathematics and to choose among its formal aspects those which might improve teaching methods. [147]

The professional development courses of the CMLW had achieved this brilliantly according to him.

8.5.5 A New Curriculum The second big assignment of the CMLW, the development of a new curriculum, received an extra impulse in 1963 because of the reorganization plans of the educational system. Like his predecessor F.J.T. Rutten, minister of education J.M.L.T.

246


Cals was also a supporter of a reorganization of the secondary education [148]. On 26 February 1963 the Wet tot regeling van het voortgezet onderwijs (Law to regulate the secondary education), popularly called the ‘Mammoth Act’, was published in the Staatsblad (Bulletin of Acts). To make a transitional arrangement possible the date of introduction was fixed on 1 August 1968. This allowed experimenting with various new types of education. The most important goal of the act, which regulated all types of secondary education, was to enhance the possibilities to move to the various types of education both vertically and horizontally. The HBS was partly incorporated in the VWO (preuniversity education), which comprised Gymnasium, Athenaeum and Lyceum, and partly continued as HAVO (Hoger Algemeen Voortgezet Onderwijs, Senior General Secondary Education). MULO and MMS disappeared, MAVO (Middelbaar Algemeen Vormend Onderwijs, Junior General Secondary Education) was created. New was also the general introduction of joint education for the first year(s) and differentiated final examinations as a result of the pupil’s choice of subjects. An important starting point in all these changes was the promotion of external democratization: to set the conditions in which the individual possibilities for all pupils to develop themselves would be optimally used and in which more pupils would have the opportunity to participate in higher types of education. For the CMLW, this development resulted in a practical sense in an even greater urge to realize new curricula (for all types of education mentioned in the Mammoth Act), so that these curricula could take effect at the same time as the Mammoth Act. This led to great turmoil among the teachers as appeared from the words of the chairman of WIMECOS: In het achter ons liggende verenigingsjaar constateren we duidelijk een gevoel van gespannenheid, veroorzaakt door de onzekerheid, waarin ons gehele onderwijs en in het bijzonder ons wiskundeonderwijs verkeert. (The past year the association clearly observes a sense of tightness caused by the uncertain state in which all our educations and in particular our mathematics education finds itself.) [149]

In 1966 the reorientation of the MULO and LBO teachers was also added to the range of duties of the CMLW. Later Freudenthal admitted that the CMLW had underestimated the modernization of the curricula: We faalden toen omdat we meenden dat we naast het verrichten van ons gewone werk ook nog het onderwijs konden vernieuwen. (We failed at the time because we thought that we, besides doing our normal jobs, could in addition modernize education.) [150]

The fact is that there was not much time to experiment with the subject matter to be introduced. In 1963 Freudenthal sketched in an article the ridiculous practice with the experiments in this field in the Netherlands. He observed that in such experiments the question to be investigated was not posed explicitly beforehand, but that in retrospect, after investing much extra time and with the cooperation of specifically selected teachers and classes, it was concluded that ploughing through the subject matter had succeeded and thus that the experiment was successful. Such practices indicated according to Freudenthal that


247

man […] absolut nicht darauf vorbereitet [ist], ein großes Objekt, wie es ein ganzes neues Programm ist, zu prüfen. Man verfällt dann leicht in eine Experimentiertechnik, die keine ist. (people […] are absolutely not prepared to test such an extended concept as a completely new curriculum. You quickly relapse into an experimental technique that isn’t an experimental technique.) [151]

Just the same on 28 October 1966 the CMLW submitted an interim report with a rough draft of the new curriculum to the State Secretary of OK&W. To motivate the proposed plans the committee mentioned among other things traditional factors, the transfer of useful knowledge and the social meaning of mathematics as a result of the many applications in various fields. Apart from that the committee in composing the curricula had of course taken into account the new types of secondary education in place after the Mammoth Act, such as the joint first years of education and the possibilities to switch between the various school types. But the most important aspect according to the committee was really “to draw up a curriculum for all school types motivated by the higher education and the interest of society” [152]. The ‘modern’ elements in this proposition that were the most eye-catching were the use of terms such as vector, set, relation, representation, transformation and function, the introduction of probability calculus and statistics and the algebraic treatment of geometry. Attention for a logical-deductive system was limited to the higher classes [153]. ‘A modest renovation’ was how Krooshof characterized the propositions of the CMLW, but De Bruyn’s criticism was sharper and he blamed the CMLW for hardly tuning the curriculum to the needs of society [154]. Freudenthal reacted by writing an article in which he explained his belief concerning the applicability. For in his opinion a curriculum that guaranteed applicability did not exist. Toegepaste wiskunde is de minst toepasbare, omdat zij de grootste deugd van de wiskunde, de flexibiliteit, mist. Als het onze bedoeling is, de leerling met een wiskunde vertrouwd te maken, die hij kan toepassen, dan moeten we ons niet op de programma’s blind staren. […] Nieuwe programma’s zijn geen wondermiddel. Er de nadruk op leggen kan betekenen dat men afleidt van waar het bij het onderwijzen van toepasbare wiskunde op aankomt. (Applied mathematics is the least applicable since it misses the greatest virtue of mathematics, the flexibility. If it is our intention to familiarize the pupil with a type of mathematics that he can apply, we should not overestimate the importance of the curricula. […] New curricula are not magic potions. Emphasizing them can imply a distraction of the crucial point in teaching applicable mathematics.) [155]

This crucial point was, according to him, fostering a good attitude towards mathematics. This is what he wrote in De Groene Amsterdammer in 1962 about the function of (modern) mathematics in school: Voor vraagstukken, die men volgens schema kan oplossen, hebben we geen mensen nodig. Die kunnen we gerust aan machines toevertrouwen. Het is het kenmerk van echte wiskunde, dat elk probleem nieuwe, originele denkarbeid vereist. Dit zou dan de voornaamste ervaring zijn die de leerling op school omtrent de wiskunde opdoet. […] Moderne wiskunde is eenvoudiger, en moderne schoolwiskunde hoort in eerste instantie eenvoudiger te zijn dan de oude. Maar in de wiskunde impliceert grotere eenvoudigheid ook een grotere mate van originele denkarbeid. Aan een rekenmachine, die geen denk-initiatief vertoont, moeten we alles haarfijn uitleggen. Haar wiskunde is onnoemelijk omslachtig. Eenvoudige wiskunde vereist denk-initiatief, en moderne wiskunde zou een geschikt middel zijn, om dit initiatief

248


aan te wakkeren. (We do not need people to solve problems that can be solved by following a scheme. Those we can safely trust to machines. The characteristic of real mathematics is that every problem requires new original thinking. This should be the most important experience pupils get at school with respect to mathematics. […] Modern mathematics is simpler, and to begin with modern school mathematics should be simpler than the old one. But in mathematics a greater simplicity also implies a larger amount of original thinking. To a calculating machine, which does not display thinking initiatives, we have to explain everything down to the minutest detail. Its mathematics is inexpressibly time-consuming. Simple mathematics requires thinking initiative, and modern mathematics would be a suitable means to stimulate this initiative.) [156]

Since Freudenthal was a member of the CMLW and nowhere commented critically on the concept curriculum of 1968, he appeared to deem the new curriculum good enough for his purposes. But soon it became clear how he had underestimated the consequences of the introduction of this curriculum, or at least had estimated them too optimistically. In Schrijf dat op, Hans he wrote: Maar laat ik eerlijk zijn: wij—ik incluis—werden overrompeld. Met de feitelijke invoering van de Mammoetwet moesten de nieuwe leerplannen wiskunde op tafel liggen. Auteursgroepen die in het buitenland hun licht hadden opgestoken, waren ons te vlug af geweest. Voor de leerplannen er kwamen waren ze al geïnterpreteerd. Een beetje New Math, niet veel, maar toch té veel en in elk geval gespeend van elke relatie met de realiteit, van elk inzicht in de dienende taak van de wiskunde. (But let me be honest: we— including me—were taken by surprise. With the actual introduction of the Mammoth Act the new mathematics curricula had to be on the table. Groups of authors who had gone abroad for information had been a sight too clever for us. Before the curricula were there they had already been interpreted. A bit of New Math, not much, but anyway too much and in any case utterly lacking any relation with reality, any insight into the supporting task of mathematics.) [157]

One of these groups of authors who anticipated the imminent changes in the curricula was that of the Moderne wiskunde method, published by Wolters Noordhoff. The authors (among whom Krooshof and Jacobs) had in writing that method relied on the so-called ‘Scottish method’ of the Scottish Mathematics Group, in Scotland published with the title Modern mathematics for schools [158]. The initiative was a follow-up from a working group that had been established by the three Pedagogische Centra (Pedagogic Centres) [159]. This working group studied the mathematics education for the future HAVO. Krooshof wrote in Euclides about the choice and the realization of the text book: Een leerboek schrijven, vroeger dikwijls het werk van één of twee man, kan in de huidige situatie niet anders meer zijn dan het werk van een team. Bovendien kan men nieuwe leerstof pas invoeren na een toetsen in de klas. Aan beide voorwaarden voldoet de Schotse methode. Een team van vier inspecteurs en 17 docenten schreef de boeken. Deze werden in een uitgave, bekostigd door het Schotse departement van onderwijs (zou zoiets in Nederland mogelijk zijn?), op 7000 leerlingen getoetst. De Nederlandse bewerking zal op ongeveer 2700 leerlingen getoetst worden. (Writing a textbook, previously often a job for one or two people, can in the present situation only be the work of a team. And what is more, the new subject matter can only be introduced after it is tested in the classroom. The Scottish method meets both conditions. A team consisting of four inspectors and 17 teachers wrote the books. They were, paid by the Scottish department of education (would something like that be feasible in the Netherlands?), tested in printed form on 7000 pupils. The Dutch version will be tested on about 2700 pupils.) [160]


249

In Moderne wiskunde modern and traditional subject matter was “approached in a new way, among other things with help of set theory and logic” [161]. There was also attention for “referring to and using phenomena from life outside the school”. The method, in which didactical modernizations such as self-motivation were also introduced, would become a great success [162]. With hindsight Freudenthal felt himself taken by surprise by ‘modern’ teaching methods. But perhaps he had trusted too optimistically and frivolously that the professional development for teachers would be a sufficient ‘defence’ against the absurd attempts to modernize—such as the author groups mentioned above. In an article in Euclides Krooshof blamed the CMLW itself for contributing implicitly to the absence of interest for the modernization of didactics through the professional development courses: Kijken we naar de didaktiek, dan moeten we zeggen, dat we met de modernisering van het wiskunde-onderwijs nog maar aan het begin staan. Men kan zich afvragen, hoe het komt, dat zo weinig docenten echte belangstelling hebben voor de didaktiek, terwijl toch hun dagelijks werk hen telkens met didaktische vragen confronteert. Er zijn daarvoor m.i. twee redenen. De eerste is, dat de meeste wiskundedocenten zich beschouwen als vakdocent met de klemtoon op vak. Ook in deze tijd wordt dat ongewild in de hand gewerkt door de Commissie Modernisering Leerplan, die wel cursussen heeft uitgeschreven in de nieuwere onderdelen van het vak, maar daarnaast geen cursussen heeft georganiseerd ter vernieuwing van de didaktiek. (Looking at the didactics, we have to conclude that in modernizing mathematics education we are only at the beginning. You may wonder why not many teachers are really interested in didactics even though their daily work confronts them over and over again with didactical problems. In my opinion there are two reasons for this. The first is that mathematics teachers think of themselves as subject teachers with the emphasis on subject. In our times too the Commissie Modernisering Leerplan unintentionally paves the way to do this; they have organized courses for the newer subjects of mathematics, but they haven’t also organized courses to modernize the didactics.) [163]

His enthusiasm about the effect and the positive results of the courses in modern mathematics first and foremost prevented Freudenthal from recognizing this clearly at the time. But even more important was that he did not think that the way to a better teaching methodology was to be found in giving ‘courses in didactics’. On the contrary, according to him the key to improving the didactics was to be found in the knowledge of the nature of modern mathematics, which would serve for a teacher as a personal handle to command a view of his subject matter, to analyse it and teach it to the pupils so that they could practice mathematics in a way that suited them. But soon after the introduction of the new curriculum little was left of Freudenthal’s jubilant enthusiasm and great trust in the effects of the professional development courses—whatever success they might have had. It seemed an overly simple argument that ‘it would sort itself out’ provided that the teachers were introduced to the modern mathematics in a correct way. In Schrijf dat op, Hans he wrote with hindsight: Had het anders gekund? Misschien wel, als we enkele jaren respijt hadden gehad. De spoed van toen heeft ons vijftien tot twintig jaar gekost. Het nieuwe bleek praktisch van begin af aan mis. (Could it have gone otherwise? Maybe, if we had been given some years grace. The rush then cost us ten to twenty years. The new things appeared to be wrong almost from the start.) [164]

250


8.6 Geometry Education One of the subjects in secondary schools that would change radically under the influence of New Math was geometry. Since the 1958 curriculum the idea of geometry education with a strictly axiomatic structure had been abandoned and room was prudently made for a geometry with an intuitive introduction that was more based on the experience of the child. Nevertheless, this introduction too was still directed at introducing the deductive method as soon as possible. The value of geometry education was still mainly found in introducing the pupil to the logic of the traditional deductive system [165]. The greatest change the introduction of New Math would imply for geometry education was the emphasis that was put on transformation geometry. Vector and coordinates played a large role in this and the algebraic elements of geometry emerged more emphatically. Because of this geometry would no longer be an isolated part of mathematics education but would be treated starting from the more general structure of mapping. In the Netherlands too, people were thinking along those lines when they were considering modernization plans. In the ‘Interimrapport van de commissie modernisering leerplan wiskunde’ (‘Interim report of the committee for modernization of the mathematics curriculum’), published in 1966 in Euclides, the most important goal of geometry education in the first year was “at first to familiarize the pupils with geometrical figures, both in two and three dimensions, and as far as possible acquaint them with the meaning and the nature of proofs” [166]. In the 1968 curriculum as proposed by the CMLW using Euclidean geometry to learn in the traditional way to axiomatize was abandoned in favour of transformation geometry. The explanation accompanying the curriculum ran: Hoewel in Royaumont niet iedereen zo ver ging als prof. Dieudonné, die de kern van zijn betoog samenvatte in de slagzin: “Weg met Euclides”, was men het er toch over eens dat het meetkundeonderwijs een radicale wijziging zou moeten ondergaan. Tot nu toe nam de driehoek met de congruentiegevallen een centrale plaats in. Dit bood de mogelijkheid spoedig tot deductief redeneren te komen. Het verkrijgen van enig inzicht in de deductieve opbouw van een theorie is voor de leerlingen van het voortgezet onderwijs belangrijk. Algemeen wordt echter ingezien, dat aan de traditionele behandeling van de planimetrie zoveel nadelen kleven, dat deze voor dit doel niet gehandhaafd kan blijven. Tegenover een statische behandeling in de geest van Euclides, geeft het gebruik van transformaties aan de behandeling een dynamisch aspect, dat veel leerlingen intuïtief zal aanspreken. (Even though in Royaumont not everyone went as far as Professor Dieudonné who summarized the core of his argument in the slogan: “Away with Euclid”, in general people agreed that geometry education would have to change radically. Until now the triangle with the congruence cases had taken up a central place. This offered the opportunity to arrive quickly at deductively reasoning. Acquiring some insight in the deductive construction of a theory is important to pupils in secondary education. However, it is generally accepted that the traditional treatment of plane geometry has so many disadvantages that it cannot be maintained for this purpose. As opposed to a static treatment in the spirit of Euclid, the use of transformations gives the treatment a dynamic aspect which will appeal to many pupils intuitively.) [167]

An example to illustrate the difference in the approach of the subject matter can be seen in the treatment of the theorem:

8.6 Geometry Education

251

Fig. 8.3 In isosceles triangles the angles at the base equal one another

C

b

A

a

D

B

In Met passer en liniaal ( With compass and ruler) from 1957 written by Bijpost and Timmer (a textbook plane geometry for the ULO) this theorem was proven in the classical way with the help of congruence cases [168]. The theorem that two triangles are congruent if the sides of one triangle are equal to those of the other was posed earlier in the book. Next it was proven that the median from the top of an isosceles triangle divides it in two congruent triangles. It was proven as follows (Fig. 8.3): Gegeven: ∆ABC ; a = b; AD = BD . Te bewijzen: ∆ACD ≅ ∆BCD . Bewijs: Uit het gegeven volgt: AC = BC. Wegens de constructie van D geldt AD = BD. Vanzelf spreekt: CD = DC. Dan volgt met behulp van het congruentiegeval direct ∆ACD ≅ ∆BCD. Met de gevolgtrekking ∠A = ∠B was de stelling vervolgens bewezen. ( Given: ∆ABC ; a = b; AD = BD . To prove: ∆ACD ≅ ∆BCD. Proof: From the given data it follows: AC = BC. Because of the construction of D it holds that AD = BD. It goes without saying that: CD = DC. With the help of a case of congruence it follows immediately that ∆ACD ≅ ∆BCD . With the conclusion ∠A = ∠B the theorem was subsequently proven.) [169]

The theorem about the equal base angles in an isosceles triangle is treated in the chapter ‘Triangles’ in the already mentioned Moderne Wiskunde. Deel 1 voor de brugklas [170]. It is striking (and characteristic for these more modern methods) that this method does not treat the geometry in a separate part but in one and the same book alternated with arithmetic and algebra. The standard ‘given—to be proved—proof’ construction was also abandoned and the plausibility of the theorem was demonstrated using problems. Congruence was no longer used in these problems, reflection was: the pupil as it were lifted the triangle, turned it around (which amounted to reflecting the triangle in the bisector of the apical angle) and concluded that the triangle matched the original (and therefore had angles equal to one another) [171]. Some of the objectives of geometry education which could also be recognized in traditional geometry education would, for that matter, still have to hold in the new curriculum for geometry. a. acquiring insight in the meaning of a mathematical proof b. deriving results that can be of use to those who have to apply mathematics later on in life. [172]

252


In addition the CMLW wanted the pupil to acquire insight in the structure of plane and space something that was not expressed well in the traditional approach to geometry. So both acquiring more knowledge of the practical, tangible side of geometry and acquiring more insight in more general mathematical phenomena, such as ‘the proof’(even though that was in practice reserved for the higher classes only) were on the wish list of the CMLW.

8.6.1 Freudenthal and Geometry Education Freudenthal first expressed his opinion on geometry education in the early 1950s when this subject received a lot of attention in the Wiskunde Werkgroep. He referred to this early work in ‘Initiation into geometry’, the lecture he held at the conference on mathematics education in Bombay in 1956 [173]. As a result of the design of a new curriculum for the VHMO in the early 1950s, discussions about the motivation of the choice of subject matter were regularly held in the Wiskunde Werkgroep. The WW members saw that the much-used argument of the formative value was defeated by the criterion of ‘usefulness’, but for the greater part of the subject matter of mathematics education it would not imply radical changes. For geometry things were different. In geometry, in particular, a lot would change if the argument of formative value were to be abandoned. Freudenthal sketched in his lecture of 1956 what such a geometry education would look like and he looked back at the discussions in the WW of the early 1950s: Een pragmatisch meetkunde-programma zou beperkt kunnen blijven tot een zeer kleine groep stellingen (b.v. Pythagoras), enkele evidente eigenschappen van gelijkvormige figuren, en een stel formules voor omtrekken, oppervlakken en inhouden. We zouden kunnen afstand doen van het logische systeem, dat ons meetkunde-onderwijs dankzij de Euclidische traditie kenmerkt. Toch dachten we er nooit aan deze conclusie te trekken en meetkunde als onderwijsvak af te schaffen. Afkerig van formele argumenten, slaagden we er toen niet in, de wortels van ons geloof in het meetkunde-onderwijs bloot te leggen. Ik denk, dat we het heden met meer succes zouden kunnen doen. (A pragmatic geometry curriculum could be limited to a very small set of theorems (for example Pythagoras), some obvious properties of isosceles figures, a set of formulas for circumferences, areas and volumes. We could renounce the logic system, which characterizes our geometry education due to the Euclidean tradition. Nevertheless we never considered drawing this conclusion and abolishing geometry as a subject in education. Loathing formal arguments we did not succeed in baring the roots of our belief in geometry education. I think that we could do that today with more success.) [174]

Within the WW the conviction ruled that, despite the ‘usefulness’ of the logical system of the Euclidean geometry, which could not be substantiated, it still had its value in mathematics education. But where was this value? Freudenthal had already regularly contended that he hardly saw a good point in the naked argument of ‘formative value’ where justifying subject matter, and in particular that of geometry education, was at stake [175]. Nevertheless he was seriously convinced in 1965 that geometry education could offer more than practical knowledge alone. According to him the excess value of the geometry was to be


253

found in the conversion of the reality the pupil could observe with the power of the mathematical system, caused by the overlap of mathematics with science, which is inherent to geometry. Meetkunde is de wetenschap van de ruimte, waarin we leven, en het is een wonderbaarlijk feit, dat we met gesloten ogen, door ons vernuft alleen, eigenschappen van deze ruimte kunnen voorspellen. En dit wonderbaarlijke feit moet voor hen, die meetkunde leren, een wonderbaarlijke gewaarwording kunnen worden. Ik denk, dat we heden zouden kunnen zeggen: meetkunde, als logisch systeem, is een middel—en wellicht het meest doeltreffende middel—om kinderen te doen voelen de macht van de menselijke geest—dat is van hun eigen geest. (Geometry is the science of the space in which we live and it is a miraculous fact that we, with closed eyes and only our intellect, can imagine properties of this space. And this miraculous fact ought to become a miraculous experience for those who learn mathematics. I think that we could say today: geometry as a logical system is a means—and perhaps the most effective means—to make children feel the power of the human intellect—that of their own mind.) [176]

However, the traditional geometry education which started from the logical-deductive system achieved the opposite according to Freudenthal: it more likely convinced children of their intellectual inferiority. The objective of the introductory geometry education at the secondary school was according to him reaching “maturity for the logical system”. Whether this ‘maturity’ could be reached by all pupils was another question. Reaching this maturity had according to Freudenthal nothing to do with the child’s age but could be tested reliably with the help of the following points: …ten eerste, het kind moet het feit van de samenhang der meetkundige waarheden zelf hebben ontdekt, ten tweede moet het hebben begrepen, dat deze samenhang iets dwingends is en ten derde moet het door dit besef in die mate zijn gegrepen, dat het er naar verlangt, om de weg van het logische systeem te bewandelen. (…firstly, the child should have discovered by themselves the fact that the geometrical truths cohere, secondly it should have understood that this coherence is compelling and thirdly it should by realizing this be moved to such an extent that it longs to walk the path of the logical system.) [177]

In the step-by-step plan he sketched here the influence of the level theory of Van Hiele can be recognized clearly. It also resembles closely the objectives Freudenthal formulated for the CMLW experiment with geometry for the lower classes [178]. Freudenthal deemed the role of geometry in mathematics education very important as becomes clear from the following statement. Here he emphasized the vital role the introductory geometry education played in educating ‘mathematicians’, a role that went much further than being only a preparation for the (axiomatic) geometry education: There is an ever increasing need for people who can put into practice mathematics with insight. It is a fact that good mathematical masters have always tried not only to develop algorithmic abilities, but also intellectual insight. How this can be done, becomes a more and more urgent problem. Initiation into geometry is the first opportunity to confront the pupils with true mathematics. In many cases it will also be the crossroads on which it is decided whether a pupil will like or dislike mathematics. [179]

254


8.6.2 Freudenthal on the Initial Geometry Education: Try It and See In his articles and lectures from the 1950s to 1960s about geometry Freudenthal mainly wrote about the importance of a well-considered introduction for geometry education during the first years of the VHMO. This was particularly obvious when he wrote about the level theory of Van Hiele, which had been given a practical interpretation in the work of Mrs. van Hiele concerning the initial geometry education. The fact that Freudenthal had no objections against learning axiomatics at the secondary school more or less faded into the background in his publications about introductory geometry education (and the importance of its practical and intuitive character). But as already shown here, Freudenthal actually had the logical system in view as an essential part of geometry education and it was mainly the traditional way of reaching this that he criticized [180]. This criticism was particularly noticeable in his enthusiastic descriptions of the less common initiatives in this field. Being a strong supporter of an intuitive method as an introduction to geometry education Freudenthal was, quite besides his enthusiasm for the work of the Van Hieles, very enthusiastic about Tatiana Ehrenfest’s Uebungensammlung (Exercise collection)—although he made the comment that these had never been tested in practice—and also about Emma Castelnuovo’s geometry course Geometria intuitiva, in which, just as in Mrs. Ehrenfest’s book, there was a striving for a link with the child’s perception of its environment [181]. In an introductory geometry course Freudenthal valued arousing the pupil’s curiosity for a problem that stirred the imagination most. This would urge the pupil to want to solve the problem. According to Freudenthal this was reached most easily by starting not in the plane but in the three-dimensional space. That was less abstract and would offer the pupil a better opportunity to use his intuition and his imagination. Working in three-dimensional space would offer the pupil literally the opportunity to ‘experience’ geometry by trying and constructing. He wrote about the advantages of this point of view in reaction to the workbooks of the Van Hieles: It has been established by experience that nobody who has started geometry with space, will meet any serious obstacles, when passing to stereometry in the systematic course. […] The notions of area and volume have been introduced neither by abstract reasoning nor by bluntly appointing that the area is to be length-times-width, and the volume length-timeswidth-times-height, but by fitting a number of things into one thing—the only psychologically justified manner to introduce them. [182]

The only intuitive manner to start with geometry in the plane was according to Freudenthal by the so-called tiling of a surface: how and why will (tiles with) specific shapes cover a surface? [183] This approach also had an element of ‘try it and see’, something that would appeal to children very much according to him. In particular for children of that age “the motor experience of ‘turning this way and that’ with its strong motor apprehension and motor memory was important psychologically”. The greatest pedagogical virtue was according to Freudenthal to have patience: the patience to wait for the moment at which the child itself asks why this trying to cover a surface is useful and only then change over to the logical system.


255

De wonderen van het passen moeten het kind doen rijpen voor het meetkundig-logische systeem. Maar tevens zal het kind in dit stadium de grondstoffen moeten vergaren èn de problemen die het op een telkens hoger peil zal moeten aanpakken. Het kind moet zijn eigen ontwikkeling kunnen zien en beoordelen, als iemand die in een fotoalbum bladert of die een berg beklimt en terugblikt over de weg, die hij is gegaan. (The miracles of fitting have to ripen the child for the geometric-logical system. But the child will also have to collect the raw materials and the problems which he has to approach at a higher level in each case. The child must be able to see and judge his own development, as someone who thumbs through a photo book or climbs a mountain and looks back on the way he came.) [185]

The job the introductory method had to perform was stimulating and developing this ‘maturing’—guiding the learning process.

8.6.3 Axiomatizing Instead of Axiomatics—But Not in Geometry In 1956 Freudenthal held the opinion that if the pupil experienced ‘the power of his own mind’ this would be a sufficient reason to maintain learning axiomatics in geometry education. However, with New Math on the stage he was stimulated to scrutinize the subject ‘learning to axiomatize’ once again and his ideas about it took a more distinct shape. In his lecture in 1956 he seemed to take for granted that children in geometry education would eventually familiarize themselves with the logical system of the Euclidean geometry—albeit after a well-considered intuitive introduction—, but 6 years later he found this far from evident. The fact that axiomatics was inextricably connected to modern mathematics in particular and thus deserved a place in mathematics education in some way, was for Freudenthal (still) indisputable. And by this he did not mean offering a “readymade system of axioms […], on which consecutively was deductively built”, but (learning) to axiomatize [186]. Learning to axiomatize did imply formative value to a certain extent according to him. With all the attention for (the axiomatics in) modern mathematics in particular the formative value of axiomatizing needed extra attention according to him. He wrote jokingly about the traditional attitude towards learning to axiomatize in education: Es ist bedauerlich, daß in der Mannigfaltigkeit der Publikationen über Axiomatik auf der Schule der bildende Wert des Axiomatisierens nicht einmal erwähnt wird. Es scheint mir das ein Mangel an Vertrauen in die Mathematik und in die Persönlichkeitsstruktur des Heranwachsenden: Mathematik als Aktivität bleibe dem erwachsenen Mathematiker reserviert, dem Manne, der alle Werkzeuge kennt und weiß, wie er sich ihrer zu bedienen hat. Das Axiomatisieren überlasse man den Meistern. Die Schüler haben Axiomatik zu lernen; der Meister weiß, welche ihnen angemessen ist. (It is regrettable that in the majority of the publications about axiomatics in school, the formative value of axiomatizing is not mentioned once. It looks to me like a lack of trust in mathematics and in the character structure of the adolescent: mathematics as an activity is reserved for the grown-up mathematician, the man who is familiar with all tools and knows how to use them. Leave axiomatizing to the expert. The pupils have to learn axiomatics: the expert knows which are suited for them to learn.) [187]

256


Gradually Freudenthal let go of the idea of learning to axiomatize in geometry, even when he still saw some obvious reasons to maintain it. One possible reason in favour of learning to axiomatize was that in geometry in particular the connection between the logical system and the pupil’s concrete experience or imagination was very close and therefore difficult to let go of. That could make it an extra useful exercise [188]. But Freudenthal did not consider this an urgent argument for maintaining axiomatics in geometry. He wrote about the other advantages for axiomatics in geometry education: De meetkunde is een bijzonder geschikt gebied, om in het mathematisch ordenen van een premathematische stof (ruimtelijke ervaringen) in te leiden; ook voor het telkens weer ordenen op een hoger niveau van de mathematische ervaringen, die op een lager niveau zijn opgedaan (opstellen van definities, herkennen van logische samenhangen) is de meetkunde een vruchtbaar veld. (Geometry is a field exceptionally suited to introducing pupils to mathematically arranging pre-mathematical subject matter (three-dimensional experiences); geometry is also a fruitful field for again and again arranging mathematical experiences, which have been acquired at a lower level (formulating definitions, recognizing logical connections), on a higher level.) [189]

But in the end this was not enough for him to plea for maintaining axiomatics as a part of geometry education. Freudenthal rejected learning to axiomatize in geometry because according to him there was no suitable modern axiomatics available that could be used in teaching. Axiomatiseren is echter moeilijker dan dit “lokaal ordenen”. Het is een “globaal ordenen”, dat ruim overzicht over het te axiomatiseren gebied veronderstelt. Meetkundige axiomatiek op school is alleen dan mogelijk, wanneer de leerling in het heruitvindende lokale ordenen dit overzicht heeft verkregen. Bovendien moet het beoogde axiomatische systeem zo eenvoudig zijn, dat geleide heruitvinding van het systeem denkbaar is. De van diverse kanten voorgestelde axiomatieken voor de meetkunde zijn zo gecompliceerd, dat slechts een geoefend wiskundige de zin van de diverse axioma’s kan vatten. De vraag ‘Axiomatiek van de meetkunde in het V.H.M.O.?’ moet derhalve om practisch redenen met “neen” worden beantwoord. (However, axiomatizing is more difficult than this “locally ordering”. It is “globally ordering”, which presupposes ample overview of the field to be axiomatized. Geometrical axiomatics in school is only possible when the pupil has acquired this overview in re-inventing locally ordering. In addition the intended axiomatic system must be so simple that guided re-invention of the system is feasible. The axiomatics proposed for geometry by various parties are so complicated that only an experienced mathematician can understand the meaning of the various axioms. The question ‘Can axiomatics of geometry be part of the VHMO?’ should therefore be answered negatively for practical reasons.) [190]

According to Freudenthal this did not prevent introducing axiomatics in other parts of the education (such as measure and algebraic structures) provided that, when looking for an alternative, the principle “the pupil shall not learn axiomatics, but shall learn to axiomatize” was strived for [191]. Axiomatics on its own was not a good guideline in interpreting mathematics education contended Freudenthal in his article ‘Bemerkungen zur axiomatischen Methode im Unterricht’ (‘Remarks on the axiomatic method in education’) [192]. The pupil should also have an intuitive idea of the rules of the game of the structure of the education according to Freudenthal. The argument that axiomatics would be a synonym for ‘absolute honesty’ was also not applicable according to him: in


257

axiomatics too the extent of strictness and formalization was always a question of agreements [193]. Recognizing this was of the utmost importance for mathematics education—and for geometry education in particular: Much harm has been done in mathematics education by people who have not noticed that there are no absolute standards of rigour and formalization in mathematics, and who reject any standard lower than their own as bogus and any higher one as highbrow. Every level in the learning process asks for corresponding levels of rigour and formalization, but traditional teaching tends to move on one level of rigour and formalization from the start of teaching to the end. In actual teaching this means too high a level to start with and too low a one to finish. In geometry Euclidean rigour in the beginning prevents teaching, at the end it blocks modern rigour. An early formalization of the language operated on in geometry prevents the urgent formalization of the operating language which is still on a shamefully low level all over the world where geometry is taught. [194]

As long as the observed level of the pupil (and the accompanying degree of exactness) were taken as the starting point, Freudenthal did not have any objection whatsoever to the introduction of modern geometry in mathematics education; not even if this geometry were to be based on the concept of ‘groups’, something which, it is worth noting, matched completely with the New Math-trend: Das besagt […] nicht, daß ich moderne Geometrie für die Schule in den Schornstein schreibe. Ganz im Gegenteil. Geometrie, und dann vom Anfang bis zum Ende vom Gruppenbegriff durchsetzt, gehört auf die Schule, und ich bin davon überzeugt, daß man mehr und bessere Geometrie unterrichten kann, wenn man seinem eignen Gewissen für mathematische Strenge folgt, als wenn man sich seine Gewissensbiße vom stirnrunzelnden Axiomatiker vorschreiben läßt. (That does not mean that I throw modern school geometry in the fireplace. Just the opposite. Geometry, from the start to the end enforced, at that, by the concept of groups, belongs to the school, and I am convinced, that one can teach more and better geometry, if one follows one’s own conscience for mathematical strictness, than when one lets frowning axiomaticians prescribe one’s scruples.) [195]

8.6.4 Modern Geometry in the Education According to Freudenthal Geometry in which the mathematical concept of group was used from start to finish—was what Freudenthal considered a modernized geometry education to be. But then of course with the proviso that the degree of exactness and the accompanying language would match the observed level of the pupil. He thus argued for a geometry that was all roughly based on Klein’s Erlanger Programm, something which more or less also held true for the new curriculum. So at first sight Freudenthal’s ideas did not deviate that much from the modernization tendencies of the time. True, the plans for the 1968 curriculum departed from reflection, translation and rotation, but Freudenthal was not negative about that either. But for the use of translations in geometry education he saw only a possibility if the starting point was the two-dimensional grid: “the most (perhaps the only) natural infinite figure”, as it was indeed proposed in the 1968 curriculum [196]. Anyway his preference was that ‘symmetry’ and therefore reflection would be taken as the starting point.

258


Symmetries are more interesting than translations and rotations. To a young child congruent figures are the same. It will not hit upon the idea that something has happened if a figure is carried to another place. To an unsophisticated mind movement is not a transformation. In this regard rotation is somewhat better than translation. If a cube is translated, nothing has happened; if it is turned and put upon an edge or a corner, something has been changed. But mirror reflection gives the strongest feeling of an important event. Symmetry as a transformation is more attractive, more abundant, and more problematic than translation and rotation. [197]

Freudenthal only deviated from the prevailing views when the attitude to the subject matter and the corresponding final goal of the (geometry) education was involved. For years Freudenthal was dead set against imposing the axiomatics of the Euclidian geometry on pupils. Now that after the advent of New Math Euclidian geometry seemed to disappear from the curriculum, he was aware that this would not indiscriminately be replaced by another—more modern—axiomatics. For in his view this would not guarantee any improvement whatsoever and completely ignore what he considered the only guideline in the interpretation of the mathematics education: learning to do mathematics. According to him this included for the pupil: learning to organize and eventually also learning to axiomatize. The casualness with which the axiomatics was up till then only treated in the geometry was unfounded according to him. What counted was not the result that could be measured in concrete knowledge—such as the knowledge of Euclidian geometry—, but the way in which the pupil had learned mathematics. Freudenthal’s attitude in the discussion about the modernization of geometry education seemed inconsistent from time to time. On the one hand this was caused by the change in his attitude towards axiomatics in geometry. In 1956 he still saw it as a part of geometry education, whereas in 1962 he preferred to see axiomatics and axiomatizing to be taught in another part of mathematics. On the other hand the—in this case apparent—contradiction in his attitude was primarily caused because at one point he seemed to take a barely deviating stance— for example when he spoke up in favour of modern geometry in education and teaching of (an) axiomatics in general—, then on another occasion lashed out at modern axiomatics proposed in geometry education in New Math (mainly in other countries). This contradiction, however, disappeared as he nuanced it as he always did with such statements: not the state of mathematics as a science but that which fitted best to reach the intended educational goal had a decisive importance to him. Modern geometry could be fitted excellently in education, but then in such a way that it matched the level of the pupil—as far as the perception of the environment, the learning level and the strictness were concerned—and that it would serve the eventual goal—learning to do mathematics.

8.7 Logic An almost inevitable subject of mathematics education in this period was logic. Inevitable because modern mathematics was just tied up in the building up from the concept of ‘sets’ and the accompanying logic. An extra reason was that Freudenthal

8.7 Logic

259

was particularly interested in geometry education and therefore the traditionally accompanying logic system regularly popped up in his lectures and publications. His enthusiasm for the level theory of the Van Hieles was for him more than once a reason to devote attention to logic in the VHMO—albeit in that context not for the pupil, but as a didactical aid for the teacher. The publication of his book Exacte logica (Exact logic) in 1961 has to be seen in that light (Fig. 8.4) [198]. Freudenthal himself took the initiative to write this work—published in the same series as his book Waarschijnlijkheid and Statistiek from the Volksuniversiteitsbibliotheek— that would be prescribed to the participants in the professional development course of the CMLW.

8.7.1 Exact Logic In his preface Freudenthal mentioned his rationale for writing the book. First of all, there were the newest developments in mathematics that have “re-created” the “ancient formal logic into an exact science”. Freudenthal saw an important role for the mathematical method, particularly in the logic-linguistic area of the linguistic research. Apart from that there was the outdated mathematics education of the VHMO: Hetgeen het meest verouderd is in onze VHMO-wiskunde, is niet het wàt, maar het hòè, de formalisering en de verwoording. Ons wiskunde-onderwijs heeft behoefte aan exacte logica, niet als nieuw onderwijsvak, maar als achtergrond, en wie in dit onderwijs werkzaam is, moet in staat zijn, die achtergrond te verkennen—niet alleen terwille van de wiskunde, maar evenzeer terwille van de didactiek van het vak. Bij didactisch onderzoek, juist in ons land, is gebleken, dat de logische structuur van het leerproces beheerst wordt door de voor de logica en linguistiek fundamentele relatie van systeem en meta-systeem. (The most outdated in our VHMO mathematics is not the what, but the how, the formalization and the choice of words. Our mathematic education needs exact logic, not as a new educational subject, but as background, and those who work in this education will have to be capable of exploring this background—not only because of mathematics but just as much because of the didactics of the subject. In didactical research, in particular in our country, it has become clear that the logical structure of the learning process is controlled by the relation between system and meta-system that is fundamental for logic and linguistics.) [199]

It was obvious that Freudenthal in mentioning “didactical research, in particular in this country” referred to the work of the Van Hieles, as he would also do in a similar context in various later publications when considering the use of logic foreducation [200]. Just as his previous book Waarschijnlijkheidsrekening en statistiek, Exacte logica too was primarily meant for the mathematics teacher (who participated in a professional development course). Its tone and structure were comparable to those in the statistics book: in a narrative tone the reader was guided through the theory via ‘Sets and mapping’, ‘Propositions’, ‘Subject-predicate’ and ‘Formal logic’. The final chapter of the book was about ‘Language and meta-language’. By his own account Freudenthal had kept this, contrary to the other chapters, popular and had written it “in a slapdash manner” [201]. All chapters were interlarded with exercises

260

Fig. 8.4 Dust jacket of Freudenthal’s Exacte logica (1961)


8.7 Logic

261

with all answers listed at the end of the book. Freudenthal also regularly, be it very concisely, gave some historical background of the theories he discussed. In the chapter ‘Formal logic’ the reader was presented a short but clear-cut introduction into the axiomatic method, which was subsequently used for proposition calculus to finally come to the axiomatics of the subject-predicate-language. The chapter ended with the announcement of Gödel’s Incompleteness Theorem. Freudenthal sketched the proof of that theorem to conclude the chapter ‘Language and meta-language’. Concise, extensive and in some aspects quite difficult as it was, the tone and pace of the book could easily be followed by a mathematically trained person. In introducing the structure of an axiomatic system Freudenthal obviously considered his own principle—“the pupil has to feel the need of ordering and has to learn to axiomatize”—not applicable for his readers (whom he thought were sufficiently mathematically schooled): everything was demonstrated and the reader was taken by the hand and led past the various stages of the arguments. Yet Freudenthal had the feeling that he had really met the reader more than half way by using apart from the indispensable mathematical context of the exact logic many examples from everyday speech as well, which were according to him “until now hardly or not at all found in this type of book” [202]. An example of the use of everyday language were the following exercises with which the reader could practice the use of logic symbols (in particular the quantifiers ∧ and ∨): Maak gebruik van de afkortingen Z( x, t): ik zie het ding x op het tijdstip t, P( x, t): ik pak het ding x op het tijdstip t, t’ < t: het tijdstip t’ is eerder dan het tijdstip t en schrijf symbolisch neer: 22. Ik zie altijd iets. 23. Soms zie ik niets. […] 39. Als ik eens een keer een ding gelijktijdig heb gezien èn gepakt, dan doe ik het later na elkaar of in het geheel niet. (Use the abbreviations Z(x, t): I see object x at time t, P(x, t): I pick object x up at time t, t’ < t: time t’ is before time t and write in symbols: 22. I always see something. 23. Sometimes I see nothing. […] 39. If once I have seen a thing, and have picked it up at the same time, I will do this later after each other or not at all.) [203]

The exercises in ‘everyday language’ did not imply much more than being elements of puzzles. The other exercises in using logic, which concentrated more on algebra and analysis, probably contributed more to the reader’s development of concept and applicable knowledge.

262


8.7.2 The Application of Modern Logic in Education Apart from the remarks he made in the preface, Freudenthal completely skipped the use of logic in education in his book Exacte logica. In a lecture from that same period he did devote attention to it [204]. Here he showed that logic was only an explicit subject in geometry education and that in algebra—even though it was equally present there—it remained a suppositious subject: Hoe moeten we deze band tussen meetkunde en logica verklaren? Blijkbaar als uitvloeisel van de mening, dat het in de meetkunde niet, maar in de algebra wel zonder logica gaat. In de algebra wordt de leerling van een goed werkend systeem van regels voorzien, om gegeven uitdrukkingen te vervormen met als doel een precies te voorspellen resultaat. In de meetkunde beschikken we niet over een even betrouwbaar systeem van schema’s. Het is wel kenmerkend, dat voetangels in de algebra (denkt u maar aan het vermenigvuldigen van vergelijkingen met 0 en van ongelijkheden met negatieve getallen) niet uit de weg worden geruimd door een logische analyse, maar dat zij door nieuwe regels plegen te worden geblokkeerd. (How should we explain this connection between logic and geometry? Apparently as a consequence of the opinion, that geometry cannot, but algebra can, do without logic. In algebra the pupil is supplied with a well-functioning set of rules to transform given expressions with the goal of accurately predicting a result. In geometry we do not have an equally reliable set of schemes at our disposal. But it is characteristic that pitfalls inalgebra (you can think of multiplying equations by zero and of inequalities of negative numbers) are not disposed of by a logical analysis, rather that they are usually blocked by new rules.) [205]

This algebra education based on ‘schemes’ had an effect even on the notation: Het woord ‘algebra’ zou in de schoolpraktijk kunnen worden vertaald als “aanwijzingen voor het invullen van schema’s—laten we het nog iets krasser zeggen—van voorgedrukte formulieren”. De voorgedrukte tekst staat in het geheugen van de leerling gegrift, die zich ertoe beperkt, neer te schrijven, wat bij het invullen tussen de gedrukte tekst in komt te staan. […] Ik bewonder dan ook elk jaar bij het nazien van het schriftelijk eindexamenwerk de scherpzinnige leraar, die weet, welk voorgedrukt formulier de leerling net aan het invullen was. Want dat is de vloek der algebra: dat er verschillende soorten voorgedrukte formulieren bestaan, en dat de verstandhouding eronder lijdt, als de een meent, dat het over kinderbijslag ging, terwijl de ander aan een girobiljet dacht. (In the school practice the term ‘algebra’ could be translated by “instructions for completing schemes—let’s put it even stronger—or pre-printed forms”. The pre-printed text is engraved in the memory of the pupil, who restricts himself to writing down what should be inserted between the pre-printed lines. […] Every year I therefore admire the shrewd teacher who, when he is correcting the written exam work, knows which pre-printed form the pupil had been completing. For such is the curse of the algebra: there are several kinds of pre-printed forms and the relations suffer when one of them thinks that it is about the child allowance, while the other had been thinking about a transfer slip.) [206]

According to Freudenthal there was only one remedy: to have the pupil write down exactly what he meant. And to do that the most obvious means ought to be used: logic symbolics. “With the greatest emphasis” he argued in favour of systematic application of logic symbolics in schools. The use of the quantifiers ∧ and ∨ should also be taught [207]. Freudenthal next illustrated it with examples resembling the

8.7 Logic

263

ones he had described previously such as ‘if I see something, I immediately pick it up’, continuing [208]: Misschien zult u zeggen: wat hebben deze uitgekookte voorbeelden met ons wiskundeonderwijs te maken? Ik zal u het antwoord niet schuldig blijven. Stapelingen van kwantoren “er is” en “alle” zijn de schrik van de jongelui, die de schoolbanken voor de collegebanken verwisselen. […] Hoe komt het, dat de schoolwiskunde zulke stapelingen van kwantoren kan vermijden? Deze vraag stelde ik me enkele jaren. Het antwoord heeft mij buitengewoon verrast. Wat bleek? Dat ook de schoolwiskunde zulke gevaarlijke kwantoren-torens kent, en nog wel in de algebra, die logisch simpeler heet te zijn dan de meetkunde. (Perhaps you will say: what is the relation between these shrewd examples and our mathematics education? I will give you the answer. Nesting the quantifiers “there exists” and “all” is what young people who exchange the school desks for the lecture-halls fear. […] How come that school mathematics can avoid such nesting of quantifiers? I have asked myself that question for several years. The answer has surprised me exceptionally. What was the case? In school mathematics too such dangerous quantifier nests occur; to be more precise they do so in algebra, which in a logical sense is even supposed to be simpler than geometry.) [209]

The fact that prospective students nevertheless had such a problem with the quantifiers was according to Freudenthal caused by the way in which quantifiers were used in the algebra education. As an example of such a use of quantifiers in mathematics education Freudenthal presented the following exercise: a. Aan welke voorwaarden moeten a en b voldoen, als − x 2 + ax + a + b voor elke waarde van x negatief is? b. Aan welke voorwaarde moet b voldoen, opdat er waarden van a zullen zijn die aan de gevonden voorwaarde voldoen? Symbolisch: ?b ∨ a ∧ x ( − x 2 + ax + a + b < 0). […] De gevaarlijke kwantoren in deze voorbeelden zijn leeuwen zonder tanden. Van de kwantor ∧ x in ∧ x ( − x 2 + ax + a + b < 0) zijn de tanden getrokken met behulp van de discriminant. De leerling weet machinaal dat bovenstaande uitdrukking gelijkwaardig is met a 2 + 4(a + b) < 0, en hiermee is het vraagstuk gereduceerd tot ?b ∨ a (a 2 + 4(a + b) < 0). Maar de volgende kwantor, ∨a is al even seniel. De discriminant reduceert het probleem tot ?b (16 − 16b > 0), en nu heeft de kandidaat de eerste kans om te stranden, want na het trainen op kwadratische en gebroken ongelijkheden kun je er zeker van zijn, dat hij niet weet, wat hij met een lineaire ongelijkheid moet doen. (a. What are the conditions a and b have to meet if − x 2 + ax + a + b is negative for all values of x?

