CAS or MATLAB in 1 st year collegiate math?

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