WDS'12 Proceedings of Contributed Papers, Part I, 66–71, 2012.
ISBN 978-80-7378-224-5 © MATFYZPRESS
Josef Úlehla and His Mathematics Textbooks for Secondary Schools L. Vízek Charles University Prague, Faculty of Mathematics and Physics, Prague, Czech Republic.
Abstract. In the introduction the author of mathematics textbooks Josef Úlehla (1852–1933) is introduced and the attention is turned to his textbooks he wrote for secondary schools. The survey of his works is presented, material is evaluated and compared and so far unpublished methodical appendix introduced. Finally possible influence of Úlehla’s work on contemporary teaching is considered.
Introduction Josef Úlehla was born on 16 March 1852 in Podivín near Břeclav. He studied at high schools in Strážnice and in Brno. He passed his school leaving exam at teacher’s college in Brno in 1873. He never studied at university. All his professional life he taught mathematics and science at elementary and later secondary schools in Moravia. Since 1897 he was a headmaster at secondary school in Klobouky near Brno. The same position he held in Jaroměřice nad Rokytnou in 1905 and in Strážnice from 1905 to 1912. He was the leading personality of Moravian teachers and actively contributed to forming of teachers associations. He died in Lipov on 22 December 1933. During his life he published more than 150 works, wrote articles, paid attention to didactics and methodology of sciences, studied the history of mathematics and philosophy. He also translated the works of English reformist teachers into Czech. He published two-part monograph Dějiny mathematiky (The History of Mathematics), 1 in higher mathematics the textbook Počet infinitesimální (The Infinitesimal Calculus). 2 For secondary school he wrote and published textbooks on natural science and mathematics. Now we are going to deal with the last mentioned books. 3
Mathematics textbooks for secondary schools Before we start analyzing Úlehla’s textbooks we would like to point out what the position of secondary school in the school system was during his life. At the same time we will explain who the textbooks were for. In 1869 the system of education in Austro-hungarian monarchy was changed according to a new law. Under the summarizing name obecné školy (basic schools) there were obyčejné obecné školy (primary schools) and so called měšťanské školy (secondary schools) established. They were usually not mixed ones, but just for boys or girls (single-sex schools). Obecné školy that are similar to our today’s Czech základní školy had eight grades. They were obligatory and were attended by children from the age of 6 to 14. Obyčejné obecné školy (founded mainly in villages) and měšťanské školy (established above all in towns) had both either eight grades. Sometimes obyčejné obecné školy had five years and were followed by měšťanské školy, which had three years. The system was inhomogeneous. In 1883 all měšťanské školy were changed into three-year měšťanské školy. They corresponded to today’s second level of Czech základní školy. 4 Josef Úlehla wrote mathematics textbooks for all three grades of měšťanské školy and he prepared special versions for girls’ and boys’ schools. The first edition of all of them appeared in 1909. To analyze the text we will make use of the textbook for boys, because the one for girls was, as we would like to prove, derived from it. Početnice pro měšťanské školy chlapecké, stupeň I. a II. The textbook Početnice pro měšťanské školy chlapecké, stupeň I. a II. (Arithmetic book for secondary boys’ schools, grades I and II) for the first and the second grades of měšťanské školy (similar to contemporary 6th and 7th grades) forms single volume. It has 75 pages. The first part, intended for the first grade, contains the 1
Úlehla J.: Dějiny mathematiky. Dědictví Komenského, Praha, part I, 1901 and part II, 1913. Úlehla J.: Počet infinitesimální. Dědictví Komenského, Praha, 1906. 3 About life and work of Josef Úlehla see Vízek L.: Josef Úlehla and His Calculus Textbook. In Šafránková J. and Pavlů J. (eds.): WDS'11 Proceedings of Contributed Papers: Part I – Mathematics and Computer Sciences. Matfyzpress, Prague, 2011, p. 91–94, or Vízek L.: Josef Úlehla (1852–1933) a jeho Dějiny mathematiky. In Bečvář J., Bečvářová M. (eds.): 32. mezinárodní konference Historie matematiky. Matfyzpress, Praha, 2011, p. 275–284. 4 For more see Kádner O.: Vývoj a dnešní soustava školství. Sfinx Bohumil Janda, Prague, part I, 1929. 2
