1. 1 Eno Tõnisson University of Tartu Estonia Answers Offered by Computer Algebra Systems to Expression Transformation Exercises

2. 2 Plan Background –School –Computer algebra systems Scope Is the answer as expected? –No How to ask? Issues on Domain –Yes Branches Comparison with equations Summary, Future work

3. Motivation Unexpected answers –according to school mathematics Could be differ in different countries and schools Answers to equations from school textbooks offered by Computers Algebra Systems –CADGME Pecs, Tonisson & Velikanova 3

4. 4 Transformation of expressions at school Mathematics lessons, textbooks and tests frequently require transformation of expressions Polynomials Algebraic fractions Exponents, Radicals Trigonometry … Dozens of exercises

5. 5 May or may not be mentioned explicitly in the text Explicitly (same words as in CAS) –Simplify –Factor –Expand Explicitly (even in more detail) –Multiply, using a distributive property –Take out factors common to all terms –Remove parentheses and simplify –Express in simplest radical form –Simplify, leaving answers with negative exponents As a part of more complicated exercise –Solve equation by factoring –Solve equation??? –Equivalence checking Simplify(expr1-expr2)  0 ????

6. 6 CAS era CASs have a wide range of commands for expression transformation. Questions about the use of the CASs in transformation exercises in schools. A reappraisal of the time and effort –Dedicated in the curriculum to the training of manual transformation skills. –Indispensable manual skills An overview of the answers offered by the CASs is necessary 6

7. 7 Classification of exercises with respect to the role a CAS plays in the solution. J. Böhm, I. Forbes, G. Herweyers, R. Hugelshofer, G. Schomacker: The Case for CAS Lokar M. & M.: CAS and the Slovene External Examination. In "The International Journal of Computer Algebra in Mathematics Education", 2001. Vol. 8, No 1.

8. C0 Exercises where the use of CAS is of little or no help. More importantly, the typing would take more time than solving the problem by-hand (this of course depends on the students' skill). C1 Traditional exercises (by-hand or using scientific calculators) that are solved faster or even trivialised by CAS (emphasis usually on manual skills). C2 Exercises that essentially test the ability of using CAS competently. C3 Exercises starting from traditional ones that are extended to CAS-exercises (e.g. by including formal parameters or using realistic data). C4 Exercises that are difficult or time consuming to solve without CAS or those that can only be solved with the aid of CAS. 8

9. 9 Example of C4: Cebatorev Assumption From The Case for CAS Factor x n −1 for n = 1, 2,..., 10 with integer coefficients. All coefficients of the polynomial factors are 1. Is this true for all exponents n? Solution: The problem was set by the Russian mathematician N.G. Cebotarev in 1938 in the Russian journal “Progress in mathematical sciences (volume IV)” and solved by V. Ivanov in 1941. The first exponent which leads to integer coefficients > 1 is n = 105. The voyage 200 factors the expression in 8 seconds, but the problem cannot be solved by hand by high school students.

10. 10 CASs. Background CASs –In the beginning were designed mainly to help professional users of mathematics –Nowadays more suitable for schools There are still some differences. How do different CASs solve problems? Michael Wester. Computer Algebra Systems. A Practical Guide. 1999 –542 problems –68 as usually taught at schools –another 34 advanced math classes

11. 11 Wester, page 45

12. 12 Scope Transformation (simplifying, expanding or factoring) of expressions from school mathematics –mainly from textbooks Immediate solving (student enters the expression and the program gives the answer) –basic commands (simplify, expand, factor) and some advanced ones. The aim is not to analyze the performance of particular CASs but to discover general trends. –In presentation only some issues Derive, Maple, Mathcad, Mathematica, Maxima, MuPAD, TI-92 Plus, TI-nspire, WIRIS

13. 13 Is the answer as expected? CAS = SCH? Yes! –In very many cases –Is the expected school-like answer good enough? Yes? –Simplest form? –Order of terms? –… No –Ask differently, more precisely? –Explain –Beware

14. 14 How to ask? Command –Basic commands (simplify, expand, factor) and some advanced ones simplify (automatic???) –Commands, menus, buttons … Expression –notation questions division :  / –

15. 15 Commands 15

16. 16

17. Some examples Different CASs Factor Trigonometric expression Combine commands 17

18. factor( ) 18

19. Factor(x^105-1) 19

20. C1 - are solved faster or even trivialised by CAS 20

21. In textbook 12(x-3)(x+3) 2 21

22. 22

23. Simplest? 23

24. 24 What is the required answer?

25. Combine commands tcollect(texpand(expr)) combine(simplify(expr)) simplify(combine(expr)) 25

26. 26

27. trigreduce, ratsimp, and radcan may be able to further simplify the result 27

28. Some steps manually 28

29. 29 Validity of transformation rules could depend on the domain

30. 30 Issues of Domain variable to be a real number variable to be in the certain interval DeriveAuthor Variable Domain MapleAssume RealDomain Assume MathematicaAssuming MaximaDefaultAssume MuPADAssume TI-92+Default| “with” TI-nspireDefault| “with” WIRISDefaultNot applicable

31. 31 Is the expected school-like answer good enough? Wester All CASs: success! (hurrah!) x = -2 ???

32. 32 Forbidden branches Operations and functions that are impossible (at least at school) in the case of some values of arguments. –Division by zero is undefined. –There is no square root for a negative number. –The domain of a logarithmic function is the set of all positive real numbers. –Tangent function is not defined if,. Demonstrate all “forbidden branches” separately in expression transformation exercises? Actually, the distinguishing of forbidden branches is discarded, and the practice is even “legalised”. For example, a textbook says: –Even though not always explicitly stated, we always assume that variables are restricted so that division by 0 is excluded; Unless stated to the contrary all variables are restricted so that all quantities involved are real numbers.

33. 33 CASs and Forbidden branches The CASs do not show “forbidden branches” either. Simplification of –all computer algebra systems give the answer 97, without recording the peculiarity of x = 0. TI-nspire adds a warning message: Domain of the result may be larger CAS=SCH<MATH

34. How? Like in Gentzen-type calculi in mathematical logic??? Only main branch with the condition(s) All branches separately and 34

35. 35 Types of obtained answers Equations (CADGME, Pecs) TypeEquivalent?Easily transformable? 1. Answer is not equivalent to the answer required at school noyes/no Anyway keeps non-equivalence 2. Answer is equivalent but can not be easily transformed to the required form yesno 3. Correct answer that is easily transformed to the required form yes okyesNot needed Already in suitable form In case of transformation exercises What is the required form? Combine commands – Type 3 Order of terms – 2? 3? ok? Trigonometry – 2? 3? Easily?

36. Some highlights Commands are simple –But sometimes we need special parameters or special commands Notation What is the required answer? –If not  equivalence Combination of commands Domain Forbidden branches 36

37. Material for different types of exercises –Indispensable skills? –Why? –Is the answer equivalent to textbook answer? –Other domain? –… C0 use of CAS is of little or no help C1 are solved faster or even trivialised by CAS C2 test the ability of using CAS competently C3 extended to CAS-exercises C4 difficult or time consuming to solve without CAS or only be solved with the aid of CAS 37

38. 38 Papers are to be submitted by 15 September 2009 More exercises –Absolute value –.. Automatic simplification Suggestions?

39. 39 Is the answer as expected? CAS = SCH? Yes! –In very many cases –Is the expected school-like answer good enough? Yes? –Simplest form? –Order of terms? –… No –Ask differently, more precisely? –Explain –Beware