264


b. What is the condition that b has to meet so that there are values of a that meet the condition that is found? Symbolically: ?b ∨ a ∧ x ( − x 2 + ax + a + b < 0). […] The dangerous quantifiers in these examples are lions without teeth. The teeth of the quantifier ∧ x in ∧ x ( − x 2 + ax + a + b < 0) are extracted with the help of the discriminant. The pupil knows by rote learning that the above expression is equal to a 2 + 4(a + b) < 0 , and with this the problem is reduced to ?b ∨ a (a 2 + 4(a + b) < 0). But the next quantifier, ∨a is equally senile. The discriminant reduces the problem to ?b (16 − 16b > 0), and now the candidate has the first chance to fail, for you can be sure that he, after his training for quadratic and broken inequalities, does not know what he has to do with a linear inequality.) [210]

The difference with the academic mathematics education was that the pupil could immediately eliminate the “quantifier nests” in the mathematics exercises at the VHMO by using quick and easy algorithms. In that way the practicing value of learning to use quantifiers was of course zero. The non-algorithm quantifiers— those that were hidden in the definitions—remained beyond the pupil’s range. This practicing with merely algorithmic quantifiers held according to Freudenthal the danger of “training for a logical seeming depth” and blocked the understanding of real quantifiers. This was a danger not only for the pupil but also for the teacher, “and that is perhaps worse, since a youthful pupil has a better chance to struggle out of this blockade”. Apart from learning to use quantifiers correctly Freudenthal also had outspoken ideas about the (function) notation to be used. Traditional notations such as f ( x), y = f ( x) or F ( x, y ) = 0 appeared to be unsatisfactory in modern mathematics according to him. Problems arose as soon as a set of functions A came up and it needed to be indicated that a specific function belonged to that set. To be able to do this unequivocally and correctly Freudenthal suggested after a discussion on function notations in Euclides the following function notation, in this case for the function on the interval (3,4]: Υ 3< x ≤ 4 ( x 2 − 3 x + 2) But here too Freudenthal refused to commit himself when a possible introduction of this notation at schools was concerned: Tenslotte, hoort deze of een dergelijke functie-notatie op de school thuis? Het antwoord is ja, indien de behoefte eraan zich voordoet. Of dit het geval is, zal van het programma en van zijn uitwerking afhangen. (To conclude, does this or a similar function notation belong to

8.7 Logic

265

the school mathematics? The answer is yes, if the need is felt. Whether this is the case will depend on the curriculum and how it is implemented.) [211]

Again Freudenthal’s point of view with respect to modernization can be recognized here: the introduction of modern elements was to him only valuable if these served a goal in education, never as a modernization in itself. Apart from that it was not the actual amount of mathematics that a pupil had learned at school at stake, but what he had learned. This had to be real mathematics and not a collection of thoughtlessly learned and therefore useless tricks. He expressed his point of view in the article ‘Logica als methode en als onderwerp’ (Logic as a method and as a subject)’: Ik heb tijdens deze lezing ervoor gepleit, dat symbolisch logische methoden door de leerling—ik zeg niet zoveel mogelijk, maar zoveel noodzakelijk—worden toegepast. Dit betekent uiteraard niet, dat de logica als leerstof haar intrede in de schoolwiskunde doet. Ik sta over het algemeen sceptisch tegenover nieuwe onderwerpen, en in ’t bijzonder lijkt mij logica als leerstof op zijn minst prematuur. Bij het woord “methode” in de titel van deze lezing heb ik echter niet zozeer de leerling voor ogen gehad, die zich bij het neerschrijven van zijn werk van logische symbolen bedient, maar veeleer de leraar die met modern logische methoden de leerstof en zijn vorm benadert en zijn eigen methodiek analyseert. (In this lecture I have argued in favour of applying—I do not say as much as possible, but as much as needed—symbolic logical methods by the pupil. This obviously does not mean the advent of logic as a subject matter in school mathematics. In general I am sceptical about new subjects, and logic as subject matter, in particular, seems to me at least premature. With the word “method” that I used in the title of this lecture I did not have the pupil, who uses logical symbols for writing down his work, in mind, but rather the teacher who approaches the subject matter and its outline with modern logical methods and analyses his own methods.) [212]

Then again Freudenthal is back on his hobbyhorse: modern mathematics, and in this case logic, was in the first place useful as a teacher’s aid to analyse the subject matter and the accompanying learning process, in order to arrive at well-considered didactics and methodology. Not only should the teacher be able to oversee the subject matter, he should also be able to oversee the logical form of that subject matter: Men zou nog een stap verder kunnen gaan, door van de leraar ook te eisen, dat hij boven de door hem gebezigde methode van aanbieding van de stof staat en zichzelf deze methode bewust kan maken. Ook hierbij kan de logische analyse diensten bewijzen. Niet in die triviale zin, dat de methode door de logische structuur worden [sic] bepaald, maar om, naar het voorbeeld van de Van Hieles, de niveau’s [sic]van inzicht te herkennen èn hun onderlinge relatie, die volstrekt logisch van aard is. (One could go even a step further by requiring that the teacher should also be able to oversee the method of offering the subject matter that he uses and that he can make himself aware of this method. Here too the logical analysis can be of use. Not in the trivial sense, that the method is determined by the logical structure, but, after the example of the Van Hieles, to recognize the levels of insight and the relation between them, which has a completely logical character.) [213]

And with this he again drew a parallel between reflecting from a higher level (of understanding) to a lower level (of understanding) and the logic of theory and metatheory which occupies such a prominent place in the modern mathematics.

266


8.8 Freudenthal and New Math: Conclusion 8.8.1 A Lonely Opponent of New Math? How New Math would develop and which consequences, intended but also unintended, it would ultimately have for the development of the mathematics curriculum, could not be foreseen in the period between 1957 and 1966. So it is not very useful to speak of camps ‘in favour of’ or ‘against’ this development. In the first few years the New Math modernization tendency in the Netherlands did not seem much more than the umpteenth attempt to revise an unmanageable curriculum that was mainly determined by tradition. But this time more was at stake: a curriculum had to be designed that was to be not only modern, but also satisfactory in the new educational structure that was part of the Mammoth Act. Since this would not only be an adaptation of the existing curriculum (as the 1958 curriculum reform was), but a really drastic change, it was easier to let go of a relatively large part of the old curriculum. Euclidean geometry and with it teaching the accompanying logic-deductive structure disappeared, for example. An intuitive introduction to geometry—which had been a discussion point for decades—was now generally accepted. At the same time the revision of the curriculum offered the opportunity to introduce modern mathematics (and modern mathematical language) in mathematics education. In his Schrijf dat op, Hans Freudenthal repeatedly pictured himself with justifiable pride as the lonely opponent of New Math: an “enfant terrible” at the time taken seriously by no one. But in what precedes in this chapter, a more nuanced picture is painted (Fig. 8.5). His really severe criticism of New Math initiatives was predominantly levelled at foreign tendencies. A Bourbakian structure of school mathematics such as came about in Belgium never happened in the Netherlands. What did come up in the Netherlands, had Freudenthal’s support and cooperation. His enthusiasm about the work in the CMLW displays this most obviously. At the end of the 1960s Freudenthal was still very hopeful that the curriculum the CMLW had designed would turn out all right. In an interview with HP Magazine he said: De leraren zijn hier de eerste jaren verschrikkelijk aan het schrikken gemaakt met alle nieuwe begrippen die zouden worden ingevoerd. […] Maar het nieuwe leerplan is in feite maar een zeer gematigde vernieuwing. Voor leraren die de cursussen hebben gevolgd, is het niet zo’n probleem. Ze moeten het met vallen en opstaan leren, maar ik heb de indruk, dat het betrekkelijk goed gaat. (The first few years the teachers in the Netherlands have been given a terrible fright with all the new concepts that were to be introduced. […] But the new curriculum is in fact only a very modest modernization. For the teachers who have attended the courses it is no problem at all. They have to learn by trial and error but I have the impression that it goes relatively well.) [214]

Yet in the mid-1980s Freudenthal wrote about the effect of the new curriculum: Het nieuwe bleek praktisch van begin af aan mis. Heel duidelijk was dat in de bovenbouw, maar je mocht het niet hardop zeggen om de lerarengemeenschap niet nodeloos ongerust te maken. Wat aan de bovenbouw schortte, is nu, drie lustra later, ten dele gerepareerd. (The

8.8 Freudenthal and New Math: Conclusion

267

Fig. 8.5 Freudenthal poses (with pipe as yet!) for a photographer of the newspaper Het Centrum (1963) new things turned out to be wrong almost from the start. That was very clear in the higher classes, but you were not supposed to say that aloud in order not to worry the community of teachers needlessly. What was wrong in the higher classes has now, fifteen years later, been partially remedied.) [215]

With the phrase “almost from the start” he probably meant the early 1970s, since it only then became really clear what the effect of the new curriculum and the new educational structure of the Mammoth Act was in practice. At the end of the 1960s anyway Freudenthal’s optimism about a positive effect of the new curriculum was still abundantly noticeable. Also the image of a lonely opponent, taken seriously by no one, was not true of Freudenthal in this period. He actually did play a role in all committees and institutions that had a say in the imminent modernizations: the NOCW, the CMLW and

268


also at an international level in the ICMI. As one of Euclides’ editors since 1956, who also published regularly in it, he could hardly be called “the odd one out” [216]. Whereas Freudenthal still epitomized the Wiskunde Werkgroep in the 1950s, his commitment to and identification with the ‘modernizers’ who increasingly acted away from the limelight were hardly noticeable in the 1960s. What was going on? Why did Freudenthal apparently go with the times? Was this at the cost of his ideals? Was he not alert enough? Or was it a question of ‘setting his sails to all—and in particular to the most influential—winds’?

8.8.2 Cooperate in Order to Adjust It indeed looked as if Freudenthal chose consciously for those parties that actually had a say in the imminent modernizations. That is how he himself explained his role in for example the NOCW and the ICMI. On the one hand this was a cunning move to seize all opportunities to engage in those committees that could have an active influence on the course of events. This matched better with his motto of preferring to cooperate—even when he did not always agree with the way of working and the ideas—in order to adjust the plans, rather than just being ‘an opponent’ and standing aloof, or remaining active in a group that had played its role. On the other hand this change of interest also happened partly unconsciously: his sojourn in the United States and his rectorship forced him to distance himself from things, also from the Wiskunde Werkgroep, for example. Because of this more of his other work, such as his purely mathematical work and besides that writing the book Lincos, also came to the fore. In addition, Freudenthal accepted almost every invitation that crossed his path, in particular when these concerned mathematics education. Underlying the (many) invitations was of course the reputation as a ‘mathematics educator’ that he had acquired in the course of the years. It logically resulted in the invitation to become a member of committees such as the CMLW. The times of being an odd man out, as he described himself in Schrijf dat op, Hans, were certainly over. But to consider him as a result of his involvement in the field as a New Math advocate is taking things to great lengths, without even taking into account the previously mentioned fact that such a conclusion was not relevant for this period. He sometimes seemed to be an advocate of New Math ideas (for example with respect to his ideas about geometry education), but he always added the essential nuance that characterized him, that learning to do mathematics was of greater importance than learning mathematics in itself. Above all it has to be said that Freudenthal did not allow himself the liberty of a laissez-faire attitude in a single committee he was involved in when it came to attaining and proclaiming his own ideas. In any company he tried to get his own vision and ideas on the table—and even more importantly: on the agenda. Indefatigably he kept hammering away at what can rightly be called his hobbyhorse in that period: method above content. Modernizing the educational method was more essential than modernizing the content of education.


269

8.8.3 Knowledge as a Weapon in the Struggle for a Better Mathematics Education For a long time—certainly until the early 1970s—Freudenthal considered professional development for teachers necessary to bring about an improvement in the educational methods. In his view this was not only a necessary but also a sufficient condition. Apart from his proposal in 1959 to establish an international research institute for the development of didactics (of mathematics) in the following years he seldom showed that he also deemed other measures necessary in addition to the professional development for mathematics teachers [217]. It was not before the end of the 1960s that within the CMLW some—for that matter less concrete—ideas arose about starting a didactics course as a supplement to the professional development for mathematics teachers, as it became clear from a letter written by Freudenthal (in his function as chairman of the ICMI) to a representative of the UNESCO about the professional development for mathematics teachers in secondary education. Le 6ème cours vient d’être achevé. A partir du 7ème on traitera des sujets de méthodique et de didactique au lieu de matière mathématique, mais on n’a pas encore trouvé la méthode de réaliser cette idée. (The sixth course has just ended. Starting with the seventh, methodological and didactical subjects instead of mathematical subject matter will be treated, but the method to realize this has as yet not been found.) [218]

A striking example of Freudenthal’s enthusiasm for the enormous success of the professional development for mathematics teachers in the Netherlands was his praise given one section above in the same letter: Je tiens à souligner que dans les Pays Bas nous avons des expériences de cinq années avec des cours très intensifs de recyclage où ont participé chaque fois la majorité des professeurs de lycée. Les résultats des cours (conférences + exercises) étaient stupéfiants. 90 % des participants pouvaient résoudre les problèmes posés dans les exercises. La plupart ont déclaré que leur enseignement a gagné beaucoup de profondeur par la connaissance de matière supérieure acquise pendant les cours. (I have to underline that in the Netherlands we have five years’ experience with very intensive professional development courses for mathematics teachers in every one of which the majority of the teachers in the Lycea have participated. The results of these courses (conferences and workshops) have been astonishing. Ninety per cent of the participants could solve the problems stated in the exercises. The majority has indicated that their teaching has gained in depth because of the knowledge of a higher level acquired during this course.) [219]

Implicitly Freudenthal expressed with this positive attitude with respect to the effect of these courses the expectation, that a teacher with sufficient knowledge of higher and modern mathematics—and besides this the necessary knowledge of the logical analysis of the learning process—should be able to develop the necessary methods and didactics himself. This was a point of view that matched well his trust in the power of the ‘higher point of view’ that the teacher should have, something he had proclaimed already since the 1940s [220]. This may have sounded slightly naive but in itself it was not such a strange thing to think at the time. In his thesis about the modernization of the mathematics education for example Brandenburg also expressed the opinion that professional development for mathematics teachers would be enough to make a success of the new

270


curriculum [221]. This was also the common opinion within the CMLW and given the idea and the effect of the professional development that had been organized this was generally accepted. Teaching methodology as a scientific discipline at the universities was discreetly growing in the Netherlands in the early 1960s. Apart from didactical theses, such as those of Mooi (1948), the Van Hieles (1957) and Brandenburg (1968), there was, often within the framework of the education of teachers, research into mathematics education and didactics of mathematics by subject-specific didacticians from academic pedagogical Institutes. There were for example the studies of Bunt in Utrecht and the education studies (under the supervision of) of psychologist-mathematician A.D. de Groot in Amsterdam [222]. The first part of Wansink’s Didactische oriëntatie voor wiskundeleraren was published in 1966. It emanated from his experiences when he was teaching certificate students didactics of mathematics and it was the first mathematical-didactical review for mathematics teachers in the Dutch language. If you wanted to become a teacher in the 1960s there still were two ways: via obtaining MO certificates or by taking the academic path [223]. The pedagogicaldidactical education of prospective teachers had been regulated in the period between 1952 and 1961 when stipulations about this were enacted in the law [224]. In the 1960s with the then imminent Mammoth Act the education of teachers was again scrutinized. The 1963 secondary education act distinguished qualifications of the first (for teaching at the VWO and the higher classes of the HAVO), the second (for teaching in the first class of the VWO and the second and third classes of the HAVO) and the third (for teaching at the MAVO) degree, a distinction opposed by many teachers. The appointment of the Commissie Opleiding Leraren (Committee for Teacher Education) (COL, chaired by J.B. Drewes, council advisor of the minister) in 1964, was the start of the organization of a professional training for teachers. What form this should have, remained subject of discussion for a long time. An important point of discussion was, apart from the question what the connection between such an institute and the university should be, the ancient discrepancy between pedagogical-didactical education versus education in the scientific discipline [225]. Euclides hardly mentioned a similar discussion about teacher education. The minds of the mathematics teachers were primarily occupied by the imminent modernization of the curriculum and the accompanying professional development in modern mathematics. At the Mathematisch Instituut in Utrecht Van der Blij and Freudenthal also gave courses for teachers, among those the course ‘Achtergronden van de schoolwiskunde’ (‘Background to the school mathematics’) [226]. Freudenthal’s enthusiasm for the professional development could also be explained from the point of view he had proclaimed years before that an academic study does not ‘end’ with obtaining the degree certificate but that a student—and therefore also a prospective teacher—should have to continue learning his whole life [227]. The fact that Freudenthal expected that a teacher, apart from having knowledge of modern mathematics, should also be capable of analysing the learning process, as an ‘extra’ requirement, was characteristic of his ideas and not often heard elsewhere.


271

With both hands Freudenthal seized the opportunity offered by the rising modernization tendency to have a chance to see at least some of his ideals realized— such as the professional development for teachers and the thorough revision of aim and content of the mathematics education at the VHMO. According to himself he made two big mistakes in that process: not availing himself of the possibility to be present in Royaumont with what would turn out to be the launch of New Math, and after that the fact that he did not take that new tendency seriously. Whether and if so, to what extent, Freudenthal could have had an influence on what happened in Royaumont is doubtful. On the other hand, the fact that he overestimated the effects of the professional development for teachers is a clear indication of his underestimation of the influence of New Math on the education in the Netherlands. The courses were indeed instrumental in initiating and enthusing a large group of teachers in modern mathematics. But Freudenthal overlooked an important link in the education that would only become obvious in the years to come: the influence that textbooks would eventually have on the education. The textbooks, admittedly, had to follow the curriculum, but the groups of authors had freedom in the interpretation. The CMLW and its professional development did not have an influence on that.

8.8.4 Freudenthal About the Aim of Mathematics Education The difficulty of placing Freudenthal in a camp for or against New Math was in the end due to the fact that with the ideas of his articles and lectures he usually put himself above that discussion. For him it was never the introduction—and if so what exactly—of modern mathematics in itself that was at stake. Krachtige pogingen worden in het werk gesteld, om de inhoud van het programma te vernieuwen. Ik sta er enigszins sceptisch tegenover. Ik meen, dat men de mathematische methode aan elk mathematiseerbaar materiaal kan leren. (Powerful attempts are made to modernize the content of the curriculum. I am a bit sceptically disposed towards these attempts. I think, that one could learn the mathematical method from any subject matter that can be mathematized.) [228]

According to Freudenthal there was not much use in discussing the concrete content of a curriculum if the intended aim of the mathematics education was not the prime matter of importance. And learning to do mathematics was his aim—the only way to learn mathematics that would appear to be ‘useful’ and applicable for the pupil. He put this aim above the traditionally determined subject matter, such as the Euclidean geometry in the geometry education, but also above the importance of the introduction of any modern axiomatics: the subject matter was subordinate to the higher purpose. When the CMLW started its activities in 1961 he wrote to Monna about this purpose of education: Ik zie geen noodzaak, om wiskundige stof af te wentelen van universiteit en hogeschool naar het VHMO. Evenmin zie ik enige noodzaak, om leerstof met het oog op practisch nut te kiezen. De leerstof hoort in het VHMO fundamenteel te zijn, hetgeen niet wegneemt, dat toepassingen van de leerstof met het oog op practisch nut gericht kunnen zijn. Een

272


voorbeeld: een behoorlijk begrip van evenredigheid zal practisch meer nut opleveren dan enige met directe blik op de praktijk gekozen leerstof. (I do not think it is necessary to shift mathematical subject matter from the university and the polytechnic to the VHMO. Neither is it necessary to select subject matter in view of practical use. The subject matter in the VHMO needs to be fundamental, which does not prevent the applications of this subject matter being directed at practical use. An example: a decent understanding of proportionality will be more useful in a practical sense than any subject matter that is chosen, guided by practice.) [229]

‘To become acquainted with the mathematical method through mathematics that pupils can apply with insight’ could for example be a way to summarize Freudenthal’s point of view in this period. If mathematics at school could be modern mathematics, it would be a welcome bonus. Freudenthal did not have concrete propositions for subject matter in this respect. However, he would gradually discover that his goal was less easily reached than he originally thought: airing his opinion in lectures and articles and giving courses were not enough. This observation would be the basis for a more vigorous action.

References 1. Freudenthal looking back at his description of his involvement in mathematics education of the 1980s. The quotation is taken from the poem ‘Huttens letzte Tage’ (1871) by the Swiss author Conrad Ferdinand Meyer (1825–1898)’; Freudenthal, Schrijf dat op, Hans 359. 2. In composing this and following paragraphs, information has been taken from the following studies: Boekholt and De Booy, De geschiedenis van de school in Nederland vanaf de middeleeuwen tot aan de huidige tijd; Saskia Grotenhuis, Op zoek naar middelbaar onderwijs. hbs, gymnasium, mms en lyceum in discussie tussen 1900 en 1970 (Amsterdam 1998); Dodde, Het Nederlandse onderwijs verandert; Wansink, Didactische oriëntatie voor wiskundeleraren. Deel I, II, III; Brandenburg, Modernisering van het wiskundeonderwijs; De Moor, Van vormleer naar realistische meetkunde; Euclides jrg. 32–43 (1956–1968); Goffree e.a., eds., Honderd jaar wiskundeonderwijs; Goffree, Ik was wiskundeleraar. 3. Freudenthal, Schrijf dat op, Hans 348, 360. 4. Ibid., 362. 5. Ibid., 349. 6. See also paragraph 2.2.2. 7. Lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished); RANH, Hans Freudenthal Papers, inv.nr. 540. 8. For a retrospect on Bourbaki's work see also: A. Borel, ‘Twenty-five years with Nicolas Bourbaki, 1949–1973’, Notices of the American Mathematical Society 45 (1998) 373–380; M. Senegal, ‘The continuing silence of Bourbaki—an interview with Pierre Cartier’, Mathematical Intelligencer 20 (1998) 22–28. 9. Among the founding fathers were the mathematicians Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean A.E. Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil. 10. See also the discussions of Bourbaki's work in Euclides in this period. 11. Freudenthal, Schrijf dat op, Hans 348–349. 12. For a description of this discussion, see also paragraph 2.2.2. 13. As an experiment statistics were given as an optional subject at the Gymnasium-α under the guidance of Bunt. See also paragraph 6.4; L.N.H. Bunt, Statistiek als onderwerp voor het

References

273

gymnasium A. An investigation into the possibilities of teaching descriptive and elementary mathematical statistics in secondary schools (Groningen 1956); L.N.H. Bunt, Statistiek als onderwerp voor het gymnasium A. Verslag van een proefneming. A Inrichting en resultaten van het onderzoek (Groningen 1957). 14. Hans Freudenthal, ‘En nu… de wiskunde’, De Groene Amsterdammer (2 December 1950); RANH, Papers of Hans Freudenthal Papers, inv.nr. 1439. 15. For a review of this curriculum and Freudenthal's comments see also paragraph 6.3; Hans Freudenthal, ‘Naar een nieuw wiskunde-onderwijs’, De Groene Amsterdammer (2 April 1955); RANH, Hans Freudenthal Papers, inv.nr. 1513. 16. The following studies were used to derive data for this paragraph: Morris Kline, Why Johnny can't add: the failure of the New Math (New York 1973); J. Kilpatrick, ‘Five lessons from the New Math era’ and R.W. Bybee, ‘The Sputnik era: why is this educational reform different from all other reforms?’, lectures at the symposium Reflecting on Sputnik. Linking the past, present and future of educational reform by the Center for Science, Mathematics, and Engineering Education, 4 October 1997, digitally available and consulted; Organisation for European Economic Cooperation, New thinking in school mathematics (1961); Brandenburg, Modernisering van het wiskundeonderwijs; Wansink, Didactische oriëntatie voor wiskundeleraren. Deel I, II, III; Goffree e.a., eds., Honderd jaar wiskunde-onderwijs. 17. Morris Kline was professor at the New York University from 1938 to 1975. Among other things he published on the history of mathematics and philosophy. In his articles and books about mathematics education he emphasized learning to apply mathematics. He was strongly opposed to New Math; Kline, Why Johnny can't add: the failure of the New Math 4–8. 18. As quoted in: D. Klein, ‘A brief history of American K-12 mathematics education in the 20th century’, in: J. Royer, ed., Mathematical cognition (Greenwich 2003). 19. Later Begle, and with him the complete SMSG project, would move over to Stanford. 20. ‘On the mathematics curriculum of the high school’, The Mathematics Teacher (1962) 191– 195; see also: Kline, Why Johnny can't add: the failure of the New Math 112–118. 21. Kline, Why Johnny can't add: the failure of the New Math 3. 22. For the role Piaget's experiences played in New Math, see also paragraph 9.5.3. 23. Such improper use of psychology he already raised in his ‘Rekendidaktiek’ from 1944; Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 18. 24. Jean Piaget (1896–1980) was a developmental psychologist from Switzerland, whose ideas— in particular in the didactical quarter these were not completely undisputed—about the development of the thinking power of children had a strong influence on many mathematicaldidactical theories, especially in the New Math quarter. In the 1970s Freudenthal regularly criticized the interpretation of Piaget's studies. 9.5.3 See also paragraph 9.5.3 and 9.6.3. 25. Freudenthal, Schrijf dat op, Hans 349. 26. F. Loonstra en P.G.J. Vredenduin, eds., Modernization of mathematical teaching in the Netherlands (Groningen 1962); W.J. Thijssen en J.H. Wansink, ‘Het internationale mathematische congres in 1958’, Euclides 34 (1958/1959) 225–245. 27. On the effect of the New Math development in, among other countries, Belgium, see also paragraph 8.5.1. 28. In the Netherlands OEES: Organisatie voor Europese Economische Samenwerking. 29. The OEEC at the end of the 1950s consisted of: Belgium, Canada, Denmark, Germany, France, Greece, Great-Britain, Ireland, Italy, Iceland, Luxemburg, the Netherlands, Norway, Austria, Portugal, Turkey, the United States, Sweden and Switzerland. 30. In the Netherlands OESO: Organisatie voor Economische Samenwerking en Ontwikkeling. 31. Freudenthal probably means OEES with OEEO; Freudenthal, Schrijf dat op, Hans 349–350. 32. Ibid., 351. 33. Ibid., 350. 34. Ibid., 350. 35. J.R. Gass to Freudenthal, 20 October 1958; RANH, Hans Freudenthal Papers, inv.nr. 1829. 36. Organisation for European Economic Cooperation. Office for Scientific and Technical Personnel, New thinking in school mathematics (1961) 4.

274


37. Discussion paper for the meeting of 13–14 April in Paris of the ‘group of experts’, 3 pages, q.v. 2–3; RANH, Hans Freudenthal Papers, inv.nr. 1829. 38. Freudenthal to R.C. Kwantes, 25 March 1959; RANH, Hans Freudenthal Papers, inv.nr. 1829. 39. Ibid. 40. Ibid. 41. See also paragraph 6.6. 42. Ibid. In this comment Freudenthal's criticism of unscientifically based (mathematics education) research can be recognized. He would later regularly return to this, among other things in his book Weeding and sowing, but also in his criticisms on other people's research (methods). 43. Metzger to Freudenthal, 16 April 1959; RANH, Hans Freudenthal Papers, inv.nr. 1829. 44. Letter written by A.A. de Vos van Steenwijk, 23 May 1959; ibid. In this letter Mrs. De Vos van Steenwijk mentioned that in composing the plans the opinion of ‘another well-known professor’ had also been asked besides that of professor Stone. 45. Discussion paper ‘International seminar in mathematical education’ 4 pages, q.v. 2; RANH, Hans Freudenthal Papers, inv.nr. 1829. See also: M.A.J.M. Matthijsen, De elite en de mythe. Een sociologische analyse van de strijd om onderwijsverandering (Deventer 1982) 146 et seq. 46. Discussion paper ‘International seminar in mathematical education’ 4 pages, q.v. 4; RANH, Hans Freudenthal Papers, inv.nr. 1829. 47. Ibid., 3. 48. Ibid. 49. Freudenthal to Metzger, 21 April 1959; ibid. 50. Ibid. 51. Ibid. Here the title and the tenor of his later articles can already be recognized, such as: Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’. 52. Freudenthal to Metzger, 21 April 1959; RANH, Hans Freudenthal Papers, inv.nr. 1829. 53. Gass to Freudenthal, 28 April 1959; ibid. 54. Freudenthal to Mrs. De Vos van Steenwijk, undated, probably of 4 May 1959; ibid. 55. Ibid. 56. Ibid. 57. Freudenthal to Kwantes, 8 May 1959; ibid. 58. Ibid. 59. Freudenthal to A.J. Piekaar, 16 July 1959; ibid. 60. Ibid. 61. Ibid. 62. H. Th. M. Leeman, L.N.H. Bunt, P.G.J. Vredenduin, ‘Verslag van het seminarium “New thinking in school mathematics” van de OEES’, Euclides 35 (1960) 218–229; Organisation for European Economic Cooperation, Office for Scientific and Technical Personnel, New thinking in school mathematics (1961). 63. J. Dieudonné, ‘New thinking in school mathematics’, in: Organisation for European Economic Cooperation, Office for Scientific and Technical Personnel, New thinking in school mathematics (1961) 31–49, q.v. 35. 64. Leeman et.al., ‘Verslag van het seminarium “New thinking in school mathematics” van de OEES’, Euclides 35 (1960) 218–229, q.v. 220. 65. Hans Freudenthal, ‘Algèbre linéaire et géométrie élémentaire By Jean Dieudonné’, The American Monthly 74 (1967) 745–748, q.v. 745. 66. Leeman et.al., ‘Verslag van het seminarium “New thinking in school mathematics” van de OEES’, Euclides 35 (1960) 218–229, q.v. 228. 67. Interview with Vredenduin in: Goffree, Ik was wiskundeleraar 163. 68. Freudenthal, Schrijf dat op, Hans 349–350.

References

275

69. Hans Freudenthal, ‘De begrippen axioma en axiomatiek in de Wis- en Natuurkunde’, Simon Stevin 39 (1955) 156–175; q.v. 164. See also: Hans Freudenthal, ‘Axiom und Axiomatik’, Mathematisch-physikalische Semesterberichte 5 (1956) 4–19. 70. Hans Freudenthal, ‘De begrippen axioma en axiomatiek in de Wis- en Natuurkunde’ 165– 166. 71. Ibid., 166. Also in his lecture ‘ Axiomatiek in het wiskunde-onderwijs bij het V.H.M.O.’, delivered during the summer course of the MC in 1962 Freudenthal spoke about axiomatizng as the essence of modern mathematics; Hans Freudenthal, ‘Axiomatiek in het wiskundeonderwijs bij het V.H.M.O.’ summer course MC 1962, 4 pages; RANH, Hans Freudenthal Papers, inv.nr. 537. 72. Hans Freudenthal, ‘Trends in Modern Mathematics’, ICSU Review 4 (1962) 54–61, q.v. 61. 73. Hans Freudenthal, ‘Het staartje van de mammoet’, De Groene Amsterdammer (28 juli 1962); RANH, Hans Freudenthal Papers, inv.nr. 1570. 74. Freudenthal, Van sterren tot inlegzolen. 75. Hans Freudenthal, Wiskunde in wetenschap en dagelijks leven (Hilversum 1967). The international series ‘Wereldakademie’ was the result of a cooperation of seven European and one American publishers and an international Advice Council. The purpose of the Wereldakademie was publishing a series of informative books that would allow both ‘specialists' and ‘interested laymen’ to familiarize themselves with the most modern scientific insights and were written by authors with ‘a worldwide reputation’. Freudenthal's Wiskunde in wetenschap en dagelijks leven was translated and published in English, German, Swedish, Spanish, Italian, French, Portugese, Japanese and Russian; ibid., 1. 76. Uitgeverij Meulenhoff to Freudenthal, 6 September 1963; RANH, Hans Freudenthal Papers, inv.nr. 42. 77. Hans Freudenthal, Wiskunde in wetenschap en dagelijks leven (Hilversum 1967) 9–10. 78. Freudenthal, ‘Opvoeding tot denken’ (unpublished 1945); RANH, Hans Freudenthal Papers, inv.nr. 466; Freudenthal ‘Oude en nieuwe universiteiten’ (unpublished 1945); RANH, Hans Freudenthal Papers, inv.nr. 467. 79. Freudenthal, ‘Trends in modern mathematics’, ICSU Review 4 (1962) 54–61, q.v. 54. See also Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 21. 80. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 13. 81. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 29. 82. Freudenthal, ‘Logical analysis and critical survey’, in: Hans Freudenthal, ed., Report of the Relations between Arithmetic and Algebra 20–41, q.v. 23. 83. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 16. 84. Freudenthal, ‘De algebraïsche en de analytische visie op het getalbegrip in de elementaire wiskunde’ 106–121. 85. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 22. Freudenthal followed the same line of thought in his article ‘Logica als methode en als onderwerp’ from 1960, ‘Logical analysis and critical survey’ from 1962, and also in ‘Enseignement des mathématiques modernes ou enseigement modernes des mathématiques?’ from 1963; Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255, q.v. 255; Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22; Freudenthal, ‘Logical analysis and critical survey’ 25–26; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 30–31. 86. Freudenthal was familiar with the importance of the thought experiment in particular via the studies of G. Polya; Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 14–15; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 32; Freudenthal, ‘Logica als methode en als onderwerp’ 255.

276


87. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 34. 88. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 13, and also 21. 89. Hans Freudenthal, ‘De betekenis van de wetenschappelijke basis voor de leraar’, De wetenschappelijke basis van de leraarsopleiding, mede in verband met de ontwikkeling van de exacte wetenschappen in de twintigste eeuw. Verslag van het zestiende congres van leraren in de wiskunde en de natuurwetenschappen, gehouden te Utrecht op 18 april 1966 (1966) 5–11, q.v. 9. 90. Ibid. 91. Hans Freudenthal, lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished) 7 pages; q.v. 1q; RANH, Hans Freudenthal Papers, inv.nr. 1831. 92. Freudenthal described this example in the following articles: ‘Integratie achteraf of vooraf’ (1961), ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ (1963) and ‘L’intégration après coup ou Ã la source’ (1969). Freudenthal used the example of the counter clerk some time before in his lecture ‘Opvoeding tot denken’ (unpublished 1945); Hans Freudenthal, ‘Integratie achteraf of vooraf’ (unpublished) 14 pages, q.v. 1, lecture at the WVO conference ‘Op zoek naar integratie’, 11 November 1961; RANH, Hans Freudenthal Papers, inv.nr. 536; also in: Freudenthal, ‘L’intégration après coup ou à la source’, Educational Studies in Mathematics 1 (1969), 327–337, q.v. 327; Freudenthal, ‘Opvoeding tot denken’ (unpublished 1945) 29 pages; RANH, Hans Freudenthal Papers, inv.nr. 466; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 36. 93. Freudenthal, ‘Integratie achteraf of vooraf’ (unpublished 1961) 14 pages, q.v. 1–2; lecture at the WVO conference ‘Op zoek naar integratie’, 11 november 1961; RANH, Hans Freudenthal Papers, inv.nr. 536. 94. Hans Freudenthal, lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished) 7 pages; q.v. 4; RANH, Hans Freudenthal Papers, inv.nr. 1831. 95. See also paragraph 9.5.3. 96. More about the use of set theory in education in the paragraphs 9.5.3 and 9.6.1. 97. Hans Freudenthal, lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished) 7 pages; q.v. 4; RANH, Hans Freudenthal Papers, inv.nr. 1831. 98. Ibid. 99. Hans Freudenthal, ‘Het staartje van de mammoet’, De Groene Amsterdammer (28 juli 1962); RANH, Hans Freudenthal Papers, inv.nr. 1570. 1570. 100. J.C.H. Gerretsen, ‘Doelstelling van het wiskundeonderwijs’, Euclides 34 (1958) 90–94, q.v. 93–94. 101. J.H. Wansink, ‘Didactische revue’, Euclides 34 (1958) 165–175, q.v. 165. 102. ‘Uit de openingstoespraak van de voorzitter van WIMECOS tot de algemene vergadering van 28 december 1961’, Euclides 37 (1962) 264–267, q.v. 266. 103. Hans Freudenthal, ‘Naar een nieuw wiskunde-onderwijs’, De Groene Amsterdammer (2 April 1955); RANH, Hans Freudenthal Papers, inv.nr. 1513. 1513. 104. There were, apart from the Royaumont Seminar in 1959, for example; another international colloquium in Aarhus in 1960 followed by a meeting of experts in Yugoslavia and annual Belgium congresses for teacher in Arlon and several courses for teachers in Brussels and Amsterdam; F. Loonstra en P.G.J. Vredenduin, ed., Modernization of mathematical teaching in the Netherlands (Groningen 1962) 5. 105. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel II 405, Deel III 20. 106. Freudenthal, Schrijf dat op, Hans 137. In a letter to Howson Freudenthal wrote that he liked neither Gattegno nor Papy and that his active involvement in the CIEAEM only started when the Polish Anna Zofia Krygowska (1904–1988) became chairwoman; Freudenthal to Howson, 19 July 1983; RANH, Hans Freudenthal Papers, inv.nr. 38.

References

277

107. But Beth's cooperation with Piaget would be the result of the activities of this group. 108. R. Holvoet, ‘Over de opbouw van de methode Papy’, in: Wansink, Didactische oriëntatie voor wiskundeleraren. Deel II 405–449, q.v. 445. 109. G. Noël, ‘Pourquoi, pour qui enseigner les mathématiques? Une mise en perspective historique de l’évolution des programmes, au xxe sciècle, en Belgique’, Zentralblatt für Didaktik der Mathematik 34 (2002) 110–119. 110. G. Krooshof, ‘Moderniseren—nieuwbouw of verbouw?’, Euclides 42 (1966) 193–203, q.v. 194. 111. P.G.J. Vredenduin, ‘Het experiment Papy’, Euclides 42 (1966) 167–182, q.v. 171. 112. From 1968 onwards the Centre also had its own periodical: NICO: Revue Périodique du Centre Belge de Pédagogie de la Mathématique. 113. Interview with Vredenduin in: Goffree, Ik was wiskundeleraar 163. 114. P.G.J. Vredenduin, ‘Een opzienbarend boek’, Euclides 39 (1963/64) 237–247; Euclides 42 (1966/67) 90–94, 161–166. Papy's Mathématique moderne would eventually consist of six volumes. 115. F. Loonstra en P.G.J. Vredenduin, eds., Modernization of mathematical teaching in the Netherlands (Groningen 1962) 5. 116. Freudenthal was co-author of the volumes II ‘Geometrie’, III ‘Analysis' en IV ‘Praktische Methoden und Anwendungen der Mathematik’; H. Behnke, ed., Grundzüge der Mathematik für Lehrer an Gymnasien sowie für Mathematiker in Industrie und Wirtschaft I-V (Göttingen 1958–1966); RANH, Hans Freudenthal Papers, inv.nr. 1747. 117. Announcement in Euclides; Euclides 35 (1959) 112. At this conference Freudenthal lectured on ‘Logica als onderwerp en als methode’, Euclides 35 (1959) 241–255. 118. Morsch, Met de moed van de hoop 222–258. 119. S.J.C. Freudenthal-Lutter, De Jenaplanschool. Een leef- en werkgemeenschap (Utrecht 1970), 5. Langeveld wrote the preface. 120. M. van Essen en J.D. Imelman, Historische pedagogiek. Verlichting, romantiek en ontwikkelingen in Nederland na 1800 (Baarn 1999) 112. 121. In the Netherlands in 1975 there were about 120 schools organized according to Jenaplan Education. The most striking characteristic of these schools was the lack of the usual division in classes. Instead there were the ‘multi-age groups' (the so-called vertical division) which made ‘repeating a class' non-existent. Furthermore developing community spirit, ‘educating to critical thinking’ and paying attention to the child as an individual were considered of paramount importance. 122. In Assen at present is the national Jenaplan library, the ‘Suus Freudenthal-Lutterbibliotheek’, mainly a collection Mrs. Freudenthal built up via her work as secretary of Pedomorfose. 123. See also paragraph 9.5.3. 124. Oral tradition evidences that Freudenthal was initially asked as an advisor for this series, but he would have passed this on to De Iongh; De Moor, Van vormleer naar realistische meetkunde 503. 125. An invitation to Berkeley came to nothing, since there was no money available. But preceding his stay in New Haven he held a ‘summer course’ there; RANH, Hans Freudenthal Papers, inv.nr. 392. 126. Freudenthal, Lincos. Design of a language for cosmic intercourse. Part 1 (Amsterdam 1960). 127. ‘H. Freudenthal’, Vrij Nederland (1976) 232–247, 244. Freudenthal says about this: “We can explain nature, the universe with the help of mathematics. So when there is another nation which has also developed science, than they also have to know mathematics, it cannot be otherwise.” 128. Vredenduin reviewed Lincos in Euclides 36 (1961) 252–254. During his year in the United States Freudenthal lectured among other things about Lincos at Brown University (Rhode Island) and Brandeis University (Massachusetts). 129. Documents from the ‘Yale’- period; RANH, Hans Freudenthal Papers, inv.nr. 316, 392, 393, 1884.

278


130. Documents related to Freudenthal's rectorship; RANH, Hans Freudenthal Papers, inv.nr. 394. 131. ‘Commissie modernisering leerplan wiskunde’, Euclides 37 (1961) 144–151, q.v. 144. 132. Dodde, Het Nederlandse onderwijs verandert 139, 167 et seq. 133. Grotenhuis, Op zoek naar middelbaar onderwijs 157 et seq. 134. Mathematician A.F. Monna worked from 1961 at the Utrecht University. In 1965 he became professor of pure mathematics. From 1963 to 1974 he was also a member of the NOCW; Kleijne, ‘NOCW vijftig jaar’, Nieuw Archief voor Wiskunde 5 (2004) 308–313, q.v. 313; Adriaan Monna and Marius van der Put, ‘Ambtenaar en wiskundige’, Nieuw Archief voor Wiskunde 5 (2004) 136–146, q.v. 141, 144. 135. Freudenthal, Schrijf dat op, Hans 350–351. 136. Freudenthal to Monna, 23 August 1961; RANH, Hans Freudenthal Papers, inv.nr. 1723. 137. Ibid. 138. Ibid. 139. ‘Uit de openingstoespraak van de voorzitter van WIMECOS tot de algemene vergadering van 28 december 1962’, Euclides 38 (1963) 251–253, q.v. 252. 140. `Mededeling van de WW’, Euclides 39 (1964) 119. 141. `Mededeling van de WW’, Euclides 40 (1964) 95. 142. A.F. Monna, ‘Bericht over de werkzaamheden van de commissie modernisering leerplan wiskunde’, Euclides 41 (1965) 104–107. 143. Hans Freudenthal, ‘De betekenis van de wetenschappelijke basis voor de leraar’, De wetenschappelijke basis van de leraarsopleiding, mede in verband met de ontwikkeling van de exacte wetenschappen in de twintigste eeuw. Verslag van het zestiende congres van leraren in de wiskunde en de natuurwetenschappen, gehouden te Utrecht op 18 april 1966 (1966) 5–11, q.v. 8. 144. Lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished); RANH, Hans Freudenthal Papers, inv.nr. 540. 145. This application was eventually turned down in 1968 referring to the imminent plans for a nationwide organization for curriculum development, in which the CMLW would have to be absorbed. See also paragraph 9.3.1: E. Wijdeveld, H. Verhage, G. Schoemaker, ‘Van CMLW tot Freudenthal Instituut’, in: Goffree et al., eds., Honderd jaar wiskundeonderwijs 355–374, q.v. 359 et seq. 146. Lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished); RANH, Hans Freudenthal Papers, inv.nr. 540. 147. Hans Freudenthal, ‘New mathematics’, The New Era in Home and School 45 (1964) 189– 191 + 188, q.v. 191, 188. 148. Minister Rutten had already proposed a reorganization of the secondary education in 1951, the so-called ‘Plan-Rutten’; Grotenhuis, Op zoek naar middelbaar onderwijs 155. 149. ‘Openingstoespraak van de voorzitter van WIMECOS, Dr.Ir. B. Groeneveld op de algemene vergadering van 28 december 1966’, Euclides 42 (1967) 184–187, q.v. 185. 150. Freudenthal, Schrijf dat op, Hans 351. 151. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 19. 152. A.F. Monna, ‘Bericht van de commissie modernisering leerplan wiskunde’, Euclides 42 (1966) 129–134, q.v. 131. 153. Commissie Modernisering Leerplan Wiskunde, Toelichting op het leerplan wiskunde (1968). 154. N.G. de Bruyn, ‘Modernisering leerplan wiskunde’, Euclides 43 (1968) 260–261. 155. Hans Freudenthal, ‘Modernisering leerplan wiskunde. Antwoord aan Prof. dr. N.G. de Bruyn’, Euclides 43 (1968) 321–322, q.v. 322. 156. Hans Freudenthal, ‘Het staartje van de mammoet’, De Groene Amsterdammer (28 July 1962); RANH, Hans Freudenthal Papers, inv.nr. 1570.

References

279

157. Freudenthal, Schrijf dat op, Hans 158. G. Krooshof, ‘Moderniseren—nieuwbouw of verbouw?’, Euclides 42 (1966) 193–203, q.v. 196. 159. Namely: the Katholiek Pedagogisch Bureau, the Christelijk Pedagogisch Studiecentrum and the Onderwijskundig Studiecentrum; Brandenburg, ‘De modernisering van het wiskundeonderwijs’, in: Wansink, Didactische oriëntatie voor wiskundeleraren. Deel III 15–49, q.v. 27. 160. G. Krooshof, ‘Moderniseren—nieuwbouw of verbouw?’, Euclides 42 (1966) 193–203, q.v. 196. 161. Ibid. 162. W. Kleijne, ‘Leerboeken, hun uitgevers en auteurs’, in: F. Goffree at al., eds., Honderd jaar wiskundeonderwijs 149–161, q.v. 159–160. 163. G. Krooshof, ‘Moderniseren—nieuwbouw of verbouw?’, Euclides 42 (1966) 193–203, q.v. 200. 164. Freudenthal, Schrijf dat op, Hans 351. 165. For the history of the initial geometry education, the controversy around it and movements that played a role see: De Moor, Van vormleer naar realistische meetkunde. 166. ‘Interimrapport van de commissie modernisering leerplan wiskunde’, Euclides 42 (1966) 129–134, q.v. 130. 167. Commissie Modernisering Leerplan Wiskunde, Toelichting op het leerplan wiskunde (1968) 8. 168. M. Bijpost, J.K. Timmer, Met passer en liniaal 1. Eenvoudig leerboek der vlakke meetkunde (Zutphen 1957) 52, 76. 169. Other methods such as Inleiding in de meetkunde by Brandenburg and Schrier from 1961 also followed this approach; W.J. Brandenburg, L. Schrier, Inleiding in de meetkunde 1 (Groningen 1961) 100. 170. H.J. Jacobs e.a., Moderne wiskunde voor voortgezet onderwijs. Deel 1 voor de brugklas (Groningen 1972) 126–130. 171. For that matter, this approach to the theorem appeared in the more progressive text books even before 1968, see for example: D.N. van der Neut, A. Holwerda, Meetkunde. Met de beginselen der goniometrie. Eerste deel (Groningen 1959) 28; P. Wijdenes, Planimetrie. Eenvoudig schoolboek voor het eerste onderwijs in de vlakke meetkunde (Groningen 1931), second edition 1933, 22–23. 172. Commissie Modernisering Leerplan Wiskunde, Toelichting op het leerplan wiskunde (1968) 19. 173. This lecture has been published both in English and in Dutch (in a somewhat adapted version); Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956) 83–97; Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v.14– 15. 174. Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 14–15. 175. The most striking example is perhaps his contribution to the discussion with Ehrenfest; Ehrenfest-Afanassjewa and Freudenthal, Kan het wiskunde-onderwijs tot de opvoeding van het denkvermogen bijdragen? 176. A similar passage Freudenthal also wrote in ‘Traditie en opvoeding’ and in ‘Tradition and education’; Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15; Freudenthal, ‘Traditie en opvoeding’, Rekenschap (3) (1957) 95–103; Freudenthal, ‘Tradition and education’, The New Era in Home and School 37 (1956) 127–132. 177. Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15. 178. E. Wijdeveld, ‘Omzien in verwondering. Ontstaan, werkwijze en effecten van de Commissie Modernisering Leerplan Wiskunde en het Instituut Ontwikkeling Wiskunde Onderwijs (1960–1970)’, Euclides 78 (2003) 218–225, q.v. 223. See also the printed version of the HKRWO lecture with the same title, 25 May 2002.

280


179. Freudenthal, ‘Preface of the editor,’ in: Hans Freudenthal, ed., Report on methods of initiation into geometry 5. 180. Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15. 181. In ‘Initiation into Geometry’ he also mentioned the studies of Bos and Lepoeter, and the study of Van Albada; Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956) 83–97; E. Castelnuovo, Geometria intuitiva (Firenze 1948, second revised version 1952); Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15–18. 182. Freudenthal, ‘Initiation into geometry’, The Mathematics Student 24 (1956) 83–97, q.v. 91–92. 183. Ibid., 87–93. A good example of working with the concept of ‘tiling’ in the initial geometry education could be found in the studies of the Van Hieles, who elaborated on the original concept of Van Albada. 184. Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 16. 185. Ibid., 16. 186. Hans Freudenthal, ‘Axiomatiek in het wiskunde-onderwijs bij het vhmo’, lecture during the summer course of the MC, 1962 (unpublished) 4 pages, q.v. 2; RANH, Hans Freudenthal Papers, inv.nr. 537. 187. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 26. 188. Ibid. 189. Hans Freudenthal, ‘Axiomatiek in het wiskunde-onderwijs bij het VHMO’, lecture during the summer course of the MC, 1962 (unpublished) 4 pages, q.v. 3; RANH, Hans Freudenthal Papers, inv.nr. 537. 190. Ibid. 191. Ibid., 4. 192. Hans Freudenthal, ‘Bemerkungen zur axiomatischen Methoden im Unterricht’, Der Mathematikunterricht (Stuttgart 1966) 61–65. This is a summary of the discussion after Freudenthal's lecture ‘Was ist axiomatik…’ at the ICMI Seminar in Aarhus in 1960. 193. With this point of view he gained the support of Streefkerk who referred to Freudenthal's statements about axiomatics in education in 1964; H. Streefkerk, ‘Problemen rondom de vernieuwing van het wiskunde-onderwijs’, in: Stemmen uit de praktijk. Vernieuwing van het wiskundeonderwijs. Inleidingen en verslagen van de conferentie wiskunde, georganiseerd door de afdeling VHMO van het Christelijk Paedagogisch Studiecentrum (werkgroep wiskunde) op 6 en 7 maart 1964 (Den Haag 1964) 7–19, q.v. 18. 194. Freudenthal, ‘Logical analysis and critical survey’ 32. In somewhat other wording he also mentioned this subject in his lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held from 19 to 23 December 1964 in Utrecht. 195. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 28. 196. Freudenthal, ‘Report on a comparative study of methods of initiation into geometry’, Euclides 34 (1957) 289–306, q.v. 299. 197. Ibid. Good examples how to realize this could be found in the studies of Van Albada and those of the Van Hieles according to Freudenthal. 198. Correspondence between Erven F. Bohn and Freudenthal, 1955; RANH, Hans Freudenthal Papers, inv.nr. 29; Hans Freudenthal, Exacte logica (Haarlem 1961). 199. Freudenthal, Exacte logica V. 200. Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255, q.v. 255; Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22; Freudenthal, ‘Logical analysis and critical survey’ 20–41, q.v.25–26; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 28–44, q.v. 30–31.

