Unexpected answers offered by computer algebra systems to school equations

1. CADGME 2010 Hluboká nad Vltavou, near České Budějovice Unexpected answers offered by computer algebra systems to school equations Eno Tõnisson University of Tartu Estonia 1

2. Plan • Background • Unexpected answers • CASs • Equations • Quadratic • Trigonometric • Could the unexpected answer be useful? How? 2

3. 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. 3

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5. Unexpected answer • Differently than • student/teacher/textbook • expects/waits for/presents • Expectations could vary • curriculum • teacher • textbook • Not incorrect but according to different standards • Classification and mapping of the unexpected answers • What are the equations/answers that have more didactical potential? 5

6. CASs • (Relatively) easily available • OpenAxiom • Maxima • Sage (Maxima??) • WIRIS • WolframAlpha • Quite different • Computer Algebra System • Open Scientific Computation Platform • Computational Knowledge Engine • … • If necessary it is possible to use some of them • We do not compose the rating. We do not focus on shortcomings. 6

7. Commands • Command solve • first choice for solving equations • equation  solution process  answer • solution process (answer) is impressionable by change of command, additional arguments, form of argument • solveradicalSolve • small difference in the expression could change the situation • 1  1.0 7

8. Equations • linear • quadratic • fractional • equations that contain an absolute value • irrational • exponential • logarithmic • trigonometric • literal equations 8

9. Plan for particular equation type • Initial set of examples • textbook etc. classification • sometimes simple, sometimes more complicated • simpler non-trivial examples • sometimes a bit more complicated • expressive examples from literature • Solving the example equations by all CASs • Tentative mapping • "zoom" if needed • detail the boundaries if needed • Special focus on the phenomena that are (could be) more meaningful to students and teachers 9

10. Classification of Quadratic Equations • Natural (textbook) classification is suitable as a base. • Classification by • (manual) solution process • number of (real) solutions 10

11. Quadratic Equations 11

12. Phenomena from Quadratic Equations • Form of the solution, equivalence of solutions • decimal fraction – common fraction • form of solution • Imaginary numbers, domain • Equal solutions, multiplicity of solutions • or • Choice of command • solve  12

13. Form • Radical 13

14. Imaginary • no solution • includes i • includes • includes decimal numbers and i 14

15. Trigonometric Equations • Different range • only sin, cos, tan • or also cot • or even sec, cosec • how complicated? • Different order (in textbooks) • all basic equations at first, then more complicated • basic equations with sine at first, then more complicated with sine, then basic with cosine etc etc • General solution, one solution or solutions in the interval • Find all solutions in the interval [0;2π) • Radians or degrees 15

16. Classification of Trigonometric Equations • Different classifications are possible • Basic equations • "nice" answer • "not-so-nice" answer • impossible (in school) • Advanced equations ("one-function") • more complicated argument • factorization • quadratic equations • biquadratic equations • More advanced ("function-change") • change function • homogeneous • … 16

17. Basic trigonometric equations 17

18. Phenomena from Basic Trigonometric Equations • choice of solution • number of solutions • 1 / 2 / infinitely • approximate-exact • when inverse function is in the answer 18

19. Number of solutions • one solution • one solution and warning • two solutions • general solution 19

20. Added by advanced trigonometric equations • Solutions are more complicated, checking correctness is more difficult • Textbook CAS • Biquadratic (trigon.) equation could be too complicated for the CAS • could be possible to solve by parts 20

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23. From other equations • Mainly same phenomena • equivalence • number domain • approximate-exact • branches • Symbolic expressions in case of literal equation • Sometimes a CAS could not solve the equation 23

24. So? • What phenomena could appear? • When the phenomenon appears? • So what? • ignore • avoid • explain • use • even evoke • unexpected  didactic, instructive 24

25. Why equations at all? • The 12th ICMI Study The Future of the Teaching and Learning of Algebra • The activities of school algebra can be said to be of three types: • generational • forming of expressions and equations • transformational • rule-based activities: collecting like terms, solving equations, simplifying expressions etc, etc • global/meta-level • problem solving, modelling, noticing structure, justifying, proving etc 25

26. Pilot Study / Pilot Course??? • Course for • students • pre-service teachers • in-service teachers • Topics • equivalence • number domain • approximate-exact • branches • The topics are very important but could be somewhat behind the scenes • Detailed mapping gives good examples • Something for everyday maths teaching? 26

27. Unexpected answer in instrumentation • Instrument = Artifact + Schemes and Techniques • Unexpected answer? • instrumental genesis • orchestration • As a base for discussion? • "Real life" example • Computer tells that … 27

28. There could be more than one (correct?) answer! • In mathematics???!!!! 28