CAS or MATLAB in 1st year collegiate math?

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CAS or MATLAB in 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.
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Professional user of both CAS an MATLABe.g.
MAPLE curvature of optimal control,MATLAB
simulate ½ conductor industry supply chains
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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

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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)

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CAS example
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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)

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MATLAB example
Most simple academicapplicationimage
processing..(e.g. basic .gif imageoff the WWW,
say a 60 x 80 pixel image)
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Institutional background

• Arizona State Universitypublic 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

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

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Professional technology for math
Courses under considerationprep 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

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Calculus is an algebra course ?

• (algebra of ) limits, especially rational
  functions(LHopitals 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!

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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)gf(Dg)

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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)gf(Dg)
• The only thing that that matters is what is on
  the final exam Traditional calculus is a course
  in algebra!

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Our engineers

• every year they come back asking more loudly
  why we dont 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
  articleThe 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

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Easy way out

• MATLAB is useless on traditional calculus
  exams,i.e.no problems with exams when using
  MATLABMATLAB becomes an add-onfor
  explorations, plotting, some checking,but is
  certainly not integrated

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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

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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

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Expressions versus functions in CAS
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Functions in CAS

• Traditional language
• find a function that
• find the aniderivative of

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Derivations in CAS
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Derivations in CAS
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MATLAB basic functions
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MATLAB advanced functions 1
externally defined usual trouble pathnames,
write-protected networked
environmentsfunction handles
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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, .

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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?

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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

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A c y, c ???clearly an undetermined
linear system ofn 4 equations in m 1
unknown(s).What should division by coefficient
matrix mean?
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Summary

• Dont take your engineers 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.