Matthias Kawski CAS or MATLAB in 1 st

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 CAS or MATLAB in 1 st 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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 that can 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 CAS example http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 MATLAB example Most simple academic application: image processing…. . (e. g. basic. gif image off the WWW, say a 60 x 80 pixel image) http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Institutional background • Arizona State University: public university in rapidly growing metro area 50 000 student total 12 000 in math each fall semester 7 000 below calculus 300 “events” of average size 40 • majority of calculus I-III, diff equns, linear algebra are engineering majors • engineering college very progressive http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Integrated curricula in 1990 s technology intensive, team-oriented, project-driven, …. . (Intro 2 Engineering, CAD, English, Physics, Calculus, …. . ) http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Professional technology for math • • Courses under consideration: …, papyrus prep for calculus, abacus mutli-var and vector calculus, slide-rule diff equations, (linear algebra) logrithm table hand-held calculator graphing calculator Computer Algebra System: MAPLE, MATHEMATICA professional numerical package: MATLAB http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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 matters is what is on the final exam: Traditional calculus is a course in algebra! http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Derivations versus derivatives • Derivatives are analytic objects, defined by limits, 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Derivations versus derivatives • Derivatives are analytic objects, defined by limits, 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 matters is what is on the final exam: Traditional calculus is a course in algebra! http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Our engineers • every year they come back asking more loudly why we don’t use MATLAB also in 1 st 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Easy way out • MATLAB is useless on traditional calculus exams, i. e. no problems with exams when using MATLAB becomes an “add-on” for explorations, plotting, some checking, but is certainly not “integrated” http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Expressions versus functions in CAS >y: = x^2; > f : = s -> s ^ 2; > subs ( x = 3 , y ); > f ( 3 ); (“plug in”, “substitute”) (“evaluate at”) > plot ( y , x = - 5. . 5 ); > plot ( f , - 5. . 5 ); > diff ( y, x ); > D(f) ; (sciences: diff. w. r. t. variable) (no x needed for derivative) http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Functions in CAS Traditional language: • “find a function that …” • “find the aniderivative of …” http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 Derivations in CAS http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 MATLAB: basic functions http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 MATLAB: advanced functions 1 externally defined: usual trouble pathnames, write-protected networked environments function handles… http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 MATLAB: function topics • Nearly ideal for numerical differentiation and numerical integration, including investigations of 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 A * c = y, c = ? ? ? clearly an undetermined linear system of n = 4 equations in m = 1 unknown(s). What should division by coefficient matrix mean? http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ? ” ICTME 2 Beirut June, 2003 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 1 st year collegiate math courses • CAlgebra. S 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/~kawski@asu. edu