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

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

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 –542 problems –68 as usually taught at schools –another 34 advanced math classes.

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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?

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.

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

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

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

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

11 Quadratic Equations TypeExamplePhenomena form (fraction) ±, form (radical) form (radical) multiplicity imaginary

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 

13 Form Radical

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

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

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 –…

17 Basic trigonometric equations TypeExamplePhenomena "Nice" answerchoice of solution number of solutions approximate-exact when inverse function "Not-so-nice" answer choice of solution number of solutions approximate-exact when inverse function Impossiblewhen inverse function

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

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

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

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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

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

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

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?

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 …

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