References

281

201. Freudenthal, Exacte logica V. 202. Ibid., VI. 203. Ibid., 56–57. 204. Lecture at the weekend conference of the Wiskunde Werkgroep of the WVO, November 1959, in 1961 published in Euclides; Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255. 205. Ibid., 244. 206. Ibid., 249. 207. ∧ Ù and ∨ are today often written as ∀ (‘for all’) en ∃ (‘there exists’), respectively. 208. Freudenthal, ‘Logica als methode en als onderwerp’ 251. 209. Ibid., 252. 210. Ibid., 252–253. Freudenthal took his example from the collection ‘250 opgaven’ (‘250 exercises’), which was published in Euclides 32 (1956) as support of the 1958 curriculum. 211. Hans Freudenthal, ‘Functies en functie-notaties’, Euclides 41 (1966) 299–304, q.v. 304. 212. Hans Freudenthal, ‘Logica als methode en als onderwerp’ 255. 213. Ibid., 255 Freudenthal followed the same line of thought in his article ‘Logical analysis and critical survey’ from 1962 and also in ‘Enseignement des mathématiques modernes ou enseigement modernes des mathématiques?’ from 1963; Freudenthal, ‘Logical analysis and critical survey’ 25–26; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 30–31. 214. Freudenthal in the interview: Bert Vuysje, ‘De wiskundeles verandert: 3 + 2 = 1’, HP Magazine (24 December 1969–6 January 1970); RANH, Hans Freudenthal Papers, inv.nr. 1847. 215. Freudenthal, Schrijf dat op, Hans 351. 216. Freudenthal's publications of the period 1956–1966 in Euclides: Freudenthal, ‘Report on a comparative study of methods of initiation into geometry’, Euclides 34 (1957) 289–306; Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255; Freudenthal, ‘Functies en functie-notaties’, Euclides 41 (1966) 299–304; Freudenthal, ‘Modernisering Leerplan Wiskunde. Antwoord aan Prof. De Bruyn’, Euclides 43 (1968) 321–322. 217. Freudenthal to R.C. Kwantes, 25 March 1959; RANH, Hans Freudenthal Papers, inv.nr. 1829. 218. Freudenthal to Albert Legrand, Institut de l’Unesco pour l’Education, 9 februari 1968; RANH, Hans Freudenthal Papers, inv.nr. 365. 219. Ibid. 220. See for example—besides similar thoughts in his unpublished article ‘Oude en nieuwe universiteiten’—the article: Freudenthal, ‘De algebraïsche en de analytische visie op het getalbegrip in de elementaire wiskunde’ 106–121. 221. Willem Jan Brandenburg (1921) studied mathematics and physics in Amsterdam and after that became a mathematics teacher. After studying pedagogy in Groningen (which he finalized in 1958) he subsequently defended his mathematical-didactical thesis, in which he described the problems arising after the modernization of the school mathematics of 1967, with Prof. J.C.H. Gerretsen and Prof. L. van Gelder in 1968. From 1963 onwards he worked as a senior lecturer at the department Lerarenopleiding (Teacher education) of the Pedagogisch Instituut in Groningen; Brandenburg, Modernisering van het wiskunde-onderwijs. 222. Adriaan D. de Groot (1914) studied mathematics and psychology. In 1946 he defended his thesis entitled Het denken van de schaker. Een experimenteel-psychologische studie. ( The thinking of the chess player, and experimental-psychological study) He had been a mathematics teacher for some years, when he from 1950 onwards was employed as professor at the University of Amsterdam, where he specialized in psychological-educational research. One of his most cited publications was Vijven en zessen. Cijfers en beslissingen: het selectieproces in ons onderwijs. At the end of the 1950s a comparative study into methods for initial geometry education was performed under his supervision (Euclidean geometry versus transformation geometry); A.D. de Groot, Vijven en zessen. Cijfers en beslissingen: het selectieproces in ons onderwijs (Groningen 1966); F. Goffree, ‘Een halve eeuw onderzoek. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 233–243.

282


223. See also: J. Vos en K. van der Linden, Waarvan akte. Geschiedenis van de M.O.-opleidingen 1912–1987 (Assen 2004); Klaske Blom, ‘Van de acten van bekwaamheid. De middelbaaronderwijs akten KI en Kv nader bekeken’ (unpublished 1998). 224. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel I 26–28. In 1952 for the academically educated teachers, 1958 for the certificate teachers and in 1961 for the engineerteacher. See also paragraph 6.5. 225. Vos and Van der Linden, Waarvan akte. Geschiedenis van de M.O.-opleidingen 1912–1987 116. 226. Freudenthal also lectured on ‘Geschiedenis van de wiskunde’; ‘Berichten Mathematisch Instituut der Rijksuniversiteit Boothstraat 1C, 6 en 17, Utrecht’, Euclides 41 (1965) 52. 227. See also Freudenthal on the education of teachers in paragraph 6.5. 228. Freudenthal, Exacte logica v. 229. Freudenthal to Monna, 23 August 1961; RANH, Hans Freudenthal Papers, inv.nr. 1723.

Chapter 9

Here’s How Freudenthal Saw It

Stel je voor wat ik een kwart eeuw geleden over zo’n gerontocraat zou hebben gezegd! Nee, geen gerontocraat - ik ben wel ambitieus maar nooit heerszuchtig geweest. Haantje de voorste, maar dan een die even gauw op de vlucht sloeg. Vooral voor gerontocraten. Imagine what I would have said about such a gerontocrat twenty-five years ago! No, not a gerontocrat—I have been ambitious but never imperious. The top dog, but then one that equally soon ran for it. In particular for gerontocrats.) Hans Freudenthal in Schrijf dat op, Hans, 1987 [1]

9.1 Introduction: Changes in the Scene of Action In 1970 Freudenthal celebrated his 65th birthday. In the chapter ‘Kwalen van de ouderdom’ (‘Problems of old age’) in his book Schrijf dat op, Hans he told how he consciously cut back his current activities and cancelled various chairmanships. Nevertheless, he did not succeed in creating an empty diary: the time he freed by dropping some activities, was taken up by other actions. “Other, in proportion heavier obligations”, that was the way he described these [2]. Many of these other activities were a direct consequence of the (culture) changes in the field of mathematics education in these years. Changes of the kind from which Freudenthal could not and would not stand aloof. In 1968 the new mathematics curriculum was introduced and the Mammoth Act had become operative in the Netherlands. What the influence on mathematics education would be was not yet clear at the time. Even when the changes in the curriculum were not so drastically ‘New Math’ as in some other countries, the danger that the introduction of modern mathematics would dominate the attention for the modernization of the teaching methods was still real. Fighting this danger remained a regularly recurring subject of Freudenthal’s articles and lectures [3]. He wrote about the need to do this: Reform continues, and it is too soon to evaluate the results. This is a period of transition and uncertainty. Without exaggerating the importance of some less acceptable tendencies, they nevertheless must be pointed out in order to be complete and perhaps to warn those who might otherwise be victimized by them. [4] © Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6_9

283

284

9 Here’s How Freudenthal Saw It

But at the end of the 1960s in general this subject was not discussed as often as it had been some years previously. In parallel with the changes in the social climate in the 1960s (in which the possibilities of self-fulfilment and the confirmation of the individuality became important) a (prudent) cultural shift in mathematics education set in. This involved, albeit initially still very sparsely, increasingly more attention for the didactics of mathematics and therefore for the pupil. The time in which mainly the mathematics as such was taken as the starting point for the interpretation and the realization of mathematics education, seemed past. With this increasing attention for the didactics the importance of the practical experiences in the class room also received more recognition. In Euclides this turning point is clearly perceptible when the subjects that were treated in the articles are considered. The proportion of the practice of teaching mathematics to the (higher) mathematics as such increased [5]. In 1969 the editors even explicitly expressed this in their intention for the journal: Toen de redactie aan de uitgever verzocht Euclides een nieuw jasje aan te trekken, was dat niet alleen met het doel dit tijdschrift een meer eigentijds uiterlijk te geven. Het vernieuwde omslag moet ook een symbool zijn voor een verandering van de inhoud. Al dikwijls werd er van verschillende zijden opgemerkt, dat Euclides maar weinig waar maakte van de ondertitel op het omslag: Tijdschrift van de didactiek van de wiskunde. Veel van de artikelen waren van vaktechnische aard, interessant, maar niet in de eerste plaats gericht op de didactiek. De redactie wil ernstig proberen daarin verandering te brengen. […] Het spreekt vanzelf dat de redactie zijn doel slechts kan bereiken in samenwerking met de lezers. Vragen, suggesties, artikelen, lesverslagen, enz. zijn daarom niet alleen van harte welkom, ze zijn een noodzakelijke voorwaarde om van Euclides een tijdschrift voor de didactiek van de wiskunde te kunnen maken. (When the editors asked the publisher to give Euclides a new look the reason was not only to give the journal a more contemporary appearance. The revamped cover must also be a symbol for a change of the contents. Before this, several people had often remarked that Euclides hardly lived up to what the subtitle on the cover implied: Journal of didactics of mathematics. Many of the articles were of a specialist nature, interesting, but not primarily aimed at the didactics. The editors seriously want to try changing this. […] It is obvious that the editors can only attain their goal in cooperation with their readers. Questions, suggestions, articles, class reports etc. therefore are not only very welcome, they are a prerequisite to be able to turn Euclides into a journal of the didactics of mathematics.) [6]

With the growing recognition for the importance of practical experience, the ‘millimetre perspective’ in mathematics education would also receive more attention. At first Euclides paid this extra attention in addition to the traditional areas of attention such as purely mathematical subjects, but in the 1970s the micro-didactics got a chance. ‘Looking at the child’ won ground as an aid in mathematics teaching and teacher education. The increased attention for didactics not only resulted in more opportunities for practical experiences, it also worked the other way round: experiences with the practice of education had an influence on the development of didactics and the organization of the classes and the curriculum. With the introduction of the Mammoth Act there was also more attention for mathematics at other non-secondary schools. Even kindergarten and elementary schools were considered—which was unprecedented.

9.1 Introduction: Changes in the Scene of Action

285

Fig. 9.1 Group photograph of the participants in the colloquium ‘How to teach mathematics so as to be useful’ in Utrecht (1967), with among others: Pollak ( back left), Wansink (with his hand on his lapel), Monna ( to the right of Wansink), Freudenthal ( front centre with bow tie), Servais ( behind Freudenthal), Van der Blij ( third row, fourth from the right with striped tie), Behnke ( second row, left of Van der Blij) and Mrs. Krygovska ( far right)

In addition to and because of this tendency the didactics of mathematics as a teaching methodology grew. This development could also be ascertained from the growing number of (international) congresses and meetings around the theme of mathematics education. Including the ICMI colloquium initiated by Freudenthal around the theme ‘How to teach mathematics so as to be useful’, held in Utrecht from 21 to 25 August 1967 (Fig. 9.1). It was a sequel to the colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht from 19 to 23 December 1964. ‘How to teach mathematics so as to be useful’ was the first international colloquium in which the central discussion point of an international panel were the differences in opinion about the role of the use of mathematics. The first ICMI congress—‘Premier Congrès International de l’Enseignement Mathématique’—was also chaired by Freudenthal. This ICMI congress (ICME) took place in Lyons from 24 to 31 August 1969 and a sequel took place in 1972 in Exeter. It was to be repeated every 4 years after that. Apart from that there were didactics meetings with physicists such as the one held in Lausanne in 1967. That meeting preceded the ICMI colloquium later that year. Freudenthal was present at the majority of such congresses. The ‘Congress on the Integration of Science Teaching’ of 1968 in Varna was in Freudenthal’s recollection one of the best congresses he ever attended: “I think because of its scientifically mixed public—a typical non-nonsense [sic] conference” [7].

286


In the Netherlands the creation of the IOWO would appear to be an important factor in these developments [8]. As an institute it was the only one of its kind in the Netherlands and therefore an indication of the increased importance of teaching methodology—in any case within the school subject mathematics. The new didactical climate apparently offered opportunities for such an initiative. From an organizational standpoint the founding of the institute was not exactly plain sailing all the way. It narrowly escaped having to wait for an umbrella curriculum institute just as the other subjects had. The IOWO was not only a result of this new interest in didactics of mathematics, at the same time it was also a substantial part of it. On top of that it was an instrument to feed this new attention, to grow and shape the teaching methodology. The influence of the mathematics education that was developed there would appear to be unprecedented [9]. In addition—and in particular—‘Looking at the child’ would become the motto of the IOWO’s way of working. What was Freudenthal’s position in all this? Even though he had almost reached his retirement he was anything but a critical outsider. In addition to his, over the years ever increasing, flow of articles and lectures on mathematics education Freudenthal’s book Mathematics as an educational task had been published both in English and (in two volumes) in German in 1973 [10]. In one bulky book of almost 700 pages almost all the previous articles he had written had been assembled. It would not serve as a book to be read in its entirety (even when it was composed as such), but rather as an important and impressive reference book for mathematics didacticians. With respect to content it did not add much to the ideas that Freudenthal had already disseminated in his articles and lectures, but as a sign that he was now almost completely devoted to mathematics education and didactics of mathematics it had great importance [11]. Apart from these publications he was also actively and completely involved in the field: ranging from activities at the organizational level, such as establishing an international journal for the didactics of mathematics, to observing practice in class rooms [12]. As professor-director of the IOWO he presented himself as the mathematics didactician in the Netherlands. But internationally too he played a leading role in the didactics of mathematics. At the same time Freudenthal continued to develop, accentuate and extend his own ideas. It is obvious to attribute this development to the influence of the abovementioned tendencies—and notably to the influence the IOWO work had. Or was Freudenthal rather the trendsetter and catalyst of the new developments in the didactics of mathematics? What was his role in the developments of the IOWO and which role did the IOWO play in his’? How did his ideas develop? How did Freudenthal present himself in the national and international field?

9.2 Educational Studies in Mathematics Among all of Freudenthal’s new activities and his accompanying indefatigable dedication at the end of the 1960s, the launch of the journal Educational Studies in Mathematics occupied a special place. The fact is that there were only a few projects

9.2 Educational Studies in Mathematics

287

in which he so clearly took the lead and the initiative all by himself. Freudenthal’s role in launching the journal was not only more explicit compared to that when the IOWO was realized, the whole project was also more of a soloist job.

9.2.1 Not Exactly Bursting with Enthusiasm: The Launch The reason for the launch was clear. At the ‘Colloquium on modern curricula in secondary education’, in December 1964 in Utrecht, a resolution was adopted in which it was decided to launch a new international journal on mathematics education [13]. As Freudenthal recalled it was Monna in his role as secretary of the colloquium who came up with this suggestion [14]. Four months later Freudenthal informally attempted to organize an ‘editorial board’ for the first time. He selected a number of participants to the colloquium and sent them a letter [15]. All reactions were positive. Beberman was the only one who indicated that he was too busy to join the editorial board. For vague reasons Freudenthal then detached himself from the process. It was not until 1967 that he again tried to carry out the resolution. The reason at the time was that he had heard that the American mathematician Peter Hilton was also minded to launch such a journal. Freudenthal suggested joining forces. Hilton took to this enthusiastically, but he declined Freudenthal’s proposition to become coeditor-in-chief [16]. Together they composed a new list of potential editors [17]. The most important criterion for this list was to approach in as many as possible different countries representatives who had a leading position within the mathematics education of their own respective countries. But Freudenthal did not try to disguise his personal preferences either. I did not insert Papy; he is an impossible man to work with; in any case from Belgium, Servais is much better, and he is a professeur de lycée. [18]

The two gentlemen soon reached an agreement on the name of the journal. ‘Mathematics Education’ and suchlike were too meaningless for Freudenthal, and apart from that he was almost certain that there already were periodicals of a similar name [19]. Hilton proposed ‘Educational Studies in Mathematics’ [20]. In his letter dated 1 November 1967 Hilton wrote about possible plans for launching two other journals about mathematics education. ‘The Chicago Center for International Mathematical Education Information’ would not only have very substantive ideas in that respect, but would also be planning to ask Freudenthal to join the editorial board of one of these [21]. Freudenthal did not think that a problem. He expected that there was enough room for three or more periodicals. However, had he known earlier about these plans he probably would not have taken on the organization of a new journal. Nevertheless I must say if I had known about the Chicago projects, I would not have started. I did it because no other seemed to do [sic]. It is with reluctance that at my age I take new charges on my shoulders. However, now I cannot withdraw. [22]

Freudenthal had as such no objection against consultation with the Center in Chicago, but he did not want to waste his time. Exactly at that moment there was an

288


important reason to put a lot of work into a journal about didactics of mathematics. The manuscripts of the colloquium in Utrecht ‘How to teach mathematics so as to be useful’ earlier that year were in his words an “excellent collection of studies”, which he would like to see published integrally. He did not want to wait long with this: not to be the first on the market (this would be ridiculous), but because I have good manuscripts and can show people that good manuscripts can be published. [23]

So that was what happened. The Reidel Publishing Company published all lectures of the colloquium, including the panel discussion, in the first issue of Educational Studies in Mathematics (ESM) [24]. Freudenthal received many congratulations when his journal was published. The only public criticism concerned the relatively high price of the journal [25]. In Euclides the advent of ESM was described as a very welcome and necessary supplement to the journals on offer at the time, since: In de huidige periode van snelle evolutie op het gebied van modernisering van het wiskundeonderwijs in alle landen ter wereld is er aan zo’n tijdschrift stellig behoefte. (In the present period in which the modernization of the mathematics education evolves so quickly in all countries all over the world there would certainly be a need for such a journal.) [26]

Because of Freudenthal’s international activities the reviewer did not question the fact that he was editor of the journal. The international character did not make ESM a direct threat for Euclides. That was different for l’Enseignement Mathématique, the official publication of the ICMI. In October 1968 Freudenthal spoke with Cartan about the position of both journals and among other things about the competition (between them), which ICMI took seriously. Henri Cartan was a member of the Bourbaki group and was an advocate of modern mathematics in education. For a long time Freudenthal was very close to Cartan, until a severe difference of opinion as a result of Freudenthal’s lecture ‘Enseignement des mathématique modernes ou enseignement moderne des mathématiques?’ occurred in 1961. Freudenthal wrote about this: In Bologna […] I seriously clashed with H. Cartan, who had been my friend for many years. These people could not understand that a mathematician of my rank did not hail Bourbaki at school but saw deep problems in such silly things as word problems in arithmetic, though he could not yet be excused by senility. [27]

In the years following this incident Cartan, as a prominent figure within the IMU, and Freudenthal, as a member of the ICMI, would regularly be opponents. This animosity was, with hindsight, also the cause of the following incident according to Freudenthal: At some opportunity I might unintentionally have insulted Cartan. On 5 September 1966 I asked him an explanation about a proof he had given (or sketched) of a so-called Weil Statement (on invariants of Lie groups). I should have known, though I can swear I didn’t, that everybody knew the proof was wrong. Cartan never answered that letter. And that was that. [28]

Later Freudenthal wrote about the conversation with Cartan about ESM: He complained about ESM, which he feared would be a deadly competitor to l’Enseignement Mathématique. In order to prove the excellence of the latter, he showed me an issue with,

9.2 Educational Studies in Mathematics

289

indeed, excellent survey papers by excellent mathematicians, though none about education. If I had shown him an issue of ESM, he would probably only have noticed the minor importance of the authors in mathematics. He was unable to grasp the difference. [29]

The anecdote clearly shows how ESM had a unique function.

9.2.2 Freudenthal as Guardian of the Level The journal strove to publish “high level material on mathematical education of more than national interest” [30]. Contrary to for example the new intention of Euclides to be a (more accessible) journal for mathematics education, there was more at stake with ESM. The goal was to form a forum for publication and exchange of research in the field of mathematics education. On top of that the contributions had to be of international significance. In his opening speech at the colloquium ‘How to teach mathematics so as to be useful’ Freudenthal emphasized the importance of the exchange of experiences at an international level with a series of rhetorical questions: Is the variety of national educational philosophies really a drawback to international talks on mathematical education or should I say that there is no better opportunity to test them than to have them bump against each other? Are not we too often and too readily inclined, when reading or hearing about the educational experiences in another country, under another educational system, to sigh: it is just a pity, but this does not apply to our situation? I would say whenever this happens, then something is wrong either in the one system or in the other, or, most likely, in both. [31]

This opening speech was printed as the first article in ESM more or less as an editorial statement. With mainly American and West-European contributions the journal did live up to the promise of its international character [32]. Freudenthal single-handedly took the responsibility to guard the level of the articles. He often invited colleagues to write something for ESM, sometimes as a result of one of their articles he had read before, sometimes as a result of a lecture [33]. He was also offered articles for publication spontaneously. He tried to get many articles revised, more often he rejected them if they did not meet his standard and criteria. In his rejection letter he often used the following excerpt: Almost every day the editor receives papers on mathematical subjects which show no or only weak relations to education. Such papers cannot be considered for publication. In particular, “educational” is not meant to include all elementary mathematics, nor are subjects from the history of mathematics to be considered as educational as such. Likewise mathematical studies in education (such as discussing mathematical techniques of processing educational data) are no subject of Educational Studies in Mathematics. [34]

If an article was based on experiments, Freudenthal demanded to inspect the complete material in the event of it being published. By his own account that had appeared to be necessary and effective. It looked as if Freudenthal in his function as editor performed in a ‘one-man-show’ with ESM. Yet he tried his utmost not to take

290


only his own opinion as a standard. To Schoenfeld, whose article Freudenthal had rejected, he wrote: Finally, what matters is not writing a paper “mere [sic] consisting with my taste”. A large number of papers published in ESM were not my taste at all. I had a lot of papers published which I did not like. There are other tastes than mine and I try to respect them. It is the same as in mathematics itself. There are subjects and methods I never liked but I would not daresay they are bad mathematics. Education is different. There is certainly no unanimity about what is heuristics, problem solving, good teaching, and so on, but I am convinced that in principle you are closer to my interpretation than you show in “Heuristics Do make a difference”, which I consider a mistake. [35]

Despite his ever critical attitude Freudenthal was never satisfied with the level of the articles in ESM. In 1977 he wrote in a letter to Pollak: I can tell you that the majority of the papers submitted never reach the point to be announced in the periodical circulars. Most of the papers are nonsense or hoax. Some people think that bad mathematics can be published as education. (Even good mathematics is not yet education.) The greatest danger is the so-called empirical work, processed with statistical nonsense methods. Even, in spite of my criticism the level of ESM is still far below what I would like it to be. [36]

According to Freudenthal other education journals accepted one out of ten articles that were submitted. Freudenthal accepted—usually after an intensive revision— about one out of three articles. He would have preferred to reject more articles to raise the general level but the low number of submissions prevented that. In 1977 Freudenthal actively started looking for a successor for the position of editor-in-chief “like a father who looks for somebody I can trust, if I am not able to care for the journal” [37]. The first one he asked was Pollak, according to Freudenthal the only editor who was really interested [38]. In a letter he explained the requirements for that position: As long as I can, I will work for ESM but I feel I have to look for a successor who will and can do this work. I wonder whether you feel you can or know somebody who could. The first demand is criticism. At this point I believe, I can trust you. The second is some universality with respect to mathematical education (from k[indergarten] to college), which is more difficult, the third some acquaintance with general education and its jargon. It is difficult to combine this. [39]

When Pollak did not answer twice over, Freudenthal assumed that it was because he did not think himself capable enough. A few months later he asked Fletcher, who refused [40]. Finally Freudenthal ended with Alan J. Bishop, a British mathematics educator in Cambridge, who accepted the job [41]. Freudenthal provided him with the following wise words: The third: do not trust the members of the Board. Except one they have never been active. […] I will support him [the new Chief] as long as I can, even without mentioning my name on the cover, say as a secret referee. [42]

So Freudenthal did not want to detach himself from ESM completely. The journal was and remained in some way his ‘baby’ for whom he kept feeling responsibility. With the—in his eyes, and more often than he cared for—poor material at his disposal he always did his utmost to make the best of it. The journal would (in

9.3 The Institute for the Development of Mathematics Education

291

p articular in the 1970s) be held in considerably high esteem. Articles of important and well-known (mathematics) didacticians such as Begle, Z.P. Dienes, F. and G. Papy, Revuz, Servais, Stone and Behnke filled up the pages [43]. Despite his strong personal ties Freudenthal knew how to keep his distance to produce a neutral journal. Even though the connection was strong and regularly showed up in the form of publications, ESM never became for example a veiled means of promotion for the IOWO. As in many other levels in Freudenthal’s professional life here also applied: he would rather publish a journal which did not always completely meet his preference and requirements with respect to its level, than be a critical outsider without a journal—and influence.

9.3 The Institute for the Development of Mathematics Education 9.3.1 From CMLW to IOWO Within the framework of the CMLW too, Freudenthal always upheld his motto ‘cooperate in order to adjust’. After the great success of the courses in modern mathematics for teachers it was already evident in 1964 that the work of the CMLW could best be done by an institute that belonged to the organization. In 1967 mathematics teacher and teacher educator Edu Wijdeveld was appointed as the first academic staff member of the CMLW, soon followed by others, among whom mathematics teacher Adri Treffers in 1969 [44]. The most important arguments for applying for an institute were at the time mainly the importance of a permanent curriculum development, professional development and guidance of teachers, but also the establishment of a documentation and information centre for mathematics education. A first request for an institute of its own with the Ministerie van Onderwijs en Wetenschappen was rejected (it took till 1968!) referring to the imminent plans for a national organization for the development of curricula. Such an organization would have to absorb the CMLW in due course [45]. After the introduction of the new curriculum in 1968 the necessity for institutionalization was felt even more keenly. The fact is that it very soon appeared that professional development courses for teachers were not an adequate preparation for the classroom practice. In the following years the teachers increasingly urgently missed didactical handles to treat the new subject matter. Starting with the academic year 1968–1969 the professional development courses, besides lectures on scientific mathematical subjects, also contained group discussions about didactical problems, but this did not solve that problem [46]. Gradually the CMLW could not cope any more with the work that had to be done to make an effective success of the task that the CMLW had undertaken—the modernization of mathematics education. This was in particular the case when on 30 May 1968 the development of mathematics in primary school (WISKOBAS) and in technical education were added to these tasks [47].

292


By 1970 the activity of our Commission, carried forth by a small number of fulltime and a few hundreds [sic] part time collaborators and an administrative office, had developed into a chaotic many millions business. [48]

The urgency of an institute to accommodate the employees and the services continually increased. As a compromise the Ministry proposed taking on the employees of the CMLW in the Pedagogische Centra and from there second them to the CMLW [49]. The CMLW would not settle for that. Other modernization committees (such as the Commissie Modernisering Moderne Talen (Committee to Modernize Modern Languages) and the Commissie Modernisering Klassieke Talen (Committee to Modernize Classical Languages) for that matter did agree to this construction [50]. The CMLW, with its seat in Utrecht, had apart from an executive committee no less than 11 subcommittees at the time. The need for a professor-director was also felt, as appeared from the minutes of the executive committee’s meeting in September 1969. Na enige discussie is men het erover eens, dat het “bedrijf”, zoals dat binnen de CMLW is uitgegroeid, gecoördineerd zou moeten worden door een direkteur op het niveau van een hoogleraar (in een vrijwel volledige dagbetrekking). Het lijkt echter vrijwel uitgesloten op dit moment een dergelijke figuur aan te trekken. (After some discussion the committee agreed that the “company”, such as has grown within the CMLW, would have to be coordinated by a director at professor’s level (as an almost completely fulltime job). However, it appears to be practically impossible to employ someone like that.) [51]

In that same month Freudenthal was officially appointed chairman of the CMLW by the State Secretary of the O&W Ministry J.H. Grosheide. Whenever the opportunity arose, Freudenthal, hammered away at the importance of the institutionalization of the CMLW. At a press conference at the University Centre De Uithof in Utrecht 1 month later he raised for example the matter how important the establishment of an institute for the professional development of mathematics teachers would be for the activities of the CMLW. Toch staan we voor problemen. We vragen ons af hoe lang we nog zo door kunnen gaan, met massacursussen een paar keer per jaar, met telkens een beroep doen op honderden die er naast hun gewone werk aan medewerken. We hebben enkele jaren geleden een plan ontworpen voor een instituut voor de heroriëntering van wiskunde-leerkrachten, dat continu het hele jaar doordraait en waar kleine groepen leraren elk een week heroriënterings onderwijs ontvangen en dat tevens een centrum voor onderwijskundige informatie zou zijn. Zoiets bestaat op grote schaal in Denemarken en in Hongarije en op kleinere schaal in Westduitsland. Er is tot nu toe bij ons nog niets van gekomen, maar we geven de moed niet op. Het is een dringende noodzaak en zal komen omdat het er moet komen. (Yet we are confronted with problems. We wonder how long we will be able to continue with mass courses a few times per year, appealing to hundreds who, in addition to their normal jobs, cooperate in these courses. A few years ago we designed a plan for an institute for the professional development of mathematics teachers, which functions the whole year through and which provides reorientation education to small groups of teachers every week and which at the same time would be a centre for information on education. Something like that exists in Denmark and in Hungary and on a smaller scale in West-Germany. In our country it has not yet materialized, but we will not lose courage. It is urgent and the institute will come because it has to.) [52]


293

In December 1969 Monna, Wijdeveld and Freudenthal proposed a design for an institute for mathematics education. Apart from the professional development courses Freudenthal mentioned, it was in particular the WISKOBAS project which—caused by the broad scope of the 10-year plan as designed in 1968—would threaten to get stuck without the support of an institute and money to employ people [53]. The initially reluctant attitude of the Ministry was due to the fear of having to grant such requests to other disciplines as well. Later they realized that the activities in the field of mathematics education could be a good example, but still they preferred to wait for the creation of an umbrella organization for all disciplines. Freudenthal wrote about this: Dem Schneckentempo jener gründlichen Plänenschmiede konnten wir uns aber nicht anpassen; wir waren ja viel weiter als die anderen Fächer. Mitte 1970 kam es zu einem Bruch met dem Ministerium, der schon lange gedroht hatte. (But we cannot adapt to the snail’s pace with which such thorough scheming proceeded; after all we were much ahead of other disciplines. Mid 1970 the break-up with the Ministry, which had been looming for a long time, finally happened.) [54]

The problem with WISKOBAS was, according to the minister, primarily a financial one: the salaries were too high. Meeting the requirement of an institute would “completely disturb the salary relations between employees of pedagogical centres, academic staff of the universities etc” [55]. In the meantime the CMLW had a budget of four and a half million Dutch guilders in 1970 [56]. In January 1971, after a long struggle with the Ministry, Secretary of State Gros heide eventually established the desired institute: the Instituut voor Ontwikkeling van Wiskundeonderwijs (IOWO), as an independent department for mathematics education at the State University Utrecht [57]. Freudenthal became professor-director, Wijdeveld was general manager. The IOWO had a staff of 37 people (among whom 21 academics) [58]. The IOWO had five departments: AVO, HBO & informatics, LBO, elementary school and special subjects [59].

9.3.2 Freudenthal and the IOWO To this very day the IOWO is closely associated with the figure of Freudenthal [60]. Officially the establishment of the IOWO was the granting of CMLW’s request for an institute, originally and primarily meant to coordinate and conduct the professional development courses. At Freudenthal’s parting as director of the IOWO Secretary of State Grosheide referred to the institute in the following way: Als vervolg op het voortreffelijke werk dat de Commissie Modernisering Leerplan Wiskunde heeft verricht, is het instituut onder leiding van Prof. Freudenthal in belangrijke mate verantwoordelijk voor dit succes. Dat een hoogleraar van zijn formaat en standing zich aan dit werk heeft willen wijden heeft denk ik in belangrijke mate tot de goede resultaten bijgedragen. Het lijkt mij meer dan billijk bij het afscheid van Prof. Freudenthal dit nog eens uitdrukkelijk te vermelden. (As a sequel to the excellent work the Commissie Modernisering Leerplan Wiskunde has done, the institute led by Prof. Freudenthal is to a considerable extent responsible for this success. The fact that he, as a professor of his

294


Fig. 9.2 Freudenthal in the IOWO (1975)

stature and standing, wished to devote himself to this work has according to me contributed considerably to the good results. It seems to me more than fair to mention this explicitly on the occasion of Prof. Freudenthal’s farewell.) [61]

The IOWO was looked upon as the institute of the CMLW with professor Freudenthal as ‘the man behind the institute’ (Fig. 9.2). But it would be too black-andwhite to say that it was expressly Freudenthal’s institute. Freudenthal had, wherever he could, declared himself in favour of establishing an institute and went to great lengths to realize it, but a more substantive first set-up of that institute was not directly his doing. According to a CMLW Proposal-working paper for the project ‘Modernisering wiskundeonderwijs in de basisschool’ (‘Modernization of mathematics education in primary school’) WISKOBAS—albeit on personal initiative of a threesome in 1967—was realized, officially as a working group initiative of the CMLW “to investigate the possibilities to bring a project about to modernize the elementaryschool mathematics education in the Netherlands” [62]. In 1972 Freudenthal wrote about this:


295

1968 zog eine Gruppe junge, der Kommission mehr oder weniger verbunderener Mathematikpädagogen den Grundschulunterricht ins Blickfeld. (In 1968 a group of young mathematics pedagogues more or less connected to the committee drew the attention to the elementary education.) [63]

This committee of initiators consisted of Brandenburg, Wijdeveld and F. Goffree, mathematics and didactics of arithmetic teacher at the Rijkskweekschool in Hengelo, in March also joined by mathematics teacher B. van der Krogt [64]. The memorandum of this working group was adopted by the IOWO and in 1969 WISKOBAS—by that time Treffers had also joined—became an official sub-committee of the CMLW. The way of working used in and by the WISKOBAS project would become characteristic for the IOWO. The IOWO annual report of 1971/1972 mentioned the following about the effect of this way of working: … het ziet er naar uit dat het IOWO bezig is een brug te slaan tussen het abstrakte nivo [sic] van de leerplantheorie en het konkrete [sic] nivo van de onderwijspraktijk. Enerzijds de fundamentele doordenking van onze eigen doelstelling, annex leerplantheorie, anderzijds de konkretisering van een ontwerp-onderzoek en -overdrachtsmodel, wijzen duidelijk in die richting. (… it seems that the IOWO is in the process of bridging the gap between the abstract level of the curriculum theory and the actual level of the educational practice. On the one hand the fundamental reflection of our own objective, and curriculum theory, and on the other hand the actualization of a draft investigation and transfer model clearly point in that direction.) [65]

What was that way of working? The basic principle was that the institute did not fix the educational goals and the curricula in advance. In close cooperation with the field a path was determinedly taken in which a continual exchange was maintained with all involved in the elementary education: pupils of the elementary education, their teachers and teachers at the Pedagogische Academies. Es ist, wenigstens auf kurze Sicht, eine Strategie enger Verknüpfung von Lehrstoffentwicklung und Fortbildung, und zwar gleichzeitig auf allen Niveaus: des Schülers, Grundschullehrers und derjenigen, die die Grundschullehrer ausbilden, fortbilden und begleiten sollen. Um auszuloten, was an Erneuerung möglich sei, und wie es zu geschehen habe, ist das unserer Meinung nach unumgänglich. (It is, at least in the short term, a strategy of closely linking the development of the subject matter and the training, and it should be at all levels at the same time: the pupils, the elementary-school teachers and those who should educate, train and guide them. To find out to what extent and how modernization would be possible, this is unavoidable in our opinion.) [66]

Here the issue was the philosophy behind it and the way of working in the realization of mathematics education, didactics of mathematics and curricula of mathematics, rather than their actual interpretation, as Freudenthal emphasized in a lecture in 1975: Een schoolwerkplan, zoals WISKOBAS het nu verstaat, moet vrij gedetailleerd ingevuld zijn, ingevuld met voorbeelden kwa [sic] leerstof én metode [sic]. Gedetailleerd ingevuld— ja, maar niet om gedetailleerd te worden nagevolgd. De details van de invulling dienen niet om de gebruiker aan banden te leggen, maar om hem iets van de geest te laten proeven die de samenstellers bezielde. Het goede voorbeeld wordt ontkracht als het slaafs wordt nagebootst; in de geschiedenis van onderwijsvernieuwing is dat[:] telkens uit een ruim

296


hart vrijheid van interpretatie toestaan; sterker, deze vrijheid moet in het goede voorbeeld zijn ingebouwd. (A school curriculum, such as the current one in WISKOBAS, has to be interpreted rather in detail, with examples both with respect to subject matter and method. Interpreted in detail—yes but not in order to be followed in detail. The interpreted details do not function to restrain the user but to let him get a taste of the spirit that the composers inspired. The edge is taken off a good example when it is imitated slavishly; in the history of modernization of education that means[:] again and again from an open heart allow freedom of interpretation; furthermore this freedom has to be integrated in the good example.) [67]

Freudenthal rarely made remarks about the course or the way of working of the IOWO. One of the few times he did this was in the following hand-written note. This was probably a reaction to an article of IOWO employee G. Schoemaker, director of the IOWO working group WISKIVON (WISKunde In Voortgezet ONderwijs, mathematics in secondary education), and meant for internal use only (Fig. 9.3):

Fig. 9.3 Text written by Freudenthal about the way of working within WISKOBAS and WISKIVON (1970s)


297

Wat hier gewoon is wordt ten onrechte als algemeen bekend ondersteld. Na tegenwoordige toestand en opdracht allereerst uiteenzetten: Principes van te creëren onderwijs • • • • • • • • •

wiskunde vanuit en binnen de realiteit, derhalve in context eigen activiteit van de leerling, anders dan naäap- en invuloefeningen, wél: mathematiseren, derhalve rijke context en reflexie, vergemakkelijkt door interactief onderwijs, met nadruk op interactie van leerlingen, derhalve werken in kleine groepen homogeen—heterogeen? voor- en nadelen

Dit is maar een losse aanwijzing. Ontwikkelingsproces Pas daarna behandelen • nadruk op ontwikkelingsonderzoek • gedurige feedback ontwikkeling ↔ onderwijs door • reflexie (What we at the IOWO find normal is (by us) often mistakenly considered general knowledge. According to the current situation and (our) task, first explain: Principles of the education to be realized • mathematics from and within reality, so in • context • it must be an activity of the pupil, if it isn’t it would be mimicking and blanks exercises, but what is desired is: • mathematizing, thus • a rich context and • reflection, made easy through • interactive education with the emphasis on interaction of pupils, and therefore • working in small groups • homogeneous—heterogeneous? Advantages and disadvantages This is nothing more than a stray hint. Development process To be dealt with after that • emphasis on development research • continuous feedback development ↔ teaching through • reflection) [68]

This text is nothing more than the written form of the ideas that already had been made explicit within the IOWO, and not an action plan devised by Freudenthal personally. This was for that matter a characteristic of the atmosphere and the way of working within the IOWO. It was not a step-by-step plan, imposed by the management, but a development strategy that became clear in the process, the way things often went at that time. The subjects that came into view via this ‘democratic’ procedure to develop a curriculum had gone through a path that started with the teachers’ reorientation to

298


small parts of the subject matter. The resulting reorientation material was tested on a larger scale by teachers and their pupils—the so-called field exploration. WISKOBAS only ended this exploration phase in 1975 by presenting a concept curriculum in which vision, didactical approach and material were presented as a first step on the road to a new method for teaching of arithmetic. Freudenthal readily agreed to this ‘democratic’ way of working. His petty conservative attitude was the reason that he generally fitted well into the democratic tendencies that caught on by the end of the 1960s (also within the university) [69]. Apart from that he greatly admired and respected the projects the (young!) practical members of the WISKOBAS working group dared to take on [70]. It was not only the realization of the projects and the way of working within IOWO that was just in his line, content-wise it also matched very well with his ideas. In a lecture about curriculum development in 1973 he expressed it as follows. In de afdeling WISKOBAS is een aanzet tot een integrale benadering wat betreft kleuteren basisonderwijs. […] Onder Moderne Wiskunde verstaat men hier niet, zoals veelal in ’t buitenland, zogenaamd moderne onderwerpen zoals verzamelingenleer en propositielogica, waarvan zin en bedoeling nooit duidelijk zijn, maar het inzichtelijk werkzaam zijn en het bewust mathematisch structureren van de realiteit. (In the department the WISKOBAS initiative has been taken to approach the education in both kindergarten and elementary school integrally. […] Modern Mathematics in the Netherlands is not, like in many other countries, so-called modern subjects such as set theory and proposition logic, with their never clear meaning and sense, but working to gain insight and consciously mathematically structuring of reality.) [71]

Taking as much time and room as possible for the development of the curriculum was, of course, a completely different starting point than that of, for example, a committee like the CMLW—not even 8 years earlier—when in just a few years a complete curriculum for the entire secondary education was designed and presented. Freudenthal said about this in 1974: Tegenover een reglementerende leerplanontwikkeling van achter het bureau hebben we voor de democratische participatie van het veld gekozen. In alle bescheidenheid mag ik zeggen dat dit instituut internationaal een reputatie geniet om trots op te zijn. (As opposed to a regulatory development of the curriculum at the desk we chose for the democratic participation of the field. With all due respect I may say that this institute has an international reputation to be proud of.) [72]

The comparison between the IOWO and the Wiskunde Werkgroep, at any time very precious to Freudenthal, also fell short in his view. In a letter to Van Hiele he wrote about an attempt by Van Hiele in the mid 1970s to enliven the working group: Wat ik ook in de Werkgroep heb geleerd (in mijn boek heb ik ervan getuigd) de Werkgroep was voor mij een gepasseerd stadium—ik deed niet meer één keer per maand aan wiskunde didactiek, maar dagelijks. In het begin van het IOWO hebben we nog een wekelijks colloquium op ’t IOWO met buitenstaanders geprobeerd, maar dat was geen succes. (Whatever I learned in the Werkgroep (I spoke about it in my book), to me the Werkgroep was past time—I was no longer just involved one day a month in didactics of mathematics, but on a daily basis. In the early days of the IOWO we tried to organize a weekly colloquium with outsiders, but that was not successful.) [73]


299

In his lectures and articles Freudenthal always mentioned ‘we’ if he spoke of the work and the working method of the IOWO. He was very proud of what the IOWO did and achieved and was right behind the institute and his co-workers. Nevertheless,—at least in the early phase of the institute—he regularly had to operate wearing two different and apparently conflicting hats. The fact was that since 1971 Freudenthal had acted as the representative of the CMLW in the Commissie Organisatie Leerplan Ontwikkeling (COLO, Committee Organization Curriculum Development), a movement which was in effect completely the opposite of the discipline-specific movement of the CMLW and the IOWO. The COLO was appointed at the end of the 1960s by the Ministry with the goal of establishing an umbrella organization in which complete—meaning for all disciplines—curricula could be developed. In 1975 the COLO finished its work with the installation of the Stichting Leerplan Ontwikkeling (SLO, Foundation Curriculum Development) and after that the CMLW became an advisory committee of the SLO. How the IOWO would fit in, was not yet clear at the time [74]. Because of the support of the Ministry the COLO-SLO path seemed the one with the greatest chance of success from the start and the IOWO initially did not look much different from an aberration of the mathematicians from the path the minister had chosen. Apparently Freudenthal did not see a contradiction in ‘covering his bets’. It was very soon clear that the IOWO and its people were very dear to him [75]. Of course he was also critical and gave his opinion just as well. But this was never of a fundamental nature and his admiration and enthusiasm always predominated [76]. That enthusiasm was not only for the content of the work of the IOWO, but—and perhaps in particular—also for the atmosphere and the working method at the institute. Freudenthal flourished in the IOWO team of primarily young people, in which he functioned as a valuable team member: open to new ideas and insights, critical, but also expecting criticism. IOWO co-worker Treffers characterized Freudenthal’s presence and influence on WISKOBAS as follows: … his personableness [sic], his intriguing off-hand remarks, his elaborations on seemingly futile observations of learning processes, his passion, his ability to use simple examples to show that mathematics is everywhere, his attention for both ‘simple’ and ‘complex’ forms of mathematics, his respect for practitioners of teaching, his admiration for a point well argued, his style of writing, his undisguised disapproval of developers and researchers who do not do justice to mathematics nor to the child, do no justice to mathematics as a human activity, to ‘mathematics as an educational task’… [77]

Freudenthal always attended the weekly meetings of both WISKOBAS and WISKIVON even though, and perhaps exactly because of it, he did not have a welldefined function. Freudenthal’s role within these teams and the IOWO in general is difficult to determine. Undeniably he left his mark on the group but how and where is difficult to assess exactly. He linked his great name and reputation to a current initiative already in function, inspired with his knowledge and vision and accelerated the whole gigantically. The main thing therein was that he did his utmost to disturb the team spirit of the IOWO as little as possible [78]. His inapproachability, often experienced by others—for the greater part due to social awkwardness—largely

300


disappeared in the atmosphere almost void of hierarchy at the IOWO. Van der Blij strikingly characterized that atmosphere by saying that “you almost had to be able to play a musical instrument”, a completely different atmosphere than reigned at, for example, the Mathematisch Instituut. Above all, Freudenthal seemed to develop and extend his ideas with lightning speed because of the IOWO. He found an outlet for his almost forgotten interest (for which he hardly got a chance in the 1950s and 1960s): observing children, looking at learning processes and sharing those experiences with others to incorporate them in education.

9.4 Exploring the World from the Paving Bricks to the Moon 9.4.1 Observations as a Father in ‘Rekendidaktiek’ In his ‘Rekendidaktiek’ from 1944 Freudenthal already mentioned briefly the importance of observing. In the context of developing learning methods for the teaching of arithmetic he valued the power of the little, everyday things of life as a source of scientific research: […] onder de onderwijzers zijn er misschien toch sommigen, die zonder theoretisch opzet, geheel zakelijk, in een dagboekachtig relaas stukje bij beetje achter elkaar kunnen aantekenen, hoe er feitelijk les wordt gegeven—schijnbare onbenulligheden zouden voor iemand, die zulk een relaas wetenschappelijk gaat ontleden, misschien meer verrassen en te denken geven dan een diep aangelegd theoretisch onderzoek. ([…] among the school teachers there are perhaps a few who could make notes of the actual manner of teaching, without a theoretical structure, completely business-like, bit by bit in a diary-like account— apparent trifles would surprise someone who analyses such an account scientifically perhaps more than a deeply oriented theoretical study, and give him more to think about.) [79]

During the war Freudenthal made such diary-like notes of observations of his children. It was not much, of the 352 pages with notes eight were devoted to observations of his children [80]. He wrote in his manuscript about observing children: Er is veel verlokkelijks in het indivueel [sic] observeren en experimenteren op psychologisch terrein. […] Zelfs de slechtste onder deze publicaties mag men, zonder voorbarig te zijn, niet overbodig noemen, maar toch mag men de waarde van dergelijk onderzoek niet overschatten. Want het zo vergaarde materiaal bezit bij al zijn goede eigenschappen, die niemand zou willen loochenen, de slechte zijde, dat het immers uitgesproken individueel materiaal is. Aan het individuele karakter verdankt [sic] dit materiaal echter ook zijn aantrekkelijkheid—de levensgeschiedenis van een een kind van 0–6 jaar te lezen is zelfs, wanneer de verslaggever zou trachten, alle bekoorlijkheid te elimineren, aantrekkelijker werk dan het bestuderen van een catalogus van cijfers, percentages, gemiddelden, correlatiecoëfficiënten en wat er meer is aan statistische hulpmiddelen in de moderne psychologie. (It is very tempting to observe individuals and experiment in the psychological field. […] Even the worst among these publications should not be called superfluous without being premature, but still the value of such research should not be overestimated. Because the material collected in this way possesses besides all its good properties, which nobody would want to deny, the bad side that it is by nature definitely individual material. But this material

9.4 Exploring the World from the Paving Bricks to the Moon

301

also owes its attractiveness to the individual character—the story of the life of a child between 0 and 6 years of age is, even when the reporter would try to eliminate all charm, more attractive to work on than studying a catalogue of numbers, percentages, averages, correlation-coefficients and what have you in statistical aids in modern psychology.) [81]

In ‘Rekendidaktiek’ he gave an example of a conversation during a walk with one of his sons. It was an amusing example that shows how difficult it was for Freudenthal—in the role of a school teacher—to arrive at a ‘spontaneous’ Socratic interview: Ik: Waar komt de wind vandaan? A.: Uit het Westen. Ik: Neen, dat bedoel ik niet. Steek je vinger op! A: Daar komt-ie vandaan. Ik: Juist! Hoe heet die richting? A: Noord. Ik: Neen. Waar staat de zon? A: In het zuiden. Ik: Neen. Wijs even, waar de zon staat! A: Daar! Ik: Staat de zon dus aan dezelfde kant, als waar de wind vandaan komt, of aan een andere kant? (I: Where does the wind come from? A: From the west. I: No that is not what I mean. Put your finger in the air! A: It comes from there. I: OK! What is that direction called? A: North. I: No. Where is the sun? A: In the south. I: No, please indicate where the sun is! A: There! I: So is the sun at the same side as where the wind is coming from, or at the other side?) [82]

Freudenthal halted here for a short intermezzo with the reader: De lezer bemerkt, dat ik niet de moed bezit te vragen “welke betrekking bestaat tussen de richting, waar de zon staat, en de richting waar de wind vandaan komt? ”, een vraag, die fraaier zou staan en beter in het schema der socratische methode zou passen. (The reader may have noticed that I do not have the courage to ask “which relation is there between the direction where the sun is and that where the wind is coming from? “, a question that would look better and would fit better in the scheme of the Socratic method.) [83]

Next he continued the dialogue with his son: A: Aan dezelfde kant. Ik: En waar staat de zon nu in de ochtend? A: Daar! Ik: Neen, je hebt net gezegd, dat-ie dààr stond. Hoe heet die richting? A: Zuid. Ik: Neen, ik bedoel, hoe die kant heet, waar de zon altijd in de ochtend staat. A: Oost. Ik: Dus waar komt de wind vandaan? A: Daar! Ik: Neen, hoe heet de richting, waar de wind nu vandaan komt?