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VÍZEK: JOSEF ÚLEHLA AND HIS MATHEMATICS TEXTBOOKS FOR SECONDARY SCHOOLS following chapters: I. Počítání (Counting), II. Základní čísla, společná míra, společný násobek (Natural numbers, greatest common divisor, least common multiple), III. Soustava desetinná (Decimal numeral system), IV. Číslice římské (Roman numeral system), V. Základní úkony početní (Basic numerical operations), VI. Počítání s čísly vícejemnými (Counting with plural numerals), VII. Počítání zlomkové (Counting with fractions), VIII. Počítání úsudkové (Counting by judgement), IX. Počet procentový (Percent calculus) and X. Jednoduchý počet úrokový (Simple interest calculus). The first part consists of the following parts: I. O poměru a srovnalosti (On proportion and comparison), II. Trojčlenka jednoduchá (Simple crossmultiplication), III. Trojčlenka složená (Compound cross-multiplication), IV. Umocňování a odmocňování dvěma (Second power and square root), V. Počet procentový (Percent calculus), VI. Jednoduchý počet úrokový (Simple interest calculus), VII. Počet diskontový (Discount calculus), VIII. Počet lhůtový (Periodical calculus), IX. Počet průměrný, směšovací a spolkový (Average, mixing and associational calculus), X. Příklady k opakování (Revision exercises) and Míry, váhy a peníze (Measures, weights and money). It is not possible and reasonable to analyze all the chapters in detail within this short article. Our aim is to give an idea of the style in which the book was written. So we will choose just several suitable parts. First let us follow the first part of the book. The first chapter entitled Počítání (Counting) contains above all motivational exercises and arithmetical problems to revise the topics of previous grades. Let us show (p. 1 and 2): 1. Otec vysypal na stůl hrst drobných peněz a počítal; co chtěl věděti. (Father emptied a handful of small change at the table and counted; what did he want to know.) 13. Vypozorujte, za kolik minut projdete km při klidné chůzi, za kolik minut jej proběhnete; za kolik minut jej projede cyklista, automobil, kočár, vůz s těžkým nákladem. (Observe how long does it take you to walk a kilometer, how long does it take you to run it; how long does it take the cyclist, the car, the carriage and a heavily loaded van to go it through.) 16. Změřte, jak jste silni, kolik kg utlačíte na siloměru. (Measure how strong you are, how many kilograms do you push on the dynamometer.) Úlehla put emphasis on relationship between studied topics and real life and based his exercises on realistic information. As a proof we can mention exercise 67 from chapter V. Základní úkony početní (Basic numerical operations), in which the basic arithmetical operations – addition, subtraction, multiplication and division – are dealt with (p. 17): Z nového Yorku vyvezlo se 12. července 1902 do Evropy 222.000 bušlů pšenice. Kolik je to angl. liber, počítá-li se bušl za 60 lb? Kolik kg, je-li angl. libra 453⋅598 g? (From New York to Europe there was imported 222.000 bushels of wheat on 12 July 1902. How many British pounds does it equal if a bushel is 60 pounds? How many kilograms if British pound equals 453⋅598 g?) During Úlehla’s times mathematics was more devoted to counting than today. It is understandable as there were no calculators available. Let us have a look how Úlehla explained square root in the second part of the book (p. 45): 22. Sestroj čtverec, jenž má 2116 cm2. 16 dm2 jest plocha čtverce, jehož strana jest dlouhá 4 dm = 40 cm; 516 cm2 jest plocha obdélníku, který se k tomuto čtverci připojuje po dvou jeho stranách jako pás a s ním činí nový čtverec. Ten jest pak delší než 80 cm. Přiměříme-li 80 k 516, vypočteme, že obdélník jest široký 6 cm, dlouhý 86 cm, plocha jeho jest 86 cm x 6 cm = 516 cm2. Výpočet tento upravujeme takto: √(22|16) = 46 ‒ 16 516 : 86 ‒ 86 x 6 Žádaný čtverec má stranu dlouhou 46 cm. (Draw a square which has the area of 2116 cm2. 16 dm2 is the area of the square, its side is 4 dm long = 40 cm; 516 cm2 is the area of rectangle attached to the square at both sides as a stripe and form a new square with it. This is longer than 80 cm. Comparing 80 to 516 we will get that the rectangle is 6 cm wide, 86 cm long, its space is 86 cm x 6 cm = 516 cm2. This calculation we adjust like this (etc). Required square has a side 46 cm long.) Approximately one third of the second part deals with the basis of financial mathematics. Included exercises are closely connected with real life and prepares the pupils for the themes of the final grade which is mainly focused on financial mathematics.
