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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. Plan. Background Unexpected answers CASs Equations Quadratic Trigonometric
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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
Plan • Background • Unexpected answers • CASs • Equations • Quadratic • Trigonometric • Could the unexpected answer be useful? How? 2
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
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
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
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 • solveradicalSolve • small difference in the expression could change the situation • 1 1.0 7
Equations • linear • quadratic • fractional • equations that contain an absolute value • irrational • exponential • logarithmic • trigonometric • literal equations 8
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
Classification of Quadratic Equations • Natural (textbook) classification is suitable as a base. • Classification by • (manual) solution process • number of (real) solutions 10
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
Form • Radical 13
Imaginary • no solution • includes i • includes • includes decimal numbers and i 14
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
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
Phenomena from Basic Trigonometric Equations • choice of solution • number of solutions • 1 / 2 / infinitely • approximate-exact • when inverse function is in the answer 18
Number of solutions • one solution • one solution and warning • two solutions • general solution 19
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
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
So? • What phenomena could appear? • When the phenomenon appears? • So what? • ignore • avoid • explain • use • even evoke • unexpected didactic, instructive 24
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
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
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
There could be more than one (correct?) answer! • In mathematics???!!!! 28