302


A: West. Ik: Waar is West? A: Waar de zon staat. Ik: Neen! Wanneer staat de zon in het westen? A: ’s Avonds. Ik: Ja. Is het nu avond? A: Neen. Nu is [het] ’s ochtends. Ik: Dus waar moet de zon nu staan? A: Waar de wind vandaan komt. Ik: Ja, daar staat de zon. Maar ik bedoel: hoe heet de streek, waar de zon elke ochtend staat—wind of geen wind. A: Het Oosten. Ik: Goed. En waar komt de wind nu vandaan? (A: At the same side. I: And where is the sun in the morning? A: There! I: No you just said that it was there. What is that direction called? A: South I: No, I mean how is this side where the sun always is in the morning, called. A: East. I: So, where does the wind come from? A: There! I: No, what is that direction called where the wind is coming from now? A: West. I: Where is west? A: Where the sun is. I: No. When is the sun in the west? A: In the evening. I: Yes. Is it evening now? A: No. Now [it] is morning. I: So where should the sun be now? A: Where the wind is coming from. I: Yes, there is the sun. But I mean: what is the direction called where the sun is every morning—wind or no wind. A: The east. I: Good. Where does the wind come from now?) [84]

The Socratic dialogue then stopped because the wind in the meantime had changed direction according to Freudenthal. Next he concluded: Leermeesters zijn meestal even hardnekkig in het ondervragen als leerlingen in het niet weten (of niet willen weten), maar zoveel er ook aan de socratische methode prijzenswaardigs is—toch zal de leermeester soms beter doen, om minder hardnekkig socratisch te zijn en eenvoudigweg te vertellen: ‘Kijk, daar staat de zon, daar komt de wind vandaan, en dat heet Oosten, want Oosten noem je ’t, waar de zon altijd in de ochtend staat’. (Instructors usually are just as persistent in questioning as pupils are in not knowing (or not wanting to know), but as much of the Socratic method is praiseworthy—the instructor nevertheless will do better if he is a bit less persistent in his Socratic manner and simply states: “Look, there is the sun, from that direction comes the wind, and this is called east, since east is what you call where the sun always is the in the morning’.) [85]

In his ‘Rekendidaktiek’ it was clear, according to Freudenthal, that the value of such observations could not lie in the individual case, but would only result in something scientifically valuable when the exercise was repeated and the results


303

compared with those of large groups of children. Not only was the small sample an impediment, but also the fact that such observations in the literature were primarily done with children from an intellectual environment, namely that of the experiment leader, was a disadvantage in Freudenthal’s eyes [86]. The observations of his children during those years were in general rather unspontaneous. Freudenthal himself played as the school teacher an explicit role in the dialogues he noted down. Unprejudiced ‘looking at the child’ had not yet come up, but enforcing learning moments had, as becomes clear from the following example: Tom 3,6. * Pak vijf suikerparels. Hij pakt een voor een en gaat onbeperkt door. Hoeveel zijn dat[? ]: “3” (juist), dat[? ] “4” (juist) dat[? ] geen antwoord (vijf; wordt hem gezegd), dat? Geen antwoord (2), ondanks herhaalde pogingen geen antwoord. Zijn dat twee? “Ja” Waarom zeg je dat niet? Geen antwoord. Zeg eens 2. Geen antwoord. Het blijkt dat hij het woord 2 niet uitspreken kann [sic]. Spraakoefening. Hij benoemt de hoeveelheden 1, 2, 3 over het algemeen goed; de hoeveelheid 4 met 30 fouten. […] Enkele dagen later: Zeg “tik tik tik”. Hij zegt onbeperkt tik. Zeg tik tik. Idem. Zeg tik tik tik. (Met zwaar accent op de eerste tik.) Lukt. Zeg tik tik met accent. Hij zegt weer tik tik tik. Na herhaalde pogingen kan hij zoowel tik tik als ook tik tik tik zeggen. Het lukt niet hem een viervoudige tik te zeggen ondanks alle rythme. Zeg eens tik tik tik. Hoeveel heb je tik gezegd. Goed. ” ” tik tik. ” ” ” ” ”.Goed. Analoge uitkomsten met tak en boem. (Tom 3,6. * Take five sugar pearls. He takes them one by one and goes on without stopping. How many did you take [?]: “3”, (correct), that [?] “4” (correct) that[?] no answer (five, he is told), that? No answer (2), despite repeated attempts no answer. Are there two? “Yes”. Why don’t you say so? No answer. Tell me 2. No answer. It appears that he cannot pronounce the word two. Pronunciation practice. He identifies the quantities 1, 2, 3 in general well; the quantity 4 has 30 a of mistakes. […] A few days later: Say “tick tick tick”. He continues saying tick. Say tick tick. Idem. Say tick tick tick. (With a heavy emphasis on the first tick.) This works. Say tick tick with emphasis. He says again tick tick tick. After repeated attempts he can say both tick tick and tick tick tick. He does not succeed in saying a fourfold tick despite all the rhythm. Say tick tick tick. How many times did you say tick? Good. Say tick tick. How many times did you say tick? Good. Similar result with tack and boom.) [87]

But even then Freudenthal stated that he, as a teacher, learned a great deal by observing and manipulating his pupils consciously during the arithmetic lessons and from the mistakes which they—and he—made in the process. In the following years Freudenthal was not much engaged in such observations, or in any case, he did not make similar notes (that have survived). Life and work took up all his time in Utrecht after the war and made ‘Rekendidaktiek’ into an exceptional excursion. In the early 1950s—in view of his articles in De Groene Amsterdammer—he did sympathize with the school adventures of his children, but there was no longer time for daily walks and observations [88]. In the 1950s and 1960s he was not so much engaged in observing the learning child. His involvement and activities in the field of education were in those years primarily on a didacticaltheoretical or an organizational level.

304


9.4.2 Observing as Grandfather Walking with the Grandchildren The fieldwork done within the scope of WISKOBAS and later also within the IOWO stirred Freudenthal’s old interest in observing learning processes. The fact is that ‘observing learning children’ has from the start played a prominent role in the way of working of the IOWO [89]. But at the same time his interest in observing was renewed in a more playful manner. In the 1960s Freudenthal became a grandfather. He was in close contact with, in particular, the children of his daughter Mirjam, who lived within walking distance [90]. Freudenthal went for walks with Bastiaan, the eldest who was born in 1970, and later also with Monica, his sister 2 years his junior, during which he made many observations—not necessarily directly related to mathematics. As a grandfather he consciously acted less as a teacher as becomes clear from the following words. I gave a 9 years old girl the problem to double a square. Of course she did not succeed. If it had been my daughter, I would have led her safely to find the solution. But she was my granddaughter so I said: ‘I see, it is too hard, let us try it another time’. [91]

It was in particular the more relaxed and uninhibited attitude of a grandfather as opposed to that of a teacher that gave Freudenthal the opportunity to see the mathematics in the world of the child (Fig. 9.4). When Bastiaan was about 28 months old Freudenthal started writing down the adventures with the little boy in a notebook [92]. He did this not only as a proud grandfather, but in the first place with the intention of reproducing Bastiaan’s cognitive development as objectively as possible [93]. Twenty-five years later Bastiaan recalled the walks in an article in Zentralblatt für Didaktik der Mathematik. He did not remember in detail what his grandfather had written about these walks. He wrote: I do not remember, e.g., that we walked so often together. What I do know is, that I liked to walk to distant places like the DE Coffee factories. They all seemed very far away then, today I walk to the railway station or to town in fifteen minutes, in my memory it seems far away like Australia. The park where we went to most of the time was […] a place so common to me, I could draw a plan by memory. [94]

Fig. 9.4 Walking with his grandchildren


305

During the walks he did not interview his grandson in the same coercive Socratic way as he did his own children. His descriptions lacked emotion and a value judgement as much as possible. In his notebooks he seldom added a concluding or an analytic remark. The intention seemed to be to reproduce as exactly as possible the minor events. From time to time he accompanied some of Bastiaan’s sayings with an explanation about his (speaking) habits. A striking example of the style in which he wrote down these conversations and happenings is the following fragment: 5-2-74 Enkele dagen geleden: Hij vindt een glazen knikker. “Als ik hard duw, zou hij op straat rollen” (wat hij niet wil). De irrealis wordt meestal alleen in de hoofd- óf de bijzin gebruikt. In antwoord op een vraag: “Anders zou…”. De knikker rolt onder een band van een geparkeerde auto. Hij kan hem niet te pakken krijgen. Ik wijs hem een stokje. Hij, op het gezicht: “Dat is niet hard genoeg.” Inderdaad een slap stokje, maar het lukt er wel mee. (5-2-74 Some days ago: He finds a glass marble. “If I push hard, it would roll on the street” (something he does not want). The irrealis (subjunctive) is usually only used in the main sentence or in the subordinate clause. In answer to a question: “Otherwise it would …” The marble rolls under the tire of a parked car. He cannot get hold of it. I show him a stick. He, at first sight: “That is not strong enough.” Indeed it is a limp stick, but it works.) [95]

For various reasons such an account of the adventures of a (small) child can of course never be called objective. Even when he did not try explicitly hard to teach Bastiaan something—the way he did with his own sons 30 years before—Freudenthal often could not resist sending Bastiaan in a certain direction via his questions and hints. In the next example the Socratic questioning method pops up again: 15-9-74. Op straat vindt hij een wieldop van een Fiat. ‘Ik heb er al zo een’. • ‘Hoeveel heb je er dan nu?’ • ‘Twee.’ • ‘Hoeveel heb je er voor een hele auto nodig?’ • ‘Dat weet ik niet.’ • ‘Hoeveel wielen heeft een auto?’ • ‘Vier.’ • ‘Hoeveel heb je er dan nu nog nodig?’ • ‘Vier.’ • ‘Nee, hoeveel moet je er nog bij vinden?’ • ‘Twee’. Thuis, opgewonden, roept hij al voor de deur [zijn vader] Frans toe: ‘Ik moet er nog twee bij vinden, dan heb ik er vier! ’ (15-9-74. He finds a Fiat’s hubcab on the street. ‘I already have one like that’. ‘How many have you got then?’ • ‘Two.’ • ‘How many do you need for a complete car?’ • ‘I don’t know.’ • ‘How many wheels has a car?’ • ‘Four.’ • ‘How many do you still need then?’ • ‘Four.’ • ‘No, how many do you still have to find then?’ • ‘Two.’ At home, excited, at the door he calls out to [his father] Frans: ‘I still have to find two, then I will have four!’) [96]

306


More often Freudenthal tried to keep the phrasing of his questions as open as possible to the let little boy draw his own conclusions as much as possible. In that way the observations had the greatest value for Freudenthal. All the same Freudenthal was well aware that the spontaneity in his observations was a relative notion. In his book Appels en peren he wrote about his influence as an observer of such casual observations: Het kon weleens spontaan starten, maar een woord, een blik, een geste van de waarnemer is voldoende om het spontane een bepaalde kant op te buigen. Is dit erg? Welneen. Er is spontaniteit, maar voor alle ontwikkeling is er ook een sociale context. Dwarsdoorsneden zijn onvoldoende, om ontwikkeling te achterhalen, zolang er geen theorie is om de dwarsdoorsneden te verbinden. Longitudinale observatie is praktisch onmogelijk. Wat heb je dan aan die terloopse observaties, in de wandeling? (It could start spontaneously, but a word, a look, a gesture by the observer is enough to bend the spontaneity in a certain direction. Is this bad? No. Spontaneity exists, but there is a social context in all developments. Cross sections are not enough to discover developments, when there is no theory to connect these cross sections. Longitudinal observation is practically impossible. What use are these casual observations during a walk then?) [97]

Yes, what use are these casual observations? Freudenthal himself subsequently gave the answer: in casual observations, in particular, he recognized an important tool for developing mathematics education. He selected particularly those observations he thought or expected would help him to draw more conclusions later, as shown in his remark about which of his adventures with Bastiaan he would choose to write down: Wel, uit de wijze waarop ik ze heb verteld, zal u duidelijk zijn geworden, dat het voor mij meer waren dan anekdoten. Ik heb niet wat ik heb waargenomen te kust en te keur genoteerd, maar alleen datgene wat me te denken gaf. Gedachten bij elkaar vormen nog geen theorie, maar elke gedachte kan aanzet tot theorie zijn. (Well, from the way I told them, it will have become clear to you that these were more than anecdotes for me. I did not note down what I observed in plenty, but only that which gave me food for thought. Thoughts put together are not yet a theory, but every thought on its own can initiate a theory.) [98]

9.4.3 Granddad Hans: A Critical Comment With this new line of approach Freudenthal explicitly crossed the borders of his own discipline, mathematics, and entered the field of pedagogy, developmental psychology and linguistics. He published his walks with Bastiaan in Pedomorfose, the journal for Jenaplan education with Mrs. Freudenthal as an editor (Fig. 9.5) [99]. Apparently a journal such as Euclides was not the right place for such not so mathematical observations. The response to his articles primarily came from linguists for that reason. Philologist E.M. Uhlenbeck responded for example to the more linguistic aspects of Freudenthal’s observations [100]. He blamed Freudenthal for speaking and judging too easily about linguistic phenomena as a non-linguist. Wat altijd weer verbazing wekt, is het gemak waarmee niet-taalkundig geschoolden over taalverschijnselen spreken en oordelen. Zijn zulke personen professor in een ander vak, dan worden hun pseudo-taalkundige uitspraken door een aantal onkritische lezers nog geloofd


307

Fig. 9.5 Celebration issue of Pedomorfose on the occasion the Mrs. Freudenthal’s seventieth birthday (1978) ook. (It is always surprising how easily non-linguistically educated people speak and judge about linguistic phenomena. And when such a person is professor in another discipline, a number of uncritical readers even believe his pseudo-linguistic statements.) [101]

308


Freudenthal immediately wrote a reaction. For him it was not the quality of the Dutch language that Bastiaan used in his conversation with his grandfather, but the linguistic levels of the communication that were at stake. And since that was not Uhlenbeck’s discipline either, Freudenthal retorted: ‘Wat altijd weer verbazing wekt is het gemak waarmee niet-taalkundig geschoolden over taalverschijnselen spreken en oordelen. Zijn zulke personen professor in een ander vak…’ eindigt E.M.U. zijn verhaal. Ik had er eigenlijk een grapje op willen maken, maar het zou zonde zijn. Deze passage spreekt beter voor zichzelf. (It is always surprising how easily non-linguistically educated people speak and judge about linguistic phenomena. And when such a person is professor in another discipline …’ E.M.U ends his story. I really wanted to joke about this, but that would be a pity. This passage speaks better for itself.) [102]

Freudenthal received an elaborate critical reaction to his walks with Bastiaan from Bruce Pelupessy, a co-worker of the Project Leerpakket Ontwikkeling Natuurkunde (Project Curriculum Development Physics) in Utrecht. Ik wil proberen, hier en daar, wat opmerkingen, twijfels en vragen te plaatsen bij hetgeen door opa Hans is geschreven. Ik doe dit omdat ik het—bescheiden—gevoel heb dat er uit de vele akties [sic] die Bastiaan onderneemt, hier en daar heel andere konklusies [sic] te trekken zijn dan opa Hans doet. We zullen er wel nooit achterkomen welke konklusies de juiste zijn. Of zou Bastiaan ons—nog—kunnen helpen… (I will try to express, here and there, a few remarks, doubts and questions to what granddad Hans has written. I do this because I have the—modest—feeling that in some places other conclusions than those granddad Hans draws, can be drawn from the many actions Bastiaan undertakes. We will probably never find out which conclusions are correct. Or would Bastiaan—still—be able to help us …) [103]

Pelupessy next showed with examples how Freudenthal unconsciously directed the conversations with his grandson. According to him Freudenthal did not immerse himself enough in the fantasy world of the little boy. In particular the question “How do you know that?” that Freudenthal praised and used so much, and the one from a similar category “What do you think?” put the toddler on the wrong track. According to Pelupessy Freudenthal had to descend figuratively more to Bastiaan’s level in their conversations. A good example was Freudenthal’s analysis of counting mud pies in the sandpit: 10-9-’74. In de zandbak in het park. Bastiaan maakte—voor het eerst—een hele rij taartjes op de rand. Ik telde mee. Het waren er 18. Toen begon hij zelfstandig van het begin af aan te tellen: 1, 2, 3, 4, 5, 8. Hij zei: “Ik wou dat het er zes waren.” (“zes” betekende tot nu toe “veel”). Ik wees hem tot waartoe het er zes waren; er was daar toevallig een gleuf in de rand van de zandbak. Hij wees op de rest en zei zoiets als “dat moeten er ook zes zijn”. Ik wees dienovereenkomstig, maar hij, door iets anders afgeleid, hield ermee op. We hadden vlierbessen geplukt en in een mandje gedaan. Hij droeg het en zei: “6 pond”. Ik: “Hoe weet je dat?” Hij: “Mijn moeder kan het wegen.” —“Hoe? ”—“Met de weegschaal”. (10-9-’74. The sandpit in the park. Bastiaan made—for the first time—a whole row of mud pies on the verge. I counted with him. There were 18. Then he began independently to start counting from the beginning. 1, 2, 3, 4, 5, 8. He said: “I wish there were six.” (“six” was until now “many”). I pointed out until where there were six; and by accident there was a groove in the verge of the sandpit. He pointed at the rest and said something like “that have to be six as well”. I pointed accordingly, but he, distracted by something else, stopped. We had picked elderberries and put them in a basket. He carried it and said: “six pounds”. I: “How do you know that?” He: “My mother can weigh it.”- “How?”—“With the scales”.) [104]


309

Peluppessy assumed that here too Freudenthal aimed too high with his question: “How do you know that?” and on top of that became involved as an instructor. Wat had Bastiaans opmerking over het mandje met vlierbessen te betekenen? “6 pond” zei hij. Het mandje is dus behoorlijk zwaar, volgens Bastiaan. Zo maar, een ‘emotionele’ opmerking. Net zo’n opmerking als “Het is vandaag behoorlijk koud”. Dat voel je gewoon. Wat moet je dan met zo’n vraag als “Hoe weet je dat?”. Nadenken, en er proberen een zo goed mogelijk antwoord op te geven. Waarom stelt Hans Freudenthal die vraag eigenlijk? Een reaktie op Bastiaans opmerking? Of wil hij hem iets leren? (What did Bastiaan’s remark about the basket with elderberries mean? “Six pounds” he said. So the basket is pretty heavy, according to Bastiaan. Just an ‘emotional’ comment, resembling “It is pretty cold today”. You just feel it. What are you then supposed to think of a question “How do you know that?” Think, and try to answer the question as well as possible. Why did Hans Freudenthal ask it in the first place? Was it in response to Bastiaan’s comment? Or did he want to teach him something?) [105]

Freudenthal completely skipped this in his response to Pelupessy. He was only interested in Bastiaan’s answer: Elk kind schijnt een zekere tijd lang een zeker telwoord voor ‘veel’ te gebruiken. Bij Monica is dat nu nog ‘zeven’, hoewel zij veel verder kan tellen. Bastiaan had op dat ogenblik zeker geen notie hoe zwaar 6 pond weegt. Het verbazingwekkende, het volstrekt ongewone in deze situatie is dat hij op mijn vraag “hoe weet je dat? ” een terzake doend antwoord geeft. Dat hij in ’t geheel een verband legt tussen ‘6 pond’ en ‘iets wegen’. Dit is op het punt waar het mij op aankomt. Bastiaan is op dat ogenblik 4 ½ jaar! (Every child seems to use one of the numerals instead of ‘many’ for a certain time. Monica now still uses seven, even though she can count much further. At that time Bastiaan certainly did not have a clue how heavy six pounds was. The amazing, the absolute unusual thing in this situation is that he answered my question “How do you know that?” adequately. That he links ‘six pounds’ and ‘the weight of something’. That is the point I want to make. Bastiaan’s age at that moment is four and a half.) [106]

When he emphasized Bastiaan’s age of four and a half the proud grandfather popped up for a bit. He explained that Bastiaan’s phantasy world was completely different from that which the writer suspected. Asking questions was to Bastiaan simply a game, his way to communicate. He was not role-playing and the phantasy world was for him, aged four, “nothing but an intellectual game” [107]. According to Freudenthal this justified the unconventional questions he asked the toddler. But he was the first to admit how difficult it was to interpret comments of children. Het doen en laten, zelfs het zich taalkundig uiten van kinderen—van wie dan ook die niet onze gelijke is—is moeilijk te interpreteren. Ik heb me wel honderden malen vergist. Conclusies trekken is nog lastiger—wie wandelingen met Bastiaan leest, merkt op dat het vaak relaas en zelden conclusie is. Misschien worden door deze dictie juist conclusies gesuggereerd, die vanzelfsprekend lijken. Ook wel door het feit dat Bastiaan veelal hardop denkt en dus zelf de conclusies schijnt te trekken. (All children’s actions—and for that matter all actions of people who are not our equals—, even expressing themselves linguistically, are difficult to interpret. I have made mistakes hundreds of times. Drawing conclusions is even more difficult—he who reads Walks with Bastiaan will notice that it is often an account and seldom a conclusion. Maybe this diction suggests conclusions that seem obvious. Caused also by the fact that Bastiaan thinks aloud and thus seems to draw conclusions himself.) [108]

In this way Freudenthal parried the criticism. The situations were so much his private ones, so personal, that no one could check whether, and if so to what extent, he

310


manipulated and controlled the situations with his grandchildren. But, Freudenthal stated, that did not matter anyway. Every observation was subjective to a certain extent. The interference of a grown-up always influenced the spontaneity of the child. But that was not important. Provided that one analysed the situation with due modesty at the back of one’s mind, it was what happened that was important. Analysing was difficult and easily criticized. Nevertheless it was small events that could lead to great mathematics according to Freudenthal, even though he did not always see it himself at once.

9.4.4 Walking on the Railway Track: The Mathematics of a 3-Year Old One of the anecdotes most quoted from Freudenthal’s notebooks was that in which he walked with Bastiaan and Monica on an abandoned railway track. It was striking that he so frequently used exactly this one from the many anecdotes in his notebooks. Striking, because it was an event that he added to the notebook in 1975, long after it had happened. This anecdote turned up for the first time in a lecture at the WISKOBAS conference in Noordwijkerhout in 1974. Later he used it again in a lecture in Poland in 1975 and at a conference in 1977 [109]. The walk in question must have happened around 1973, but Freudenthal only wrote it down in his notebook 2 years later (Fig. 9.6). Vandaag voor ’t eerst Monica naar de rails meegenomen. Bastiaan was iets ouder toen ik het eerst met hem deed. Ik geloof dat het de tweede keer tussen de rails was dat de geschiedenis gebeurde die ik in allerlei artikels heb verteld: ik zie dat hij iets opzettelijk probeert en hij ziet dat ik het zie. Hij: “Opa, kijk hoe ik het doe, een schoen erop, een schoen erin”—namelijk op en tussen de biels. Maanden later: “Opa, nu doe ik op de grote mensen manier”—van biel tot biel. Monica ontdooide pas toen ze zich tussen de biels van mijn hand had losgemaakt. Ik zag, dat ze bewust deed: één voet erop een voet ertussen in. Ik riep Bastiaan toe: “Moet je eens kijken wat Monica doet”. Hij luisterde niet, maar Monica ging direct tot inspanningen over om het op de grote mensen manier te doen. (Today I took Monica to the railway track for the first time. Bastiaan was a bit older when I took him there for the first time. I believe it was the second time walking between the rails that what I have described in several articles happened: I see him trying to do something deliberately and he sees that I see it. He: “Look granddad how I do it, one shoe on and one shoe in”—that is to say on and between the sleepers. Months later: “Granddad, now I do it in the grown-up manner”—from sleeper to sleeper. Monica only relaxed when she had extricated herself from my hand between the sleepers. I saw her doing it consciously: one foot on and one foot in between. I called to Bastiaan: “Come and see what Monica is doing”. He did not listen, but Monica immediately started efforts to do it in the grown-up manner.) [110]

Later Freudenthal blamed himself for not interpreting the remark of the little boy correctly enough. In his lecture in 1975 he said that at home his wife had explained to him that Bastiaan’s cry “Granddad, look how I do it …” had been an invitation to imitate him: I was stupified [sic] by this example of feminine intuition, and I was sorry I had disappointed the boy by my stupidity. [111]


311

Fig. 9.6 a and b The ‘sleeper anecdote’ from Freudenthal’s notebook (1975)

he confessed to his audience. The second mistake he made in relation to this incident in the first instance was not to have taken it seriously enough in view of Bastiaan’s development mathematically. Several times I told this story at lectures and I added: This is the first mathematics in this boy’s life: one foot therein, one foot above. It is mathematics not because he did it, this periodical activity, but because he became conscious of it and formulated it in a way that resembles mathematical intuition. I called it mathematics in a slightly joking way but people would believe I am silly. I was wrong. [112]

Bastiaan’s remark about the ‘grown-up’ manner a few months later was the reason that Freudenthal understood what he had said in the first place: there was even more mathematics involved in the event than he originally thought. It was not just recognizing the periodic activity and formulating it that could be interpreted as mathematics: I now know that my first interpretation was wrong and that what the boy said a year before was mathematics in an even deeper sense than I understood then. Probably without knowing it I had always made steps from one sleeper to the next. He would imitate it but he knew it was too long a step so he replaced my period by half of it: because isomorphism was difficult to attain he was then satisfied with a homomorphism. Only a year later he was able to perform the isomorphism, but in both cases he was conscious about the two phenomena

312


and about their relatedness. This is the reason why I was right to call it his first mathematicism and wrong to do it jokingly. [113]

This also immediately explained why Freudenthal did not note this event when it had happened: apparently it took some time for him to see the significance of the moment. This was also evident from the fact that in the years after the event he told the story as an example of the first mathematical action of a toddler, and at the same time implicitly as an example of how difficult it was to do casual observations. He did not express any doubt about interpretation long after the event took place, he was only surprised that he had never understood it in the right way before. “Observing and creating repetitive structures—this is an activity with countless variations that often has a mathematical bias”, that was the lesson Freudenthal learned from this event. The walk-on-the-railway-track anecdote was unique in the sense that it was one of the few in which Freudenthal applied a fairly far-reaching mathematical analysis and spoke about it in public. No comments were made on his observation and his analysis. For him it was not so much the example as such and the accompanying conclusion that was at stake, but the example as an illustration of the role and the power of observations of small children in their own world. And in particular in mathematics it was necessary to start with the everyday, little things. In 1974 Freudenthal said about this: Der Fehler dieser stark formalisierten Mathematik ist, daß sie darüber hinweggeht, wieviel wirkliche Mathematik Kinder vom ersten Schuljahr an, vielleicht schon im Kindergarten lernen können. Aber wirkliche Mathematik, die so einfach ist, daß wir Erwachsenen, die wir uns so sehr an sie gewöhnt haben, nicht mehr sehen, daß es sich um lernenswerte Dinge handelt, die Kinder lernen müssen. (The mistake of this strongly formalized mathematics is that it overlooks how much real mathematics children from the first year at school, can perhaps already learn in kindergarten. But real mathematics, that is so simple, that we grown-ups, who have got used to it so much, that we no longer see that these things worth learning, are things children have to learn.) [114]

9.4.5 Observing and the IOWO In a letter to Goffree about the mathematics education development work Freudenthal declared, on being asked, that observing children and students was “his richest source of experience” [115]. Observing thus played a prominent role in the development work of the IOWO and was, applied on this scale, perhaps one of the most innovative aspects of the development work of the IOWO. The role of observations was of course very clearly expressed in the experimental schools and the observation practice in the Pedagogische Academies [116]. But there was also much time and attention for the discussion of the observations during the weekly meetings of the working groups. Freudenthal stimulated this and emphasized the importance of looking at children open-mindedly. At the same time he for his part was inspired and enthused by what he saw realized within the IOWO [117].


313

Fig. 9.7 Scribbled note of Freudenthal to WISKOBAS co-workers (1976)

Working within the IOWO was very fruitful for Freudenthal. He could for example test his own ideas within the context of the IOWO. He did this very explicitly by scribbling his observations on pieces of paper and distributing these in the pigeon holes of his co-workers (Fig. 9.7). On these scraps of paper he sketched short stories, rough ideas, anecdotes, but above all little exercises. He asked all co-workers to solve these little exercises in writing. Als volgt een opgave. Wilt u beschrijven hoe U het zelf opgelost hebt. Wilt U het zoveel mogelijk kinderen geven en beschrijven, hoe zij het oplosten. (Here is an exercise. Please describe how you solved it yourself. Please give it to as many children as possible and write down how they solved it). [118]

Apparently he was very interested in solution strategies and used the IOWO coworkers as guinea pigs. These informal actions came closest to what could be called systematic research, but Freudenthal did not directly use them for some explicit purpose. Such activities and notes mainly contributed implicitly to the formation of Freudenthal’s ideas, so there is no literal reference to be found in his work. Just like so many things that crossed his path, he absorbed it in his ideas. Sporadically these scribbled notes in the pigeon holes did contain a more crystallized idea, such as the following: Het blijkt bij proeven dat kinderen 1e klas basisschool, die sommen als 3 et2 en 4 e3 enz.. al behoorlijk kunnen, op de vraag “wat prefereer je, die twee stapels met snoepjes van 3 en 2 of die ene van 5” enz. in ’t algemeen foutief antwoorden. Geen wonder indien ze nooit geleerd hebben optelsommen op preferenties toe te passen. Het lijkt mij een vruchtbaar gebied—ook met vragen erbij waar het optelmodel niet op de preferentie past. Dus ook bijv. een touwtje zo lang of hetzelfde in twee stukken. Een taart of dezelfde in parten. Een sinaasappel als geheel of in parten. Drie appels en twee peren, of drie peren en twee appels. Enz. (From tests it becomes clear that children of the first year of the elementary school who can make sums such as 2 ok3 and 4 a3 reasonably well, in general answer

314


the question “what would you like better, these two piles with three sweets or one of five sweets” etc. wrongly. This is not surprising if they have never learned to apply addition sums to preferences. This seems to me a fruitful research area—also with questions where the model of the addition does not match the preference. So for example a string this long or two pieces of half its length. One piece or the same pie in parts. An orange as a whole or in segments. Three apples and two pears, or three pears and two apples. Etc.) [119]

Many of the Bastiaan anecdotes ended up in the pigeon holes in that way too, often even before Freudenthal used them in his articles and lectures. Just like in his articles Freudenthal did not dare to explain or analyse these anecdotes in the rough version in which he presented them here. He threw them as open-mindedly as possible in the group to get a response as objectively as possible. He made an exception in the description of the following walk, in which he allowed himself in the closing sentences a lashing out at the teaching of arithmetic in level groups (as opposed to year classes: 28-11-76 Met Bastiaan en Monica bij de Leijen. Een groot aantal amazones passeert ons. Ik vraag: “Hoeveel zijn het er wel geweest? ” Monica meteen: “Zeven”. Bastiaan: “40”. Ik: “Neen”. Bastiaan: “20”. Ik: “Ik denk 15 à 20”. Ik, voor de grap: “We hoeven toch alleen maar de hoeven in het zand te tellen.” Bastiaan telt van 1 tot 20 terwijl hij telkens in een andere afdruk stapt. “Dus 20”. Ik: “Maar een paard heeft vier poten”. Hij, springend met grote stappen: “4, 8, 16, (aarzelend) 20, 24, (aarzelend) 27, (aarzelend) 30, 34, (aarzelend) 37, (aarzelend) 40, Dus 20 paarden”. Ik: “Maar 40 hoeven zijn 10 paarden. Een paard heeft 4 poten. 10 paarden hebben tien linkervoorpoten, 10 rechtervoorpoten, dit zijn er al twintig”. Bastiaan: “Dus 80; 20 paarden hebben 80 poten”. De fout in de tafel van 4 ontstaat blijkbaar door tellen 24, 25, 26, 27. Conclusie: leve het niveaurekenen. (28-11-76 With Bastiaan and Monica at the Leijen (a recreation ground in Bilthoven, close to Utrecht). A large number of horsewomen pass by. I ask: “How many would that have been?” Monica immediately: “Seven”. Bastiaan: “40”. I: “No”. Bastiaan: “20”. I: “I think 15 or 20”. I, jokingly: “We only have to count the hoof prints in the sand.” Bastiaan counts from 1 to 20 while stepping from one hoof print into another. “So 20”. I: “But a horse has four legs”. He leaping with great steps: “4, 8, 16, (hesitatingly) 20, 24, (hesitatingly) 27, (hesitatingly) 30, 34, (hesitatingly) 40. So 20 horses”. I: “But forty hoofs are ten horses. A horse has four legs. 10 horses have ten left forelegs, 10 right forelegs, that is already twenty”. Bastiaan: “So 80; 20 horses have eighty legs”. The mistake in the four-times table apparently occurs by counting 24, 25, 26, 27. Conclusion: Long live the level groups for arithmetic.) [120]

9.5 Observations as a Source 9.5.1 Professor or Senile Grandfather? So by the time of his ‘Rekendidaktiek’ in 1944 the seed of Freudenthal’s interest in observations had already been sown. When he observed his grandchildren 30 years later Freudenthal rediscovered the power of observing: observation as the source of inspiration and information. However, for Freudenthal this was not limited to learning to assess the utterances of a toddler during a walk on their true merit. More than that: he considered observing the starting point for the development of mathematics

9.5 Observations as a Source

315

education. He did realize that this was neither an obvious nor a very accepted point of view as became clear in his speech in 1977: I agree it is not easy to observe individual developments, and the laboratory of the psychologist is the least appropriate place to do it. The easiest place is the family and the next easiest the classroom. So if there are so many families and so many classrooms in the world, why do we know so little about learning processes? The answer is that you need a strategy of observing—a highly sophisticated strategy but which can be expressed in one principle: look and listen with an open mind and have the courage to notice and to report events that most people would consider as too silly to be noticed and to be reported—there might be a minority who can appreciate them, and this minority will be right. [121]

Outwith the IOWO, in lectures and speeches Freudenthal more and more often hammered away on the interest of observations. With this he risked his neck in the field of education. For according to Freudenthal you needed courage to attach importance to the little everyday things and tell the world about them. Just as when he had cited the example of the walk on the rail track with Bastiaan, his observations were greeted with mirth. Freudenthal also expressly took advantage of this by painting himself as a ‘silly old master’ and more than once stressing the so-called senselessness of the events. We rijden met hem door de Veluwe waar een geweldige storm veel bomen heeft geveld. Het meest indrukwekkende voor hem is, dat dit hetzelfde hout is als waarmee zijn vader thuis timmert. Hij raakt er niet over uit gepraat—een hele week niet—‘Bastiaan bomen zagen’. Ik vertel hier heel onbenullige verhaaltjes—ik waarschuwde u al eerder. Misschien vindt u me kinds. Inderdaad, ik leef mee met een jongetje van nog geen drie, voor wie dit—de gelijksoortigheid van hout in het bos en houten planken—een geweldige ontdekking is en ik ben niet te oud om zijn Aha-Erlebnis mee te voelen. (Of ben ik er net weer oud genoeg voor?) (We drive with him through the National Park De Veluwe where an enormous storm has blown down many trees. The most impressive thing to him is that this wood is the same his father uses to build at home. He never tires of talking about it—for the rest of the week—‘Bastiaan cutting trees’. I tell here very silly stories—I have warned you before. Maybe you think I am senile. Indeed, I feel for a boy not yet three years old to whom this—the similarity between wood from a forest and wooden planks—is an exciting discovery and I am not too old to feel his aha-experience as well. (Or is it that I am just old enough?)) [122]

He took a seemingly vulnerable position as a grandfather, astonished and endeared about childish trifles. Seemingly—since because of the authority he had already acquired as a professor of mathematics and in the field of mathematics education at that,—he was able to do this without losing his credibility. More than that, his message: back to the small things, the domestic events, to learn about learning mathematics, came across even better because of it. Ich erzählte einmal versammelten Pädagogen, wie sich eine Viereinhalbjährige aufregte, als sie merkte, daß sie auch unsichtbare Dinge (die Beine unter dem Tisch der Herumsitzenden) zählen konnte, von einem Achtjährigen, der sich ebenso aufregte, als er entdeckte, daß ein Hundertstel mehr sei als ein Tausendstel. Es regte keinen der Zuhörer auf, es langweilte sie. Und doch sind das die wahrhaft tiefen mathematischen Erlebnisse, wichtiger als das Einmaleins und algebraische Formeln—wichtig nicht nur psychologisch, auch als Mathematik! (I once told the assembled pedagogues how a four year old became excited when she noticed that things that could not be seen (the legs of those around the table)

316


could also be counted, how an eight-year old was equally excited when he discovered that one hundredth was more than one thousandth. No one in the audience was excited, they were bored. And yet these are the really deep mathematical results, more important than multiplication tables and algebraic formulas—important not only psychologically but also mathematically.) [123]

In addition he also indirectly implied in this message the long road that still had to be travelled in developing mathematics education. After all, if someone of Freudenthal’s prestige declared a search for the source in the little, everyday things squatting on his haunches—and even then found no more than indications for the direction in which he had to look for answers—it was clear that a ready-made solution for the interpretation of mathematics education could not be expected for a long time. And that was exactly Freudenthal’s message: we are only at the beginning of a long road. And that beginning was the small child and the way it was part of his world. And then it was obviously the small, apparently silly events that were important, particularly when you were looking for handles to develop mathematics education. Because in these silly little things mathematics could be found.

9.5.2 The Paradigm: The Ultimate Example If the observations he made while writing ‘Rekendidactiek’ were primarily intended to refute existing ideas on education and the formation of knowledge, the observations now had a more active function: a starting point and an instigation for theory, that was what observing children constituted to Freudenthal. Observations leading him to smaller insights, such as the first mathematical activity of a 3-year-old, but also observations leading to bigger more general insights and guidelines that kept him occupied in the 1970s and 1980s. Freudenthal had already proclaimed himself an advocate of the principle of guided reinvention in mathematics education for some time. Now something was added. Finding the key example, the paradigm, was essential for the principle ‘learning mathematics by being allowed to actively do mathematics’ according to Freudenthal [124]. Science historian and philosopher Thomas Kuhn (1922–1996) introduced the notion of paradigm in his book The structure of scientific revolutions from 1962 [125]. In it he sketched a development model for science in which a period of ‘normal science’ was alternated with a period of ‘revolution’. The first was characterized by the agreement found between scientists. In the normal period the scientific community considered the same achievements of predecessors normative. These achievements (and the accompanying problems and research method) functioned as characteristic, illustrative example for the scholarship at work: the paradigm. When enough problems occurred that could no longer be solved within this framework, the agreement ceased to exist and a time of ‘crisis’ began. A scientific revolution was needed to again reach consensus about a new paradigm and a period of normal science.


317

Kuhn’s book and the terms he introduced (including paradigm) were much discussed in the 1960s and 1970s and were particularly popular in social science circles. Freudenthal referred nowhere to Kuhn’s paradigm in a didactical context. Yet ‘paradigm’ and ‘paradigmatic instruction’ would form an important part of Freudenthal’s didactical terminology in the 1970s. The way in which this happened was characteristic of Freudenthal’s gift for seizing upon current developments and assimilating them in his own ideas by giving them a twist of his own. One of the first times that Freudenthal used the term paradigm was in his article ‘Trends in modern mathematics’ from 1962. In it he spoke of the mathematical tendency to adapt the (mathematical) language to the increasing complexity of the mathematical structures. In this tendency he saw “a paradigm which shows how mathematics can be applied in chaotic working fields like human language, namely by remodelling the field rather than simply by describing it” [126]. In 1972 Freudenthal used the term in a more didactical context in a lecture [127]. As in his article of 1962, Freudenthal emphasized the discontinuity in the learning process in his use of the term paradigm: the paradigm as an example for the pupil of another closely related branch of mathematics and thus as a starting point to extend the pupil’s mathematics insight. The underlying ideas had already been formed earlier, of course, but it was not before 1972 that he brought them up for discussion in this way: look in the learning process for an example that will be crucial for reflection and making the pupils aware of what they do or know unconsciously. Such an example served as model and starting point for generalizations to other situations, for reflections on one’s own actions and finally for a jump in the learning process to a certain increase of the level. Freudenthal’s starting point remained, for that matter, that the mathematical learning process was not a gradual one, but a discontinuous one. In order to explain what we like to consider as mathematics, I will stress two main features: mathematizing reality, and generality by paradigms with a vast scope. […] “Generality by paradigms” aims at a philosophy of interpreting mathematics for the primary level which is opposed to the prevailing ones. We do not believe that abstraction is being achieved by accumulating a large number of concrete examples, nor generality by adducing a collection of special cases, in particular if behavioural responses have to be reenforced, if some technique has to be settled and improved by excercises and drill, but this is not characteristic of mathematics. True mathematics has to be found in the discontinuities of the learning process. [128]

In this way Freudenthal’s regenerated attention for observing brought all important characteristics in his vision on the development of mathematics education together: analysing the learning process in order to look for discontinuities in it and finding paradigmatic situations—in which for him the situations should be: the more realistic, the better. In 1972 the event took place which to him was ‘the paradigm of all paradigms’: I begin with the event that for me, if my memory does not deceive me, was the paradigm of the paradigm. Certainly I had pressed for paradigmatics instruction for long, and I could sprinkle others and myself with examples of paradigms, but not until a talk with 8-year old (8;2 to 8;6) did I discover the one paradigm that I really found convincing as an example of a paradigm. The area to which it belongs is not included in traditional arithmetic and

318


Fig. 9.8 How many roads lead from A to C by way of B? A

B

C

mathematics teaching—this made my experience possible—though it would fit at least into the second grade, but even kindergarten teachers who participated in our refresher courses tried it convincingly. Meanwhile I expounded this paradigm many times: I drew a map with three towns A, B, C, where A and B are joined by three roads and B and C by two. The question is: In how many ways can I travel from A through B to C? […] It is characteristic of the fundamental weaknesses of our traditional arithmetic instruction that 8–9 year olds have difficulties with this problem, and that even adults, including people who have enjoyed a higher education in the arts or social sciences, often do not know how to tackle such problems. [129]

In Schrijf dat op, Hans Freudenthal explained why this question in particular was the ‘paradigm of all paradigms’ to him, and in doing so recapitulated the essence of a good paradigm (Fig. 9.8): De vraag—en dus het antwoord—staat model voor een rits van vragen, gelijksoortige, ingewikkeldere, als beginpunt van een groot stuk wiskunde, van de onderbouw van de lagere school als je wilt tot heel hoog boven, een vaak door mij beproefde weg. (The question—and therefore the answer—serves as a model for a series of questions, similar, more complex, as a starting point of a large part of mathematics, from the lower classes of the primary school if you want, to very high upwards, a road often tried and tested by me.) [130]

In his lecture in 1972 Freudenthal reported about his experiences with children tackling this problem. He told how it took them an effort to solve it. Subsequently he gave the children a new problem: a mouse has to go through two walls to get to a piece of cheese. The first wall has three holes and the second two holes: The children answered the new question without any hesitation. They even answered the question of how many paths I must draw to join four houses directly with each other, that is to say, they answered it by 12 which shows that rather than counting the number of paths they used a paradigm, which in the present case was not refined enough. This is true mathematics. It does not matter that the original problem was formulated with special numbers. One seemingly special case was sufficient to show the essentials of the structure. It was acquired by a hard but effective learning process, and it led to mastering a host of isomorphic problems. Do children grasp consciously this isomorphy? In general they do not, though it happened, that after such a lesson an intelligent seven years old girl told her parents: “Granddaddy is stupid, is he not? He gave me ten times the same problem and did not notice it.” [131]

This example showed how children could make themselves aware of a solution strategy that they had (unconsciously) developed and used in a specific situation, by extricating it from this situation and using it in other situations that are related to the original one. This made the ‘how many roads lead from A to C via B’-question a good example of the paradigm. A year later Freudenthal wrote about of learning mathematics which is characterized by leaps: Und Mathematik ist nicht Routine, sonder das Einmalige, aber doch allgemein Bedeutungsvolle. Herauszufinden, was paradigmatisch sein könnte, ist die Kunst der Lehrplanentwicklung in der Mathematik. (And mathematics is not routine, but rather the one-off,


319

nevertheless still generally meaningful. To find out what could be paradigmatic is the trick in developing mathematics curricula.) [132]

Looking for paradigms was for Freudenthal the way to go in education development. The concept paradigm was also a characteristic example of a term, picked up by Freudenthal from his (extra-mathematical) environment, which in the interaction between him and the IOWO was interpreted further. It was closely related to another of IOWO’s/Freudenthal’s hobby-horses: that of a rich context. Een rijk paradigma. Rijke contexten—ik weet niet wie op het IOWO het eerst deze term heeft gebruikt. In elk geval stamt het idee uit het concreet ontwikkelingswerk van het IOWO en het zal eventjes geduurd hebben voor het bewust gemaakt en verwoord werd— er is inmiddels veel op gereflecteerd. Oorspronkelijk was het alleen maar ‘context’.‘Rijk’ kwam erbij toen de rijkdom zich onder het ontwikkelen manifesteerde. (A rich paradigm. Rich contexts—I do not know who used this term for the first time at the IOWO. In any case, the idea originated in the actual development work of the IOWO and it will have taken a while before someone was aware of it and used the word—in the meantime many people have reflected on it. Initially it was only ‘context’. ‘Rich’ was added when the richness manifested itself in the development.) [133]

Later Freudenthal apparently could not remember who used such terms in this context for the first time. Observing, analysing learning processes, looking for jumps and paradigms: it all came back both in Freudenthal’s work and in that of the IOWO co-workers. In his ideas Freudenthal had already laid the foundation for this in the 1950s and 1960s, but he now saw it brought into practice, detailed and applied to the education development of the IOWO. If only one learning process benefitted from this combination, it was surely Freudenthal’s own. With the IOWO as a means to descend as often as he liked to the essence—that of the reality of the class room—, he could for the first time sharpen and concretize his, until then mainly theoretical considerations about the mathematics education and underpin them with real-life experiences. The combination with his regained enthusiasm for the (grand-)children’s world produced subject matter for many lectures and articles, and with that, also for his books [134].