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VÍZEK: JOSEF ÚLEHLA AND HIS MATHEMATICS TEXTBOOKS FOR SECONDARY SCHOOLS Početnice pro měšťanské školy chlapecké, stupeň III. The textbook Početnice pro měšťanské školy chlapecké, stupeň III. (Arithmetic book for secondary boys’ schools, grades III) has 91 pages and consists of the following chapters: I. Umocňování a odmocňování třemi (Third power and third root), II. Složitý úrok (Compound interest), III. Výpočty pojišťovací (Insurance counting), IV. Počet mincovní (Coin calculus), V. Cenné papíry (Securities), VI. Řetězový počet (Chain calculus), VII. Obchod a knihy obchodní (Trade and accounting books), VIII. O číslech protivných (On opposite numbers), IX. O číslech obecných (On general numbers), Rovnice (Equations), X. Příklady k opakování (Revision exercises), Tabulky (Tables), Kursovní list (Exchange rates), Míry, váhy a peníze (Measures, weights and money) and Ukázka obchodních knih (Excerpts from accounting books). Similarly to the previous book we will choose several illustrative examples. Financial mathematics is represented by the second chapter entitled Složitý úrok (Compound interest). Let us demonstrate how Úlehla explained it (p. 7): Hochovi zbylo po rodičích K 10.000. Tyto peníze uložil poručník do spořitelny na 4%. Kolik peněz vyplatila by spořitelna po 10 letech? Kdo nechá peníze ve spořitelně a nevybere úroku, tomu se připočítává úrok na konci úrokové lhůty k jistině a pak dostane úrok už i z tohoto úroku. Nyní připočítávají spořitelny úrok po půl roce; když se připočítává úrok po roce jest ve spořitelně na konci roku prvního: 4% z K 10.000 = 400⋅‒ K 10.400⋅– na konci roku prvního: 4% z K 10.400 = 416⋅‒ K 10.816⋅‒ ... a t. d. na konci roku desátého: K 14.802⋅44. (The son inherited 10.000 crowns from their parents. His guardian paid this sum into account in the savings bank with the interest of 4%. How much money would be paid back by the bank ten years after? If somebody keeps money in the bank and does not collect interest, this interest is credited to the principal sum at the end of interest period and then he gets interest from this interest (compound interest). Nowadays saving banks credit interest twice a year; if the interest is credited a year after, in the saving bank is at the end of the first year: 4% of 10.000 = 400⋅‒ 10.400⋅‒ crowns at the end of the first year: 4% of 10,400 = 416⋅‒ 10.816⋅‒ crowns ... etc. at the end of the tenth year 14.802.44 crowns.) In the following example we will concentrate on equations and word tasks solved by them. We would like to point out that the textbook deals only with linear equations with one unknown. First, we will show how the equations were explained and then how the author worked with so called equivalent equation adjustment (p. 45): 18. 5 = 5, 7 = 7, 4 + 5 = 9, 5 ‒ 1 = 4, x = 9. Spojíme-li rovnítkem dvě rovná čísla, vznikne rovnice. (If we connect two equal numbers with a sign of equation, we will get an equation.) 