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CAS or MATLAB in 1 st year collegiate math?

This article explores the use of Computer Algebra Systems (CAS) and MATLAB in 1st year collegiate math courses. It discusses the differences between CAS and MATLAB, their applications in calculus, and the integration of these tools into the curriculum. The author also addresses the challenges of using either choice and provides insights into their distinct natures.

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CAS or MATLAB in 1 st year collegiate math?

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  1. CAS or MATLABin 1st year collegiate math? Matthias Kawski  Arizona State University Tempe, U.S.A. http://math.asu.edu/~kawski  This work was partially supported by NSF grants DMS 00-72369 and DMS 01-07666. http://math.asu.edu/~kawskikawski@asu.edu

  2. Professional user of both CAS an MATLAB:e.g. MAPLE: curvature of optimal control,MATLAB: simulate ½ conductor industry supply chains http://math.asu.edu/~kawskikawski@asu.edu

  3. Outline • Brief intro-contrast: CAS versus MATLAB • Brief survey: Matt K and his environment why this question? • (Traditional) calculus is just algebra ! • MAPLE and calculus ??? • MATLAB and calculus ??? • The next courses: MAPLE versus MATLAB http://math.asu.edu/~kawskikawski@asu.edu

  4. Computer ALGEBRA Systems • MATHEMATICA, MAPLE, DERIVE, …. • $ 1000 professional, $ 150 student version • can do state-of-the-art numerics, graphics, ….but main data structure is symbolic expressions(NOT numbers). • Can do virtually all symbolic calculations thatcan be done by hand, but faster, much more reliably (fewer mistakes), and more systematically • can call MATLAB from inside CAS (inconvenient) http://math.asu.edu/~kawskikawski@asu.edu

  5. CAS example http://math.asu.edu/~kawskikawski@asu.edu

  6. MATLAB • $ 1000 professional, $ 150 student version • state-of-the-art numerics, graphics, ….main data structure is matrices of floating point numbers • professional use in sciences, engineering, math,… • fast ! • can “call CAS” from inside (“symbolic toolbox”) http://math.asu.edu/~kawskikawski@asu.edu

  7. MATLAB example Most simple academicapplication:image processing…..(e.g. basic .gif imageoff the WWW, say a 60 x 80 pixel image) http://math.asu.edu/~kawskikawski@asu.edu

  8. Institutional background • Arizona State University:public university in rapidly growing metro area50 000 student total12 000 in math each fall semester 7 000 below calculus 300 “events” of average size 40 • majority of calculus I-III, diff equns, linear algebraare engineering majors • engineering college very progressive http://math.asu.edu/~kawskikawski@asu.edu

  9. Integrated curricula in 1990s technology intensive, team-oriented, project-driven,…..(Intro2Engineering, CAD, English, Physics, Calculus,…..) http://math.asu.edu/~kawskikawski@asu.edu

  10. Professional technology integrated • All students have at (almost) all times access to professional computer software • especially during the exams! i.e. exams needed to change (usually including internet access) • typically, one or two computers at each table,but do not teach in traditional computer lab set up in rows where studenst hide behind screens…. http://math.asu.edu/~kawskikawski@asu.edu

  11. Professional technology for math Courses under consideration:prep for calculus, calculus, mutli-var and vector calculus, diff equations, (linear algebra) • …, papyrus • abacus • slide-rule • logrithm table • hand-held calculator • graphing calculator • Computer Algebra System: MAPLE, MATHEMATICA • professional numerical package: MATLAB http://math.asu.edu/~kawskikawski@asu.edu

  12. Calculus is an algebra course ? • (algebra of ) limits, especially rational functions(L’Hopital’s rule …. everything via Taylor expansion) • derivatives versus derivations • antiderivatives (as opposed to integrals) • “proof”: CAS can get 90% right on almost any final exam • The only thing that that matters is what is on the final exam: Traditional calculus is a course in algebra! http://math.asu.edu/~kawskikawski@asu.edu

  13. Derivations versus derivatives • Derivatives are analytic objects, defined bylimits, approximability by linear objects… • Derivations are algebraic objects that are defined as linear maps that “satisfy the Leibniz (product) rule”: D(fg)=(Df)g+f(Dg) http://math.asu.edu/~kawskikawski@asu.edu

  14. Derivations versus derivatives • Derivatives are analytic objects, defined bylimits, approximability by linear objects… • Derivations are algebraic objects that are defined as linear maps that “satisfy the Leibniz (product) rule”: D(fg)=(Df)g+f(Dg) • The only thing that that matters is what is on the final exam: Traditional calculus is a course in algebra! http://math.asu.edu/~kawskikawski@asu.edu