9.5.3 Here is How Freudenthal Saw It: Concept of Number and Didactical Phenomenology When observing young children through ‘mathematical spectacles’ the development of the number concept is one of the things that first catches the eye. That is what happened when Freudenthal observed his own children. The notes he made for ‘Rekendidaktiek’ already contained observations of counting sons [135]. And also during the walks with Bastiaan counting and the number concept regularly came up. The number concept was also a regularly recurring subject in Freudenthal’s articles and books in the 1970s. For a number of reasons it played an important role when he developed his ideas. First of all, the way New Math approached the number concept was one of the main points Freudenthal criticized in the New

320


Math d evelopments for the teaching of arithmetic. Secondly, for Freudenthal the development of the number concept was closely related to observing as an aid in developing mathematics education. Finally, it would be an indirect reason to develop an important didactical handle for Freudenthal: didactical phenomenology. 9.5.3.1 The Number Concept in New Math Under the influence of New Math and the related wish to modernize teaching of arithmetic the number concept received due attention. At the end of the 1960s the emphasis in teaching of arithmetic in the Netherlands was mainly on the rote learning of arithmetic techniques [136]. Under the influence of New Math acquiring (mathematical) insight by the pupil himself was emphasized more, in primary schools too. The connection with the development of children’s thinking was sought in modern mathematics, in particular in logical structures. As a result set theory became the new basis for teaching of arithmetic. The supporters found a justification for this point of view in Piaget’s work. This development was a thorn in Freudenthal’s flesh. In an interview by the Haagse Post of 1969 he spoke about this: Prof. Freudenthal heeft vooral bezwaar tegen de in sommige leerboeken toegepaste benadering van de wiskunde, geheel vanuit de verzamelingsleer. ‘Dat is een grote vergissing’, meent hij, ‘je vindt dat in veel van de Amerikaanse literatuur, maar het is allemaal fout wat daar gebeurt. Dat hele gedoe is afkomstig van de Zwitserse psycholoog Piaget, waar sommige mensen in het wiskundeonderwijs wiskunde van schijnen te leren. Ik heb het al aan diverse vakmensen uitgelegd en die zeiden dan steeds weer: wat stom dat we die fout niet eerder hebben gezien.’ (Prof. Freudenthal particularly objects [to] the approach of some textbooks based completely on the set theory. ‘That is a big mistake’, he thinks, ‘you see that often in the American literature, but it is all wrong what happens there. All that originates from the Swiss psychologist Piaget, of whom some people in mathematics education seem to learn mathematics. I have already explained this to various people of the field and they said over and over again: how stupid that we have not seen that mistake before.’) [137]

What was the issue? In 1941 Piaget and his assistant Szeminska wrote a book about the development of the number concept: La génèse du nombre chez l’enfant (The origin of number in the child), in which children of various ages were tested for their notion of the cardinal number (the total number of elements). From Piaget’s findings it was—mistakenly—deduced that the notion of the cardinal number in the development of children took up the most important position, more important than the ordinal number (the number indicating the position in a linear list). Mistakenly, as Piaget himself also indicated in a footnote in an article from 1973: Several authors (Freudenthal, etc.) seem to have understood that I think the ordinal number is more primitive than the cardinal number, or the opposite. I have never made such a statement and have always considered these two aspects of finite numbers indissociable and psychologically reinforcing one another in a synthesis that goes beyond both the inclusion of classes and the order of asymmetrical transitive relations. [138]

The results of Piaget’s research suggested that the spontaneous development of the number concept with children closely resembled the logical structure of modern mathematics, in which the one-one mapping of two sets on each other would take


321

up a particularly important place. The conclusion that logic and structure of modern mathematics had to be taken as the basis of teaching of arithmetic, was subsequently easily drawn. In their designs for teaching of arithmetic the French Nicole Picard and the Canadian Zoltan Dienes, for example, started with logic reasoning in kindergarten. In the Netherlands, Picard’s work has been taken as the starting point of the structuralistic arithmetic method Denken en Rekenen (Thinking and calculating), in which Mrs. Freudenthal was also involved [139]. According to the Belgian pedagogue Valeer van Achter, editor of Denken en Rekenen, one should look for the “fundamentally and mathematically well-considered preliminary exercises on the number” [140]. In doing so he emphasized, just like Dienes did, the use of the one-one mapping between two sets to link up to the number concept of young children. Editor De Iongh described in the accompanying Tijdschrift Denken en Rekenen how set theory could contribute to the insight in mathematics. He spoke of the ‘undercomplex’: “the entirety of the, not easily put into words, “imaginations” and “memories” given to us by mathematical insight”. In registering and learning exact use of language and coupling this to the already present ‘undercomplex’ set theory was necessary according to him: Als we op deze wijze pogen het ‘ondercomplex’ voor het gewone rekenen te ontwikkelen, blijkt dat dit een deel vormt van wat de wiskundigen beschrijven in de elementaire verzamelingsleer en wel het deel, dat alleen de eindige verzamelingen bevat. Dit is de wezenlijke rechtvaardiging ervoor, dat we deze elementaire verzamelingsleer—neen, niet apart gaan bestuderen—maar wel gaan gebruiken om tellen en rekenen van de goede ‘plaatjes’ te voorzien. Dat we daardoor reeds heel vroeg de leerlingen vertrouwd maken met dat fundamentele deel van de wiskunde, de verzamelingsleer, dat de woorden en begrippen levert, waarmede tegenwoordig het gehele bouwwerk van de wiskunde wordt opgetrokken, is een extra voordeel, nu ook de wiskunde van de middelbare school zich meer en meer rondom de verzamelingsleer gaat herstructureren. (If we try to develop the ‘undercomplex’ for the normal arithmetic, it appears that this is part of what mathematicians describe in elementary set theory, in the part that contains only finite sets. This is the essential justification for the fact that we will—no, not study separately but—use the elementary set theory to provide counting and calculating with good ‘pictures’. In doing so we familiarize the pupils at a very early stage with that fundamental part of mathematics, set theory, that provides us with the words and notions that at present are used to raise the complete structure of mathematics. This is an extra advantage now that the mathematics of secondary school is also being restructured around set theory.) [141]

In R.A. de Jong’s study from 1986 the arithmetic method Denken en rekenen was characterized thus: “offers ample opportunity to mathematically rich education”, but with the comment “whether they can be realized is the question”. Ultimately the method would not take root in the Netherlands. In 1974 the last volumes were published [142]. 9.5.3.2 Number Concept and Bastiaan’s Currants Such a use of set theory at the primary school was characteristic for the New Math approach of the teaching of arithmetic. In this way a start could already be made with mathematics in primary school. Characteristic for this approach with sets, was

322


then to emphasize the number concept as a quantity (cardinal number or ‘numerosity aspect’) and not as a counting number (ordinal number, embodied by the infinitely running series of numbers). Here a reference was then made to the cardinal number (or: cardinality) as it was defined by Cantor at the end of the Nineteenth century. It seemed that the foundations of the number theory and the accompanying arithmetic laws could then easily be proven. In this approach counting (down) was replaced by the cardinal aspect of the number and the one-to-one mapping and comparing of two finite sets. In several articles Freudenthal resisted this approach [143]. “Bragging of clever authors” he called it, bragging that had in his view nothing to do with mathematics [144]. Neither in a mathematical nor a didactical sense did he approve of taking the aspect of quantity as a base for the natural numbers and in doing so neglecting the principle of complete induction. Mathematically this approach was ‘going round in circles’, because in the development of a theory of finite sets, which would then have to serve as the basis of the natural numbers, complete induction was eventually needed too: The majority of the manuals suggest a concept of the natural number based purely on set theory and ignore or obscure the mathematical fact that this is possible. Natural numbers are introduced as the cardinals of finite sets; but to explain what a finite set is, the familiar sequence of natural numbers is used. This is simply going around in a circle. [145]

And also in a didactical sense the ‘etcetera …’ of the infinite series of numbers was, according to Freudenthal, indispensable for children for the development of the number concept. The more primitive concept of ‘quantity’ on which some of the methods was based, was according to Freudenthal also present in animals and certainly was not sufficient as a starting point. Ik heb leerboekjes voor de lagere school gezien, waarin de kinderen na twee jaar nog niet konden tellen. […] Ze leren dan alleen om te zien dat vijf meer is dan drie enzovoort. Dat kunnen dieren ook. Een vogel ziet of een van de vijf eieren uit zijn nest verdwenen is, maar een vogel kan daarmee nog niet tellen! Die benadering is de puurste waanzin. (I have seen textbooks for the primary school in which children still could not count after two years.[…] They only learn to recognize that five is more than three etcetera. Animals can do that too. A bird notices when one of his five eggs has disappeared from its nest, but with that a bird still cannot count! That approach is the purest nonsense.) [146]

After this first primitive quantity aspect Freudenthal’s “psychological etc.” was very important for the subsequent development of the number concept. He followed the same line of thought as in his first Euclides article from 1948 [147]: Het kind leert niet rekenen door Venn-diagrammen [sic] met elkaar te vergelijken, maar door te tellen. Het telt op, niet door onoverzichtelijke verzamelingen bij elkaar te gooien, maar door verder te tellen. Het trekt af, door terug te tellen, het vermenigvuldigt de 3 door bij drieën verder te tellen. Het primitieve aantal, dat ook dieren bezitten, is spoedig door het veel effectievere telgetal overschaduwd. (The child does not learn arithmetic by comparing Venn diagrams with each other, but by counting. Children count, not by throwing cluttered sets together, but by continuing counting. They subtract by counting backwards, they multiply three by continuing counting from threes. The primitive quantity, which animals know as well, is soon eclipsed by the much more effective counting number.) [148]


323

Of course the ‘quantity’ had to be treated in the teaching of arithmetic, but “it is a mathematical and didactical illusion to think that the quantity on its own will be enough” [149]. According to Freudenthal the counting number played an important role in the formation of the number concept, and that should also be recognized in the teaching of arithmetic [150]. Counting was according to Freudenthal the first conscious mathematical activity of a child. It started as a rhythmic game [151]. In its play the child subsequently experienced the invariant properties of this counting. This is what Freudenthal again saw when he observed his grandchildren. He described for example how his granddaughter, at the time aged five, had much fun counting all the eyes, ears, feet etc. of a company. To her it was the fun of the game to reach the same number each time she counted anew [152]. In particular counting (and not the one-to-one mapping of two sets onto each other) was at her level the ultimate proof of being ‘equal’. The mathematical concept ‘set’ was according to Freudenthal of a higher level than the formation of numbers [153]. In spite of this the mental formation of sets was needed for the number concept, but for another reason than was urged in New Math circles: What matters is forming certain mental objects of which the numbers are properties; I call these objects sets because this is what we do with mathematical objects of this kind. Several times I signalled the confusion, which is all the fashion now: rather than forming sets, which is important for the number concept, one attempts to instil the set concept in the children. And one stap [sic] further, rather than numbers one wishes to teach the children the number concept. It is ever and again the skipping of levels in the learning process. In the genesis of the set and number concept the lowest level is constituting sets and numbers, and only at far higher levels is the constitution of the set concept and the number concept at home. [154]

In Bastiaan Freudenthal saw the exception to the rule in the development of the number concept. Contrary to most children, as a toddler Bastiaan did not have fun in counting-for-the-sake-of-counting. His ‘acquisition of the number concept’ took place in one clear jump in his learning process which Freudenthal witnessed. He spoke of it at a conference in 1975: Dat kind weigert te tellen, de moeder werd wanhopig omdat het kind nooit zou leren tellen en het begon pas te tellen nadat het volledig het getalbegrip verworven had. De verwerving van het getalbegrip was heel duidelijk één ogenblik. We zaten te eten, aan het dessert. Hij had aalbessen op zijn bord. Hij hield het lepeltje omhoog en zei: dat is zoveel als wij hier aan tafel zitten. We waren met zijn zessen aan tafel en er lagen zes aalbessen op de lepel. Ik vroeg hem waarom, toen zei hij dat zie ik zo, en toen zei hij: er zijn twee kinderen, twee volwassenen en twee opa en oma. (That child refuses to count, the mother was desperate because the child would never learn to count and he only started counting after he had completely acquired the number concept. The acquisition of the number concept was very clearly at one specific moment. We were at dinner, eating the dessert. On his plate were currants. He held his spoon high and said: that is as much as we here around the table. There were six of us at the table and there were six currants on his spoon. I asked him why, and he said here is how I see it and then he said: there are two children, two grown-ups and two granddad and grandma.) [155]

324


It was only a few days after this event that Bastiaan started counting. According to Freudenthal it was exceptional for a child to firstly acquire the number concept and only after that start counting, like Bastiaan did. He emphasized in this lecture the importance of the mental link Bastiaan made at the crucial moment [156]. Both the persons and the currants were explicitly present, but the most essential aspect in acquiring the number concept in this example was the mental aspect, breaking with the explicit. Just like the walk on the railway Freudenthal would use the ‘currant’ observation in many articles and lectures. To him it was the ultimate example of a discontinuity in the learning process. The crux in this observation was according to Freudenthal: It shows that rather than sets it is structures that are fundamental in the mathematical development of children. Indeed how can an unstructured chaos, which a set is supposed to be, attain interest? The child likes structure and tries to put structures into the world around it in order to measure it. [157]

Structure was the key word in developing mathematics in young children. Quantity was a property of sets without structure. Therefore to experience quantity as a mental object, the child had to be able to break it loose from the structure of the sets to be considered. That was the discontinuity in the learning process when a child acquired the number concept. In the case of the currants it was clear: The sets to be compared bear heavy structure, a structure by generations and by sides of the table of the one set, and the geometrical arrangement of the six [currants] in the second. By these heavy structures comparing the sets is achieved. Number is a property of sets without structure, and in order to constitute number as a mental object, the child has to eliminate—not necessarily explicitly but certainly implicitly—structures from sets to be compared. [158]

To acquire the number concept one had first of all to search for structure. The oneto-one mapping between two sets, which was emphasized in New Math, wronged this insight according to Freudenthal. Onze verzamelingstheoretische vooroordelen eisen, dat we de relatie, die Bastiaan tussen de aalbessen en de personen schijnt te constitueren, als één-één-afbeelding interpreteren, maar het is een veel globalere relatie, waarbij verzamelingen niet in elementen geatomiseerd, maar in groeperingen gestructureerd worden. (Our preconceptions of set theory require that we interpret the relation, which Bastiaan seems to constitute between the persons and the currants, as a one-to-one mapping, but it is a much broader relation, in which sets are not atomized in elements, but are structured in classifications.) [159]

9.5.3.3 “Here is how I see it” and Didactical Phenomenology According to Freudenthal in observing young children it amounted to being alert to statements such as Bastiaan’s “here is how I see it” in the currant example. That was not only true in acquiring the number concept. Freudenthal went as far as defining geometry at the primary school as “what can be characterized in terms of behaviour of the pupil by the reaction “here is how I see it” to the question “why” “[160]. The


325

trick subsequently was to get children to reach the point “to let their internal visions so to speak condensate externally”, without forcing them to use a language that did not match their level [161]. Wie wiskunde onderwijst, ziet gaarne dat zijn leerlingen hun gedachten in onberispelijke formuleringen kunnen uiten; te schatten hoe ver de onderwijzende mag gaan met deze eis, is iets waarvoor tact en wijsheid van hem gevraagd worden. Wanneer het ‘ik zie het zo’ van een leerling door stamelen of het opleggen van formuleringen door de onderwijzende wordt vervangen, is men didactisch geen stap verder. Wie onderwijst, ga eens bij zich zelf na, hoe moeilijk het zijn kan te beredeneren hetgeen men al helder en duidelijk ziet. (Anybody who teaches mathematics, is pleased to see his pupils expressing their thoughts perfectly formulated; in order to estimate how far the one teaching may go in requiring this, is something for which tact and wisdom are demanded. When the ‘here is how I see it’ of a pupil is replaced by stammering or enforcing formulations by the one teaching, didactically seen that has not taken us one step further. If you are a teacher, work out for yourself how difficult it can be to reason out what you already see clearly and plainly.) [162]

The kind of geometry that was developed in the 1970s by the WISKOBAS team was, according to Freudenthal, geared to the domain of “here is how I see it”: “pupils reason with their hands and their eyes, in a model and on paper, not in a linguistically established system of concepts” [163]. The question to what extent it was allowed or obligatory that the subject matter was pre-structured was important in this respect. Freudenthal argued against starting with pre-structured material, but recommended choosing phenomenologically richer material in a vertical planning of the subject matter, in which the same subject matter was treated at an increasingly higher level. But what was the way to go? There was still a great deal of work to be done in developing mathematics education starting from Freudenthal’s vision (that was shared by the WISKOBAS team). Offering a rich context was only the beginning. One had to look for more level structures and decisive steps in the learning process by observation of learning processes. In his articles Freudenthal described innumerable examples taken from his own observations of his (grand)children and from the designs that were experimented with in WISKOBAS. But he also gave a theoretical method for developing mathematics education: this he called “didactical phenomenology”. Without the term ‘phenomenology’ actually being mentioned, the seed for this theoretical aid could already be found in Freudenthal’s article ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’ from 1948. In it he argued that the teacher should think through the subject matter from a (not necessarily mathematical, but certainly also didactical!) higher point of view. Next he made a start with what he would be calling the didactical phenomenology of the number concept in the 1970s, even though he was far from doing it as thoroughly as he would do this 30 years later [164]. A first real initiative to didactical phenomenology followed in Freudenthal’s article ‘Phaenomenologie van het getalbegrip’ (‘Phenomenology of the number concept’) [165]. The article provided a mathematical analysis of the number concept that could be used to arrive at a better approach of the accompanying didactics. Freudenthal’s reason for this article was to be able to combat the abuses he saw in

326


the way the development of the number concept was approached. At the same time he provided a sound base on which a better approach could be built. In 1984 Freudenthal devoted the book Didactische fenomenologie van wiskundige structuren. Deel I (Didactical phenomenology of mathematical structures. Part I) to ‘didactical phenomenology’ [166]. The origin of this book was an article from 1973, entitled ‘Lernzielfindung im Mathematikunterricht’ (‘Finding learning goals in mathematics education’) [167]. This article was included in the book as one of its chapters. In it he described among other things the way the IOWO worked when it developed mathematics education and searched for learning goals by observing learning processes. “Didaktische Phänomenologie eines mathematischen Gebietes” (“Didactical phenomenology of a mathematical field”) was according to him an important theoretical model for all those involved in this process: Niveaus moeten niet in de leerstof, maar in het leerproces gezocht worden. Om die te vinden, moet men leerprocessen observeren. Dit kan worden voorafgegaan en begeleid door een analyse van de leerstof, in denkbeeldige processen geplaatst, die ik didactische fenomenologie heb genoemd. (Levels should be looked for in the learning process rather than the subject matter. To find them, one has to observe learning processes. This can be preceded and guided by an analysis of the subject matter, positioned in imaginary processes, which I have called didactical phenomenology.) [168]

Here he did not have a phenomenology with a philosophical slant in mind (contrary to the role ‘phenomenology’ for instance played in Langeveld’s work), but “eine Analyse der Mathematik als Phänomen, nicht nach ihren Grundlagen” (“an analysis of mathematics as a phenomenon, not according to its foundations”), a detailed analysis of the sources of mathematical ideas in reality [169]. The addition “didactical” indicated that it had to be done from a didactical perspective: directed at acquiring all that was needed to be able to organize the ‘thing’, the ‘phenomenon’ mathematically. In his book of the early 1980s Freudenthal provided an extended description: Onze wiskundige begrippen, structuren, denkbeelden zijn uitgevonden als werktuigen om de verschijnselen van de fysische, sociale, mentale wereld te ordenen. Fenomenologie van wiskundige begrippen, structuren, denkbeelden betekent: ze in hun relatie tot de fenomenen te beschrijven waarvoor zij geschapen en waartoe zij uitgebreid werden in het leerproces van de mensheid; en voorzover deze beschrijving met het leerproces van de jongere generatie te maken heeft, wordt zij didactische fenomenologie, wegwijzer voor de onderwijsgevende naar de instap van de lerende in het leerproces van de mensheid. Niet in haar geschiedenis, maar in haar nog voortdurende leerproces—sloppen en omwegen vermijdend en levende wortels sparend en versterkend. (Our mathematical concepts, structures, and ideas have been invented as tools to sort out the phenomena of the physical, social and mental world. Phenomenology of mathematical concepts, structures, and ideas means: describe them in relation to the phenomena for which they have been created and to which they have been extended in the learning process of mankind; and in so far as this description has to do with the learning process of the younger generation, it becomes didactical phenomenology, a guide for those who teach to the entry in the learning process of mankind of those who learn. Not in mankind’s history but in its continuous learning process—avoiding blind alleys and detours and saving and reinforcing living roots.) [170]

However important he considered didactical phenomenology as a starting point in developing mathematics education, to Freudenthal such a theoretical analysis was


327

indeed never enough for the development of mathematics education. He presented it in the 1970s only as an aid, made behind the desk and for the time being producing not more than a list (constantly subject to improvements) to support the observing education developer.

9.5.4 The Right to Sound Mathematics for All With the emphasis on the observation of learning processes in children the starting point of the development of mathematics education drifted further and further away from mathematics as a science. Not only within the IOWO but also in Freudenthal’s work this aspect came up more and more. By in the first place observing the learning child, the world of that child presented itself as a new starting point automatically. This so-called realistic teaching methodology was extremely characteristic for the 1970s [171]. IOWO’s democratic procedure to develop education also paved the way for it. Not a “regulatory curriculum development from behind a desk” as Freudenthal called it, but “democratic participation of the field”, in which developers closely involved the teachers, the teachers’ trainers and their students and schools [172]. It was obvious that in this process the target group had a very direct influence on the process. In the curriculum of WISKOBAS it was described as follows: Het kader van de algehele doelstelling en van wiskundeonderwijs, van waaruit dit ontwerp ontwikkeld is, is: • begeleiding van de kinderen in hun benadering van de wiskundige aspecten van de werkelijkheid, zoals men zich die nu en in de (nabije) toekomst voorstelt • wiskundige vorming in harmonie met persoonlijke aanleg en met redelijke eisen van samenleving en voortgezette opleiding (The framework of the general objective and of mathematics education, from which this curriculum has been designed, is: • supporting children in their approach of mathematical aspects of the reality, as it is imagined now and in the (near) future • mathematical formation in harmony with personal talents and with reasonable requirements from society and advanced schooling) [173]

And at the same time with and through this starting point the goal of the mathematics education that had to be developed, also changed: the subject matter had in the first place to be useful to the pupil. Freudenthal explained in the WISKOBAS Bulletin what the criterion was to him: Je kunt je leerlingen niet uitleggen wat er allemaal met de wiskunde gedaan kan worden, en dat hoeft ook niet. Je kunt wel—op elk nivo [sic]—laten zien dat er iets met wiskunde kan worden gedaan en dat iedereen er iets mee kan doen. Iemand opzadelen met een wiskunde waar hij niets mee kan doen, is te veel van het goede. Iemand verstoken doen zijn van die wiskunde die hij wel kan gebruiken, is hem geestelijk uithongeren. […] Laat ik dan dichter bij de grond blijven, heel dicht bij de grond. Ik dacht dat we alle leerstof moeten weren, waarvan we, als de leerling vraagt ‘waarvoor dient het?’ met een goed geweten alleen maar kunnen zeggen: ‘nergens voor’. (You cannot explain to pupils all the things that can be done with mathematics, and you do not need to. What you can do—at

328


every level—is show that something can be done with mathematics and that everybody can do something with it. Burdening someone with mathematics that is of no use to him is too much of a good thing. Depriving someone of the mathematics that he can use, is starving him mentally. […] Let me stay closer to the ground, very close to the ground. I thought that we have to exclude all subject matter of which, if a pupils asks ‘what good it is?’ we can only say with a clear conscience: ‘none’.) [174]

The criterion of usefulness therefore reigned supreme. With this the argument that mathematics had ‘formative value’ for the pupil lost its attention in this period. This was based on the developers’ attitude which was directed more to the child, and on the great social relevance of mathematics. Freudenthal contributed by organizing the colloquium ‘How to teach mathematics so as to be useful’ of 1967. In his lecture at this colloquium Freudenthal devoted a few lines to ‘why’ teaching ‘useful mathematics’: Since mathematics has proved indispensable for the understanding and the technological control not only of the physical world but also of the social structure, we can no longer keep silent about teaching mathematics so as to be useful. In educational philosophies of the past, mathematics often figures as the paragon of a disinterested science. No doubt it still is, but we can no longer afford to stress this point if this keeps our attention off the widespread use of mathematics and the fact that mathematics is needed not by a few people, but virtually by everybody. [175]

Freudenthal had the answer to the question ‘how to attain that the mathematics taught is useful?’ ready. Not by teaching applied mathematics, not by teaching pure mathematics to apply later, but: The problem is not what kind of mathematics, but how mathematics has to be taught. In its first principles mathematics means mathematizing reality, and for most of its users this is the final aspect of mathematics, too. For a few ones this activity extends to mathematizing mathematics itelf. [176]

By learning the pupil to mathematize you did justice to the pupil and to mathematics. Simply training in mathematical techniques was of no use at all according to Freudenthal: Nobody can teach and nobody can learn enough prefabricated mathematics to meet all possible mathematizable situations. Moreover a mathematical subject that has reached the state of a technique, can with more efficiency be handled by machines than by man. […] Mathematics as an aim in itself counts for a very small minority only. For all others mathematics is important enough to play a part in their education. [177]

‘Mathematics for all’, that was his message. Approaching this by learning to mathematize in the real world would be a way that was functional in the world of the user: the child. Freudenthal preferred to make as few concessions as possible to that reality in which mathematics had to be discovered. He often found the ‘fairy tales’ in which the teaching packages of the IOWO were wrapped up, too farfetched and too artificial: Ik vind dergelijke dingen kun je ook realistisch doen en niet met kabouters. Die kaboutermentaliteit… Maar goed, misschien wat ouderwets van me of zo. Ik erger me wanneer ik mijn kleinkinderen naar Paulus de boskabouter zie kijken. Ze vinden het verschrikkelijk leuk, maar het stoort me op één of andere manier. (My opinion is that you can also do such things in a realistic way instead of with gnomes. That gnome mentality … Anyway, maybe


329

I am a bit old-fashioned or something. I am annoyed when I see my grandchildren looking at ‘Paulus de boskabouter’ (a Dutch children’s TV programme). They tremendously like it but in one way or another it annoys me.) [178]

It was more in his line to stay as close as possible to the reality of everyday life when looking for mathematics education. Moreover, in diverting to fairylands the danger of a certain laziness and wandering off was looming in his opinion. Maak je het jezelf niet bijzonder makkelijk met die sprookjeswerelden? Zou je niet, als je die niet had, naar veel reëlere werelden zoeken om de kinderen aan te bieden dat ze daarin werken? […] Beïnvloed je de stof niet al te zeer misschien te ongunstig door nu alles in zo’n wereldje te stoppen? (Don’t you make things especially easy for yourself with those fairylands? Wouldn’t you, if did not have those, look for more realistic worlds for the children to work in? […] Don’t you influence the subject matter too much and maybe in a disadvantageous manner by putting everything in such a world?) [179]

According to Freudenthal mathematics strongly tied to the reality was the kind that sunk in best: I espouse the philosophy of teaching mathematics related to reality. Mathematics is important for many people because it admits of multifarious applications. I do not trust teachers of other disciplines to be able to tie bonds of mathematics with reality which have been cut by the mathematics teacher. Moreover, I do not believe that mathematics, not tied to lived reality with strong bonds, can have a lasting influence in most individuals. […] People usually forget what is not related to the world in which they live. For most people mathematics cannot be an aim in itself; if they have learned it in an unrelated way, they will never be able to use it. [180]

Freudenthal wanted the mathematics a pupil learned at school to be something which he had acquired in an active way: mathematics which the pupils could employ, manipulate and apply in relevant situations. Another requirement for the mathematics to be taught that regularly popped up in his article was that of being ‘sincere’. The mathematics to be taught had to be real, ‘honest’ mathematics. Pedagogic ideals that caught on in this period were in this way also expressed in the mathematics education. In the ‘critical’ or ‘emancipatory pedagogy’ (a pedagogic movement that arose in Germany in the 1960s and 1970s) the critique on the establishment and the injustice of the social inequality was an important aspect [181]. Putting the child in a central position (‘vom Kinde aus’) in the reformed pedagogy and the individual development of the child remained important, but at the same time the critical pedagogy aimed at emancipation and improving society. Education could not be detached from the society in which it was provided and when this society was criticized this had to reveal itself in the education. In the education this could, for example, be found in the development of curricula containing subject matter that was relevant to the pupil. When socially relevant curricula were established children from the working classes would profit more from the education and the educational system would no longer benefit the upper classes alone. Such general pedagogic ideals could be found in a ‘translated form’ in the mathematics education. The ‘educate to become critical individuals’ and ‘honesty’ also became a criterion to select the subject matter in mathematics education. Apart from

330


the social relevance that was required, no concessions could be made to the mathematical correctness of that subject matter: as mathematics it had to remain correct and be justified to the pupil at any moment. Mathematical honesty’ was an important requirement to Freudenthal Wherever he found this requirement infringed upon, he took up his pen. In a series of articles in Euclides, for example, he accused the Nomenclatuurcommissie (Nomenclature Committee) of wanting to cultivate ‘school mathematics’ instead of real mathematics in its propositions [182]. In particular where group theory and sets were used in education, the mathematical honesty was often duped according to Freudenthal. A serious business: We have to be careful and honest if we want to adapt some piece of high mathematics to a lower level. Simplifying is a good thing but wrong elementarisations are a danger, and so is imitating superficial features while destroying the great ideas of some mathematical theory. If children are taught groups they are entitled to learn genuine group theory rather than a childish version. In the past, mathematics has seriously suffered under the falsifying tendencies in adaptations of mathematical subject matter to school level. Let us be more cautious in the future. Honesty is a cardinal virtue in education. Nothing is lost if some subject matter cannot be taught prematurely and much is gained if it can in an honest way. [183]

The contradiction was not that of ‘didactical versus scientific’, argued Freudenthal. Good subject matter was chosen with a view to the efficient use of that mathematics in life [184]. And it was best not to teach anything, which only a few pupils later would later find out to be something that could have been taught better or—worse— was wrong: Er zijn volwassenen, die menen tegen kleuters in een brabbeltaaltje te moeten praten; als de kleuter dit overneemt, kunnen ze immers zeggen: zie je, hij kan niet beter. Gelukkig leren kleuters het af. In Wiskunde II zullen we ze het afleren—aldus volgens Vredenduin. En de anderen? Moeten ze maar blijven brabbelen? (There are grown-ups who think they have to talk to toddlers in baby-talk; for if the toddler copies this they can say: you see, he can’t do better. Fortunately, toddlers cure themselves. In Mathematics II we will teach them not to do so—according to Vredenduin. And the others? Do they have to keep using baby-talk?) [185]

The mathematicians themselves were responsible for teaching the pupils to think clearly and for teaching them useful mathematics according to Freudenthal. What the range of the responsibility was, became clear from his article ‘De niveaus in het leerproces en de heterogene leergroep, met het oog op de middenschool’ (‘The levels in the learning process and the heterogeneous learning group, in view of the comprehensive school’). In it he discussed his mathematics experiences with LBO pupils who were ‘traumatized’ by the arithmetic education. Als wiskundigen zijn we verplicht aan het herstel van hun gehavende persoonlijkheid mee te werken. Alleen door middel van de wiskunde—en dan goede—kunnen ze genezen worden van het trauma, dat hun door slechte wiskunde is toegebracht—als het ware een homeopatische kuur. (As mathematicians we have the obligation to cooperate in the recovery of their damaged personality. Only by means of mathematics—and now good mathematics—they can be cured of the traumas inflicted on them by bad mathematics—a homoeopathic course of treatment as it were.) [186]


331

By introducing these pupils to good mathematics again, namely at the zeroth level, it was possible to repair the damage done according to Freudenthal. And that good mathematics was “a mathematics […] in which—apart from most elementarily— one does not have to do arithmetic, a mathematics connected to reality, close to children and socially relevant”, such as that designed by the IOWO for the LBO [187]. Freudenthal’s opinion on such mathematics was “indispensable knowledge and skills for everyone, as reader of a newspaper, father or mother, citizen, consumer and producer, and also as someone who either enjoys art or creates it, you can profit from it” [188]. In the same article another for Freudenthal important aspect of ‘democratic’ mathematics education came up: his interest in the possibilities for learning in a heterogeneous group. The great interest in the Middenschool (a type of comprehensive school) in that period made this a topical subject. The core of the idea for the Middenschool was the construction of an integrated secondary education consisting of a Middenschool, followed by an upper school. In other countries such as Germany and Britain this idea had been detailed to the ‘Gesamtschule’ and the ‘comprehensive school’ respectively [189]. In the Netherlands in 1973 experiments in this direction were initiated by the government and the Innovatiecommissie Middenschool (ICM, Innovation Committee Middenschool) was appointed [190]. The most important reason to establish the Middenschool was to postpone choosing for a profession or a study and in doing so offer equal opportunities for everyone. Because of the principle of ‘equal opportunities for everyone, irrespective of social environment and background’ Freudenthal rather liked the idea of the Middenschool. The interpretation however, was exactly the opposite of what he had in mind with heterogeneous education: Er moet gedifferentieerd worden. Akkoord! Maar “differentiëren”, zoals bedoeld in de discussies omtrent Gesamtschule en Middenschool—is een eufemisme. Je zegt “differentiatie” en je bedoelt “separatie”. […] Men begint ermee het hoofd te breken over differentiatie, terwijl de integratie nog niet eens aan de orde is geweest, laat staan doordacht is, en zwelgt in het ontwerpen van geraffineerde systemen van differentiatie: hoe men de kinderen verschillende leerstof zal toedienen terwijl men de schijn van één programma ophoudt. (Differentiation is needed. OK! But “differentiation” as implied in the discussions about the Gesamtschule and the Middenschool—is a euphemism. You say “differentiation” and you mean “separation”. […] People start with racking their brains over differentiation, whereas integration has not yet been discussed, let alone thought through, and wallows in designing sophisticated differentiation systems: how to present the children with different subject matter and at the same time keep up the appearance of one programme.) [191]

But where could the power of working be found in heterogeneous groups according to him? Wat men na zou moeten streven en trachten te realiseren, is een systeem, waarbij de leerlingen op verschillende niveaus—elk op het zijne—met elkaar aan hetzelfde onderwerp werken en leren werken. Dit is het principe van heterogene leeren werkgroepen, waar de leden van lager niveau zich op het hogere niveau van hun medeleden kunnen oriënteren en die op hoger niveau inhoudelijk en sociaal kunnen profiteren van hun kijk op het lagere. Het is niet zo’n vreemd idee, die heterogene leeren werkgroep; op veel plaatsen in de maatschappij is het gerealiseerd. Het kost alleen moeite leerinhouden zakelijk en leerprocessen sociaal opnieuw te doordenken, om tot deze onderwijsvernieuwing te geraken. Wil

332


men zich deze moeite niet getroosten, dan loopt de operatie ‘Middenschool’ de kans een papieren plan met papieren gevolgen te blijven. (What one should be striving for and should be trying to realize is a system in which the pupils at different levels—each at his own— work and learn to work on the same subject together. That is the principle of heterogeneous learning and working groups, in which members of a lower level can orient themselves at the higher level of their co-members and those of the higher level can profit with respect to content and socially from their view on the lower level. It is not such a strange idea, this heterogeneous learning and working group; it has been realized in many places in society. The only trouble is to again consider fully the subject matter in a professional way and the learning processes in a social way to come to this innovation of education. If one does not want to take that trouble, then the ‘Middenschool’ operation runs the risk of remaining a paper plan with paper consequences.) [192]

In particular mathematics, with its different levels in the learning process, was very well suited for such an approach according to Freudenthal. Within a group pupils could work on the same subject matter at different levels and could profit from each other’s differences in their own learning process. In this way pupils would be able to follow the same mathematics curriculum at the Middenschool at least for the first 2 years: De samenwerking moet de leerlingen op alle niveaus de toegang naar een hoger niveau openen, die van het lagere, omdat ze zich op die op het hogere kunnen oriënteren, die van het hogere, omdat de blik naar het lagere niveau hun nieuwe inzichten verschaft. (The cooperation has to give pupils of every level access to a higher level; to those of a lower level because they can orient themselves to that higher level, to those of the higher level because looking at the lower level gives them new insights.) [193]

Freudenthal realized that interpretation of the concept of a ‘heterogeneous learning group’ was barely hatched and that the possibilities it offered were still very uncertain. Yet to him it was certainly worth investigating further, in particular whenmathematics education was concerned [194]. And with that Freudenthal returned to that with which it all had started and he was after: making observations—and in particular observations of his grandson. Freudenthal and Bastiaan as the “extremely heterogeneous learning group”. Ik kom terug op een van die leergroepen. Ik bedoel die extreem heterogene van grootvader en kleinzoon. U herinnert het zich wel? Neen, niet lachen! Het was echt een leergroep. Heeft de kleinzoon iets geleerd? Ja, doordat hij iemand had om tegen te praten. En de grootvader? Heeft hij iets geleerd? Iets, neen, heel wat. Zoveel, dat hij het gepast vond om u er hier van te vertellen. Dit ter geruststelling van hen die vrezen, dat heterogene leergroepen de leerprocessen naar onderen moeten nivelleren. (I return to one of these learning groups. I mean the extremely heterogeneous one of grandfather and grandson. You do recall that one? No, don’t laugh! It really was a learning group. Has the grandson learned something? Yes he did, because he had someone to talk to. And the grandfather? Did he learn something? Something, no, a great deal. So much that it seemed only right to tell you about it. I tell you this only to reassure those who fear that heterogeneous learning groups can only level down the learning processes.) [195]

9.6 Enfant Terrible

333

9.6 Enfant Terrible 9.6.1 Weeding In the early 1970s Freudenthal had found, in and by the work he did at the IOWO, a way to be productively active in the field of mathematics education. The fact that he was involved in the process of education development that took place there was extremely fruitful for him and results of this can be found in all his lectures, articles and books from that time onwards. But all the same he still was an active critic in many fields [196]. To many people in the circles of mathematics education he had the reputation of being a feared professor of distinction. Mathematics didactician and teacher’s trainer Van Dormolen remembers: Nu was Freudenthal in die tijd niet iemand waarbij je veilig ging vragen wat hij precies bedoelde. We waren eigenlijk een beetje bang voor hem en begrepen niet, dat diens scherpe kritieken helemaal niet betekenden dat hij je dom vond. Jaren later leerde ik dat hij het juist in iemand waardeerde als die met hem in debat wilde gaan, maar als jong leraar begon je daar niet aan. (At the time Freudenthal was not a man to whom you turned safely to ask what he precisely meant. We were, in fact, a little afraid of him and did not understand that his biting critique did not mean that he considered you to be ignorant. Years later I learned that he rather appreciated it if someone wanted to debate with him, but being a newly appointed teacher you did not think of debating with him.) [197]

Internationally too it was clear that Freudenthal did not stand for any nonsense. Bishop wrote about this: He was a tough critic and someone who did not shirk his responsibilities in helping a knowledge field to develop. He undoubtedly believed that those of us who are genuinely concerned with the development of knowledge are not as critical of ideas, methods and practices as we ought to be. His first inspiration to me therefore was as a critic. [198]

Freudenthal was well aware of this image and cherished it: Een enfant terrible—zowel nationaal als internationaal—waar ze blijkbaar toch niet omheen konden, zoals straks zal blijken. Of hoopten ze dat ik al rijzende mijn wilde haren zou kwijt raken? (An enfant terrible—both nationally and internationally—that they apparently could not ignore as will become clear soon. Or did they hope that I would settle down when I was more and more making a name for myself?) [199]

But that critical attitude and the unvarnished manner in which he often criticized people and things did not appear out of thin air. Such actions were dire necessities to Freudenthal. When in his opinion something was scientifically groundless, an erroneous statistical application or simply gross nonsense he could not help commenting. In addition, he kept emphasizing that it was not the right moment to rest on his laurels: there was still a long way to go where mathematics education was concerned. And going that way involved being on the look-out for abuses all the time. Freudenthal’s critical attitude was of course far from new. He stuck to his motto from 1944 ‘all positive action starts with criticism’. But the new curriculum of 1968, the influence of New Math and all the ideas and rising initiatives in education in general made him more than ever an active critic.

334


It is striking how sparingly Freudenthal mentioned other people’s work in a positive context [200]. He in particular mentioned other people’s work explicitly when he wanted to explain how things should not be done. In preface of Mathematics as an educational task he explained why he did not want to use footnotes in this book: On the other hand I avoided all citations with respect to textbooks, designs and lessons wherever it was feasible. I believe I had a compelling reason to do so for this material was frequently subjected to criticism, which in fact was often negative. The material could be sharply divided into serious work and trash. Citations in footnotes would have meant tarring everything with the same brush. This I would hate to do. At the same time, it would have been too much an honour for trash to be quoted along with serious literature. Therefore, I have made explicit quotations only in a few quite specific cases. [201]

On the face of it his criticism often concerned trivial details, but behind that there was always a better funded criticism. One of Freudenthal’s hobby-horses, for example, was the correct use of (mathematical) language [202]. In his article ‘Verzamelingen in het onderwijs’ (‘The use of sets in education’) he denounced the ill-chosen use of the association of the mathematical set with collecting as a hobby, and all the linguistic problems caused by that. What is a stamp collection after all in that sense: a set of pieces of paper? Or “a kind of such pieces of paper that resemble each other in the sense that they can or could be used in the same postal function”? “Detailed criticism” Freudenthal called it, “that does not affect the broad outline of the method” [203]. Is dit nu muggezifterij? Het is evenzeer muggezifterij als het verschil tussen ‘liggen’ en ‘leggen’, tussen ‘word’ en ‘wordt’. Het antwoord op uiteenzettingen als de bovenstaande luidt vaak: ‘Je kunt met kinderen niet alles zo exact doen’. 3 + 2 is voor elke leeftijd hetzelfde, een drogredenering is voor elke leeftijd een drogredenering. Als je met verzamelingen van postzegels of van letters op een bepaalde leeftijd niet redelijk kunt werken, omdat ze te veel kritisch besef vereisen, laat ze dan weg, je verliest er niets aan. (Is this splitting hairs? It is splitting hairs as much as the difference between ‘lie’ and ‘lay’ is and that between ‘your’ and ‘you’re’. The answer to explanations as the one mentioned above is often: ‘With children you cannot do everything that exactly’. 3 +2 is the same for children of every age and the same is true for casuistry. If you cannot reasonably use sets of stamps or sets of characters at a certain age because they require too much critical awareness, you had better leave them be, you will not lose anything by it.) [204]

By emphasizing the details and also—and on purpose!—mentioning the small mistakes Freudenthal hoped to prevent worse mistakes: […] laat het dan wiskunde blijven: verzamelingen en relaties, waar ze voor deugen en niet om er sommetjes mee te maken. We hebben een van de echte wiskunde gescheiden schoolwiskunde gehad en we gaan met reuzestappen weer naar zo iets toe. Toen ik in de buitenlandse literatuur zag wat voor fouten kunnen worden gemaakt, heb ik er in lezingen en cursussen op gewezen. Ik hoop dat er een preventieve werking van uit gaat. ([…] let it remain mathematics: sets and relations, use them for the purpose they are good for and not just to do sums. We have had a school mathematics that differed from the real mathematics and we are heading that way again with giant leaps. When I saw in the international literature what kind of mistakes can be made, I have pointed them out in lectures and courses. I hope that has a preventive effect.) [205]

That was exactly the point of his criticism, the preventive effect. By pulling out the weeds immediately after they have emerged, you could prevent them from

9.6 Enfant Terrible

335

s preading. This was badly needed, particularly when abuse of modern mathematics in education was at stake. The grave danger of what he called ‘deceit’ was according to him that it prevents serious attempts to improve mathematics education: All this is the consequence of a modernism that insists on sets no matter what.[…] I have no fear that it will last (it is undoubtedly a transitory phenomenon), but it tends to discredit modern mathematics and consequently delays its true development in instruction. To sustain serious efforts, it is essential to criticize the idle production of manuals, films, and other methods of mathematics instruction. [206]

9.6.2 Drumming on Empty Barrels Apart from criticizing the details, such as the use of mathematical language, Freudenthal often also criticized a discipline as a whole. In particular the science of teaching had to endure much in that respect. Mostly these were particulars that were translated wrongly from the (general) science of teaching to education in a specific discipline, but regularly also the science of teaching in general was at stake. For example in his article ‘De Middenschool’ in which he discussed the Dutch school system: […] met de splitsing van het AVO naar MAVO, HAVO, VWO erbij is het Nederlandse voorgezette onderwijs zeker het meest opgesplitste in de hele wereld. Deze opsplitsing— een kluif om te watertanden voor onderwijskundigen, die van systemen en systematisme leven—is bestendigd en verfijnd door de Mammoet-wet, in een tijd dat in heel Europa de algemene trend de andere kant op wees. ([…] with the division of the AVO in MAVO, HAVO and VWO the Dutch system for secondary education is certainly the most subdivided school system in the world. This subdivision—a bone that makes the mouths of the didacticians who live on systems and systematics water—is made permanent and refined by the Mammoth Act in a time that in the whole of Europe the general trend was in the opposite direction.) [207]

According to Freudenthal, it was characteristic of the science of teaching to work from abstract schematics and only complete these with the education in a specific discipline in the end. He was categorically against this: the content of the education and the way in which that content was taught cannot be separated. To arrive at a meaningful education it made no sense to approach this education in a purely abstract and theoretical manner. In another article about the Middenschool he phrased it as follows [208]: Men kan vanuit het algemene deduceren wanneer men al eens een keer naar het algemene toe geïnduceerd heeft, van het onderwijs in ’t algemeen naar het vak, als men de weg van het vak naar ’t algemeen al eens heeft afgelegd. Zo zou men ook dergelijke conferenties moeten opzetten: met de vakken beginnen, om vanuit de concrete ervaringen het algemene te begrijpen en naar waarde te schatten—als men onderwijstheoretische argumenten ten bate van de middenschool zoekt, kan men ze alleen vanuit de vakken vinden. (You can only deduct starting from the general principles when you have induced towards the general principles before, from the education in general to that in a specific discipline, if you have already taken the road from the discipline to the general education. That is also the way such conferences ought to be structured: starting from the disciplines, to understand and value the general principles from concrete experiences—if one is looking for arguments in

336


favour of the Middenschool in the theoretical science of teaching, one can only find these starting from the disciplines.) [209]

It was very rare to involve the expertise in the disciplines in educational research. According to Freudenthal this approach induced an “aversion that was sometimes increased to hysteria” in the science of teaching [210]. In addition, within the educational system there was “a vast cloak of charity to cover things that went wrong in the educational production and a gigantic carpet to sweep them under” [211]. There was also an enormous gap between psychology as a discipline and learning mathematics according to Freudenthal. In the preface of Mathematics as an educational task he desperately asked himself how and by whom this gap could be filled: When in an excellent modern book [R.M. Gagné, The conditions of learning (London 1965)] I tried to find out what I should understand under learning and how I should subdivide it, I felt myself very far from what I had experienced myself and with others as mathematical learning. A feeling of loneliness seized me: is mathematics really so different? I wish that someone who profoundly understands both mathematics and psychology would show us the bridge. [212]

Seen from his side the question was not that he did not want to fill the gap, but he simply did not see how it could be done with the currently available knowledge. Until a solution for this problem was found Freudenthal continued the battle against every empty theory, in particular when it could have dangerously drastic consequences for the education. A good example of this was Freudenthal’s criticism on an international comparative study of schoolchildren’s records of learning in their native language and science (IEA) [213]. In Freudenthal’s opinion it was not a very good study. It used, for example, double standards and pupils in different countries were compared in situations that were absolutely incomparable. The results—among which the conclusion that Dutch children lagged behind their peer groups in the sciences—were therefore completely unreliable, but all the same Freudenthal could not refrain from emphasizing that more attention ought to be paid to the sciences in our schools [214]. Hoe komt het dat dit soort mammoet-onderzoek—ondanks de steeds weer blijkende levensgrote gebreken—door blijft gaan? Eén van de redenen is het ontzag voor grote getallen, computers, databanken; een andere dat het gemakkelijker is om miljoenen dan om duizenden voor onderzoek los te krijgen. Het opvoedkundig bezwaar van dit soort onderzoek is het aankweken van zorgeloosheid bij de onderzoeker die geheel op de grote getallen gaat vertrouwen, en de uitnodiging aan het publiek tot het zonder kritiek aanvaarden van met deskundig gebaar geponeerde cijfers. (How is it possible that this kind of mammoth research—despite its life-size deficiencies that keep turning up—continues? One of the reasons is the awe that averages, computers and databases inspire; another is that it is easier to get millions than thousands for research. The educational objection against this type of study is that it cultivates careless researchers who will start to rely completely on averages, and that it invites the public to accept without question the figures that are presented with an expert gesture.) [215]

Freudenthal considered it detrimental that in the planning of the study no efforts whatsoever were made to gather and process information on how and why the curriculum in the sciences was structured the way it wasWhat in the end did happen

9.6 Enfant Terrible

337

was the well-known procedure with empty ‘education barrels’—with all the associated consequences for teaching the disciplines: Ik beklemtoon het nog eens: ik trek de bekwaamheid van de Nederlandse projectleiding niet in twijfel. Het onderzoek is bepaald niet slechter dan wat anders en elders op dit gebied wordt gepresteerd. Wat fout is, is het idee van dit soort onderwijskunde dat de Duitsers ‘inhaltneutral’ noemen—het werken in holle vormen, die dan maar met vakinhouden moeten worden gevuld. Dit systeem veroordeelt op zichzelf bekwame onderzoekers tot werk waarvoor ze niet opgeleid en niet berekend zijn. Met schouderophalen aan de vakkundige kant komen we niet verder. Dit oordeel moest een keer—en dan gedocumenteerd— uitgesproken worden. (I emphasize once again: I do not doubt the expertise of the Dutch project leaders. The study is certainly no worse than that presented by others and in other places in this field. Wrong with this kind of science of teaching is what the Germans call ‘inhaltneutral’—working in empty vessels which in the end have to be filled with disciplinary content. This system condemns researchers who are all in all competent, to do work for which they have not been trained and to which they are not equal. We will not get any further if we keep shrugging at the disciplinary side. This verdict just had to be announced—and then supported with evidence.) [216]

A well-chosen example of such an empty vessel which Freudenthal often used was the taxonomy of Bloom as described in Taxonomy of educational objectives. The classification of educational goals (1956) [217]. This study, which was particularly popular in the United States in the 1970s, was used as a tool to develop curricula in circles of educationalists. Bloom’s taxonomy had been developed as an aid in determining and rating exam results, in particular in the humanities, to categorize learning objectives and order them hierarchically. Its application was in Freudenthal’s words “a fad totally unsuitable for mathematics and physics that only tickles the funnybone” [218]. In his book Weeding and Sowing Freudenthal elaborately explained why such a classification would not work: the fact that aspects such as ‘observing’ and ‘experimenting’ were missing on its own made the taxonomy worthless for science [219]. In addition an important aspect was overlooked: the skill to be able to do tests was lacking in the schemes. To mention this was a taboo, but according to Freudenthal it could not be wiped out when the value of such systems was at stake. The worst thing was that in such systems the stand-alone mathematical problems were the starting point without stating which place they took up in the learning process. And that was to Freudenthal exactly the essence: learning processes and thus the subject matter ought to be the starting point. In his lecture ‘Taxonomieën—waarvoor?’ (‘Taxonomies—to what purpose?’) he made it clear in no uncertain terms how little he saw in Bloom’s taxonomy and in the science of teaching it supported: ‘Taxonomy’ is een van die modes, waaraan iedereen in de onderwijskunde verplicht wordt aan [sic] mee te doen. ‘Taxonomy’ is maar een symptoom. Het euvel is gebrek aan wetenschappelijk houvast. In de natuurwetenschappen is het ook begonnen met indelen, bij voorkeur volgens heilige getallen. Dit stadium is in de onderwijskunde nog niet overwonnen. (“Taxonomy’ is one of those fashionable things in which everybody who is involved in the science of teaching is obliged to participate. “Taxonomy’ is only a symptom. The shortcoming is that it has no scientific footing. In science it has also started with the classification, preferably according to holy numbers. In the science of teaching this stage has not yet been superseded.) [220]

338


Bloom’s taxonomy was also misused in IEA’s study to be able to validate the test for want of content measures. Freudenthal explained: The purpose of Bloom’s categories in this context was the following: tests require a validation ritual; if, as in the present case (among numerous others) no yardsticks for validation of the contents of the instruments are available, one can use as an object of validation the distribution over Bloom’s categories, which in fact as a formal criterion is more attractive to psychometrists than subject matter and educational context, and is more intelligible too. […] The attempts at validation had entirely failed, but since they had taken place, the instrument was pronounced valid. If rituals are indispensable, would it not be time to replace the Bloom categories ritual by another where for instance committee and experts unite in a silent prayer? [221]

Freudental informed the Minister van Onderwijs en Wetenschappen in a letter about the IEA study that such mala fide research was anything but unique. In his experiences with the Stichting voor Onderzoek van het Onderwijs (SVO, the Foundation for Research of Education) he had also noticed such failures more than once [222]. For opportunistic reasons he did not always make this public: Ik ken tussen de tien en twintig projecten van SVO van iets naderbij—er is er geen een onder, waarop niet ernstige aanmerkingen zijn te maken—aanmerkingen t.a.v. inhoud of t.a.v. de behandeling door SVO. Dit zou een heel toevallig ongunstige steekproef kunnen zijn, maar daar geloof ik niet in. Ik heb tot nu toe over mijn ervaringen in het onderwijskundige wereldje in bredere kring gezwegen. Ik deed dit uit opportunisme. Ik wilde door misstanden in de bestaande structuur aan te vallen niet mijn instituut, het IOWO, in gevaar brengen. Ik heb wel over de verantwoordelijkheid die ik met dit zwijgen op me laadde, wakker gelegen. Tot mijn verschoning mag ik eraan toevoegen, dat ik mijn bezwaren wel intern kenbaar heb gemaakt—zonder succes. (I am familiar in some detail with between ten and twenty SVO projects—and there is not one among them with which one cannot find faults—faults with respect to its content or with respect to its treatment by the SVO. This could be a very coincidentally unfavourable sample, but I do not think so. Until now I have remained silent about my experiences in the broader circles of the educational world. I did so because of opportunistic reasons. I did not want to jeopardize my own institute, the IOWO, by attacking abuses in the existing structure. But I have lost some sleep over the responsibility that I took on myself by remaining silent. By way of an excuse I may add that I did make my objections known internally—without success.) [223]

Freudenthal’s criticism on the SVO activities was almost always related to the fact that experts in the disciplines had not been asked for advice enough: SVO worked with empty vessels. It was remarkable that Freudenthal—who at first sight never seemed afraid to call a spade a spade—apparently weighed his words and certainly pondered on their consequences when he considered venting his criticism or to refrain from doing so. Years later, in a letter to secretary of state of O&W, Mrs. Ginjaar-Maas, Freudenthal expressed his criticism on the CITO test (a test at the end of the primary education to determine which secondary education was appropriate for a pupil). It showed how cautious he still was in expressing his criticism of educationalists: Ik hoop dat U met deze uiteenzetting iets kunt doen. Men hoeft niet te weten uit welke koker de pijl komt. Voor onderwijskundigen ben ik toch het zwarte schaap en dat komt de zaak niet ten goede. (I hope this explanation is useful to you. There is no need to make public whose bright idea this is. In the eyes of educationalists I am the black sheep anyway and this does not help the case.) [224]

9.6 Enfant Terrible

339

Despite the feeling that his criticism did not have much effect in the world of education, Freudenthal did not regret it. In Schrijf dat op, Hans he looked back: Ik heb me veel met onderwijs meer in het algemeen bemoeid, maar nooit met onderwijs als leeg vat, door anderen te vullen, zoals sommige psychologen en algemeen onderwijskundigen het opvatten, die in en door het lawaai wanneer je op dat vat trommelt, de meeste zelfbevrediging scheppen en de meeste aandacht trekken. Of veeleer: voor zover ik mij wel met zoiets heb bemoeid, was het om onwetendheid, charlatanerie en boerenbedrog aan de kaak te stellen. […] Ik heb ook getrommeld, maar niet op lege vaten, ik heb de trom geroerd om voor mijn ideeën te werven, in het onderwijs en anderszins. Ja dat is het, ik ben een tamboer geweest, ik heb de trom geroerd om mensen wakker te maken. […] Nee, ik was geen goed tamboer. Een goede tamboer gaat niet diep in de nacht, maar in de ochtend trommelen—als het tijd is om op te staan. Maar denk niet dat ik er spijt van heb. Een enfant terrible hoort trots op zijn ondeugden te zijn. (I have been active in the field of education in general, but never with education represented as an empty vessel, to be filled by others, as in the view of some psychologists and general educationalists, who in and because of the noise when they drum on that vessel create the most self-gratification and attract the most attention. Or rather: as far as I have interfered with things like that, it was to expose ignorance, charlatanism and humbug.[…] I have drummed too, but not on empty vessels, I have beaten the drum to attract attention to my ideas, in the education world and elsewhere. Yes, that is it, I have been a drummer, I have beaten the drum to awake people. […] No, I was not a good drummer. A good drummer does not drum in the middle of the night, but early in the morning—when it is time to rise. But do not think that I regret it. An enfant terrible is supposed to be proud of his mischief.) [225]

He continued drumming, in small affairs, in greater affairs, in public or internally, indefatigable and always with the hope and the expectation to produce an effect, sooner or later. In the last of three articles on the issue on nomenclature he wrote Het is nog niet, het hoofd in de schoot leggen. Zolang er leven is, is er hoop en zolang er fouten worden gemaakt, is er rood potlood. (It is not yet surrendering. While there is life, there is hope and while mistakes are made they can be blue-pencilled.) [226]

9.6.3 Freudenthal on Piaget: Admiration and Merciless Criticism If there was a single subject that appeared to be endlessly criticized by Freudenthal, it would have been the work of the Swiss psychologist Piaget. Or a bit more precisely: the use of Piaget’s work in the development of mathematics education. During the war when he studied the literature Freudenthal had not yet read any of Piaget’s studies, but he had read an introduction to his work by C. Philippi-Siewertsz van Reesema [227]. It was not until 1957 that Freudenthal first mentioned Piaget’s studies in an article: All will admit that Piaget’s research is highly interesting. But it is quite another thing to apply his results to teaching mathematics, firstly because Piaget’s mathematical background has been rather weak, but mainly because Piaget’s approach hardly reflects the teaching situation of the classroom, but the rather unusual laboratory situation of the psychologist. Mathematical teaching theory can be furthered by mathematical teachers who are able mathematicians and able educators. [228]

340


A few years later Freudenthal made a passing reference to Piaget when he noted that a technique to search for protocols of learning processes was lacking. Man könnte mir hier Piaget und seine Schule vorhalten. In der Tat haben sie den Lernprozess untersucht, aber denn in einem Milieu, das uns kaum interessiert: Das Lernen im Laboratorium, nicht in der Klasse. So kommt es, daß Piagets Erfahrungen von Schulmännern nicht bestätigt werden. Es kommt weiter hinzu, daß Piagets Schule von einem verschrobenen Bild der Mathematik ausgeht. (Here one could confront me with Piaget and his school. And indeed they have investigated the learning process, but they did it in an environment that is hardly interesting to us: learning in the laboratory, not in the classroom That is the reason that Piaget’s experiences are not confirmed by pedagogues. And on top of that, Piaget’s school starts from a strange image of mathematics.) [229]

But Freudenthal’s criticism only became really sharp and extensive when from the 1970s onwards people used Piaget and his experiences to justify the introduction of New Math. The consequences this could have for the development of mathematics education were to Freudenthal ‘weeds’ in their worst form. In Mathematics as an educational task he felt the need to criticize Piaget to such an extent that he devoted a complete appendix to it [230]. Who was Piaget and why were his ideas so dangerous in Freudenthal’s view for the developments in the mathematics education? 9.6.3.1 A Swiss Developmental Psychologist Originally a Swiss biologist, Jean Piaget (1896–1980), who gathered followers particularly after the Second World War, developed into a developmental psychologist in the early part of the Twentieth century [231]. Piaget published a large number of books and articles in which he was seeking how knowledge was produced. He divided the knowledge a child gathered on the world around him into aspects such as logic, space, time, chance and mathematics. His early publications include many observations of his own children. Later on he concentrated on more formally methodologically constructed experiments with various children, an approach that would later become known as ‘Piaget’s school’. Pivotal in Piaget’s ideas was the fact that there were phases in the development that occurred in all children in a certain order. The environment of the child provided the material and the experiences from which the child started to think. These experiences were actively processed in thinking frames or ‘schème’ as Piaget called them—each with its own specific logical structure. There was a continuous interaction between the experiences of the child and the thinking strategies it employed at that moment. Assimilation (fitting the reality in the thinking frame) and accommodation (adapting the thinking frames to the new experiences and observations) continuously alternated with each other (adaptation). According to Piaget the order of the thinking strategies a child passed through during his development was fixed in an innate logical pattern. The logic of the thinking frame that a child employed at a certain moment determined how it dealt with reality.

