cas or matlab in 1 st year collegiate math?

http://math.asu.edu/~kawski [email protected]

Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003

CAS or MATLAB in 1st year collegiate math?

Matthias Kawski

Arizona State UniversityTempe, U.S.A.

http://math.asu.edu/~kawski This work was partially supported by NSF grants DMS 00-72369 and DMS 01-07666.



Professional user of both CAS an MATLAB:e.g. MAPLE: curvature of optimal control,

MATLAB: simulate ½ conductor industry supply chains



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



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)



CAS example



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”)



MATLAB example

Most simple academicapplication:image processing…..(e.g. basic .gif imageoff the WWW, say a60 x 80 pixel image)



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



Integrated curricula in 1990stechnology intensive, team-oriented, project-driven,…..(Intro2Engineering, CAD, English, Physics, Calculus,…..)



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….



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)



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!



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)



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!



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



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”



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



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



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 = - 5 . . 5 ); > plot ( f , - 5 ..5 );

> diff ( y, x ); (sciences: diff. w.r.t. variable)

> D(f) ; (no x needed for derivative)



Functions in CAS

Traditional language:• “find a function that …”• “find the aniderivative of …”



Derivations in CAS

')'( gf



Derivations in CAS

'')'()'()''(')'( 2 ggfggfgf



MATLAB: basic functions



MATLAB: advanced functions 1

externally defined: usual trouble pathnames,

write-protected networked environments

function handles…



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, ….



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?



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



A * c = y, c = ???clearly an undetermined linear system ofn = 4 equations in m = 1 unknown(s).

What should division by coefficient matrix mean?



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.

cas or matlab in 1 st year collegiate math?

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