  15. Our engineers • every year they come back asking more loudly why we don’t use MATLAB also in 1st year calc. • so far held them at bay, compromise: calculus w/ CAS, introduce MATLAB in some DE sections, LA mostly w/ MATLAB…. • main motivation for this presentation and article:The clients seem to be very ill-informed about • the very distinct natures of either alternative • of how they mesh w/ the requested curriculum • and how problematic it is to INTEGRATE either choice http://math.asu.edu/~kawskikawski@asu.edu

  16. Easy way out • MATLAB is useless on traditional calculus exams,i.e.no problems with exams when using MATLABMATLAB becomes an “add-on”for explorations, plotting, some checking,but is certainly not “integrated” http://math.asu.edu/~kawskikawski@asu.edu

  17. CAS gives trouble • CAS by itself earns A on traditional calculus exam.i.e. either • need to completely redesign exams,or • prohibit CAS on exams. • Choice I is very hard, but it can be done (10 +years …) • Choice II is again just an “add-on”, no integration. • but neither one makes my engineers happy at this time http://math.asu.edu/~kawskikawski@asu.edu

  18. A closer look at the divergence • Calculus as “mathematics of continuous change”… changing objects = functions • Calculus as the study of • differentiable functions, and • integrable functions • Take closer look at functions in CAS / MATLAB http://math.asu.edu/~kawskikawski@asu.edu

  19. > y : = x ^ 2 ; > f : = s -> s ^ 2; > subs ( x = 3 , y ); (“plug in”, “substitute”) > f ( 3 ); (“evaluate at”) > plot ( y , x = - 5 . . 5 ); > plot ( f , - 5 ..5 ); > diff ( y, x ); (sciences: diff. w.r.t. variable) > D(f) ; (no x needed for derivative) Expressions versus functions in CAS http://math.asu.edu/~kawskikawski@asu.edu

  20. Functions in CAS • Traditional language: • “find a function that …” • “find the aniderivative of …” http://math.asu.edu/~kawskikawski@asu.edu

  21. Derivations in CAS http://math.asu.edu/~kawskikawski@asu.edu

  22. Derivations in CAS http://math.asu.edu/~kawskikawski@asu.edu

  23. MATLAB: basic functions http://math.asu.edu/~kawskikawski@asu.edu

  24. MATLAB: advanced functions 1 externally defined: usual trouble pathnames, write-protected networked environmentsfunction handles… http://math.asu.edu/~kawskikawski@asu.edu

  25. http://math.asu.edu/~kawskikawski@asu.edu

  26. MATLAB: function topics • Nearly ideal for numerical differentiation and numerical integration, including investigationsof the limiting processes… indeed, an almost perfect match for very reformed calculus course • incl. even functions defined as antiderivatives… • hard: function composition, inverse functions, …. http://math.asu.edu/~kawskikawski@asu.edu

  27. Summary and conclusions • MATLAB is a very easy add-on, useless in exams • CAS, but not on exams, is just another add-on • CAS, incl. on exams, requires dramatic rethinking • For traditional course CAS is much easier match • Major challenge: use MATLAB (w/ “integration” demand) as “vehicle” to implement (next step of) true calc reform? http://math.asu.edu/~kawskikawski@asu.edu

  28. Compare linear algebra • We now have two parallel linear algebra courses: • target: Jordan canonical form = MAPLE course • target: Singular Value Decomposition = MATLAB • Similar with differential equations • Maybe next, two calculus courses… • one course that is essentially algebra of mappings • one course that studies continuous change of numerically defined functions http://math.asu.edu/~kawskikawski@asu.edu

  29. http://math.asu.edu/~kawskikawski@asu.edu

  30. http://math.asu.edu/~kawskikawski@asu.edu

  31. http://math.asu.edu/~kawskikawski@asu.edu

  32. A * c = y, c = ???clearly an undetermined linear system ofn = 4 equations in m = 1 unknown(s).What should division by coefficient matrix mean? http://math.asu.edu/~kawskikawski@asu.edu

  33. http://math.asu.edu/~kawskikawski@asu.edu

  34. Summary • Don’t take your engineer’s wishes lightly: Either way it will be a lot of work, and true integration will necessarily may radically change the course – how much is wanted? • Both CAS and MATLAB offer to greatly expand students’ horizons beyond the traditional algebra-oriented 1st year collegiate math courses • CAlgebraS are considerably easier to integrate into more traditional courses • MATLAB most typically is only an “add-on”, – unless the courses are dramatically reformed. http://math.asu.edu/~kawskikawski@asu.edu

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