9.6 Enfant Terrible

341

As a developmental psychologist Piaget could be placed on the constructivism camp. This term expresses how he emphasized children’s reinvention of ideas and logical ways of thinking, instead of dishing up self-contained ideas to them. According to Piaget in their development children needed to acquire new ideas actively. This more general aspect of Piaget’s developmental theory in itself was in line with Freudenthal’s own ideas. But in Piaget’s theory it was subsequently complemented by a specific logic which defined the stage of development in every child. This aspect was expressed using the term genetic epistemology in the broader characterization of Piaget’s theories. In an interview with Vredenduin, Freudenthal said of Piaget’s view on the development of a child: De hele binding van verschillende verschijnselen aan een leeftijdsperiode, dat schijnt me nu weer onjuist. […] Het is een psychologisch onderzoek en hij [Piaget] denkt dat bepaalde door hem logisch geconstrueerde ontwikkelingen daar in zitten. Er zit natuurlijk ook een ontwikkeling in, maar dat je die logisch kunt construeren, dat geloof ik niet.[…] Hij heeft een bepaald epistemologisch beeld van hoe kennis is opgebouwd, gestructureerd is en dienovereenkomstig stelt hij zich de ontwikkeling voor. […] Uit de mathematische structuur heeft hij een epistemologie afgeleid, die zeker niet ontwikkelingstheoretisch is. (That bond between different phenomena and age, that does not seem correct to me at all. […] It is a psychological study and he [Piaget] thinks that certain developments he constructed in a logical way can be found in it. There is of course a development but I do not believe that you can construct that in a logical way […] He has constructed a certain epistemological image of the way in which knowledge is produced and he imagines the development accordingly. […] From the mathematical structure he has derived an epistemological one, which is certainly not in accordance with the developmental theory.) [232]

In Piaget’s work there was a great deal about specific mathematical aspects of the child’s development to be found [233]. This made his findings even more attractive for use in didactics and education of mathematics. Van Hiele, for instance, had studied Piaget’s work during his doctorate and had been inspired by it when he developed his level theory. Later he would publish, among other things, a study on Piaget’s psychology in relation to mathematics education in the series Acta Paedagogica Ultrajectina of the Paedagogisch Instituut van de Universiteit Utrecht (the Pedagogic Institute of the Utrecht University). This series contained more studies on Piaget’s work such as that of A.H. Syswerda ( De ruimtevoorstelling bij het kind volgens J. Piaget en B. Inhelder, (The representation of space of the child according to J. Piaget and B. Inhelder)) and Bunt ( The development of the ideas of number and quantity according to Piaget) [234]. Van Hiele’s sharpest criticism of Piaget’s work was that in his description of the jumps to a higher level he assumed a specific genetic maturation process at a certain period. Based on his didactical observations Van Hiele’s view was that these jumps did not occur in the genetic process but were discontinuities in the learning process. He also pointed out that, based on Piaget’s findings, it was not correct to develop mathematics education according to the structure of mathematical logic—such as the logical structure he descried in the cognitive development of the child. In the circles of general didacticians and pedagogues Piaget’s theories were greatly admired but at the same time also criticized to a certain extent [235]. This was also evident from the preface to Van Hiele’s study, written by Langeveld, in

342


which he spoke of the publications that had appeared on Piaget’s work in the institute over the previous years. It will appear from these publications that we are critical of several of Piaget’s theories, methods and conclusions in many respects. This, however, does not interfere with our deep respect for this scientific worker who devoted the greater part of his life to genetic psychology. [236]

With the advent of New Math at the end of the 1950s its supporters accepted Piaget’s theories with open arms as an implicit psychological approval of the decision to start from the logical structure in mathematics education. In Belgium, for example, interest for Piaget’s work was great, partly as a result of his membership of the CIEAEM in which Papy was also active [237]. In Schrijf dat op, Hans Freudenthal described it as “the blessing of a psychologist, an epistemological psychologist who took it for granted that mathematics developed according to epistemological lines in the individual, true enough” [238]. This was the motivation for Freudenthal to commence critical action. 9.6.3.2 About Flimsy Experiments and Pseudo Mathematics Freudenthal’s preoccupation with Piaget and his work went so deep that he returned to it and further expounded his criticism even in his final book of 1991 [239]. In Revisiting mathematics education de did not blame Piaget for unintentionally contributing to the impact of New Math with his theories. Just as 20 years before in Mathematics as an educational task he put the responsibility on the mathematicians themselves. He very much blamed the mathematicians that they had never expressed their criticism on Piaget’s work loudly enough (in particular on what Freudenthal called the pseudo-mathematical part): A mathematician should be able to indicate in Piaget’s problems what is adequate and what is distorted, and with a bit of common sense he could be the person best placed to uncover the numerous errors in Piaget’s experiments and interpretations. [240]

Freudenthal himself took this responsibility as a mathematician in an extensive appendix in Mathematics as an educational task entitled: ‘Piaget and the Piaget School’s investigations on the development of mathematical notions’ [241]. In his preface he wrote: To be honest I should say that I feel there is no need to embellish low-key education using high-brow psychology, in particular if the cited literature is far removed from educational preoccupations. If others prefer this procedure, then indeed, I feel the need to oppose it. Misusing Piaget’s name has become quite a habit in didactical literature. This led me to discuss in passing, and finally in a more connected form in the Appendix, what Piaget’s investigations could mean for mathematical education. [242]

Freudenthal deemed a more detailed observation in this form necessary to further support his own “somewhat summary criticism” on Piaget’s work from his articles. Nevertheless he also felt it incumbent on him to express his appreciation for Piaget’s work:

9.6 Enfant Terrible

343

Before going into this I would like to stress the wealth of ideas in his work, his originality, not to say genius, while not excluding the negative undertones which often reverberates this word. [243]

There were several reasons for Freudenthal’s criticism of Piaget. First of all there was the obvious difference in Freudenthal’s and Piaget’s approaches. At the beginning of his career Piaget did report on observing his own children, but later in his more well-known studies he relied on what Freudenthal called laboratory situations: an artificial set-up of an experiment, subject and experiment leader who accomplished well-defined experiments. Of course this contrasted sharply with the casual observations and the relaxed walks of Freudenthal and his grandson Bastiaan. But it also had very little in common with teaching situations in a class room, according to Freudenthal. To Freudenthal the experiments in Piaget’s studies epitomized the desk science which had lost all contact with reality and which he hated so much. The only time that Freudenthal consciously conducted experiments was a series of intentional and therefore more or less deliberate physics experiments with Bastiaan, but in these he allowed the boy to do and plan as much as possible by himself. He wrote about these experiments—which he called “Bastiaan’s lab”: Het is niet het Lab van de ontwikkelingspsycholoog, waar de kinderen—in ’t Engels—subject, d.w.z. onderwerp of onderdaan, zijn, of—in ’t Nederlands—proefpersoon. Bastiaan doet proeven. (It is not the lab of a development psychologist where children are subjects, in the sense of undergoing an experience. Bastiaan conducts experiments.) [244]

Freudenthal’s suspicion of Piaget’s laboratory situations was based on his own experience with observations. Once he saw clearly how the theory acquired in experiments in the laboratory was refuted by the casual practice. At stake was the term conservation, a term with which toddlers were not yet familiar according to Piaget. Freudenthal recounted how he and Bastiaan walked along a canal that was in the process of being widened. Bastiaan excitedly remarked that water would have to be added to replace the dredged ground—Freudenthal could hardly believe what he heard: Het hier gebeurde is zo verbazend, omdat het een teken is van wat Piaget conservatie noemt, ver beneden de leeftijd die volgens de psychologische laboratoriumproeven hiervoor bepaald is—ontdekkingen in het vrije veld contra de dwangbuis van het laboratorium. Meermalen in de volgende maanden geeft hij—met de dezelfde opwinding—blijk van het ontdekken van conservatieverschijnselen, wanneer iets overgegoten wordt van een breed in een smal vat, wanneer bij het onderdompelen van zijn hand het water in een weckglas stijgt. Wanbegrip ten aanzien van conservatie doet zich, volgens mijn ervaringen, bij kinderen alleen in kunstmatige laboratoriumsituaties voor. (What happened here is so amazing since it is a sign of what Piaget calls conservation, far below the age psychological laboratory experiments state is the specific age for it—discoveries in the open countryside as opposed to the straitjacket of the laboratory. In the months thereafter—with the same excitement— he displays the discovery of conservation phenomena, when something is poured from a narrow into a wide vessel, when the water level is raised once he immerses his hand in a jar. Misunderstanding conservation occurs in my experience only when children are in artificial laboratory situations.) [245]

But the mistakes Piaget made in conducting the experiments were to Freudenthal worse and more serious than their artificial set-up: not only mathematical mistakes,

344


but words too were misinterpreted and this not uncommonly led to erroneous conclusions. By translating a chapter from Piaget’s La répresentation de l’espace chez l’enfant (The representation of space of the child) from 1948 in his appendix, Freudenthal sketched Piaget’s mathematical mistakes. He warned the reader that the translation was not very smooth, as he was not a “professional translator who is obliged to translate things he does not understand”, and with this already indicated what the level of the article was [246]. Next he translated, adding dozens of footnotes with explanations and questions, Piaget’s almost incomprehensible discussion of his eight “infralogical operations” which determined the Euclidian space as he perceived it. After that Freudenthal concluded: After our short analysis this extract speaks so loudly for itself that there is no need to add criticism. It is, however, not too far-fetched to ask whether such ideas have influenced Piaget’s work in a decisive way. They did influence it, but I doubt whether they were in any way decisive influences. [247]

Despite Piaget’s repeatedly wrong use of mathematical terms, his mathematically ramshackle theories and ditto experiment set-up, Freudenthal apparently deemed it necessary to devote much attention to this. He did this because Piaget’s findings— despite all the wrongs—were still mathematically interpreted and had their influence on didactics of mathematics and on the content of mathematics education, where they acted as the psychological justification and foundation for taking the mathematical structure as the starting point. The errors in the linguistic interpretations of the test results were even more of a thorn in Freudenthal’s flesh than the mathematical errors. It was of vital importance to Freudenthal that subjects in Piaget’s test set-ups had understood the question linguistically, but the experimenter never ascertained this during the tests [248]. In his appendix he described a conservation experiment from Piaget’s book La génèse des structures logiques élémentaires (The origin of logical elementary structures) indicated by Freudenthal with N: N starts with investigating whether and to which degree children understand that pouring fluid from one vessel into another or into a number of vessels does not influence the quantity; the subjects are asked whether this is more or less than that or as much as that. According to the author’s intention the question aims at the quantity rather than at the number of vessels or the height of the level of the fluid. This fact, however, is not included in the information provided to the children; the experimenters’ questions are sometimes even misleading, to which contributes the linguistic circumstance that in French both “is this more?” and “are there more?” can be translated by the same phrase “y en a-t-il plus?” [249]

When Piaget concluded that a child of a certain age did not have a command of a certain concept, according to Freudenthal he proved nothing more than the fact that the child had not understood the question. According to him, Piaget’s results did not relate to the cognitive development but to the linguistic development. Often it was a matter of communication problems between the experimenter and the children: an answer was not understood, misinterpreted or simply erroneously considered wrong. The only things that were tested according to Freudenthal were the communication abilities.

9.6 Enfant Terrible

345

There is no real interpretation of mathematics to which Piaget’s research can be related. Though the subjects of his experiments are set mathematical tasks, he almost never tests whether they understood the language in which the task was formulated. It is rather the linguistic than the mathematical content or the task which is object of the investigation. In other cases, rather than testing the reaction of the child to a mathematical problem, the leader of the experiment tests how the child manages to disentangle irrelevant complications of the problem, or whether you can cheat the child by misleading suggestions. A still sadder sight is the attempt by the experiment leader to induce the experiment subject, who might hesitate between a right and a wrong answer, to choose the wrong one. Even worse are the manifest blunders of the experiment leader who marks correct answers as wrong. [250]

Quite apart from the mathematical errors, the faulty set-up of the experiments and the misinterpretations of its results, Freudenthal criticized Piaget’s work in other ways. However, he did not blame Piaget himself for that, but the mathematicians who got carried away with Piaget’s work when new teaching material was introduced, even omitting to check the mathematics Piaget used [251]. In an epilogue to the book Zoals een kind het ziet. Wegwijzer in de theorieën van Piaget (As a child sees it. Handbook for Piaget’s theories) Van Hiele wrote a carefully mildly formulated, but still essential critique on Piaget’s work, which on the whole matched Freudenthal’s [252]. Subsequently, Freudenthal wrote in a review of this book that he “would not want to miss … the excellent epilogue” [253]. According to Freudenthal it incited to think, to think … uiteraard ook over dit boekje, en dan niet door na te praten, maar kritisch. Hij [de lezer] mag dit best riskeren, want van vrijwel alle proeven die hij hier vindt, is de ondeugdelijkheid al erkend, van sommige door Piaget zelf. Maar zelf erachter komen is meer waard. (… indeed also about this book, and not by parroting it but by thinking about it in a critical way. He [the reader] can easily risk this since the flimsiness of practically all tests found in it has already been acknowledged, sometimes by Piaget himself. But if he finds this out himself it has more value.) [254]

For the essence of Piaget’s work—the insight that the knowledge and the experience of a child is essentially different from those of the (teaching) adult—was something that Freudenthal could also endorse of course. Years later during an interview with Piet Vredenduin about ‘Portrayal of mankind’ he admitted that his and Piaget’s image of man in itself probably would not be that different [255]. The big difference was found in the starting points. Piaget poured the knowledge man acquired into a (semi-mathematical) structure and therewith deemed it proven that this knowledge originated in that same structure. Freudenthal wrote about this way of thinking: Je maakt die kennis van die mens los, dan beschouw je de kennis die klaar gekomen is, en die orden je op de een of andere manier in een lexicon of in een wiskundige zin. Zo zit die kennis in elkaar. Maar zo is hij niet tot stand gekomen. Niet in de geschiedenis, en komt ook bij een individu niet zo tot stand, integendeel, het gaat net omgekeerd. […] Ik zie juist hoe langzaam het gaat voor je die kennis kunt losmaken van de manier waarop je hem zelf verworven hebt. Dat wil zeggen een deel van de kennis blijft op een onbewuste manier verworven en er is nog heel wat vereist om die kennis daarvan los te maken. (You dissociate that knowledge from the man, then you consider the knowledge that is ready, and order that in whatever manner in a lexicon or in a mathematical sense. That is how that knowledge is structured. But that is not the way it originated. Not in history and also not in an individual, on the contrary, it is just the other way round. […] What I see is how slowly knowledge

346


is dissociated from the manner in which you acquired it yourself. That is, that a part of the knowledge remains acquired unconsciously and it takes quite a lot to disassociate that knowledge from the way it was acquired.) [256]

Making that knowledge from unconscious conscious, that was the task one was faced with in the organization of education. The quest for education could not be considered separate from the content of a discipline—locally, at the level of the learning child. His aversion to the blind acceptance of psychological research at the cost of the professional knowledge of the mathematicians meant that Freudenthal returned to the source: observing the child. Freudenthal distinguished two important activities to be able to arrive at the development of a theory about mathematics education: careful and continuous observation of the learning process and analysing it [257]. It was better to start with the day-to-day reality and look around you than unquestioningly accept high-flown theories with open arms, as was his argumentation, since: Als we weinig van cognitieve ontwikkeling afweten, zou de reden ervan niet kunnen zijn, dat we het tot nu toe te ver, te hoog hebben gezocht, dat we over wat ons onbelangrijk leek, argeloos heen zijn gestapt? (If we know little of cognitive development, would the reason for that not be that until now we have looked for it too far deep, too high, and that we have been overlooking things that seemed unimportant to us?) [258]

9.7 The Task for the Future In 1969 Freudenthal looked back over the past years. He characterized the publication of his article ‘Instruction of modern mathematics or modern instruction of mathematics?’ of 1963 as “a solitary effort to defend the priority of pedagogy and the art of teaching over programs” [259]. “Nonconformist”, was how he characterized the article [260]. Since that moment a great deal had changed: seriously more attention was given to pedagogics and didactics in mathematics education. This was a more and more commonly accepted situation and attention was still increasing, not in the least because of the efforts of Freudenthal himself. These changes and the increase—internationally too—of the activities in the field of mathematics education made it a turbulent, but because of that also an inspiring, period. It is difficult to separate the jumps in Freudenthal’s own learning process from this background. The enormous flood of activities both nationally and internationally caused countless initiatives: ranging from congresses and conferences to the observation activities of the IOWO (Fig. 9.9). In all these ways just as many fruitful ideas originated with Freudenthal, as came to his attention. Ideas to criticize—as he did with Piaget—but also ideas to build on constructively. And now—more than in any other period—Freudenthal’s unbridled energy and indefatigable efforts are striking. He mobilized his good name and reputation for the various social functions he performed: starting Educational Studies in Mathematics and organizing the first international congress for didactics of mathematics are prime examples. He immediately jumped on existing initiatives and subsequently

9.7 The Task for the Future

347

Fig. 9.9 Freudenthal during ICME-1980 in Berkeley (1980)

gave those an unprecedented acceleration and an extra stimulus, as he did interacting with WISKOBAS and the IOWO. He kept close track of topical developments and immediately absorbed everything. In developing his ideas he assimilated and integrated the ideas of others super-fast. One important element in this stood out: (re)discovering observations as a source for the development of mathematics education. Against the background of the importance of useful, honest and sound mathematics education for all, at all levels, observing kept coming back as a footing and a guideline. Looking for paradigms was a logical consequence of this for Freudenthal. When he, over a period of a few weeks, patiently helped a granddaughter discover the principle of duplication of a square, the girl asked “How can you teach mathematics in this way to a whole class?” She was right to ask this question. School teachers cannot afford to delay solutions patiently for hours or weeks. They have to develop methods to teach 250.000 [sic] children a year mathematics in the same way as a girl is taught by a granddaddy who is a university professor in mathematics. [261]

348


And with that he summarized the assignment for the future: develop methods to make a class rediscover mathematics in the same way a grandfather did this with his grandchildren. The IOWO was made for this assignment. Freudenthal would in this tradition still perform a lot of work in the field of education, not only within the scope of IOWO but also outside it. “A mathematician […] who has always and in the past years practically exclusively been interested in education” was how he described himself in 1973 [262]. The scales of his interest and work had permanently tipped to the side of education: Freudenthal—the unapproachable mathematics professor, the senile grandfather, the indefatigable critic—was mathematics education.

References 1. Freudenthal about himself and his activities in the field of mathematics education in the 1970s and 1980s; Freudenthal, Schrijf dat op, Hans 329. 2. Freudenthal, Schrijf dat op, Hans 328. 3. On the difference between teaching modern mathematics and modernizing mathematics education, he wrote among other things, the following articles: Hans Freudenthal, ‘Why to teach mathematics so as to be useful’, Educational Studies in Mathematics 1–2 (1968) 3–8; Freudenthal, ‘Modernisering Leerplan Wiskunde. Answer to Prof. de Bruyn’, Euclides 43 (1968) 321–322; Freudenthal, ‘Les tendances nouvelles de l’enseignement mathématique’, Revue de l’Enseignement Supérieur 46–47 (1969) 23–29; Freudenthal, ‘Introduction’, New Trends in Mathematics Teaching ii (1970); Hans Freudenthal, ‘Die neuen Tendenzen im MathematikUnterricht’, Neue Sammlung 11 (1971), 146–153; Hans Freudenthal, ‘De Middenschool’, Rekenschap 20 (1973) 1 4. Freudenthal, ‘Recent tendencies in the teaching of mathematics’, in: W.E. Lamon, ed., Learning and the nature of mathematics (Chicago 1972) 11–18, q.v. 13–14; identical to: Freudenthal, ‘Les tendances nouvelles de l’enseignement mathématique’, Revue de l’enseignement supérieur 46–47 (1969) 23–29. 5. Euclides. Maandblad voor de Didactiek van de Wiskunde 43–50 (1967–1975). 6. G. Krooshof, ‘Euclides. Tijdschrift voor de didactiek van de wiskunde’, Euclides 45 (1969) 1. 7. Freudenthal to Howson, 1983; RANH, Freudenthal Papers, inv.nr. 1771, 38. 8. On (the establishment of) the IOWO, see paragraph 9.3. 9. For a glimpse behind the ‘IOWO scenes’ see for example also: De Moor, Van vormleer naar realistische meetkunde; Goffree e.a., eds., Honderd jaar wiskundeonderwijs. 10. Freudenthal, Mathematics as an educational task (Dordrecht 1973); Freudenthal, Mathematik als pädagogische Aufgabe I, II (Stuttgart 1973). 11. Some of the subjects in Mathematics as an educational task received extra emphasis (compared with Freudenthal’s articles), such as the attention he paid to the history of mathematics and the appendix he devoted to criticizing Piaget. See also paragraph 9.6.3. In a review of Van Streun’s thesis Heuristisch wiskunde-onderwijs from 1989 Freudenthal wrote about his own Mathematics as an educational task: “[Mathematics as an educational task] is little more than a collection of articles published from the mid-1950s to the end of the 1960s which are thus no longer compulsory literature after 1973.” Hans Freudenthal, ‘Boekbesprekingen’, Nederlands tijdschrift voor opvoeding, vorming en onderwijs 5 (1989) 302– 304, q.v. 302. 12. On the establishment of the international journal for didactics of mathematics, see paragraph 9.2. 13. This colloquium was organized commissioned by the ICMI; RANH, Hans Freudenthal Papers, inv.nr. 1831.

References

349

14. Freudenthal to Howson, 1983; RANH, Hans Freudenthal Papers, inv.nr. 38. 15. This letter was sent on 13 April 1965 to the following people: Prof. M. Beberman (United States), E. Castelnuovo (Italy), L. Félix (France), dr. A.Z. Krygovska (Poland), Prof. W. Servais (Belgium), Prof. P. Suppes (United States), Prof. B. Thwaites (Great Britain) and Prof. A. Wittenberg (Canada); RANH, Hans Freudenthal Papers, inv.nr. 1785. 16. Freudenthal to Hilton, 10 October 1976; RANH, Hans Freudenthal Papers, inv.nr. 1786. 17. Correspondence between Hilton and Freudenthal, October-December 1967; RANH, Hans Freudenthal Papers, inv.nr. 1786. The following names were added as suggested by Freudenthal: dr. A. Engel (Germany), Prof. A.M. Gleason (United States), Prof. H.B. Griffiths (Great Britain), C. Hope (Great Britain), dr. H.O. Pollak (United States), Prof. A. Revuz (France), Prof. S. Sobolev (Soviet-Union,) Prof. dr. H.G. Steiner (West-Germany) and Prof. I.M. Yaglom (Soviet-Union). Hilton suggested the following persons: dr. D.K. Abbiw-Jackson (Ghana), Prof. E.G. Begle (United States), Prof. G. Choquet (France), Prof. E.B. Dynkin (Soviet-Union) and Prof. W.T. Martin (United States). 18. Freudenthal to Hilton, 20 October 1967; RANH, Hans Freudenthal Papers, inv.nr. 1786. During a discussion at a conference of Belgian teachers in 1961 Freudenthal was offended by Papy to such an extent that he received a letter with apologies from the organizers after the conference. According to Freudenthal Papy would have called out: “Even Professor Freudenthal [according to Freudenthal ‘-with the undertone of that gaga-’] confesses to teach linear algebra”. Freudenthal disliked Papy so much, that he only started attending the CIAEM meetings after Papy resigned as chairman of the CIAEM in 1970. This situation would improve a bit later on. The fact that Freudenthal’s relation with Servais—despite the enormous differences of opinion with respect to mathematics education—was considerably better was evidenced by his statement in that same letter to Howson: “Never in my life, with people I met, was friendship and profound disagreement more closely knitted than in my relation with Servais”; Freudenthal to Howson, 1983; RANH, Hans Freudenthal Papers, inv.nr. 38. 19. Freudenthal to Hilton, 20 October 1967; RANH, Hans Freudenthal Papers, inv.nr. 1786. 20. Freudenthal to Hilton, 1 October 1967; RANH, Hans Freudenthal Papers, inv.nr. 1786. 21. A ‘Journal of International Mathematical Education’ would be at stake here, the other journal was ‘Mathematical Education Abstracts’; RANH, Hans Freudenthal Papers, inv.nr. 1786. 22. Letter to Hilton, 9 October 1967; RANH, Hans Freudenthal Papers, inv.nr. 1786. 23. Ibid. 24. Educational Studies in Mathematics 1 (1968) 1–243. The ‘panel discussion’ was a literary reflection of the discussion about the aim of mathematics education by the panel consisting of Behnke, Hammersley, Krygovska, Pollak, Revuz, Servais and Sobolev, led by Freudenthal. 25. Correspondence with D. Reidel Publishing Company, the publisher of Educational Studies in Mathematics, review in: The New Era. The World Education Fellowship Journal 51 (1970), review in: Euclides 45 (1969); RANH, Hans Freudenthal Papers, inv.nr. 1789, 1790. 26. Book review by Joh. H. Wansink, Euclides 45 (1969) 38–39, q.v. 38. 27. Freudenthal to Howson, 1983; RANH, Hans Freudenthal Papers, inv.nr. 38. 28. Ibid. 29. Ibid. 30. RANH, Hans Freudenthal Papers, inv.nr. 1790. 31. Freudenthal, ‘Why to teach mathematics so as to be useful’, Educational Studies in Mathematics 1 (1968) 3–8, q.v. 4. 32. France and the United States, in particular, were well represented. An exception was a Polish contribution by Mrs. Krygovska. In later issues Russian articles were published more frequently; Educational Studies in Mathematics 1 (1968). 33. Freudenthal requested something like that in letters to among others Van Hiele (14 August 1968), Van Albada (15 August 1968), Revuz (29 October 1968), Van Dormolen (11 November 1968) and Vredenduin (11 November 1968); RANH, Hans Freudenthal Papers, inv.nrs. 1790–1794. 34. ESM-correspondence; RANH, Hans Freudenthal Papers, inv.nr. 1790. 35. Freudenthal to Alan Schoenfeld, 22 August 1978; RANH, Hans Freudenthal Papers, inv.nr. 1806.

350


36. Freudenthal to Pollak, 1 April 1977; RANH, Hans Freudenthal Papers, inv.nr. 1807. 37. Freudenthal to Fletcher, 5 November 1977; RANH, Hans Freudenthal Papers, inv.nr. 1807. 38. According to Freudenthal Pollak was the only editor who had asked for almost all manuscripts in his field for approval; RANH, Hans Freudenthal Papers, inv.nr. 1807. 39. Freudenthal to Pollak, 1 April 1977; RANH, Hans Freudenthal Papers, inv.nr. 1807. 40. Freudenthal to Fletcher, 5 November 1977; RANH, Hans Freudenthal Papers, inv.nr. 1807. 41. In 1991 Bishop would write the preface to Freudenthal’s posthumously published book; Freudenthal, Revisiting mathematics education. China lectures. 42. Freudenthal to Bishop, 25 November 1977; RANH, Hans Freudenthal Papers, inv.nr. 1807. 43. Educational Studies in Mathematics 1–6 (1968–1975). 44. E. Wijdeveld, ‘Omzien in verwondering’, Euclides (2003) 218–225. 45. The Commissie Organisatie Leerplan Ontwikkeling (COLO) was established a year later. 46. CMLW folder, 1968–1969; RANH, Hans Freudenthal Papers, inv.nr. 418, 1725. 47. RANH, Hans Freudenthal Papers, inv.nr. 1725; Wijdeveld e.a., ‘Van CMLW tot Freudenthal Instituut’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 355–374, q.v. 359. 48. Freudenthal in the lecture ‘The Institute on Developing mathematical Education (IOWO) in the Netherlands’, Argentina 1972; RANH, Hans Freudenthal Papers, inv.nr. 571. 49. The three national Pedagogic Centres are the APS (Algemeen Pedagogisch Studiecentrum), CPS (Centraal Pedagogisch Studiecentrum) and the KPC (Katholiek Pedagogisch Centrum). They support the education, advise schools and the government and organize courses, conferences and workshops. 50. Minutes of a plenary meeting of the CMLW, 6 June 1969; RANH, Hans Freudenthal Papers, inv.nr. 1725. 51. Minutes of the meeting of 10 September 1969; RANH, Hans Freudenthal Papers, inv.nrs. 418, 1725. 52. Freudenthal at a press conference of the CMLW, 22 October 1969. In a press conference of 12 November 1975 he mentioned the curricula developers in England and Italy as examples of developers of mathematics education “from and close to reality”; RANH, Hans Freudenthal Papers, inv.nr. 603, 1725. 53. RANH, Hans Freudenthal Papers, inv.nrs. 603, 1725. 54. Freudenthal, ‘Strategie der Unterrichtserneuerung in der Mathematik’, Beiträge zum Mathematikunterricht (1972) 41–45, q.v. 42. 55. Correspondence CMLW, 18 September 1970; RANH, Hans Freudenthal Papers, inv.nr. 1726. 56. By way of comparison, at the start in 1961 the budget was Fl 500,000,-; Wijdeveld et.al., ‘Van CMLW tot Freudenthal Instituut’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 355–374, q.v. 359. 57. A more extensive description of the realization of the IOWO can be found in: Wijdeveld e.a., ‘Van CMLW tot Freudenthal Instituut’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 355–374. 58. R.A. de Jong, Wiskobas in Methoden (Utrecht 1986) 11. 59. Wijdeveld employs another classification, in which he distinguishes three departments with respect to content: WISKOBAS, WISKIVON (for VWO, HAVO, MAVO, LBO, MBO, HBO and special subjects) and the department general management training and curriculum development; Wijdeveld et.al., ‘Van CMLW tot Freudenthal Instituut’, in: Goffree e.a., eds., Honderd jaar wiskundeonderwijs 355–374, q.v. 360; RANH, Hans Freudenthal Papers, inv. nr. 571. 60. This association was of course affirmed when the institute—after being discontinued in 1981—changed via the Utrecht university department ‘Onderzoek Wiskundeonderwijs en Onderwijscomputercentrum’ (OW & OC) into the ‘Freudenthal Instituut’ in 1991. 61. Mr.J.H. Grosheide on the occasion of the farewell party of Freudenthal as director of the IOWO on 14 August 1976; RANH, Hans Freudenthal Papers, inv.nr. 420. 62. Proposal for a discussion paper;RANH, Hans, Freudenthal Papers, inv.nr. 1738. 63. Freudenthal, ‘Strategie der Unterrichtserneuerung in der Mathematik’, Beiträge zum Mathematikunterricht (1972) 41–45, q.v. 42.

References

351

64. RANH, Hans Freudenthal Papers, inv.nr. 1738. 65. Annual report IOWO 1971/1972; RANH, Hans Freudenthal Papers, inv.nr. 418. 66. Freudenthal, ‘Strategie der Unterrichtserneuerung in der Mathematik’, Beiträge zum Mathematikunterricht (1972) 41–45, q.v. 44; RANH, Freudenthal Papers, inv.nr. 575. He used a similar text in, among other things, his lecture: ‘Leerplanontwikkeling in de wiskunde’ of 21 September 1973. 67. Freudenthal, ‘“Stenen voor brood” of “door een rijstebrijberg naar luilekkerland”’, in: Wiskobasbulletin 1975; RANH, Hans Freudenthal Papers, inv.nr. 1371. 68. Hand written note from Freudenthal to the IOWO staff members, undated; RANH, Hans Freudenthal Papers, inv.nr. 404. 69. Compare his attitude with respect to the student revolution, as described in paragraph 3.5. 70. Oral announcement of A. Treffers, 22 April 1999. 71. Freudenthal’s lecture for Amicitia Naturae Interpres on 21 September 1975; RANH, Hans Freudenthal Papers, inv.nr. 575. 72. Lecture at the occasion of the visit of the minister of Education, J. van Kemenade, to the IOWO, 27 November 1974; RANH, Hans Freudenthal Papers, inv.nr. 589. 73. Freudenthal to Van Hiele, 4 June 1987; RANH, Hans Freudenthal Papers, inv.nr. 37. 74. Correspondence CMLW-IOWO, 1975; RANH, Hans Freudenthal Papers, inv.nr. 1731. 75. This paragraph is to a large extent based on conversations with Van der Blij, Treffers and Grootendorst, spring 1999. 76. An example of that criticism is the one on the fairylands in which the WISKOBAS-projects were situated, see also paragraph 9.5.4; Freudenthal at a WISKOBAS conference in 1975; RANH, Hans Freudenthal Papers, inv.nr. 412. 77. A. Treffers, ‘WISKOBAS and Freudenthal—realistic mathematics education’, in: L. Streefland, ed., The legacy of Hans Freudenthal (Dordrecht 1993) 89–108, q.v. 106. 78. In his book Van vormleer naar realistische wiskunde De Moor discusses the results of a poll of ex-WISKOBAS employees, which concentrates on the work done for geometry education. There was a large difference, depending on the person, in how the influence of Freudenthal on the WISKOBAS project was experienced. Freudenthal’s contributions were remembered by most of the employees as being of mainly a micro-didactical nature; De Moor, Van vormleer naar realistische meetkunde 497. 79. Freudenthal, ‘Rekendidaktiek’ (unpublished 1944) 54–55; RANH, Hans Freudenthal Papers, inv.nr. 465. 80. He wrote sometimes, for example, about Tom’s number concept and he noted statements of Thijs and Didi.; notebook from 1943, pages 9, 16, 17, 35, 43, 54; RANH, Hans Freudenthal Papers, inv.nr. 440. 81. Freudenthal, ‘Rekendidactiek’ 4; RANH, Hans Freudenthal Papers, inv.nr. 465. 82. Ibid., 70–71. 83. Ibid. 84. Ibid. 85. Ibid. 86. See also paragraph 4.4. 87. Notebook ‘Rekendidaktiek’, 1943; RANH, Hans Freudenthal Papers, inv.nr. 440. 88. See also paragraph 5.1.1. 89. Freudenthal had always been intensively involved in observing in the experimental WISKOBAS and WISKIVON schools, but also in the Pedagogische Academies; Freudenthal, Schrijf dat op, Hans 359. 90. In his books Mathematics as an educational task and Appels en peren. Wiskunde en psychologie. Gebundelde opstellen he also wrote about his elder grandchildren; Freudenthal, Mathematics as an educational task; Hans Freudenthal, Appels en peren. Wiskunde en psychologie. Gebundelde opstellen (Apeldoorn 1984). 91. Lecture ‘The Institute on Developing Mathematical Education (IOWO) in the Netherlands’, 1972 Argentina, 9 pages; RANH, Hans Freudenthal Papers, inv.nr. 571.

352


92. In two little notebooks Freudenthal wrote about his adventures with his grandson over the period from August 1972 to September 1977; RANH, Hans Freudenthal Papers, inv.nrs. 446 and 447. 93. Observing and noting activities of and with children is a frequently occurring phenomenon with pedagogues such as W. Stern and O. Decroly. Decroly for example noted events and expressions of his little daughter at an age comparable to Bastiaan’s. Freudenthal was familiar with this work as a result of his studies during the war and noted in 1944 about this “Decroly does not find his means in formal material as Mont[essori] does, but in real life”. Kohnstamm also noted an observation of his grandchild from time to time; C. PhilippiSiewertsz van Reesema, Uit en over de werken van Prof.Dr. Ovide Decroly (Groningen 1931); Kohnstamm, Keur uit het didactisch werk van Prof.dr. Ph. Kohnstamm 90; Freudenthal’s notebook 1943; RANH, Hans Freudenthal Papers, inv.nr. 440. 94. Bastiaan van der Velden, ‘Between “Bastiaan ou de l’éducation” and “Bastiaan und die Detektive”’, Zentralblatt für Didaktik der Mathematik 32 (2000) 201–202, q.v. 201. 95. Little notebook with observations of Bastiaan and Monica, 1972–1976; RANH, Hans Freudenthal Papers, inv.nr. 446. This anecdote is also part of the article: Freudenthal, ‘Wandelingen met Bastiaan. Wereldverkennen van de straatstenen tot de maan’, Pedomorfose 25 (1975) 51–64, q.v. 58. 96. Freudenthal, ‘Wandelingen met Bastiaan. Wereldverkennen van de straatstenen tot de maan’, Pedomorfose 25 (1975) 51–64, q.v. 62. 97. This article dates from 1977/1978, composed from Jaarverslag over 1977, Provinciaals Utrechts Genootschap (1978) 8–18 and Weeding and sowing; Freudenthal, Appels en peren 101. 98. Freudenthal, ‘Cognitieve ontwikkeling—kinderen geobserveerd’; ibid. 99. Freudenthal, ‘Wandelingen met Bastiaan’, Pedomorfose 25 (1975) 51–64, Freudenthal, ‘Bastiaan’s lab’, Pedomorfose 30 (1976) 35–54, Freudenthal, ‘Bastiaan meet zijn wereld’, Pedomorfose 37 (1978) 62–68. 100. Uhlenbeck gave his reaction to an interview with Freudenthal in Resonans (1975–76) 18– 22, q.v. 20–21: E.M. Uhlenbeck, ‘Professor Freudenthals kleinzoontje verdedigd tegen zijn grootvader’, Forum der Letteren 16 (1975) 253–254; RANH, Hans Freudenthal Papers, inv.nr. 616. 101. E.M. Uhlenbeck, ‘Professor Freudenthals kleinzoontje verdedigd tegen zijn grootvader’, Forum der Letteren 16 (1975) 253–254, q.v. 254. 102. Freudenthal’s reaction was not printed in full in Forum der Letteren. But he was allowed to publish a short reaction, including the offer “should the reader wish he could be sent, without extra costs” the complete text; Freudenthal, ‘Commentaar’, Forum der Letteren 17 (1976) 263; RANH, Hans Freudenthal Papers, inv.nr. 616. 103. B. Pelupessy, ‘Enkele annotaties bij “Wandelingen met Bastiaan”’, Onderwijs in Natuurwetenschap. Een nieuwsbrief over de integratie van natuurwetenschappen in het basis- en voortgezet onderwijs 2 (1975) 17–20, q.v. 17. 104. Freudenthal, ‘Wandelingen met Bastiaan’, Pedomorfose 25 (1975) 51–64, q.v. 61. 105. Peluppessy, ‘Enkele annotaties bij “Wandelingen met Bastiaan”’, Onderwijs in Natuurwetenschap 2 (1975) 17–20, q.v. 19. 106. Freudenthal, ‘Annotaties bij annotaties, vragen bij vragen’, Onderwijs in Natuurwetenschap 2 (1975) 21–22, q.v. 22. 107. Ibid. 108. Ibid., 21. 109. RANH, Hans Freudenthal Papers, inv.nrs. 606, 618. 110. Comment of 20 April 1975 in the notebook with observations of Bastiaan and Monica, 1972–1976; RANH, Hans Freudenthal Papers, inv.nr. 446. Striking and characteristic is Freudenthal’s hypercorrect use of language when he speaks about the “biel” and the accompanying plural “biels” (The Dutch word ‘biels’ (a singular Dutch word, which means sleeper) is in more popular language often used as the singular form ‘biel’). The author wishes to thank Prof.dr. A. Treffers for making this remark, 9 November 2001).

References

353

111. Lecture in Poland 1975; RANH, Hans Freudenthal Papers, inv.nr. 606. 112. Ibid. 113. Ibid. 114. Freudenthal, ‘Mathematische Erziehung oder Mathematik im Dienste der Erziehung’, Lecture university week in Innsbruck, 21 June 1974; mimeographed presentation, 18 pages, q.v. 12; RANH, Hans Freudenthal Papers, inv.nr. 1055. 115. Freudenthal to Goffree, 9 February 1986; RANH, Hans Freudenthal Papers, inv.nr. 33. 116. During an interview in 1977 Freudenthal mentioned that he visited the Pedagogische Academie in Gorinchem once a week, and went one day per week to a school where the students of the Pedagogische Academie (PA-students) did their teaching practice and also one day per week to a school for lower vocational education in Utrecht. He said about this: “The most enjoyable thing I did in my whole life was going to that PA, working with the students there and with the children of that school”; interview with Hans Freudenthal: ‘Wiskunde in andere wetenschappen’, in: De vakidioot. Maandblad van de subfaculteiten wiskunde en natuur- en sterrekunde 9 (1977) 14–20, q.v. 15; RANH, Hans Freudenthal Papers, inv.nr. 1849. 117. His lecture held at a CIEAEM conference in Bordeaux in 1974 was an introduction to film material of a teaching situation on IOWO material (entitled: “Kijk op kans”). In this lecture Freudenthal spoke about the various observations of teaching situations in the class room; RANH, Hans Freudenthal Papers, inv nr. 588. 118. Freudenthal to WISKOBAS staff members, 21 May 1976; RANH, Hans Freudenthal Papers, inv.nr. 405. 119. Freudenthal to WISKOBAS staff members, 7 February 1973; ibid. 120. Freudenthal to WISKOBAS staff members, 30 November 1976; ibid. Freudenthal’s “long live the level groups for arithmetic” probably referred to the arithmetic method of the working group led by H.M.M. Vossen, an arithmetic method strongly based on rote learning. In the study of R.A. de Jong, as described in his Wiskobas in methoden, it appeared that this method was hardly related to what was characteristic for WISKOBAS: In this method De Jong hardly found anything of the WISKOBAS characteristics as he had defined them (among which for example “many-sided approach to the number concept and operations” and “no isolated understanding-acquiring activities in particular aimed at the primary education”). This would explain Freudenthal’s lashing out at “level groups for arithmetic”: Bastiaan (who apparently was taught according to the method Niveaucursus rekenen) appeared to have difficulties with applying his arithmetic abilities in this context and understand what he was doing; R.A. de Jong, Wiskobas in methoden 180–183. 121. Freudenthal during the opening lecture of ‘The first conference of the international group for the psychology of mathematics education’ in Utrecht, 29 August–2 September 1977; RANH, Hans Freudenthal Papers, inv.nr. 1840. 122. Freudenthal, ‘Cognitieve ontwikkeling—kinderen geobserveerd’, article composed from Jaarverslag over 1977, Provinciaals Utrechts Genootschap (1978) 8–18 and Freudenthal, Weeding and sowing; Freudenthal, Appels en peren 105. 123. Freudenthal, ‘Der Ärger mit der Mengenlehre’, as article: Hans Freudenthal, ‘Was fehlt der modernen Mathematik?’, in: Deutscher Zeitung 38 (21 September 1973); RANH, Hans Freudenthal Papers, inv.nr. 1621. 124. In his later publications the paradigm would also continue to play an important role. He wrote about it, among other things in Mathematics as an educational task, but the value he attached to the concept ‘paradigm’ emerged even better in Weeding and sowing. In Revisiting mathematics education too he came back to this extensively. 125. T.S. Kuhn, The structure of scientific revolutions (Chicago 1962); G. de Vries, De ontwikkeling van wetenschap. Een inleiding in de wetenschapsfilosofie (Groningen 1984) 91–117. 126. Hans Freudenthal, ‘Trends in modern mathematics’, ICSU Review 4 (1962) 54–61, q.v. 61. See for the complete quotation also paragraph 8.4.1. 127. The concept paradigm appeared in two lectures in 1972: ‘Tendenzen zur Mathematik in der Grundschule’, gehouden op ‘Der Jahrestagung der Deutschen Mathematiker-Vereinigung’

354


and in the lecture he gave in Argentina, entitled: ‘The Institute on Developing Mathematical Education (IOWO) in the Netherlands’: Freudenthal, ‘Tendenzen zur Mathematik in der Grundschule’, Didaktik der Mathematik 1 (1973) 2–11; RANH, Hans Freudenthal Papers, inv.nr. 571. 128. Freudenthal in the lecture ‘The Institute on Developing mathematical Education (IOWO) in the Netherlands’, Argentina 1972; RANH, Hans Freudenthal Papers, inv.nr. 571. 129. Freudenthal, Weeding and sowing 202. 130. Freudenthal, Schrijf dat op, Hans 358. 131. Lecture of Freudenthal in Argentina, end 1972; RANH, Hans Freudenthal Papers, inv.nr. 571. 132. Freudenthal, ‘Was fehlt der modernen Mathematik?’, Deutsche Zeitung, 21 September 1973; RANH, Hans Freudenthal Papers, inv.nr. 1621. 133. Freudenthal, Schrijf dat op, Hans 358. 134. These themes—observing and analysing learning processes, jumps and discontinuities, paradigms and learning mathematics by doing it in a realistic context—keep returning in his later publications. That mathematics subject matter is paradigmatic for example is found in Mathematics as an educational task (in particular for the higher mathematics of the secondary and university education) and in Weeding and sowing it is treated extensively in a more abstract manner as a tool for development of education. 135. See also the ‘tick, tack and boom’ -example in paragraph 9.4.1. 136. The most common arithmetic methods were rote learning methods, comparable to Fundamenteel rekenen (see also paragraph 4.3); De Jong, Wiskobas in methoden 133 et seq. 137. Freudenthal in: Vuysje, ‘De wiskundeles verandert: 3 + 2 = 1’, Haagse Post (December 1969, January 1970); RANH, Hans Freudenthal Papers, inv.nr. 1847. 138. J. Piaget, ‘Comments on mathematical education’, in: A.G. Howson, ed., Developments in mathematical education. Proceedings of the second international congress on mathematical education (Cambridge 1973) 79–87, q.v. 82. 139. De Jong, Wiskobas in methoden 158–161. Denken en rekenen was published by Malmberg. The first edition was published in 1968. In 1976 the method was taken from the list. See also paragraph 8.5.2. 140. V. van Achter, De modernisering van het rekenonderwijs op de basisschool (’s-Hertogenbosch 1969) 23. 141. J.J. de Iongh, ‘Zeggen en zien’, in: Keur uit de eerste jaargang van het tijdschrift Denken en Rekenen (’s-Hertogenbosch 1970) 7–9, q.v. 8–9. 142. De Jong, Wiskobas in Methoden 161. 143. Freudenthal in: Vuysje, ‘De wiskundeles verandert: 3 + 2 = 1’, HP Magazine; Freudenthal, ‘Les tendances nouvelles de l’enseignement mathématique’, Revue de l’Enseignement Supérieur 46–47 (1969) 23–29; Freudenthal, ‘Phaenomenologie van het getalbegrip’, Wijsgerig Perspectief (1971) 225–235; Freudenthal, ‘Tendenzen zur Mathematik in der Grundschule’, Didaktik der Mathematik 1 (1973) 2–11; Freudenthal, Weeding and sowing. 144. Freudenthal, ‘Ontsporingen’, Wiskobas Bulletin 1 (1971); RANH, Hans Freudenthal Papers, inv.nr. 1351. 145. Freudenthal, ‘Recent tendencies in the teaching of mathematics’ 16. 146. Freudenthal in: Vuysje, ‘De wiskundeles verandert: 3 + 2 = 1’; RANH, Hans Freudenthal Papers, inv.nr. 1847. 147. See also paragraph 5.3.2. 148. Freudenthal, ‘Phaenomenologie van het getalbegrip’, Wijsgerig Perspectief (1971) 225– 235, q.v. 234. 149. Ibid., 234. 150. Freudenthal, Mathematics as an educational task 191. Van Parreren wrote in his Leren op school something similar when he mentions the incremental concept forming according to the theory of the Russian psychologist Gal’perin. He describes the need for counting down as the first step in learning to count up; C. van Parreren, Leren op school (Groningen 1976 (tenth revised edition)) 30–31.

