Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, CAS or MATLAB in 1 st year collegiate math? Matthias Kawski  Arizona State University Tempe, U.S.A.  This work was partially supported by NSF grants DMS and DMS

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, Professional user of both CAS an MATLAB: e.g. MAPLE: curvature of optimal control, MATLAB: simulate ½ conductor industry supply chains

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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)

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, CAS example

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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”)

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

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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, Integrated curricula in 1990s technology intensive, team-oriented, project-driven,….. (Intro2Engineering, CAD, English, Physics, Calculus,…..)

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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….

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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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!

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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)

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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 that matters is what is on the final exam:Traditional calculus is a course in algebra!

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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

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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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

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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, Functions in CAS Traditional language: “find a function that …” “find the aniderivative of …”

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, Derivations in CAS

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, Derivations in CAS

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, MATLAB: basic functions

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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June,

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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, ….

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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?

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June,

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June,

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June,

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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?

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June,

Matthias Kawski. “ CAS or MATLAB in 1 st year collegiate math ?” ICTME 2 Beirut June, 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 CAlgebraS are considerably easier to integrate into more traditional courses MATLAB most typically is only an “add-on”, – unless the courses are dramatically reformed.