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Functions and Operators in MAPLE and in MATLAB

Functions and Operators in MAPLE and in MATLAB. Matthias Kawski Dept of Math & Statistics Arizona State University Tempe, AZ. U.S.A. Longstanding interest: How do different new technologies “interfere” with the development of key concept images.

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Functions and Operators in MAPLE and in MATLAB

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  1. Functions and Operators in MAPLE and in MATLAB Matthias Kawski Dept of Math & Statistics Arizona State University Tempe, AZ. U.S.A. http://math.asu.edu/~kawskikawski@asu.edu

  2. Longstanding interest:How do different new technologies “interfere” with the development of key concept images. • Immediate trigger (for this article):Our engineers again asking to replace MAPLE by MATLAB in our introductory calculus • Immediate reaction:“Not again”, “do they never learn/understand?” …. but, after some reflection, …. http://math.asu.edu/~kawskikawski@asu.edu

  3. Institutional background • Large state university w/ major research focus.50 000 studentsMath: 13 000 / semester.All in small classes!!! • Calculus and above - main client: engineering students • Engineering: Aggressive reforms of instructional practices,largely driven by demanding accreditation reviews (ABET).Some key points: Teamwork, problem solving, PROFESSIONAL tools (technology) http://math.asu.edu/~kawskikawski@asu.edu

  4. The largest classes w/ engineers Calculus I Calculus II Calculus III Diff eqns Linear Algebra VectCalc / Fourier PDE MAPLE both • Some key concepts: • Functions • Derivatives and integrals (limits) • Linearity MATLAB http://math.asu.edu/~kawskikawski@asu.edu

  5. Functions • Many different points of view • abstractly: function = graph • covarying quantities (relation satisfying vertical line test) • formulas y=f(x) • mappings • computable assignments • procedure, subroutine, method • … http://math.asu.edu/~kawskikawski@asu.edu

  6. Functions and their graphs Adaptive grid – not at all transparent Making & arguing from table is so important! http://math.asu.edu/~kawskikawski@asu.edu

  7. Variables: CAS teaches precision http://math.asu.edu/~kawskikawski@asu.edu

  8. CAS: (pure) functions Math: no need for an “x” in defn of function, plot, composition , derivative The sciences are different – e.g. always differentiate W.R.T some QUANTITY http://math.asu.edu/~kawskikawski@asu.edu

  9. CAS: function composition The most important operation is function composition. – in CAS ?? – in MATLAB ?? http://math.asu.edu/~kawskikawski@asu.edu

  10. CAS gem: Chain-rule w/o variables http://math.asu.edu/~kawskikawski@asu.edu

  11. MATLAB functions • Everything is a matrix, “tables of numeric values” • Predefined functions usually heavily “overloaded”(i.e. domains are “unions”, MATLAB guesses what the user must have intended) • Functions as subroutines or external m-files,no arrow analogue or function w/o variable names • Composition OK for procedures, not for tables • Inverse only numerically as procedures http://math.asu.edu/~kawskikawski@asu.edu

  12. MATLAB function example: Division Division by matrix – if not invertible – yields, of course, least square solution! No warning message http://math.asu.edu/~kawskikawski@asu.edu

  13. Summarizing observations • CAS and MATLAB support very different aspects of the function concept. • Integration versus add-on: The students under-stand what matters is what is on the final exam.Allowing MATLAB / CAS on all exams ?is the tool useless, does it trivialize the exam, oris it a meaningfully integrated professional tool? http://math.asu.edu/~kawskikawski@asu.edu

  14. Conclusion CAS matches traditional calculus – but it trivializes the exams. MATLAB is just a cute add-on – that is next-to-useless on the exams. Traditional calculus is algebra, not analysis Limits are fundamentally topological, yet in trad calculus exams they are all algebra. Differentiation is about approximability bylinear objects, using limits.Derivations are purely algebraic operations: The traditional calc texts, and also the exams are about derivations! http://math.asu.edu/~kawskikawski@asu.edu

  15. Final thought Maybe, maybe,…when our engineers ask that we use MATLAB in calculus, and that we INTEGRATE it into the curriculum,…maybe they really mean that we should reformthe calculus, diffeq, linalg, … curriculaSO THAT MATLAB can easily be integrated“integrability of MATLAB” just an INDICATORthat the courses match modern demands….!!! http://math.asu.edu/~kawskikawski@asu.edu

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