References

355

151. Freudenthal, ‘Recent tendencies in the teaching of mathematics’ 15. 152. Freudenthal, Mathematics as an educational task 191. 153. See also: Freudenthal, ‘Kennst du deinen Vater?’, Der Mathematikunterricht 5 (1974) 7–18, 11. 154. Freudenthal, Weeding and sowing 216. 155. Freudenthal at the ‘Noordwijkerhout konferentie’ in 1975; RANH, Hans Freudenthal Papers, inv.nr. 412. This is also used and described in among other things: the lecture in Poland in 1975, Freudenthal, ‘Wat is meetkunde?’, Euclides 50 (1974/75) 151–160, Freudenthal, ‘Wandelingen met Bastiaan. Wereldverkennen van de straatstenen tot de maan’, Pedomorfose 25 (1975) 51–64, q.v. 61, Freudenthal, Weeding and sowing 281. 156. Here again a parallel can be drawn with Van Parreren’s description of theory of the Russian psychologist Gal’perin. According to Gal’perin’s incremental forming of a mental action, a mental action is learned first at a material level, then on a verbal and after that on a mental level; Van Parreren, Leren op school 32. 157. Lecture in Poland 1975; RANH, Hans Freudenthal Papers, inv.nr. 606. 158. Ibid. 159. Freudenthal, ‘Wat is meetkunde?’, Euclides 50 (1974/75) 151–160, 154. 160. Ibid., 152. 161. Ibid. 162. Ibid., 153. 163. Ibid., 160. 164. See also paragraph 5.3.2. 165. Freudenthal, ‘Phaenomenologie van het getalbegrip’, Wijsgerig Perspectief (1971) 225– 235. 166. Freudenthal, Didactische fenomenologie van wiskundige structuren 10. Deel I (Utrecht 1984). The book was also published as: Didactical phenomenology of mathematical structures (Dordrecht 1983). The German version has not been published (‘Didaktische Phänomenologie mathematischer Strukturen’, 1981); RANH, Hans Freudenthal Papers, inv.nr. 632. 167. Freudenthal, ‘Lernzielfindung im Mathematikunterricht’, Zeitschrift für Pädagogik 20 (1974) 719–738. 168. Freudenthal, ‘Taxanomieën—waarvoor?’, Faraday 45 (1976) 142; RANH, Hans Freudenthal Papers, inv.nr. 610. 169. Freudenthal, ‘Lernzielfindung im Mathematikunterricht’, Zeitschrift für Pädagogik 20 (1974) 719–738, 730. 170. Freudenthal, Didactische fenomenologie van wiskundige structuren 9–10. 171. See also: Nelissen, ‘Eeuwenlang rekenen’, Willem Bartjens 18 (1998/99) 36–39. 172. RANH, Hans Freudenthal Papers, inv.nr. 589. 173. ‘Onderwijsleerplan WISKOBAS’, as this was distributed during the WISKOBAS working week in Scheveningen, 16–19 March 1971; RANH, Hans Freudenthal Papers, inv.nr. 407. 174. Freudenthal, ‘Een pijnlijke operatie’, Wiskobas Bulletin 3 (1973) 118–122; RANH, Hans Freudenthal Papers, inv.nr. 1362. 175. Freudenthal, ‘Why to teach mathematics so as to be useful’, Educational Studies in Mathematics 1 (1968) 3–8, q.v. 4–5. 176. Ibid., 7. 177. Hans Freudenthal, ‘Allocution du premier Congrès International de l’Enseignement Mathématique Lyon 24–31 août 1969’, Educational Studies in Mathematics 2 (1969) 135–138, q.v. 136. 178. Hans Freudenthal during the interview: ‘Wiskunde in andere wetenschappen’, in: De Vakidioot. Maandblad van de Subfaculteiten Wiskunde en Natuur- en Sterrekunde 9 (1977) 14–20, q.v. 16; RANH, Hans Freudenthal Papers, inv.nr. 1849. “Paulus de Boskabouter”, a creation of Jean Dulieu (pseudonym of Jan van Oort) had started in 1946 as a cartoon and was broadcasted daily as a radio play. From 1967 “Paulus de Boskabouter” was shown on

356


Dutch television as a series with puppets. For that matter, Freudenthal called the relevant teaching package ‘Sproeteldam’ “excellent” as a mathematical article. 179. Freudenthal at the conference in Noordwijkerhout in 1975; RANH, Hans Freudenthal Papers, inv.nr. 412. 180. Hans Freudenthal, ‘Geometry between the devil and the deep sea’, Educational Studies in Mathematics 3 (1971) 413–435, q.v. 420. 181. Important representatives of this movement were K. Mollenhauer en W. Klafki; W.A.J. Meijer, Stromingen in de pedagogiek (Baarn 1996), 55 et seq. 182. Freudenthal, ‘Kanttekeningen bij de nomenclatuur’, Euclides 47 (1971), 139–140, Freudenthal, ‘Nog eens nomenclatuur’, Euclides 47 (1972), 181–192, Freudenthal, ‘Nomenclatuur en geen einde’, Euclides 49 (1973) 53–58. 183. Freudenthal, ‘What groups mean in mathematics and what they should mean in mathematical education’, in: G.A. Howson, ed., Developments in mathematical education: the proceedings of the Second International Congress on Mathematical Education (Cambridge 1973) 101–114, q.v. 114. 184. Freudenthal, ‘Nog eens nomenclatuur’, Euclides 47 (1972) 181–192, q.v. 181. 185. Ibid., 192. 186. Freudenthal, ‘De niveaus in het leerproces en de heterogene leergroep, met het oog op de middenschool’, Gesamtschule Conferentie (1973) 88–99, q.v. 89–90. 187. Ibid., 90. 188. Ibid. 189. M.A.J.M. Matthijsen, De elite en de mythe. Een sociologische analyse van de strijd om onderwijsverandering (Deventer 1982) 163. 190. Ibid. 164. 191. Freudenthal, ‘De Middenschool’, Rekenschap 20 (1973) 157–165, q.v. 164. 192. Ibid., 165. 193. Freudenthal, ‘De niveaus in het leerproces en de heterogene leergroep, met het oog op de middenschool’, Gesamtschule Conferentie (1973) 88–99, q.v. 93. Freudenthal derived the certainty that this practice offered opportunities from the experiments of the Russian mathematician Davydov, among other things. 194. In the Jenaplan Education—in which Mrs. Freudenthal was active—the heterogeneous learning group also played an important role. In his book Weeding and sowing Freudenthal considered this concept and the difficulties of such an approach for the education developer; Freudenthal, Weeding and sowing 60–64; S.J.C. Freudenthal-Lutter, Naar de school van morgen (Alphen aan de Rijn 1971) 81. 195. Freudenthal, ‘De niveaus in het leerproces en de heterogene leergroep, met het oog op de middenschool’, Gesamtschule Conferentie (1973) 88–99, q.v. 98. 196. Even more expressly than in his book Mathematics as an educational task Freudenthal was critical in his Weeding and sowing. A preface to a science of mathematical education from 1972—a book which perhaps best characterized Freudenthal’s way of working and his attitude, as the title in particular shows. In this book he systematically ‘weeded’ the state of affairs in the fields of science, education, education as a discipline, and the discipline of mathematics education; Freudenthal, Weeding and sowing. 197. J. van Dormolen, ‘Freudenthal cursief. Hoe didactiek een deel van wiskunde kan worden’, Nieuwe Wiskrant 10 (3) (1991) 19. 198. A. Bishop, ‘Freudenthal cursief. HF-an inspiration in the search for truth’, Nieuwe Wiskrant 10 (1991) 13. 199. Hans Freudenthal on himself in: Freudenthal, Schrijf dat op, Hans 347. 200. Important exceptions were the work of Tatiana Ehrenfest, Van Dantzig, the Van Hieles and the Russian researcher V.V. Davydov. In the preface to Mathematics as an educational task he also mentioned Castelnuovo, Krygovska, Servais and Revuz in a list of international friends that had taught him a great deal. In the course of the 1970s Freudenthal was, uncharacteristically, considerably more generous with his praise, in particular when the work

References

357

of the education developers of the IOWO was involved; RANH, Hans Freudenthal Papers, inv.nr. 1589. 201. Freudenthal, Mathematics as an educational task vi. 202. His discussion with Vredenduin as the chairman of the Nomenclatuurcommissie in Euclides was a good example of this; Freudenthal, ‘Kanttekeningen bij de nomenclatuur’, Euclides 47 (1971) 139–140, Freudenthal, ‘Nog eens nomenclatuur’, Euclides 47 (1972) 181–192, Freudenthal, ‘Nomenclatuur en geen einde’, Euclides 49 (1973) 53–58. 203. Freudenthal, ‘Verzamelingen in het onderwijs’, Euclides 45 (1970) 321–326, q.v. 321. It was related to criticism to: Krooshof, Jacobs, e.a., Moderne wiskunde deel 1 (Groningen 1968). 204. Freudenthal, ‘Verzamelingen in het onderwijs’, Euclides 45 (1970) 321–326, q.v. 325–326. 205. Ibid., 326. 206. Freudenthal, ‘Recent tendencies in the teaching of mathematics’, in: W.E. Lamon, ed., Learning and the nature of mathematics (1972) 11–18, q.v. 14; Translation of: Freudenthal, ‘Les tendances nouvelles de l’enseignement mathématique’, Revue de l’enseignement supérieur 46–47 (1969) 23–29. 207. Freudenthal, ‘De Middenschool’, Rekenschap 20 (1973) 157–165, q.v. 159. 208. Freudenthal preferred having a disciplinary component integrated in the teachers’ training; Freudenthal, ‘Denkbeelden over universitaire lerarenopleiding’, lecture at seminar of the ‘Vereniging Universitaire Leraren Opleiding Nederland’, 1976; RANH, Hans Freudenthal Papers, inv.nr. 609. 209. Freudenthal, ‘De niveaus in het leerproces en de heterogene leergroep, met het oog op de middenschool’, Gesamtschule Conferentie (1973) 88–99, q.v. 91. 210. Freudenthal in a review in 1975 of the book of G. Boomsma, Handboek voor het moderne rekenonderwijs in de basisschool; RANH, Hans Freudenthal Papers, inv.nr. 602. 211. Ibid. 212. Freudenthal, Mathematics as an educational task vi. 213. This study was organized by the International Association for the Evaluation of Educational Achievement (IEA); Freudenthal, ‘Mammoetonderzoek wekt wantrouwen’, Universiteitsblad U Utrecht (14 June 1974). Freudenthal also reviewed this study in: Freudenthal, ‘Een internationaal vergelijkend onderzoek over wiskundige prestaties’, Pedagogisch Studiën 52 (1975) 43–55; Freudenthal, ‘Pupils’ achievement internationally compared—the IEA’, Educational Studies in Mathematics 6 (1975) 127–186; Freudenthal, ‘Schülerleistungen im internationalen Vergleich’, Zeitschrift für Pädagogik 21 (1975) 890–910; Freudenthal, ‘Leerlingenprestaties in de natuurwetenschappen internationaal vergeleken’, Faraday 45 (1975) 58–63; Freudenthal, ‘De wereld van de toetsen’, Rekenschap 23 (1976) 60–72. 214. Freudenthal, ‘Mammoetonderzoek wekt wantrouwen’, Universiteitsblad U (Utrecht 14 June 1974). 215. Ibid. 216. Ibid. 217. Langeveld, also criticized Bloom’s study;M.J. Langeveld, ‘Over de voorbereiding van het gebruik van exacte methoden in bewerking en interpretatie van psychologische onderzoekingen’, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde 30 (Amsterdam 1967) 187–204. 218. Freudenthal, ‘Mammoetonderzoek wekt wantrouwen’, Universiteitsblad U (Utrecht 14 June 1974). 219. Freudenthal, Weeding and sowing. Freudenthal previously reviewed Bloom’s work in ‘Lernzielfindung im Mathematikunterricht’, Zeitschrift für Pädagogik 20 (1974) 719–738. 220. Freudenthal in a presentation at the symposium ‘Toepasbaarheid van taxonomieën’ 1976, entitled: ‘Taxonomieën—waarvoor?’; RANH, Hans Freudenthal Papers, inv.nr. 610. 221. Freudenthal, Weeding and sowing 143–144. 222. The Stichting voor Onderzoek van het Onderwijs was the organization that was responsible for the programme and the subsidies of the education studies in the Netherlands. Later most of SVO merged with NWO and it was finally abolished in 1997.

358


223. Freudenthal to the minister of O & W about international comparative studies into school performances, 28 June 1974; RANH, Hans Freudenthal Papers, inv.nr. 1730. 224. Ginjaar-Maas replied to this: “I do not know whether you are a black sheep in the educational field—but I do know that you often manage to put your finger exactly on the sore spots, and I greatly admire you because of that. But I have told you that already. Thanks again for your letter.” Freudenthal to Ginjaar- Maas, staatssecretaris van Onderwijs en Wetenschappen, 2 January 1989, reply from Ginjaar-Maas, 5 January 1989; RANH, Hans Freudenthal Papers, inv.nr. 33. 225. Freudenthal, Schrijf dat op, Hans 360–363. 226. Freudenthal, ‘Nomenclatuur en geen einde’, Euclides 49 (1973) 53–58, q.v. 53. 227. Namely the article: C. Philippi-Siewertsz van Reesema, ‘Inleiding tot de werken van Prof. dr. Jean Piaget’, Paedagogische Studiën 10 (1929) 1–52; RANH, Hans Freudenthal Papers, inv.nr. 440. 228. Freudenthal, ‘Report on a comparative study of methods of initiation into geometry’, Euclides 34 (1957) 289–306, q.v. 306. 229. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, 15. 230. The notebook with notes he made after reading Piaget’s La naissance de l’intelligence chez l’enfant also dated from this period. These notes were short summaries and did not contain Freudenthal’s comments on the work; RANH, Hans Freudenthal Papers, inv.nr. 445. 231. The following studies were used in composing this paragraph: A.L. Baldwin, Theories of child development (New York 1967); H.E. Gruber and J.J. Vonèche, ed., The essential Piaget. An interpretive reference and guide (1977 London); R. Kohnstamm, Kleine ontwikkelingspsychologie I. Het jonge kind (Houten 1993); Mary Sime, Zoals een kind het ziet. Wegwijzer in de theorieën van Piaget (Purmerend 1976) with preface and epilogue by Van Hiele; L. Smith, J. Dockrell en P. Tomlinson, ed., Piaget, Vygotsky and beyond. Future issues for developmental psychology and education (London 1997); Rita Vuyk, Overview and critique of Piaget’s genetic epistemology 1965–1980 (Londen 1981). 232. Interview by Piet Vredenduin about ‘portrayal of mankind’, 9 January 1984; RANH, Hans Freudenthal Papers, inv.nr. 1850. 233. Four of the earlier studies related to mathematical aspects were La génèse du nombre chez l’enfant, La géométrie spontanée de l’enfant, La réprésentation de l’espace chez l’enfant and La génèse des structures logiques élémentaires; J. Piaget et Alina Szeminska, La génèse du nombre chez l’enfant (Neuchâtel 1941), J. Piaget, Bärbel Inhelder et Alina Szeminska, La géométrie spontanée de l’enfant (Parijs 1948), J. Piaget et Bärbel Inhelder, La réprésentation de l’espace chez l’enfant (Parijs 1948) and J. Piaget et Bärbel Inhelder, La génèse des structures logiques élémentaires (Neuchâtel 1959). 234. P.M. van Hiele, Development and learning process. A study of some aspects of Piaget’s psychology in relation with the didactics of mathematics (Groningen 1959); A.H. Syswerda, De ruimtevoorstelling bij het kind volgens J. Piaget en B. Inhelder. Synthese en critiek (Groningen 1955); L.N.H. Bunt, The development of the ideas of number and quantity according to Piaget (Groningen 1951). 235. From the side of the psychologists, for example, the comments were that Piaget went too far with his abstract mathematical theories in his interpretation of the cognitive development of the child; A.L. Baldwin, Theory of child development (New York 1967) 299–300. 236. M.J. Langeveld in ‘Preface’ van: P.M. van Hiele, Development and learning process. A study of some aspects of Piaget’s psychology in relation with the didactics of mathematics (Groningen 1959) iv. 237. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel III 99. 238. Freudenthal, Schrijf dat op, Hans 349. 239. Freudenthal, Revisiting mathematics education. China lectures. 240. Freudenthal, Mathematics as an educational task 662. 241. Ibid., 662–677. 242. Ibid., v–vi.

References

359

243. Ibid., 662. 244. Freudenthal, ‘Bastiaan’s lab’, Pedomorfose 15 (1976) 35–54, q.v. 35. 245. Freudenthal, ‘Cognitieve ontwikkeling—kinderen geobserveerd’; in: Freudenthal, Appels en peren 104. Freudenthal also noted this observation in: Freudenthal, ‘Bastiaan’s lab’, Pedomorfose 15 (1976) 35–54, q.v. 35. 246. Freudenthal, Mathematics as an educational task 663. 247. Ibid., 669. 248. Ibid., 87–88. 249. Ibid., 671. 250. Ibid., 120. 251. See also paragraph 9.5.3, which describes how the introduction of the number concept, a line of approach that Freudenthal considered to be wrong, would be supported by Piaget’s study results. 252. Sime, Zoals een kind het ziet. Wegwijzer in de theorieën van Piaget. 253. Freudenthal in a book review of: Sime, Zoals een kind het ziet. Wegwijzer in de theorieën van Piaget; RANH, Hans Freudenthal Papers, inv.nr. 1714. 254. Ibid. 255. Interview by Piet Vredenduin about ‘portrayal of mankind’, 9 January 1984; RANH, Hans Freudenthal Papers, inv.nr. 1850. 256. Ibid. 257. From: ‘Verslag van het kollege van Prof.Dr.H. Freudenthal’ op de WISKOBAS-conferentie te Lochem, 1972; RANH, Hans Freudenthal Papers, inv.nr. 408. 258. Freudenthal, ‘Cognitieve ontwikkeling—kinderen geobserveerd’, article composed from Jaarverslag over 1977, Provinciaals Utrechts Genootschap (1978) 8–18 and Freudenthal, Weeding and sowing; Freudenthal, Appels en peren 105. 259. He meant his article: Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’. 260. Freudenthal, ‘Recent tendencies in the teaching of mathematics’, in: W.E. Lamon, ed., Learning and the nature of mathematics (1972) 11–18, q.v. 13. 261. Lecture of Freudenthal in Argentina, end 1972; RANH, Hans Freudenthal Papers, inv.nr. 571. 262. Freudenthal, ‘De niveaus in het leerproces en de heterogene leergroep, met het oog op de middenschool’, Gesamtschule Conferentie (1973) 88–99, q.v. 88.

Chapter 10

Epilogue

We Have Come Full Circle

Onderwijs is een geweldig systeem met zijn eigen wetten van reactie op, en immuniteit tegen, maatregelen van buiten, een systeem dat zijn eigen leven leidt. Het is allesbehalve ongevoelig, het is geweldig vatbaar voor invloeden, als je maar weet waar en hoe, maar niemand weet het. Of veeleer: zij die het weten, weten niet dat ze het weten, en zij die denken dat ze het weten, kunnen zich schromelijk vergissen. Freudenthal in de rede bij zijn afscheid als hoogleraar-directeur van het IOWO op 14 augustus 1976 (Education is a wonderful system with its own laws in reaction to, and immunity against, external measures, a system which leads its own life. It is far from being insensitive, it is enormously susceptible to influences, if you know where and how to influence it, but nobody knows that. Or rather: those who do know it, don’t know themselves that they know it, and those who think they know it, can be greatly mistaken.) Freudenthal during his speech on the occasion of his retirement as professor-director of the IOWO on 14 August 1976 [1]

The history of the Dutch educational system displays whimsical developments. Foreign influences, curriculum legacies, innovations ordered by politicians, pressure from publishing companies and their choices in publishing teaching methods: the influences were innumerable, various and not easily predictable. The following joke (probably an American one) sketches the development of an arithmetic word problem over the last 50 years: Teaching Math in 1950: A logger sells a truckload of lumber for $ 100. His cost of production is 4/5 of the price. What is his profit? Teaching Math in 1960: A logger sells a truckload of lumber for $ 100. His cost of production is 4/5 of the price, or $ 80. What is his profit? Teaching Math in 1970: A logger exchanges a set L of lumber for a set M of money. The cardinality of set M is 100. Each element is worth one dollar. Make 100 dots representing the elements of the set M. The set C, the cost of production contains 20 fewer points than set M. Represent the set C as a subset of set M and answer the following question: What is the cardinality of the set P for profits? Teaching Math in 1980: A logger sells a truckload of lumber for $ 100. Her cost of production is $ 80 and her profit is $ 20. Your assignment: Underline the number 20.


361

362

10 Epilogue

Teaching Math in 1990: By cutting down beautiful forest trees, the logger makes $ 20. What do you think of this way of making a living? Topic for class participation after answering the question: how did the forest birds and squirrels feel as the logger cut down the trees? There are no wrong answers. [2]

Even though the joke is, of course, greatly exaggerated, it is characteristic for the way in which the changes in the mathematics education were experienced: from the mechanistic straightforward approach, via the insanity of sets of New Math to the socially sound approach, which had hardly anything to do with ‘real mathematics’, from the 1990s. This sketch is of course greatly exaggerated too and not directly applicable to the Dutch situation, but in broad outlines both developments do indeed match up. In the Dutch mathematics education of the 1990s the dispute seemed definitively settled with the so-called ‘realistic mathematics education’ as the winner. The IOWO had—via the Onderzoek Wiskunde-onderwijs en Onderwijscomputercentrum department—grown into the Freudenthal Instituut and the realistic mathematics education in a Freudenthallian vein developed there was famous all over the world [3]. It looked as if the pre-war struggle between the advocates of a logically deductive approach and the advocates of an intuitive approach had been decided definitively with the latter as the winners [4]. But dissenting opinions could also be heard: children would no longer learn real mathematics, but linger on in a never ending practical-realistic introduction. In an article in NRC/Handelsblad on the occasion of his inaugural speech F.J. Keune stated that school mathematics was “as flat as a pancake” [5]. A heated discussion followed between the advocates and the opponents of the realistic mathematics education, during which Keune received support both from the side of the teachers and the academics. It was as if the logical-deductive movement showed signs of a new revival. Had we come full circle? Should we go back to Bourbaki—or 30 years before that—to Dijksterhuis and friends? Had Freudenthal not ‘won’ after all? These are the wrong questions. There was a large difference between the theory of realistic mathematics education and the educational practice—or: the impression that practice made. Realistic mathematics education does not stop at ‘teaching mathematics from examples’ as the advocates of that mathematics education had already come up with. The term ‘realistic’ certainly did not imply that there would be no room for abstraction, or that no attention would be given to proving theorems. The discussion was not about being pro or contra ‘Freudenthal’ (or: the movement associated with his ideas), but it was about the Dutch mathematics education: that is what in practice realistic mathematics education turned out to be [6]. Without pursuing in greater depth the question whether realistic mathematics education has been implemented in the correct way in the present mathematics education, one thing has to be put first and that is that the discussion hardly affects the fact whether Freudenthal was right or wrong. It is even likely that in this discussion Freudenthal would support the view that ‘more real mathematics’ should be incorporated—in any case for better pupils. After all: for him the realistic aspect was only a starting point, though admittedly for some of the pupils it was also the

Epilogue

363

end point, but ideally it would be a first step in mathematics education on the way to mathematics, to discovering logical relations, to proofs, including axiomatizing, at all levels. Yes, I told you there are so many kinds of understanding in mathematics. At every moment you may believe you have just reached ultimate understanding of some subject, such that nothing is left to be desired. But no, there is no ultimate understanding in mathematics, you can understand any problem in an ever larger context, from an ever higher point of view, and finally—it looks the lowest of all, but perhaps it is the highest—you can learn to understand it in the perspective of the learning child, and this is what happened many times to me. [7]

With practically all movements taking part in the discussion on mathematics education, disseminating mathematics always was the main thing in the end. Freudenthal too was of the opinion that to a certain level ‘learning what mathematics was’ had to be achievable for all pupils. To him ‘learning to do mathematics’ was as important as the realistic aspect as a starting point [8]. Freudenthal retired as professor-director of the IOWO on 14 August 1976. The IOWO was in its fifth year of being at that time. In his retirement lecture Freudenthal assumed the role of IOWO’s grandfather: no longer directly involved in the small and larger bothers and worries, but at a certain distance enjoying and reaping the fruits from all the work that was done. He looked back at his role in the IOWO: Als ik terugkijk op mijn activiteit bij het IOWO en U me vraagt wat ik denk dat mijn belangrijkste bijdrage was, dan zeg ik: hun met mijn gezag als wiskundige garanderen dat hetgeen zij aan het ontwikkelen waren, echte wiskunde is, dat om jezelf als wiskundige waar te maken, je geen minderwaardigheidscomplexen bij anderen hoeft te kweken door middel van verzamelingenleer, propositie-calculus, groepentheorie, vectorruimten en andere hoogdravende onverteerde theorie, dat je wiskunde overal kunt ontdekken, met je blote oog en je gezond verstand, dat het het kenmerk van wiskunde is, zo voor zich zelf te spreken, dat je je niet hoeft uit te sloven, om anderen ervan te overtuigen dat het waard is, om te kennen, te leren, te onderwijzen. (If I look back at my activities in the IOWO and you ask me what I think was my most important contribution, I can tell you: based on my authority as a mathematician guarantee that what they were developing, is real mathematics, that to proof yourself as a mathematician, you do not need to cultivate inferiority complexes with others by using set theory, proposition calculus, group theory, vector spaces and other magniloquent undigested theories, that you can discover mathematics everywhere, with your naked eye and based on common sense, that the characteristic of mathematics is to be so self-evident that you do not need to go out of your way to convince others that it is worthwhile to know about, to learn, to teach mathematics.) [9]

At first sight he frankly seemed to plea with these statements—easily interpreted as provocative—for the nuts and bolts mathematics that would be attacked so viciously in the 1990s. But Freudenthal’s motto ‘to just do it (mathematics), and do not make it more complicated than needed’ was certainly not an argument in favour of the eternal intuitive nuts and bolts introduction instead of ‘real mathematics’. For him, in the end the mathematics that had to be learned had to be proper real, correct and relevant mathematics. Furthermore, pupils should experience it as an activity, a way of thinking. Taught in that way, mathematics was an indispensable element of anybody’s education, in the same way as it was indispensable in society: “don’t ask how much mathematics

364

10 Epilogue

Fig. 10.1 Freudenthal posing for the back flap of his autobiography Schrijf dat op, Hans. Knipsels uit een leven (1986)

a child can cope with, ask how mathematics in education can contribute to the human dignity of the child,” he said in that same lecture [10]. The IOWO was to Freudenthal the epitomization of his theory on didactics of mathematics. Or even better: the epitomization of the path leading to the theory on didactics of mathematics. Soon after that he wrote his book Weeding and Sowing, ‘preface to a science of mathematics education’, about that path [11]. The years after his retirement in 1976 were not years of rest and looking back for Freudenthal (Fig. 10.1). He probably worked even more than ever on mathematics education and didactics of mathematics, meanwhile adding new elements to his ideas, revising or accentuating others. During the last 15 years of his life he wrote two books and just under 200 articles on mathematics education. He could be seen at the institute almost every day. The Freudenthal of the mathematics education of 1975 as he is presented in this study is in many ways certainly not the ‘definitive’ Freudenthal, but fundamentally little would change in the direction and the way in which he looked for mathematics education [12]. In a letter of 1977 to his Polish friend Krygovska he looked back at his work in mathematics education. He urged her to have confidence in the developments in mathematics education and in the influence of her ideas on it. Their work would certainly turn out to be effective according to Freudenthal. Voici la différence entre votre tristesse et la mienne: la vôtre est ce que vous craignez que ce que vous avez semé, n’aille germer ou soit étouffé par de mauvaises herbes—la mienne, que je ne verrai plus la moisson. Soyez sûre qu’elle sera riche. Ayez plus de confiance! Ce que nous avons créé, tant peu que ce soit, survivra. Vous vous croyez vieille? Se vêtir à la mode ne démontre pas qu’on soit jeune. À notre âge on peut être plus jeune que ceux qui dès le début n’étaient que des vieillards. (This is the difference between your sadness and mine: yours is that you fear that what you have sown will not germinate or will be choked by weeds—mine is that I will not see the harvest. You can be sure that it will be rich. Have more confidence! What we have created, however little it may be, will survive. You think you are old? Dressing according to the fashion is not a sign of youth. At our age one can be younger than those who were old-aged from the start.) [13]

Confidence! That was the way in which he inspired Krygovska with courage. And also: following the prevailing fashion does not imply that you are young. Freuden-

References

365

thal was convinced that what he sowed—recalcitrant as it may seem—would result in a harvest. And 15 years later a harvest certainly came up. At that time the Freudenthal institute, which nationally sets the trend and internationally has prestige, bears his name. His name was also given to an award conferred for “outstanding achievement in mathematics education research” established by ICMI [14]. When mathematics education is the issue Freudenthal himself is an institute which nobody involved in that education can ignore. But resting on his laurels was not in Freudenthal’s line. Some weeks before his death he wrote to the German mathematics didactician Hans Steiner (1928–2004): Ich habe weiterhin große Pläne. Mit dem Computer geht es ja rasend schnell—so schnell wie man spricht oder denkt. Allmählich wird man ein lebender Mythos.[…] Mir geht es augenblicklich so gut wie kaum je in den letzten zehn Jahren. (Furthermore I have great plans. With the computer it all goes as quick as lightning—as quick as one speaks or thinks. You gradually become a living myth.[…] At this moment I feel better than I ever did the past ten years.) [15]

A few weeks later Freudenthal died on a bench in the park in Utrecht where he was found by children at play. The same park where years before he had often walked with his grandson. Walks that were the germ for the observation of children. We have come full circle.

References 1. Hans Freudenthal, ‘Wiskunde-onderwijs anno 2000’, Christelijk Pedagogisch Studieblad (1976) 42–46. 2. The origin of this joke could not be found, primarily because it was widespread via Internet. 3. N. van Dam, ‘“Ze rekenen ons alles aan wat mis is”. De aparte status van het Freudenthal Instituut’, Het Onderwijsblad (2003); available and consulted as an electronic journal. 4. See also the final conclusion in: K. Gravemeijer and J. Terwel, ‘Hans Freudenthal: a mathematician on didactics and curriculum theory’, Journal of Curriculum Studies 32 (2000) 777–796, 792 5. ‘Pannenkoekwiskunde’, NRC/Handelsblad (25 April 1998). Keune, algebra professor in Nijmegen, in his inaugural speech pled for giving more attention to abstraction and logical reasoning to better make use of the Dutch mathematical talent that would be lost because of the realistic approach. 6. Keune too appeared to agree with the selection of ideas and principles from Freudenthal's work presented to him in a conversation. His criticism mainly concerned blowing up only one aspect (the realistic one) of the didactics of mathematics in the mathematics education; conversation with F.J. Keune, 29 April 1999. 7. Freudenthal, ‘Teacher training—an experimental philosophy’, Educational Studies in Mathematics 8 (1977) 369–376, q.v. 374. 8. Streefland described Freudenthal as being responsible for the origin of the theory of ‘mathematics as an activity’; L. Streefland, ‘Wiskunde als activiteit en de realiteit als bron’, Nieuwe Wiskrant 5 (1985) 60–67. 9. Freudenthal, ‘Wiskunde-onderwijs anno 2000’, Christelijk Pedagogisch Studieblad (1976) 42–46, q.v. 45. 10. Ibid., 46. 11. Freudenthal, Weeding and Sowing.

366

10 Epilogue

12. Geometry education and fraction education are only a few of the examples in development of the mathematics education to which Freudenthal was still contributing in the 1970s. The influences of pedagogues and learning psychologists such as Carpay, Van Parreren, Gal’perin and Davydov on Freudenthal's work could only be assessed well in the subsequent period. The same holds for his international contacts and friendships with like-minded didacticians such as Krygovska, Josette Adda, Christine Keitel and their influence on Freudenthal’s work. 13. Freudenthal to Krygovska, 24 November 1977. How much they agreed became clear from another of Freudenthal's letters to Krygovska, a few years later. “En dehors de l’IOWO il n’y a personne qui comprenne et interprète mes idées comme vous et qui y ajoute des arguments et des aspects essentiels”; (“Outside of the IOWO there is nobody who understands and interprets my ideas as you do and who adds essential arguments and aspects.”) Freudenthal to Krygovska, 29 april 1980; RANH, Hans Freudenthal Papers, inv.nr. 47. 14. This is the so-called ‘Hans Freudenthal Medal’ that is awarded in odd years as a recognition for “a major cumulative program of research”. 15. Freudenthal to H.G. Steiner, 21 September 1990; RANH, Hans Freudenthal Papers, inv.nr. 71.

Bibliography

Freudenthal’s unpublished papers, notes and complete bibliographies are to be found in his personal archive in the Rijksarchief in Haarlem, Noord-Holland. Rijksarchief in Noord-Holland, Papers of Hans Freudenthal (1905–1990), mathematician, 1906–1990.

Freudenthal’s Published Works Freudenthal, H., ‘Über die Enden topologischer Räume und Gruppen’ (Berlin 1931). Freudenthal, H., ‘Qualität und Quantität in der Mathematik’, Euclides 8 (1932) 89–98. Freudenthal, H., ‘5000 jaren internationale wetenschap’, Euclides 22 (1946) 222–241. Freudenthal, H., ‘Rekenmachines winnen den oorlog’, De Groene Amsterdammer (16 maart 1946). Freudenthal, H., ‘Redeneren en doen’, De Groene Amsterdammer (8 maart 1947). Freudenthal, H.,‘De algebraïsche en de analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides (1948) 106–121. Freudenthal, H., ‘Eikenhout is eikenhout’, De Groene Amsterdammer (13 maart 1948). Freudenthal, H., ‘Tekort aan leraren. Gebreken aan de opleiding’, De Groene Amsterdammer (3 juli 1948). Freudenthal, H., ‘Onderwijsvernieuwing, maatschappelijk gezien’, De Vrije Katheder 8 (1949) 6–8. Freudenthal, H., ‘Examen in vergiftigde toestand’, De Groene Amsterdammer (8 oktober 1949). Freudenthal, H., ‘Reorganisatie van het Hoger Onderwijs ii’, De Groene Amsterdammer (26 november 1949). Freudenthal, H., ‘De universiteit – voor wie?’, De Groene Amsterdammer (5 november 1949). Freudenthal, H., ‘De recrutering der studenten. Het plan-studieloon der Afdeling Utrecht van het Verbond van Wetenschappelijke Onderzoekers’, De toelating tot het Hoger Onderwijs (Leiden 1950) 51–74. Freudenthal, H., ‘Zwaardere eindexamens?’, De Groene Amsterdammer (23 september 1950). Freudenthal, H., ‘Practische rekenkunde’, De Groene Amsterdammer (24 oktober 1950). Freudenthal, H., ‘En nu… de wiskunde’, De Groene Amsterdammer (2 december 1950). Freudenthal, H., ‘De rol der traditie in de wiskundige wetenschappen’, De Nieuwe Stem (1950) 574–589. Freudenthal, H., ‘Driehoeken en spijkerschriftcijfers’, De Groene Amsterdammer (8 april 1950).

© Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6

367

368

Bibliography

Freudenthal, H., ‘De dwang der denkgewoonten’, Wetenschap en Samenleving (1950) (I) 177–180, (1951) (II) 12–16. Freudenthal, H., ‘Het Trippenhuis’, De Groene Amsterdammer (1951–1960). Freudenthal, H., ‘De dwarskijker i’, Euclides 26 (1951) 245–251. Freudenthal, H., ‘Escalators: en geen einde!’, De Groene Amsterdammer (9 september 1951). Freudenthal, H., J. Sittig, De juiste maat: lichaamsafmetingen van Nederlandse vrouwen als basis van een nieuw maatsysteem voor dames-confectiekleding (Leiden 1951). Freudenthal, H., T. Ehrenfest-Afanassjewa, Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen? Discussie tussen T. Ehrenfest-Afanassjewa en prof.dr. H. Freudenthal (Purmerend 1951). Freudenthal, H., ‘Mevrouw, ze groeien ook zo hard!’, De Groene Amsterdammer (4 augustus 1951). Freudenthal, H., ‘Onderwijsvernieuwers bijeen’, De Groene Amsterdammer (24 november 1951). Freudenthal, H., ‘De opkomst der statistische denkwijze in de moderne wetenschap’, Natuurkundige Voordrachten 29 (1951) 57–65. Freudenthal, H., ‘De slotsom van de eindexamens’, De Groene Amsterdammer (19 mei 1951). Freudenthal, H., ‘Weet u wat een escalator is?’, De Groene Amsterdammer (25 augustus 1951). Freudenthal, H., ‘De dwarskijker II’, Euclides 28 (1952) 58–62. Freudenthal, H., ‘Examens en leerstof’, Vernieuwing van Opvoeding en Onderwijs 10 (1952) 193–197. Freudenthal, H., ‘Gambling with a poor chance of gain’, knaw Proceedings 55 (1952) 433–438. Freudenthal, H., ‘De kinderexamens’, De Groene Amsterdammer (16 augustus 1952). Freudenthal, H., ‘Hoe oud is Maria?’, De Groene Amsterdammer (23 mei 1953). Freudenthal, H., ‘Jagerslatijn’, De Groene Amsterdammer (17 juli 1953). Freudenthal, H., ‘Het mechanica-vraagstuk’, Nieuwe wegen bij het onderwijs in de wiskunde en de natuurwetenschappen (Purmerend 1953) 22–43. Freudenthal, H., ‘Op het telraam van het toeval’, De Groene Amsterdammer (11 juli 1953). Freudenthal, H., ‘Vakgeleerde of paedagoog?’, Christelijk Schoolblad. Onze Vacatures (7, 15 en 21 mei 1953). Freudenthal, H., ‘Weekend-conferentie Wiskunde-werkgroep’: Amersfoort, 7 en 8 november 1953’, Vernieuwing van Opvoeding en Onderwijs 12 (1953) 91–92. Freudenthal, H., ‘Erziehung des Denkvermögens (Diskussionsbeitrag)’, Archimedes 6 (1954) 87–89. Freudenthal, H., Van sterren tot inlegzolen (Arnhem 1954). Freudenthal, H., ‘De begrippen axioma en axiomatiek in de Wis- en Natuurkunde’, Simon Stevin 39 (1955) 156–175. Freudenthal, H., ‘Naar een nieuw wiskunde-onderwijs’, De Groene Amsterdammer (2 april 1955). Freudenthal, H., ‘Ons weekend’, Mededelingenblad van de Wiskunde Werkgroep der W.V.O. (1955). Freudenthal, H., ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 25 (1956) 14–18. Freudenthal, H., ‘Axiom und Axiomatik’, Mathematisch-physikalische Semesterberichte 5 (1956) 4–19. Freudenthal, H., ‘Initiation into Geometry’, The Mathematics Student 24 (1956) 83–97. Freudenthal, H., ‘De leraarsopleiding’, Vernieuwing van Opvoeding en Onderwijs 133 (1956) 173–180. Freudenthal, H., ‘De opstand der horden’, Rekenschap 3 (1956) 180–190. Freudenthal, H., ‘Opvoeding tot hele mensen’, De Groene Amsterdammer (21 april 1956). Freudenthal, H., ‘Relations entre l’enseignement secondaire et l’enseignement universitaire en Hollande’, L’Enseignement Mathématique 2 (1956) 238–249. Freudenthal, H., ‘Tradition and education’, The New Era in Home and School 37 (1956) 127–132. Freudenthal, H., ‘Vrijheid en verantwoordelijkheid’, Universiteit en Hogeschool 2 (1956) 295–301. Freudenthal, H., ‘Traditie en opvoeding’, Rekenschap 3 (1957) 95–103. Freudenthal, H., Waarschijnlijkheid en statistiek (Haarlem 1957). Freudenthal, H., (ed.), Report on methods of initiation into geometry (Groningen 1958).

Bibliography

369

Freudenthal, H., ‘A comparative study of methods of initiation into geometry’, l’Enseignement Mathématique 5 (1959) 119–139. Freudenthal, H., ‘Een gevaarlijk boek’, Euclides 34 (1959) 188–192. Freudenthal, H., ‘In schandblok klemmen: geen overschrijding der normen’, Universiteit en Hogeschool 5 (1959) 259–263. Freudenthal, H., ‘Report on a comparative study of methods of initiation into geometry’, Euclides 34 (1959) 289–306. Freudenthal, H., Lincos. Design of a language for cosmic intercourse. Part I (Amsterdam 1960). Freudenthal, H., ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255. Freudenthal, H., ‘Vrijetijdsbesteding’, Wetenschap & Samenleving 14 (1960) 77–79. Freudenthal, H., Exacte Logica (Haarlem 1961). Freudenthal, H., ‘Onderwijsproblemen in de Verenigde Staten’, De Groene Amsterdammer (14 januari 1961). Freudenthal, H., ‘Viskunde’, De Groene Amsterdammer (23 december 1961). Freudenthal, H., ‘De georiënteerde ruimte’, Faraday 31 (1962) 15–16. Freudenthal, H., Inleiding tot het denken van Einstein (Assen 1962). Freudenthal, H., ‘Logical analysis and critical survey’ in: Freudenthal, Report of the relations between arithmetic and algebra (Groningen 1962) 20–41. Freudenthal, H., ‘The main trends in the foundations of geometry in the 19th century’, Logic, methodology and philosophy of science: proceedings of the 1960 International Congress (Stanford 1962) 613–621. Freudenthal, H., (ed.), Report on the relations between arithmetic and algebra (Groningen 1962). Freudenthal, H., ‘Het staartje van de mammoet’, De Groene Amsterdammer (28 juli 1962). Freudenthal, H., ‘Trends in Modern Mathematics’, icsu Review 4 (1962) 54–61. Freudenthal, H., ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’, L’Enseignement Mathématique ix (1963) 28–44. Freudenthal, H., ‘Tendenzen in der modernen mathematik’, Der Mathematische und Naturwissenschaftliche Unterricht 16 (1963) 301–306. Freudenthal, H., ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’ Der Mathematikunterricht (1963) 5–29. Freudenthal, H., ‘Die Geometrie in der modernen Mathematik’, Physikalische Blätter 20 (1964) 352–356. Freudenthal, H., ‘New mathematics’, The New Era in Home and School 45 (1964) 188–191. Freudenthal, H., ‘The role of geometrical intuition in modern mathematics’, International Counsel Scientific Unions Review 6 (1964) 206–209. Freudenthal, H., De wereld in de spiegel (Utrecht 1964). Freudenthal, H., ‘Bemerkungen zur axiomatischen Methode im Unterricht’, Der Mathematikunterricht (Stuttgart 1966) 61–55. Freudenthal, H., ‘De betekenis van de wetenschappelijke basis voor de leraar’, Verslag van het Zestiende Congres van Leraren in de Wiskunde en Natuurwetenschappen 18 (1966) 5–11. Freudenthal, H., ‘De eerste ontmoeting tussen de wiskunde en de sociale wetenschappen’, Verhandelingen van de Koninklijke Vlaamse Academie voor Wetenschappen xxvii (1966) 43–55. Freudenthal, H., ‘Functies en functie-notaties’, Euclides 41 (1966) 299–304. Freudenthal, H., ‘Met leuzen alleen komt men er niet’, De Groene Amsterdammer (28 mei 1966). Freudenthal, H., ‘Het zwaartepunt van de wetenschap ligt bij de jeugd’, De Groene Amsterdammer (24 december 1966). Freudenthal, H., ‘Algèbre linéaire et géométrie élémentaire By Jean Dieudonné’, The American Monthly 74 (1967) 745–748. Freudenthal, H., ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22. Freudenthal, H., Wiskunde in wetenschap en dagelijks leven (Hilversum 1967). Freudenthal, H., ‘Gebruik en misbruik van de statistiek’, Geloof en Wetenschap 66 (1968) 1–12. Freudenthal, H., ‘Modernisering Leerplan Wiskunde. Antwoord aan Prof. dr. N.G. de Bruyn’, Euclides 43 (1968) 321–322.

370

Bibliography

Freudenthal, H., ‘Paneldiscussion’, Educational Studies in Mathematics 1 (1968) 61–93. Freudenhal, H., ‘Réponse. Remarques’, Dialectica 21 (1968), 103–104, 164–165, 222–224, 255–256. Freudenthal, H., ‘Why to teach mathematics so as to be useful’, Educational Studies in Mathematics 1 (1968) 3–8. Freudenthal, H., ‘Braces and Venn diagrams’, Educational Studies in Mathematics 1 (1969) 408–414. Freudenthal, H., ‘The concept of integration at the Varna Congress’, Educational Studies in Mathematics 1 (1969) 338–339. Freudenthal, H., ‘Further training of mathematics teachers in the Netherlands’, Educational Studies in Mathematics 1 (1969) 484–492. Freudenthal, H., ‘icmi report on mathematical contests in secondary education (Olympiads)’, Educational Studies in Mathematics 2 (1969) 80–114. Freudenthal, H., ‘L’intégration après coup ou à la source’, Educational Studies in Mathematics 1 (1969) 327–337. Freudenthal, H., ‘Allocution du premier Congrès International de l’Enseignement Mathématique Lyon 24–31 août 1969’, Educational Studies in Mathematics 2 (1969) 135–138. Freudenthal, H., ‘A teachers course colloquium on sets and logic’, Educational Studies in Mathematics 2 (1969) 32–58. Freudenthal, H., ‘Les tendances nouvelles de l’enseignement mathématique’, Revue de l’Enseignement Supérieur 46–47 (1969) 23–29. Freudenthal, H., ‘XIth International Olympiad Bucharest, 5–20 July 1969’, Educational Studies in Mathematics 3 (1970) 128–129. Freudenthal, H., ‘The aims of teaching probability’, The Teaching of Probability & Statistics (1970) 151–167. Freudenthal, H., ‘Un cours de géométrie’, New Trends in Mathematics Teaching ii (1970) 309–314. Freudenthal, H., ‘Introduction’, New Trends in Mathematics Teaching II (1970). Freudenthal, H., ‘Verzamelingen in het onderwijs’, Euclides 45 (1970), 321–326. Freudenthal, H., ‘Samen jarig’, Euclides 46 (1970/71) 177–180. Freudenthal, H., ‘Geometry between the devil and the deep sea’, Educational Studies in Mathematics 3 (1971) 413–435. Freudenthal, H., ‘Kanttekeningen bij de nomenclatuur’, Euclides 47 (1971) 139–140. Freudenthal, H., ‘Korrel clxx’, Euclides 46 (1971) 270–271. Freudenthal, H., ‘Die neuen Tendenzen im Mathematik-Unterricht’, Neue Sammlung 11 (1971), 146–153. Freudenthal, H., ‘Phaenomenologie van het getalbegrip’, Wijsgerig Perspectief (1971) 225–235. Freudenthal, H., ‘The emperical law of large numbers, or the stability of frequencies’, Educational Studies in Mathematics 4 (1972) 484–490. Freudenthal, H., ‘Nieuwe niet-Euclidische meetkunde’, Euclides 48 (1972) 13–18. Freudenthal, H., ‘Nog eens nomenclatuur’, Euclides 47 (1972) 139–140. Freudenthal, H., ‘Recent tendencies in the teaching of mathematics’, in: W.E. Lamon (ed.), Learning and the nature of mathematics (Chicago 1972) 11–18. Freudenthal, H., ‘Strategie der Unterrichtserneuerung in der Mathematik’, Beiträge zum Mathematikunterricht (1972) 41–45. Freudenthal, H., Mathematics as an educational task (Dordrecht 1973). Freudenthal, H., Mathematik als pädagogische Aufgabe 1, 2 (Stuttgart 1973). Freudenthal, H., ‘De middenschool’, Rekenschap 20 (1973) 157–165. Freudenthal, H., ‘De niveaus in het leerproces en de heterogene leergroep, met het oog op de middenschool’, Gesamtschule Conferentie (1973) 88–99. Freudenthal, H., ‘Les niveaux de l’apprentissage des concepts de limite et de continuité’, Accademia Nazionale dei Lincei 370 (1973) 109–115. Freudenthal, H., ‘Nomenclatuur en geen einde’, Euclides 49 (1973) 53–58. Freudenthal, H., ‘Tendenzen zur Mathematik in der Grundschule’, Didaktik der Mathematik 1 (1973) 2- 11. Freudenthal, H., ‘Was fehlt der modernen Mathematik?’, Deutscher Zeitung 38 (21 september 1973).

Bibliography

371

Freudenthal, H., ‘What groups mean in mathematics and what they should mean in mathematical education’, in: G.A. Howson (ed.), Developments in mathematical education: the proceedings of the Second International Congress on Mathematical Education (Cambridge 1973) 101–114. Freudenthal, H., ‘The crux of course design in probability’, Educational Studies in Mathematics 5 (1974) 261–277. Freudenthal, H., ‘Kennst du deinen Vater?’, Der Mathematikunterricht 5 (1974) 7–18. Freudenthal, H., ‘Lernzielfindung im Mathematikunterricht’, Zeitschrift für Pädagogik 20 (1974) 719–738. Freudenthal, H., ‘Mammoetonderwijsonderzoek wekt wantrouwen’, Universiteitsblad U Utrecht (14 juni 1974). Freudenthal, H., ‘Sinn und Bedeutung der Didaktik der Mathematik’, Zentralblad für Didaktik der Mathematik 3 (1974). Freudenthal, H., ‘Soviet Research on teaching algebra at the lower grades of the elementary school’, Educational Studies Mathematics 5 (1974) 391–412. Freudenthal, H., ‘Die Stufen im Lernprozess und die heterogene Lerngruppe im Hinblick auf die Middenschule’, Neue Sammlung 14 (1974) 161–172. Freudenthal, H., ‘Waarschijnlijkheid en statistiek op school’, Euclides 49 (1974) 245–246. Freudenthal, H., ‘Wat is meetkunde?’, Euclides 50 (1974/75) 151–160. Freudenthal, H., ‘Annotaties bij annotaties, vragen bij vragen’, Onderwijs in Natuurwetenschap 2 (1975) 21–22. Freudenthal, H., ‘Een internationaal vergelijkend onderzoek over wiskundige studiepresentaties’, Pedagogische Studiën 52 (1975) 43–55. Freudenthal, H., ‘Wandelingen met Bastiaan. Wereldverkennen van de straatstenen tot de maan’, Pedomorfose 25 (1975) 51–64. Freudenthal, H., ‘Variabelen’, Euclides 51 (1975/76) 349. Freudenthal, H., ‘Bastiaan’s lab’, Pedomorfose 30 (1976) 35–54. Freudenthal, H., (ed.), L.E.J. Brouwer. Collected Works 2. Geometry, analysis, topology and mechanics (Amsterdam 1976). Freudenthal, H., ‘Preface’, Educational Studies in Mathematics 7 (1976) 189. Freudenthal, ‘Taxanomieën – waarvoor?’, Faraday 45 (1976) 142. Freudenthal, H. ‘Wiskunde-onderwijs anno 2000’, Euclides 52 (1976) 290–295. Freudenthal, H., ‘De C.M.L.-wiskunde: een interview met prof. dr. H. Freudenthal’, Euclides 52 (1976/77) 100–107. Freudenthal, H., ‘Bastiaan’s experiments on Archimedes’ principle’, Educational Studies in Mathematics 8 (1977) 3–16. Freudenthal, H., ‘Lernzielfindung im Mathematikunterricht’, Der Mathematikunterricht 23 (1977) 26–45. Freudenthal, H., ‘Teacher training – an experimental philosophy’, Educational Studies in Mathematics 8 (1977) 369–376. Freudenthal, H., ‘Bastiaan meet zijn wereld’, Pedomorfose 37 (1978) 62–68. Freudental, H. ‘Cognitieve ontwikkeling: kinderen geobserveerd’, Jaarverslag 1977 Provinciaal Utrechts Genootschap (1978) 8–18. Freudenthal, H., ‘Die Crux im Lehrgangentwurf zur Wahrscheinlichkeitstheorie’, Didaktik der Mathematik (1978). Freudenthal, H., Weeding and sowing. Preface to a science of mathematical education (Dordrecht 1978). Freudenthal, H., ‘New Math or new education’, Prospects ix (1979) 321–331. Freudenthal, H., ‘Should a mathematics teacher know something about the history of mathematics?’, For the Learning of Mathematics 2 (1981) 30–33. Freudenthal, H., Didactical phenomenology of mathematical structures (Dordrecht 1983). Freudenthal, H., Appels en peren. Wiskunde en psychologie. Gebundelde opstellen (Apeldoorn 1984). Freudenthal, H., Didactische fenomenologie van wiskundige structuren (Utrecht 1984). Freudenthal, H., Berlin 1923–1930. Studienerinnerungen von Hans Freudenthal (Berlin 1987).