21. Vypočítej x na příkladech: x ‒ 4 = 5, x + 4 = 5. Řešení. Součet, rozdíl, součin a podíl dvou rovnic jest rovnice nová. Proto jest: a) x‒4=5 b) x + 4 = 5 +4=+4 –4=–4 x ‒ 4 +4 = 5 + 4 x+4–4=5–4 x=9 x=1 ... atd. ([3], str. 44) (Find x in examples: x ‒ 4 = 5, x + 4 = 5. Solution. Sum, difference, product, quotient of two equations is a new equation. Therefore it is (etc.). Some of Úlehla’s exercises are not only closely connected to the real life, but they are in a way unusual and introduce original humor into teaching. The following exercises can serve as an illustration (p. 46): 36. Pacholek s koňmi zavláčil by pole za 15 hodin, volák za 21 hodinu; za kolik hodin je zavláčí spolu? (36. The groom with a horse can harrow the field in 15 hours, a farmer with an ox in 21 hours. How long will it take them to cultivate the field together?) 42. Dvě slečinky z města šly vedle pasačky a ptaly se: „Kolik je vás? Jistě sto!“ ‒ „Těchto hus není sto! Ale, kdyby jich byla ještě polovice, čtvrtina a vy dvě k tomu, bylo by sto!“ (42. Two young ladies from a town passed a shepherdess and asked her: “How many are you altogether? Surely a hundred.” – “There is not a hundred geese. But if there was still half, a quarter and you two, so it will make a hundred.”)
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VÍZEK: JOSEF ÚLEHLA AND HIS MATHEMATICS TEXTBOOKS FOR SECONDARY SCHOOLS Početnice pro měšťanské školy dívčí, stupeň I. a II. The textbook Početnice pro měšťanské školy dívčí, stupeň I. a II. (Arithmetic book for secondary girls’ schools, grades I and II) is similar to the issue for boys. It contains the same chapters in the same order. There are no differences in explanation of various topics, it contains the same examples. The number of exercises is lower and therefore there are fewer pages in the book. Some simpler and motivational exercises were omitted and substituted by several so-called girls. In the issue for girls the example number 16 about force measurement is absent. Also example number 13 about running, walking a driving a kilometer is substituted by the following one (p. 4): 13. Můžete-li, spočítejte, kolik polen jest metr polenového dříví. (13. If you are able to, count how many logs are there in a metre of log wood.) Typical exercises for girls are about sewing, housework and cooking. Let us introduce one original task about vegetable (p. 30): 30. Zvaž prostřední cibuli a z toho vypočítej, kolik q cibule potřebují asi do roka města Praha, Brno, Ostrava, Opava, Vídeň, potřebuje-li osoba na týden průměrně 2 cibule? (30. Weigh an average onion and count, how many quintals will need Prague, Brno, Ostrava and Vienna a year supposing that a person eats two onions a week?) Početnice pro měšťanské školy dívčí, stupeň III. This book (Arithmetic book for secondary girl schools, grades III) contains only 71 pages which is 20 less than the issue for boys. That means that there are also less chapters as several were left out. The following chapters are missing: I. Umocňování a odmocňování třemi (Third power and third root), II. Složitý úrok (Compound interest), VI. Řetězový počet (Chain calculus), VIII. O číslech protivných (On opposite numbers), IX. O číslech obecných (On general numbers) and Rovnice (Equations). The rest of the book remained unchanged.
