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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. - PowerPoint PPT Presentation
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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.
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 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/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
CAS example
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
MATLAB example
Most simple academicapplication:image processing…..(e.g. basic .gif imageoff the WWW, say a60 x 80 pixel image)
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
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/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
Integrated curricula in 1990stechnology intensive, team-oriented, project-driven,…..(Intro2Engineering, CAD, English, Physics, Calculus,…..)
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
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)
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 that matters is what is on the final exam:Traditional calculus is a course in algebra!
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 that matters is what is on the final exam: Traditional calculus is a course in algebra!
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 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/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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
MATLAB becomes an “add-on”for explorations, plotting, some checking,
but is certainly not “integrated”
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
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)
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
Derivations in CAS
')'( gf
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
Derivations in CAS
'')'()'()''(')'( 2 ggfggfgf
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
MATLAB: basic functions
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
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/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
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/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st year collegiate math ?” ICTME 2 Beirut June, 2003
http://math.asu.edu/~kawski [email protected]
Matthias Kawski. “ CAS or MATLAB in 1st 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 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.