372

Bibliography

Freudenthal, H., Schrijf dat op, Hans. Knipsels uit een leven (Amsterdam 1987). Freudenthal, H., Revisiting mathematics education. China lectures (Dordrecht 1991).

Secundaire Literatuur Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1–3 (1924–1927), voortgezet als: Euclides. Tijdschrift voor de didactiek der exacte vakken 4–56 (1927–1980). Educational Studies in Mathematics 1–11 (1969–1980). Mededelingen van de ww van de wvo 1–15 (1953–1967). Nieuwe Wiskrant. Tijdschrift voor Nederlands Wiskundeonderwijs, 1–12 (1981–1993). Vernieuwing van Opvoeding en Onderwijs 9–28 (1950–1970). ‘Adres inzake het mechanica-onderwijs op de h.b.s.-b’, Euclides 28 (1952) 123–128. ‘Adres van de Wiskunde-werkgroep van de w.v.o. inzake het ontwerp-leerplan van wimecos’, Euclides 30 (1955) 202–203. ‘Adres van wimecos aan de minister van onderwijs, kunsten en wetenschappen inzake het mechanica onderwijs’, Euclides 29 (1953) 213–217. ‘Commissie Modernisering Leerplan Wiskunde’, Euclides 37 (1961) 144–151. ‘Het conferentie-weekend op het Maarten-Maartens-huis te Doorn georganiseerd door de Wiskunde Werkgroep der w.v.o.’, Euclides 24 (1948) 81–133. Educational Studies in Mathematics, ‘In memoriam HF (’90)’ 23 (1993). Feestboek Prof. Dr. Hans Freudenthal – 70 jaar (1975) (uitgave van het Mathematisch Instituut en het iowo ter ere van Freudenthals 70e verjaardag). ‘Five Years iowo’, Educational Studies in Mathematics 7 (1976). ‘H. Freudenthal’, Vrij Nederland (1976) 232–247. ‘Interimrapport van de Commissie Modernisering Leerplan Wiskunde’, Euclides 42 (1966) 129–134. Keur uit de eerste jaargang van het tijdschrift Denken en Rekenen (’s-Hertogenbosch 1970). Kijk op Hans (1980) (bundel opstellen van iowo-medewerkers, aangeboden aan prof.dr. H. Freudenthal ter gelegenheid van zijn vijfenzeventigste verjaardag). ‘On the mathematics curriculum of the high school’, The Mathematics Teacher (1962) 191–195. ‘Openingstoespraak van de voorzitter van wimecos, Dr.ir. B. Groeneveld op de algemene vergadering van 28 december 1966’, Euclides 42 (1967) 184–187. ‘De programma’s wiskunde voor Gymnasium b en hbs b’, Vernieuwing van Opvoeding en Onderwijs 11 (1953) 216–218. ‘Rapport van de leerplan-commissie-1954 van wimecos inzake het opstellen van een ontwerpleerplan en een ontwerp-eindexamenprogramma voor wiskunde voor de h.b.s.-b’, Euclides 30 (1955) 149–176. ‘Rapport van sectie M’, in: Rapport van de Staatscommissie tot Reorganisatie van het Hoger Onderwijs (’s-Gravenhage 1949) 335–353. Stemmen uit de praktijk (1964) (conferentieverslag Vernieuwing van het wiskundeonderwijs). ‘Tweede conferentie-weekend, georganiseerd door de wiskunde werkgroep’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 174–176. ‘Uit de openingstoespraak van de voorzitter van wimecos tot de algemene vergadering van 28 december 1961’, Euclides 37 (1962) 264–267. ‘Uit de openingstoespraak van de voorzitter van wimecos tot de algemene vergadering van 28 december 1962’, Euclides 38 (1963) 251–253. ‘De universitaire opleiding tot leeraar in wiskunde en aanverwante vakken’, Euclides 4 (1927/28) 41–57. ‘Verslag, door de Vereeniging van Directeuren van Hoogere Burgerscholen met 5-jarigen cursus, in zake het rapport der Commissie-Beth, uitgebracht aan den Inspecteur E. Jensema’, Bijvoegsel 3 (1926/27) 69–72. ‘Verslag van de Varende Conferentie’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 16–21.

Bibliography

373

‘Weekend-conferentie uitgaande van de Wiskunde Werkgroep der wvo’, Vernieuwing van Opvoeding en Onderwijs 7 (1948) 134–136. De wetenschappelijke basis van de lerarenopleiding, mede in verband met de ontwikkelingen van de exacte wetenschappen in de twintigste eeuw. Verslag van het zestiende congres van leraren in de wiskundeen de natuurwetenschappen, gehouden te Utrecht op 18 april 1966 (Groningen 1966). ‘Wiskunde in andere wetenschappen’, in: De vakidioot. Maandblad van de Subfaculteiten Wiskunde en Natuur- en Sterrekunde 9 (1977) 14–20. ‘Het wiskunde-programma voor het v.h.m.o. Een ontwerp van de Wiskunde-Werkgroep van de w.v.o.’, Euclides 28 (1953) 206–226. ‘Wiskunde Werkgroep der w.v.o.’, Euclides 24 (1948) 56. ‘Wiskunde Werkgroep. Verslag samenkomst van 8 mei 1948 te Utrecht’, Vernieuwing van Opvoeding en Onderwijs (1948) 62–63. Achter, V. van, De modernisering van het rekenonderwijs op de basisschool (’s-Hertogenbosch 1969). Adda, J., ‘Freudenthal cursief. Hans Freudenthal (1905–1990)’, Nieuwe Wiskrant 10 (1991) 26. Albada, P.J. van, ‘De wiskunde voor niet-mathematische richtingen’, Vernieuwing van Opvoeding en Onderwijs 8 (1949) 154–159. Alberts, G., Jaren van berekening. Toepassingsgerichte initiatieven in de Nederlandse wiskundebeoefening 1945–1960 proefschrift (Amsterdam 1998). Alberts, G., Twee geesten van de wiskunde. Biografie van David van Dantzig (Amsterdam 2000). Alberts, G., H. Blauwendraat (red.), Uitbeelden in wiskunde (Amsterdam 2000). Alberts, G., F. van der Blij, J. Nuis (red.), Zij mogen uiteraard daarbij de zuivere wiskunde niet verwaarlozen (Amsterdam 1987). Alberts, G., H.J.M. Bos, J. Nuis (red.), Om de wiskunde. Stimulansen voor toepassingsgerichte wiskunde rond 1946 (Amsterdam 1989). Alberts, G., I. Moerdijk, ‘Interview W.T. van Est. De geometer’, Nieuw Archief voor Wiskunde 5/3 (2001) 52–56. Baayen, P.C., ‘Beleid om de wiskunde’, in: Alberts, G., H.J.M. Bos, J. Nuis (red.), Om de wiskunde (Amsterdam 1989) 61–74. Baggen, P., Vorming door wetenschap. Universitair onderwijs in Nederland 1815–1960 proefschrift (Delft 1998). Bakker, E.L., ‘The technology of instructional development’ in: Travers, R.M.W. et al (ed.), Second handbook of research on teaching (Chicago 1973) 245–285. Baldwin, A.L., Theories of child development (New York 1967). Bartels, A., Een eeuw middelbaar onderwijs 1863–1963 (Groningen 1963). Bastide-van Gemert, S. la, ‘Hans Freudenthal’, in: Biografisch woordenboek van Nederland. Deel 5 (Den Haag 2001). Bastide-van Gemert, S. la, ‘Van ‘terloopse rekenlessen’ tot dagtaak’, Euclides 81 (2005) 94–98. Beckers, D., Jacob Gelder en de wiskundige ideologie in Nederland (1800–1840) (Nijmegen 1995). Begehr, H.W.G., H. Koch et al. (eds.), Mathematics in Berlin (Basel 1998). Behnke, H., (ed.), Grundzüge der Mathematik für Lehrer an Gymnasien sowie für Mathematiker in Industrie und Wirtschaft i-v (Göttingen 1958–1966). Berkel, K. van, In het voetspoor van Stevin (Meppel 1985). Berkel, K. van, Dijksterhuis. Een biografie (Amsterdam 1996). Berkel, K. van, Citaten uit het boek der natuur (Amsterdam 1998). Berkel, K. van, A. van Helden, L. Palm (eds.), A history of science in the Netherlands: survey, themes and reference (Leiden 1999). Berkel, K. van, ‘Over het ontstaan van de twee culturen in negentiende-eeuws Europa’, Spiegel der letteren. Tijdschrift voor Nederlandse literatuurgeschiedenis en voor literatuurwetenschap 42 (2000) 83–96. Berkel, K. van, Academische illusies. De Groningse universiteit in een tijd van crisis, bezetting en herstel, 1930–1950 (Amsterdam 2005).

374

Bibliography

Bertin, E.M.J., H.J.M. Bos, A.W. Grootendorst (eds.), Two decades of mathematics in the Netherlands 1920–1940. A retrospection on the occasion of the bicentennial of the Wiskundig Genootschap. Part 1 (Amsterdam 1978). Beth, H.J.E., ‘Het “meer en meer wiskundig” karakter der H.B. School met 5-jarigen cursus’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924/25) 90–100. Beth, H.J.E., ‘Het experimenteel georiënteerde onderwijs in mechanica’, Euclides 5 (1928) 49–60. Beth, H.J.E., ‘Mechanica opnieuw examenvak’, Euclides 6 (1929/30) 44–48. Beth, H.J.E., J. van Andel, P. Cramer, E.J. Dijksterhuis, ‘Ontwerp van een leerplan voor het onderwijs in wiskunde, mechanica en kosmographie op de H.B. Scholen met vijfjarigen cursus’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 2 (1925/26) 113–139. Beukers-Arts, I., Kees Boeke: een Nederlands onderwijsvernieuwer (Tilburg 1974). Bigot, L.C.T., G. van Hees, Pedagogiek IV. Serie leerboeken ten dienste van kweekscholen. Verleden en heden. Beknopte geschiedenis van opvoeding en onderwijs (Groningen 1963). Bishop, A., ‘Freudenthal cursief. hf – an inspiration in the search for truth’, Nieuwe Wiskrant 10 (1991) 13. Blij, F. van der, ‘Professor Freudenthal zeventig jaar’, Euclides 51 (1975) 3–7. Blij, F. van der, e.a., Een kwart eeuw wiskunde 1946–1975 (Amsterdam 1973). Bijpost, M., J.K. Timmer, Met passer en liniaal 1. Eenvoudig leerboek der vlakke meetkunde (Zutphen 1957). Block, A. de, Algemene didaktiek der theoretische intelligentie (Antwerpen 1960). Boeke, K., ‘Van onderwijsvernieuwing naar mensvorming’, Vernieuwing van Opvoeding en Onderwijs 7 (1949) 298–299. Boekholt, P.Th.F.M., Cahiers voor lokale en regionale geschiedenis (Den Haag 1991). Boekholt, P.Th.F.M., E.P. de Booy, Geschiedenis van de school in Nederland vanaf de middeleeuwen tot aan de huidige tijd (Assen 1987). Boermeester, C., Over meetkunde-onderwijs en psychologie (Groningen 1955). Boermeester, C., Grepen uit de psychologie van het denken in verband met het wiskunde-onderwijs (Purmerend 1956). Bois-Reymond, M. du, J. Lamboo, M. Smits, T. Walaardt-Sacré, Onderwijzersleven (Nijmegen 1981). Borel, A., ‘Twenty-five years with Nicolas Bourbaki, 1949–1973’, Notices of the American Mathematical Society 45 (1998) 373–380. Bos, H., ‘Afscheid van Dirk Struik. Uit de ivoren toren’, Nieuw Archief voor Wiskunde 5/2 (2001) 244–247. Bosch, M., Het geslacht van de wetenschap. Vrouwen en hoger onderwijs 1878–1948 proefschrift (Amsterdam 1994). Boyer, C.B., A history of mathematics (New York 1968, tweede editie 1989). Brandenburg, W.J., L. Schrier, Inleiding in de meetkunde 1 (Groningen 1961). Brandenburg, W.J., Modernisering van het wiskundeonderwijs (Groningen 1968). Bremner, R.H.B., et al (ed.), Children & youth in America. A documentary history. Volume ii 1866–1932 (Cambridge 1971). Bruyn, N.G. de, ‘Modernisering leerplan wiskunde’, Euclides 43 (1968) 260–261. Bunt, L.N.H., ‘De keuze van de leerstof bij het onderwijs in de wiskunde’, Euclides 24 (1948) 83–105. Bunt, L.N.H., De leerstof van ons wiskunde-onderwijs. Een onderzoek naar opvattingen en gebruiken dienaangaande (Groningen 1949). Bunt, L.N.H., The development of the ideas of number and quantity according to Piaget (Groningen 1951). Bunt, L.N.H., Statistiek als onderwerp voor het gymnasium a. An investigation into the possibilities of teaching descriptive and elementary mathematical statistics in secondary schools (Groningen 1956). Bunt, L.N.H., Statistiek als onderwerp voor het gymnasium a. Verslag van een proefneming. A Inrichting en resultaten van het onderzoek (Groningen 1957). Bunt, L.N.H., The training of a mathematics teacher in the Netherlands (Groningen 1962).

Bibliography

375

Burg, F. van den, De Vrije Katheder 1945–1950. Een platform van communisten en niet-communisten proefschrift (Amsterdam 1983). Calmthout, M. van, ‘Hans Freudenthal. Kampioen van de alledaagse wiskunde’, in: P. Brill (red.), Kopstukken van het laagland. Een eeuw Nederland in honderd portretten (Amsterdam 1999). Casimir, R., Het Nederlandsch Lyceum (Groningen 1934). Castelnuovo, E., Geometria intuitiva (Firenze 1948, tweede herziene druk 1952). Commissie Modernisering Leerplan Wiskunde, Toelichting op het Leerplan Wiskunde (1968). Corput, J.G. van der, Het Mathematisch Centrum (Groningen 1946). Coster, B., ‘De ontwikkeling van het ruimteinzicht’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 3 (1926/27) 143–154. Dalen, D. van, Mystic, geometer and intuitionist – The life of L.E.J. Brouwer. Volume I: the dawning revolution (Oxford 1999). Dalen, D. van, L.E.J. Brouwer. Een biografie. Het heldere licht van de wiskunde (Amsterdam 2001). Dantzig, D. van, ‘Over de maatschappelijke waarde van onderwijs in wiskunde’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 3 (1926/27) 186–196. Dantzig, D. van, ‘Woord en werktuig. De strijd om het mechanica-onderwijs’, Euclides 5 (1928/29) 86–103. Davis, P.J., R. Hersch, The mathematical experience (Boston 1984). Deen, N., Een halve eeuw onderwijsresearch in Nederland. Het Nutsseminarium voor pedagogiek 1919-1969 proefschrift (Groningen 1969) Dekker, R., Wiskunde leren in kleine heterogene groepen proefschrift (Utrecht 1991). Dessart, C.H., H. Frandsen, ‘Research on teaching secondary-school mathematics’, in: R.M.W. Travers et al (eds.), Second handbook of research on teaching (Chicago 1973) 1177–1195. Dijksterhuis, E.J., ‘Antwoord aan mevrouw Ehrenfest-Afanassjewa’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924) 60–68. Dijksterhuis, E.J., ‘Moet het meetkunde-onderwijs gewijzigd worden? Opmerkingen naar aanleiding van een brochure van Mevr. Ehrenfest-Afanassjewa’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924) 1–26. Dijksterhuis, E.J., ‘Beschouwingen over de universitaire opleiding tot leeraar in wisen natuurkunde’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 2 (1925/26) 81–95. Dijksterhuis, E.J., ‘Nadere beschouwingen over de opleiding tot leeraar in wisen natuurkundige vakken’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 2 (1925/26) 146–157. Dijksterhuis, E.J., ‘De historische behandelingswijze van de axiomatia der mechanica van Newton’, Euclides 4 (1927/28) 245–255. Dijksterhuis, E.J., ‘De aanslag op de mechanica’, Euclides 5 (1928) 61–69. Dijksterhuis, E.J., ‘De opleiding tot leeraar in wisen natuurkunde volgens de plannen van de commissie- Sijmons’, Euclides 6 (1929/30) 185–203. Dijksterhuis, E.J., ‘Opmerkingen over het onderwijs in mechanica als onderdeel der physica’, Euclides 7 (1930/31) 76–94. Dijksterhuis, E.J., ‘Epistemisch wiskunde-onderwijs’, Euclides 10 (1933/34) 165–213. Dijksterhuis, E.J., Mechanisering van het wereldbeeld (Amsterdam 1950). Dodde, N.L., ‘Nederlandse onderwijspolitiek 1945–1980’, Pedagogisch Tijdschrift 5 (1980) 273–293. Dodde, N.L., Het Nederlandse onderwijs verandert (Muiderberg 1983). Dormolen, J. van, Didactiek van de wiskunde (Utrecht 1974). Dormolen, J. van, ‘Freudenthal cursief. Hoe didactiek een deel van wiskunde kan worden’, Nieuwe Wiskrant 10 (1991) 19. Duyvendijk, A.J. van, De motivering van de klassieke vorming: een historisch-paedagogische studie over twee eeuwen proefschrift (Groningen 1955). Dunk, H.W. von der, W.P. Heere, A.W. Reinink (red.), Tussen ivoren toren & grootbedrijf; De Utrechtse Universiteit 1936–1986 (Utrecht 1986). Duparc, H.J.A., A.W. Grootendorst, ‘Historical Survey’, in: E.M.J. Bertin, H.J.M. Bos, A.W. Grootend-orst (eds.), Two decades of mathematics in the Netherlands 1920–1940. A retrospection on the occasion of the bicentennial of the Wiskundig Genootschap. Part 1 (Amsterdam 1978) v-xxiv.

376

Bibliography

Ebel, R.L., et al (ed.), Encyclopedia of educational research (London 1969). Eeden, C. van, The scientific family tree of David van Dantzig (Amsterdam 2000). Ehrenfest-Afanassjewa, T., ‘Moet het Meetkunde-onderwijs gewijzigd worden? Een antwoord aan den heer E.J. Dijksterhuis’, Bijvoegsel van het Nieuw Tijdschrift voor Wiskunde 1 (1924) 47–59. Ehrenfest-Afanassjewa, T., Uebungensammlung zu einer geometrischen Propädeuse (Den Haag 1931). Eilander, M., ‘Caveant consules’, Euclides 22 (1947) 212–221. Essen, M. van, J.D. Imelman, Historische pedagogiek. Verlichting, romantiek en ontwikkelingen in Nederland na 1800 (Baarn 1999). Est, W.T. van, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Koninklijke Nederlandse Akademie van Wetenschappen. Jaarboek 1991 (Amsterdam 1991) 139–142. Est, W.T. van, ‘Hans Freudenthal. 17 september 1905–13 oktober 1990’, Nieuw Archief voor Wiskunde 9 (1991) 131–136. Freeman-Smulders, A., ‘Lezen als ongedeelde vaardigheid. Een Freudenthal voor het leesonderwijs gezocht’, Leesgoed: tijdschrift voor kinderboeken 24 (1997) 30–33. Freudenthal-Lutter, S.C.J., De Jenaplanschool. Een leef- en werkgemeenschap (Utrecht 1970). Freudenthal-Lutter, S.C.J., Naar de school van morgen (Alphen aan de Rijn 1972). Gelder, X. van, H. Wansink (red.) Heimwee naar de hbs (Amsterdam 1998). Gerrits, G.C., ‘Het onderwijs in de natuurkunde aan de Hoogere Burgerschool. Opmerkingen naar aanleiding van het Rapport over het natuurkunde-onderwijs, uitgebracht aan het Bestuur van de Neerlandsche Natuurkundige Vereeniging’, Euclides 4 (1928) 256–268. Gerretsen, J.C.H., ‘Doelstelling van het wiskundeonderwijs’, Euclides 34 (1958) 90–94. Glas, E., Wiskunde en samenleving in historisch perspectief (Muiderberg 1981). Goffree, F., Leren onderwijzen met wiskobas proefschrift (Utrecht 1979). Goffree, F., Ik was wiskundeleraar (Enschede 1985). Goffree, F., ‘Een halve eeuw onderzoek. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 233–243. Goffree, F., ‘De opbrengst. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 333–345. Goffree, F., M. van Hoorn, B. Zwaneveld (eds.), Honderd jaar wiskundeonderwijs – een jubileumboek (Leusden 2000). Gravemeijer, K.P.E., Developing realistic mathematics education proefschrift (Utrecht 1994). Gravemeijer, K., J. Terwel, ‘Hans Freudenthal: a mathematician on didactics and curriculum theory’, Journal of Curriculum Studies 32 (2000) 777–796. Groen, W., ‘Vier decennia wiskundeonderwijs’, Nieuw Archief voor Wiskunde 5/4 (2003) 304–308. Groot, A.D. de, Vijven en zessen. Cijfers en beslissingen: het selectieproces in ons onderwijs (Groningen 1966). Groot, A.W. de, De universiteit van Amsterdam in oorlogstijd (Amsterdam 1946). Grotenhuis, S., Op zoek naar middelbaar onderwijs proefschrift (Amsterdam 1998). Gruber, H.E., J.J. Vonèche (eds.), The essential Piaget. An interpretive reference and guide (London 1977). Hark, M. ter, Kunnen mensen denken? Honderd jaar onderzoek naar taal en cognitie (Groningen 2000). Heijnen, P.H., Methodiek van het rekenen in de lagere school, ten dienste van normaallessen en kweekscholen (Groningen 1917). Hesseling, D.E., Gnomes in the fog. The reception of Brouwer’s intuitionism in the 1920s proefschrift (Utrecht 1999). Hiele, P.M. van, ‘Een poging om de richtlijnen op te stellen voor een didactiek der wiskunde’, Euclides 24 (1948) 122–133. Hiele, P.M. van, De problematiek van het inzicht proefschrift (Amsterdam 1957). Hiele, P.M. van, Development and learning process. A study of some aspects of Piaget’s psychology in relation with the didactics of mathematics (Groningen 1959). Hiele, P.M. van, Begrip en inzicht. Werkboek van de wiskundedidactiek (Purmerend 1973).

Bibliography

377

Hiele, P.M. van, ‘Freudenthal en de didaktiek der wiskunde’, Euclides 51 (1975) 8–10. Hiele, P.M. van, D. van Hiele-Geldof, ‘Een fenomenologische inleiding tot de meetkunde’, Euclides 33 (1957) 33–47. Hiele-Geldof, D. van, De didaktiek van de meetkunde in de eerste klas van het v.h.m.o. proefschrift (Amsterdam 1957). Hilvoorde, I. van, Grenswachters van de pedagogiek. Demarcatie en disciplinevorming in de ontwikkeling van de Nederlandse academische pedagogiek (1900–1970) (Amsterdam 2002). Hoorn, M. van, ‘Interview met Albert Koldijk’, Euclides. Vakblad voor de wiskundeleraar 78 (2002) 46–50. Hollman, K.P.M.J., Mastery-learning en het wiskunde-onderwijs: exemplarisch materiaal voor gedifferentieerd onderwijs (’s-Hertogenbosch 1974). Hoving, W., J. van IJzeren, Onderzoek naar de wenselijkheid en de mogelijkheid om bij het onderwijs in de oude talen op gymnasia en lycea de lectuur met na-klassieke schrijvers uit te breiden beoordeeld in verband met de betekenis van het klassieke onderwijs in zijn geheel (Groningen 1940). Howson, A.G., (ed.), Developments in mathematical education. Proceedings of the Second International Congress on Mathematical Education (Cambridge 1973). Howson, G., ‘Hans Freudenthal and the foundations of a discipline of mathematics education’, Nieuwe Wiskrant 5 (1985) 68–72. Howson, G., C. Keitel, J. Kilpatrick, Curriculum development in mathematics (Cambridge 1981). Hulst, J.W. van, ‘Martinus Jan Langeveld. 30 oktober 1905–15 december 1989’, in: Koninklijke Nederlandse Akademie van Wetenschappen. Jaarboek 1991 (Amsterdam 1991) 154–162. Idenburg, Ph.J., Schets van het Nederlandse schoolsysteem (Groningen 1964). Idenburg, Ph.J., J. Karsemeijer, H.H. Janssen, Een eeuw middelbaar onderwijs herdacht (Groningen 1963). Imelman, J.D., W.A.J. Meijer, De nieuwe school gisteren en vandaag (Amsterdam 1986). Iongh, J.J. de, ‘Zeggen en zien’, in: Keur uit de eerste jaargang van het tijdschrift Denken en Rekenen (’s-Hertogenbosch 1970) 7–9. Jacobs, H.J. e.a., Moderne wiskunde voor voortgezet onderwijs. Deel 1 voor de brugklas (Groningen 1972). Jansen, G.H., S. Vess, ‘Hans Freudenthal, tussen de bergen en de zee’, in: Ballonnen en brood. Utrechtse portretten (Utrecht 1986) 101–107. Janssen, G.A., ‘Over onze tijd met een heenwijzing naar onze taak’, Euclides 28 (1953) 116–122. Jensema, E., Het middelbaar onderwijs en de critiek, die daarop uitgeoefend wordt (Groningen 1927). Jong, B.C. de, Jan Ligthart (1859–1916). Een schoolmeester-pedagoog uit de Schilderswijk proefschrift (Groningen 1996). Jong, R.A. de, Wiskobas in methoden proefschrift (Utrecht 1986). Katz, V.J., A history of mathematics (New York 1993). Klausmeier, H.J., J.K. Davis, ‘Transfer of learning’, in: Ebel, R.L., et al (ed.), Encyclopedia of educational research (London 1969) 1483–1493. Kleijne, W., ‘nocw vijftig jaar’, Nieuw Archief voor Wiskunde 5 (2004) 308–313. Klein, D., ‘A brief history of American K-12 mathematics education in the 20th century’, in: J. Royer (ed.), Mathematical Cognition (Greenwich 2003). Kline, M., Why Johnny can’t add: the failure of the New Math (New York 1973). Klomp, H.A., De relativiteitstheorie in Nederland. Breekijzer voor democratisering in het interbellum proefschrift (Utrecht 1997). Knegtmans, P.J., Een kwetsbaar centrum van de geest. De Universiteit van Amsterdam tussen 1935 en 1950 (Amsterdam 1998). Kohnstamm, Ph.A., Keur uit het didactisch werk van Prof. Dr. Ph. Kohnstamm (Groningen 1952). Kohnstamm, R., Kleine ontwikkelingspsychologie I. Het jonge kind (Houten 1993). Kolesnik, W.B., Mental discipline in modern education (Madison 1958). Koning, J., Enige problemen uit de didactiek der natuurwetenschappen in het bijzonder van de scheikunde proefschrift (Dordrecht 1948).

378

Bibliography

Kool, M.J.H., Die conste vanden getale. Een studie over Nederlandstalige rekenboeken uit de vijftiende en zestiende eeuw, met een glossarium van rekenkundige termen proefschrift (Utrecht 1999). Kossmann, E.H., De lage landen. Deel ii 1914–1980 (Amsterdam 1976). Krooshof, G., ‘Moderniseren – nieuwbouw of verbouw?’, Euclides 42 (1966) 193–203. Krooshof, G., ‘Euclides. Tijdschrift voor de didactiek van de wiskunde’, Euclides 45 (1969) 1. Kruit, P.C. van der, K. van Berkel (eds.), The legacy of J.C. Kapteyn. Studies on Kapteyn and the development of modern astronomy (Londen 2000). Kuhn, T.S., The structure of scientific revolutions (Chicago 1962). Lange, J. de, ‘In memoriam Hans Freudenthal’, Nieuwe wiskrant 10 (1990) 3. Langeveld, M.J., Inleiding tot de studie der paedagogische psychologie van de middelbare schoolleeftijd (Groningen 1954, eerste druk 1937). Langeveld, M.J., ‘Over de voorbereiding van het gebruik van exacte methoden in bewerking en interpretatie van psychologische onderzoekingen’, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde 30 (Amsterdam 1967) 187–204. Langeveld, M.J., Opvoedingshulp als groeiende wetenschap (Amsterdam 1972). Leeman, H.Th.M., L.N.H. Bunt, P.G.J. Vredenduin, ‘Verslag van het seminarium “New thinking in school mathematics” van de oees’, Euclides 35 (1960) 218–229. Leen, A., De ontwikkeling van het rekenonderwijs op de lagere school in de 19e en het begin van de 20e eeuw (Groningen 1961). Lenstra, H.W., ‘Boekbespreking. Prof.dr. Hans Freudenthal Waarschijnlijkheidsrekening en statistiek’, Euclides 32 (1957) 248–249. Levering, B., ‘De betekenis van M.J. Langeveld voor de naoorlogse pedagogiek (met het accent op de periode 1945–1960)’, Pedagogisch Tijdschrift 16 (1991) 147–160. Loonstra, F., P.G.J. Vredenduin, Modernization of mathematical teaching in the Netherlands (Groningen 1962). Maanen, J.A. van, Facets of seventeenth centrury mathematics in the Netherlands proefschrift (Utrecht 1987). Maanen, J.A. van, ‘In memoriam prof.dr. E.M. Bruins (1909–1990)’, Tijdschrift voor de Geschiedenis der Geneeskunde, Natuurwetenschappen, Wiskunde en Techniek 14 (1991) 116–118. Mandemakers, C.A., Gymnasiaal en middelbaar onderwijs. Ontwikkeling, structuur, sociale achtergrond en schoolprestaties, Nederland, ca. 1800–1968 proefschrift (Rotterdam 1996). Matthijsen, M.A.J.M., De elite en de mythe. Een sociologische analyse van de strijd om onderwijsverandering (Deventer 1982). Meeder, M., F. Meester, Vrouwiskundig. Meisjes in het wiskundeonderwijs (Amsterdam 1984). Melsen, A.G.M. van, ‘Herdenking van Evert Willem Beth (7 juli 1908–12 april 1964)’, Jaarboek der Koninklijke Nederlandse Akademie van Wetenschappen 1964–1965 (Amsterdam 1965). Meijer, W.A.J., Stromingen in de pedagogiek (Baarn 1996). Metselaars, H.J.A.H.G., ‘Het archief van Hans Freudenthal voor onderzoek beschikbaar’, Nieuw Archief voor Wiskunde 4/9 (1991) 211–212. Moerdijk, I., J.P. Murre, ‘In memoriam Willem Titus van Est’, Nieuw Archief voor Wiskunde 5/4 (2003) 281–283. Monna, A.F., ‘Bericht over de werkzaamheden van de Commissie Modernisering Leerplan Wiskunde’, Euclides 41 (1965) 104–107. Monna, A.F., ‘Bericht van de Commissie Modernisering Leerplan Wiskunde’, Euclides 42 (1966) 129–134. Monna, A., M. van der Put, ‘Ambtenaar en wiskundige’, Nieuw Archief voor Wiskunde 5 (2004) 136–146. Molenaar, L., ‘Wij kunnen het niet langer aan de politici overlaten.’ De geschiedenis van het Verbond van Wetenschappelijke Onderzoekers 1946–1980 (Delft 1994). Molenaar, L., De rok van het universum. Marcel Minnaert, astrofysicus 1893–1970 (Amsterdam 2003). Mooij, H., Over de didactiek van de meetkunde benevens benaderingsconstructies van een hoek in gelijke delen proefschrift (Amsterdam 1948).

Bibliography

379

Moor, E.W.A. de, ‘Het ‘gelijk’ van Tatiana Ehrenfest-Afanassjewa’, Nieuwe Wiskrant 12 (1993) 15–24. Moor, E.W.A. de, ‘Het ongenoegen van Dijksterhuis’, Nieuwe Wiskrant 15 (1996) 22–26. Moor, E.W.A. de, Van vormleer naar realistische meetkunde. Een historisch-didactisch onderzoek van het meetkundeonderwijs aan kinderen van vier tot veertien jaar in Nederland gedurende de negentiende en twintigste eeuw proefschrift (Utrecht 1999). Moor, E.W.A. de, ‘Het kistje van Albada’, Nieuwe Wiskrant 21 (2001) 13–22. Morsch, C.J.J.A., Met de moed van de hoop. Studies over de vernieuwing van opvoeding, onderwijs en maatschappij in Nederland in de periode tussen ± 1930 en 1984 (Eindhoven 1984). Nelissen, J., ‘Eeuwenlang rekenen. Historische ontwikkelingen in het rekenonderwijs’, Willem Bartjens 18 (1998) 36–39. Nepveu, R.M., et al., Universitaire flitsen: van Illustre School tot Rijksuniversiteit (Utrecht 1986). Neut, D.N. van der, A. Holwerda, Meetkunde. Met de beginselen der goniometrie. Eerste deel (Groningen 1959). Noël, G., ‘Pourquoi, pour qui enseigner les mathématiques? Une mise en perspective historique de l’évolution des programmes, au xxe sciècle, en Belgique’, Zentralblad für Didaktik der Mathematik 34 (2002) 110–119. Organisation for European Economic Cooperation. Office for Scientific and Technical Personnel, New thinking in school mathematics (1961). Parreren, C.F. van, Intentie en autonomie in het leerproces proefschrift (Amsterdam 1951). Parreren, C.F. van, Psychologie van het leren. Verloop en resultaten van leerprocessen (Zeist 1960). Parreren, C.F. van, J.A.M. Carpay (eds.), Sovjetpsychologen aan het woord. Leerpsychologie en onderwijs 2 (Groningen 1972). Parreren, C.F. van, Leren op school (Groningen 1976, tiende herziene druk). Pelupessy, B., ‘Enkele annotaties bij “Wandelingen met Bastiaan”’, Onderwijs in Natuurwetenschap. Een nieuwsbrief over de integratie van natuurwetenschappen in het basis- en voortgezet onderwijs 2 (1975) 17–20. Philippi-Siewertsz van Reesema, C., ‘Inleiding tot de werken van Prof. dr. Jean Piaget’, Paedagogische Studiën 10 (1929) 1–52. Philippi-Siewertsz van Reesema, C., Uit en over de werken van Prof.dr. Ovide Decroly (Groningen 1931). Piaget, J., B. Inhelder, La représentation de l’espace chez l’enfant (Paris 1948). Piaget, J., ‘Comments on mathematical education’, in: A.G. Howson (ed.), Development in mathematical education. Proceedings of the second international congress on mathematical education (Cambridge 1973) 79–87. Polak, F.L., ‘Mens, techniek en cultuur’, in: C.J. Drippel e.a., Mens, cultuur en techniek (Den Haag 1954). Polya, G. How to solve it (New York 1957, eerste druk 1945). Pulliam, J.D., History of education in America (Columbus 1976). Riedesel, C.A., P.C. Burns, ‘Research in the teaching of elementary-school mathematics’, in: R.M.W. Travers et al (eds.), Second handbook of research on teaching (Chicago 1973) 1149– 1176. Rupp, J.C.C., Van oude en nieuwe universiteiten (Den Haag 1997). Scheltens, J., ‘Natuurkunde en wiskunde’, Euclides 5 (1929) 268–269. Schenk, F., De Utrechtse school. De geschiedenis van de Utrechtse psychologie tussen 1945 en 1965 (Utrecht 1982). Schogt, J.H., ‘De mechanica als afzonderlijk leervak op de hoogere burgerschool’, Euclides 5 (1928) 70–79. Schuyt, K., E. Taverne e.a., 1950. Welvaart in zwart-wit. Nederlandse cultuur in Europese context (Den Haag, 2000). Senegal, M., ‘The continuing silence of Bourbaki – an interview with Pierre Cartier’, Mathematical Intelligencer 20 (1998) 22–28. Sime, M., Zoals een kind het ziet. Wegwijzer in de theorieën van Piaget (Purmerend 1976).

380

Bibliography

Smid, H.J., Een onbekookte nieuwigheid? Invoering, omvang, inhoud en betekenis van het wiskundeonderwijs op de Franse en Latijnse scholen 1815–1863 proefschrift (Delft 1997). Smith, L., J. Dockrell, P. Tomlinson (eds.), Piaget, Vygotsky and beyond. Future issues for developmental psychology and education (London 1997). Staring, A.J., ‘Doel en middelen bij het onderwijs in de mechanica’, Euclides 18 (1941) 115–124. Streefkerk, H., ‘Het leerplan voor de wiskunde op de h.b.s.’, Euclides 21 (1945) 30–39. Streefkerk, H., ‘Kan het wiskundeonderwijs tot de opvoeding van het denkvermogen bijdragen?’, Euclides 27 (1951) 57–59. Streefkerk, H., ‘Problemen rondom de vernieuwing van het wiskunde-onderwijs’, in: Stemmen uit de praktijk. Vernieuwing van het wiskundeonderwijs. Inleidingen en verslagen van de conferentie wiskunde, georganiseerd door de afdeling v.h.m.o. van het Christelijk Paedagogisch Studiecentrum (werkgroep wiskunde) op 6 en 7 maart 1964 (Den Haag 1964) 7–19. Streefland, L., ‘Wiskunde als activiteit en de realiteit als bron’, Nieuwe Wiskrant 5 (1985) 60–67. Streefland, L., Realistisch breukenonderwijs. Onderzoek en ontwikkeling van een nieuwe leergang proefschrift (Utrecht 1988). Streefland, L., (ed.), The Legacy of Hans Freudenthal (Dordrecht 1993). Streefland, L., Learning from history for teaching in the future (Utrecht 1996). Streun, A. van, Heuristisch wiskunde-onderwijs. Verslag van een onderwijsexperiment proefschrift (Groningen 1989). Struik, D.J., Geschiedenis van de wiskunde (Utrecht 1990). Syswerda, A.H., De ruimtevoorstelling bij het kind volgens J. Piaget en B. Inhelder. Synthese en critiek (Groningen 1955). Terwel, J., Onderwijs maken proefschrift (Harlingen 1984). Timmer, J.K., ‘De wiskunde werkgroep na 1945’, Vernieuwing van Opvoeding en Onderwijs 183 (1961) 185–186. Thijssen, W.J., J.H. Wansink, ‘Het internationale mathematische congres in 1958’, Euclides 34 (1959) 225–245. 376 “elke positieve actie begint met critiek” Turkstra, H., J.K. Timmer, Rekendidactiek (Groningen 1953). Treffers, A., Wiskobas doelgericht. Een metode van doelbeschrijving van het wiskundeonderwijs volgens wiskobas proefschrift (Utrecht 1978). Treffers, A., E.W.A. de Moor, ‘Het aanvankelijk meetkundeonderwijs’, Euclides 50/51 (1974) 81–99. Turkstra, H., Het veelomstreden vraagstuk van het wiskunde-onderwijs op de middelbare school (Zutphen 1938). Uhlenbeck, E.M., ‘Professor Freudenthals kleinzoontje verdedigd tegen zijn grootvader’, Forum der Letteren 16 (1975) 253–254. Velden, B. van der, ‘Between “Bastiaan ou de l’éducation” and “Bastiaan und die Detektive” ’, Zentralblad für Didaktik der Mathematik 32 (2000) 201–202. Velthuys-Bechthold, P.J.M., Inventory of the papers of Hans Freudenthal (1905–1990), mathematician. 1906–1990 (Haarlem 1999). Verbist, R., A. de Block, Experimenteel-didaktisch onderzoek naar de waarde van denkpsychologisch georiënteerd rekenonderwijs (Amsterdam 1961). Verhoeven, P., Tekstbegrip in het onderwijs Klassieke Talen. Een onderzoek naar de verwezenlijking van nieuwe doelstellingen in het schoolonderzoek Grieks en Latijn proefschrift (Leiden 1997). Vos, J., J. van der Linden, Waarvan akte. Geschiedenis van de mo-opleidingen 1912–1987 (Assen 2004). Vredenduin, P.G.J., ‘Een opzienbarend boek’, Euclides 39 (1963/64) 237–247. Vredenduin, P.G.J., ‘Het experiment Papy’, Euclides 42 (1966) 167–182. Vries, G. de, De ontwikkeling van wetenschap. Een inleiding in de wetenschapsfilosofie (Groningen 1984). Vuyk, R., Overview and critique of Piaget’s genetic epistemology 1965–1980 (Londen 1981). Vuysje, B., ‘De wiskundeles verandert: 3 + 2 = 1’, HP Magazine (24 december 1969–6 januari 1970).

Bibliography

381

Wanders, W.E., S. Bohncke, Boeiend rekenen voor de lagere school. Algemene inleiding (’s-Hertogenbosch 1950). Wansink, J.H., ‘Mathematical teaching in Dutch secondary schools’, Euclides 26 (1950) 99–114. Wansink, J.H., ‘De wiskunde werkgroep van de w.v.o.’, Euclides 28 (1953) 197–205. Wansink, J.H., ‘Didactische revue’, Euclides 34 (1958) 165–175. Wansink, J.H., Didactische oriëntatie voor wiskundeleraren. Delen i, ii, iii (Groningen, 1969). Weijers, I., ‘Mondige burgers. Een cultuurhistorische plaatsbepaling van de pedagogiek van M.J. Langeveld’, Pedagogisch Tijdschrift 19 (1994) 189–206. Wijdeveld, E., ‘Omzien in verwondering. Ontstaan, werkwijze en effecten van de Commissie Modernisering Leerplan Wiskunde en het Instituut Ontwikkeling Wiskunde Onderwijs (1960– 1970)’, Euclides 78 (2003) 218–225. Wijdenes, P., Planimetrie. Eenvoudig schoolboek voor het eerste onderwijs in de vlakke meetkunde (Groningen 1931). Wijdenes, P., H.J.E. Beth, Nieuwe school-algebra. Derde deel (Amsterdam 1947). Willoughby, S.S., ‘Mathematics’, in: R.L. Ebel et al (eds.), Encyclopedia of educational research (London 1969) 766–777. Wiskunde Werkgroep van de Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs, Nieuwe wegen bij het onderwijs in de wiskunde en de natuurwetenschappen (Purmerend 1953). Wiskunde Werkgroep van de Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs, De aansluiting van het rekenonderwijs op de lagere school tot het wiskundeonderwijs op de middelbare school (Purmerend 1958). Wolthuis, J., ‘De ontwikkeling van de algemene onderwijsvakbeweging’ (doctoraalscriptie onderwijssociologie Rijksuniversiteit Groningen) 1980. Wolthuis, J., ‘Theorievorming over de Nederlandse onderwijsgeschiedenis: een terreinverkenning’, Info. Tijdschrift voor onderwijswetenschappen 16 (1985/86) 228–251.

Index

A Abbiw-Jackson, D.K Abel, N.H. Achter, V. van Adda, J. Albada, P.J. van Alberts, G. Alders, C.J. Andel, J. van B Baan, P.H.A. Barrau, J.A. Beberman, M. Begle, E.G. Behnke, H. Berkel, K. van Berlage, H.P. Bernoullis, Johan Beth, E.W. Beth, H.J.E. Bieberbach, L. Bijpost, M. Bishop, A.J. Blij, F. van der Bloom, B.S. Bockwinkel, H.B.A. Boeke, K. Boermeester, Chr. Bolkestein, G. Boomsma, G. Bosschart, R.A.J. Bouman, P.J. Bourbaki-avant-la-lettre Brandenburg, W.J. Breughel-Vollgraff, D. Brouwer, L.E.J. Bruins, E.M.

Bruner, J. Bruyn, N.G. de Bunt, L.N.H. Buytendijk, F.J.J. C Cals, J.N.L.Th. Cantor, G.F.L.Ph. Carpay, J.A.M. Cartan, H. Casimir, R. Castelnuovo, E. Chasles, M. Chevalley, C. Choquet, G. Clay, J. Corput, J.G. van der Coster, B. Coulomb, J. Cramer, P. D Dalen, D. van Dante (Durante degli Alighieri) Dantzig, D. van Davydov, V.V. Decroly, O. Dedekind, R. Deetman, W.J. Delsarte, J. Dewey, J. Dieleman, G. Diels, P.A. Dienes, Z.P. Dieudonné, J.A.E. Dijksterhuis, E.J. Dol, mej. Dormolen, J. van

© Springer Science+Business Media Dordrecht 2015 S. la Bastide-van Gemert, All Positive Action Starts with Criticism, DOI 10.1007/978-94-017-9334-6

383

Index

384 Douwes, B.J. Drewes, J.B. Dulieu, J. Dynkin, E.B. E Ehrenfest, P. Ehrenfest-Afanassjewa, T. Ehresmann, C. Eilander, J. Einstein, A. Engel, A. Est, W.T. van F Félix, L. Fehr, H. Feigl, G. Flaubert, G. Fletcher, W.T. Floris, V. Fokker, A.D. Fröbel, F. Freudenthal, Hans Freudenthal, Jedidja Freudenthal, Joseph Freudenthal, L. Freudenthal, Matthijs Freudenthal, Mirjam Freudenthal, T. Freudenthal-Ehmann, E. Freudenthal-Lutter, S.J.C. G Gödel, K. Gagné, R.M. Gal’perin, P.J. Gangadharan, K.S. Gass, J.R. Gattegno, C. Gelder, L. van Gerretsen, J.C.H. Ginjaar-Maas, N.J. Gleason, A.M. Gravemeijer, K.P.E. Griffiths, H.B. Groeneveld, B. Groot, J. de Grootendorst, A.W. Grosheide, J.H. Gunning, J.H.

H Haantjes, J. Hadamard, J. Hammersley, J.M. Heemert, P.C. van Herbart, J.F. Heyting, A. Hiele, P. van Hiele-Geldof, D. van Hijmans van den Bergh, L.J. Hilbert, D. Hilton, P. Holwerda, A. Hopf, H. Hurewicz, W. Huygens, C. I Imelman, J.D. Iongh, J.J. de J Jacobi, C.G.J. Jacobs, H.J. Janssen, G.A. Janssen, H.H. Janssen, J.R. Jensema, E. Jong, R.A. de Julia, G. Julius, W.H. K Kühnel, J. Keitel, C. Kemenade, J. van Keune, F.J. Klafki, W. Klein, F. Kline, M. Klomp, H.A. Kloosterman, H.D. Kohnstamm, Ph.A. Koksma, J.F. Koning, J. Kossmann, E.H. Krogt, B. van der Krooshof, G. Krygovska, A.Z. Kuhn, T. Kwantes, R.C.

Index L Löwner, K. Lange, J. de Langeveld, M.J. Leeman, H.Th.M. Leeuw, G. van der Legendre, A.M. Leibniz, G.W. Lenstra, H.W. Leujes, D. Ligthart, J. Limburg, J. Lockefeer, L. Lohuizen, T. van Loo, P.J. van Luning Prak, J. M Mandelbrojt, S. Mannoury, G. Martin, W.T. Maupertuis, P.L. Meijer, W.A.J. Metzger, H. Meyer, C.F. Minnaert, M.C.J. Mises, R. von Molenaar, L. Mollenhauer, K. Monna, A.F. Montessori, M. Mooij, H. Moor, E.W.A. de Moyer, J.C. N Nauta, J. Nelissen, J. Neumann, J. von Neut, D.N. van der Newton, I. Nijhof, H. O Odysseus Oort, J. van Ornstein, L.S. P Papy, F. Papy, G. Parreren, C.F. van Pelupessy, B.

385 Pestalozzi, J.H. Petersen, P. Philippi-Siewertsz van Reesema, C. Piaget, J. Picard, E. Picard, N. Piekaar, A.J. Planck, M. Pollak, H.O. Polya, G. Pompe, W. Popken, J. Possel, R. de R Rümke, H.C. Reckendorf, A.O. Reindersma, W. Revuz, A. Roloff, H.P. Rosenthal, A. Rutten, F.J.Th. S Schaake, G. Schmidt, E. Schoemaker, G. Schoenfeld, A. Schogt, J.H. Schrier, L. Schur, I. Selz, O. Servais, W. Sijmons, B. Sirolf, V. Sittig, J. Smid, H.J. Sobolev, S. Steiner, H.G. Stern, W. Stone, M.H. Streefkerk, H. Streun, A. van Struik, H.J. Stubenrouch, G.C. Suppes, P. Syswerda, A.H. Szeminska, A. T Telemann, G.P. Terlep, J.F. Terwel, J.

386 Thijssen, W.P. Thorbecke, J.R. Thorndike, E.L. Thwaites, B. Timmer, J.K. Tol, M.G. van Treffers, A. Tucker, A.W. Turkstra, H. U Uhlenbeck, E.M. Unnik, W.C. van V Velden, B. van der Verrijp, D.P.A. Versluys, J. Vos van Steenwijk, A.A. de. Vossen, H.M.M. Vredenduin, P.G.J.

Index W Walusinski, G. Wansink, J.H. Washburne, C. Waterink, J. Weil, A. Weyl, H. Wijdenes, P. Wijdeveld, E. Wittenberg, A. Wolff, J. Y Yaglom, I.M. Z Zelm, J.C. van Ziller, T.

All Positive Action Starts with Criticism

Recommend Documents