Arithmetic books for občanské školy The textbooks named Početnice pro měšťanské školy chlapecké, stupeň I. a II. and Početnice pro měšťanské školy dívčí, stupeň III. were re-published during 1915 and 1913. Not only were they newly printed but there was also a new typesetting and different number of pages. The content remained unchanged. Whether other titles were also reprinted would be the subject of further studies. They have not been found in the Czech libraries yet. In the years 1920–1923 Početnice pro občanské školy, stupeň I., II. and III. were published. In imprint we can read “the third revised issue”. We supposed that there were no first and second editions, and that this is the third edition of Početnice pro měšťanské školy. The change of the title probably reflects the effort to change the name of that type to school from the Czech měšťanská škola to občanská škola which was not reached at the end. 5 Početnice pro občanské školy is substantively the same as Početnice pro měšťanské školy for boys, but they are not divided according to sexes, grades I and II are published separately, the sequence of chapters is different and exercises with old content were substituted by new ones. For example the following ones: 16. Rakousko mělo r. 1906 státního dluhu 9.606.400.000 K. (16. Austria had in 1906 the national debt 9.606.400.000 crowns.) Početnice pro měšťanské školy chlapecké, stupeň I. a II. 1909, p. 7. is substituted by this one: 6. Rakouská říše měla roku 1912 dluhu 12.748,894.700 franků; válkou vzrostl tento dluh na 150.000,000.000 fr. (6. The Austrian Empire had in 1912 the debt of 12 748 894 700 francs; due to war this debt increased to 150.000.000.000 francs.) Početnice pro občanské školy, stupeň I., 1920, p. 10. In closing we would like to point out that some parts of Početnice pro občanské školy were published for the fourth, respectively for the fifth time (see References).
The translation of Úlehla’s books into Slovak In the years 1923 and 1924 Úlehla’s textbooks were translated into Slovak. Similarly to the Czech issue for občanské školy those foreign language versions were not divided into two different versions for boys and girls 5
For more see Kádner O.: Vývoj a dnešní soustava školství. Sfinx Bohumil Janda, Praha, part II, 1931, p. 45.
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VÍZEK: JOSEF ÚLEHLA AND HIS MATHEMATICS TEXTBOOKS FOR SECONDARY SCHOOLS and grades I and II were published separately. The content is identical with the issue for občanské školy which was the base for the translation as the imprint says.
Reviews The first edition of Početnice pro měšťanské školy chlapecké and Početnice pro měšťanské školy dívčí was reviewed by Josef Groulík and Konrád Pospíšil (see References). Their opinions about them were mainly positive. They appreciated the comprehensibility of the texts, pointed out the importance of interdisciplinary relationships and positively evaluated Úlehla’s reformist pedagogical tendencies. The only objections they had were about some exercises that are, according to them, unrealistic. They criticized the following ones: 21. Krupař má mouku po 28 h a 35 h, žena chce mouku po 32 h; kterak jí krupař vyhoví? (21. A barley-dealer has flour for 28 and 35 hellers, a woman wants the one for 32 hellers; how can he manage it?) Početnice pro měšťanské školy chlapecké, stupeň I. a II. 1909, p. 65. or 32. Zelinářka má dvojí cibuli, po 60 g a 35 g; kterak vybere 100 cibulí, aby vážily 5 kg? (32. A greengrocer has two kinds of onion, 60 and 35 grams; how can she pick out 100 pieces of them to get 5 kilograms?) Početnice pro měšťanské školy chlapecké, stupeň I. a II. 1909, p. 67. Příklady početní, metodický doplněk k učebnicím pro měšťanské školy In the archives of Moravian Country Museum in Brno Úlehla’s manuscript called Příklady početní, metodický doplněk k učebnicím pro měšťanské školy (Arithmetic exercises, a methodology supplement to the textbooks for secondary schools) is kept and it has never been published. The manuscript contains 24 (respectively 26) parts entitled as follows: 1. Čtení a napisování čísel (Numerals reading and writing), 2. Převádění a rozvádění čísel (Numerals conversion and decomposition), 3. Nejmenší společný násobek (Least common multiple), 4. Rozklad na kmenové činitele (Prime factorization), 5. Nejvyšší společná míra dvou čísel (Highest common factor), 6. Sečítání (Addition), 7. Odčítání (Substraction), 8. Násobení (Multiplication), 9. Dělení (Division), 10. missing, 11. Trojčlenka (Cross-multiplication), 12. Počet směšovací a průměrný (Mixing and average calculus), 13. Počítání podle soustavy šedesátkové (Counting in sexagesimal system), 14. Čísla vícejemná (Plural numbers), 15. Počet procentový (Percent calculus), 16. Počet úrokový (Interest calculus), 17. Počet lhůtový (Periodical calculus), 18. Složitý počet úrokový (Compound interest calculus), 19. Počet mincovní (Coin calculus), 20. Peněžní trh (Financial market), 21. Cenné papíry (Securities), 22. Počet řetězový (Chain calculus), 23. Druhá mocnina (Second power), 24. Třetí mocnina a třetí odmocnina (Third power and third root), 25. missing and 26. Úkoly o práci (Exercises on work). Each part is provided by several solved exercises, after them there are 48 exercises to solve. In the manuscript there is the information about the plan to publish 8 test papers with 6 exercises. So this is a collection of exercises for written tests. Why it was not published will be the theme of further studies.
Conclusion Úlehla’s arithmetic books contain exercises based on real facts, they include a lot of exercises for revision and in later issues they reflect social, political and cultural changes. As it was pointed out in the reviews, some exercises are not realistic. Other negative features are the lack of solved examples and no results. Even nowadays historical textbooks can be inspiring for us. They reflect the original style of Josef Úlehla, his creativity and original approach. They can enrich the mathematics teaching not only at basic schools. Acknowledgments. This work was supported by the grant GA ČR P401/10/0690 Prameny evropské matematiky and the project Specifický vysokoškolský výzkum 2012-261-315.
References Groulík J.: Početnice pro měšťanské školy chlapecké. Pedagogické rozhledy 23 (1909–1910), p. 29–30. Pospíšil K.: Početnice pro měšťanské školy dívčí. Škola měšťanská 11 (1909), příloha odboru matematicko-technického, p. 53–55. Pospíšil K.: Početnice pro měšťanské školy dívčí. Pedagogické rozhledy 23 (1909–1910), p. 593–594. Úlehla J.: Početnice pro měšťanské školy chlapecké, stupeň I. a II. C. k. školní knihosklad, Praha, 1909, 75 pages, 2nd edition, C. k. školní knihosklad, Praha, 1915, 82 pages. Úlehla J.: Početnice pro měšťanské školy chlapecké, stupeň III. C. k. školní knihosklad, Praha, 1909, 91 pages. Úlehla J.: Početnice pro měšťanské školy dívčí, stupeň I. a II. C. k. školní knihosklad, Praha, 1909, 70 pages.
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VÍZEK: JOSEF ÚLEHLA AND HIS MATHEMATICS TEXTBOOKS FOR SECONDARY SCHOOLS Úlehla J.: Početnice pro měšťanské školy dívčí, stupeň III. C. k. školní knihosklad, Praha, 1909, 73 pages, 2nd edition, C. k. školní knihosklad, Praha, 1913, 69 pages. Úlehla J.: Početnice pro občanské školy, stupeň I. 3rd edition, Státní nakladatelství, Praha, 1921, 52 pages, 4th edition, Státní nakladatelství, Praha, 1924, 88 pages, 5th edition, Státní nakladatelství, 1930, 88 pages. Úlehla J.: Početnice pro občanské školy, stupeň II. 3rd edition, Státní školní knihosklad, Praha, 1920, 44 pages. Úlehla J.: Početnice pro občanské školy, stupeň III. 3rd edition, Státní nakladatelství, Praha, 1923, 124 pages, 4th edition, Státní nakladatelství, Praha, 1930, 138 pages. Úlehla J.: Počtovnica pre slovenské školy měštianské. Český učitel 26 (1922–1923), č. 28, 9th February 1923, p. 441. Úlehla J.: Počtovnica pre slovenské školy meštianské, díl I., pre 5. školský rok. Štátne nakladateľstvo, Praha, 1923, 38 pages. Úlehla J.: Počtovnica pre slovenské školy meštianské, díl II., pre 6. školský rok. Štátne nakladateľstvo, Praha, 1923, 94 pages. Úlehla J.: Počtovnica pre slovenské školy meštianské, díl III., pre 7. školský rok. Štátne nakladateľstvo, Praha, 1924, 48 pages. Úlehla J.: Příklady početní, metodický doplněk k učebnicím pro měšťanské školy. Unpublished manuscript, Moravian Country Museum in Brno, fond Josef Úlehla, sign. 41/82, no. S89.
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