The role of a visual language in reconnecting a compartmentalized curriculum

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The role of a visual language in reconnecting a
compartmentalized curriculum
Matthias Kawski Department of Mathematics Arizona
State University Tempe, AZ 85287 kawski_at_asu.edu h
ttp//math.la.asu.edu/kawski
Shannon Holland Ctr. for Innovation in Engin.
Educ. Arizona State University Tempe, AZ
85287 skholland_at_asu.edu http//ciee.eas.asu.edu/fc
/microscope
This work was partially supported by the National
Science Foundation through the grants DUE
97-52453 (Vector Calculus via Linearization
Visualization ) and DUE 94-53610 (ACEPT), and
through the Cooperative Agreement EEC 92-21460
(Foundation Coalition)
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Visual language ?

• Algebraic symbols are one, but not the only way
  to do mathematics, or to learn mathematics
• Why now, not at previous times?
• Before the printing press?
• Before the PC?
• Before JAVA?
• New technologies suggest to reevaluate old
  paradigms!

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Disconnected Curriculum I

• Common occurrence, cycles..
• Efficiency establish standard syllabus with
  well-delineated courses
• specialists perfect each syllabus
• while communication with original customersfades
  away
• sudden uproar asks to re-evaluate objectives
• courses adapt, or are replaced by new courses..

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Disconnected Curriculum II

• Concerns
• Waste of resources, endless duplication
• Not taking advantage of structural
  reinforcementthrough cross-links (c.f.
  A.Gleason, Samos 1998)
• Poor public image w/ all undesired consequences
• Uninspired students, math is conceived as a
  collectionof unrelated facts, rules,
  algorithms,..
• ...

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A specific case

• VC and LA have often been combined.
• In 1995 ASU FC identified an integrated coursein
  VC - DE - Circuits as desirable from
  organiza-tional point of view (registration,.).
  Lots of colleagues/students wondered/askedDo VC
  / DE share, have anything big in common?

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How badly even our knowledgeis compartmentalized
During presentation on vector calculus at
professional meeting with very good
mathematicians in audience Which of the
pictured vector fields is linear?.
Our tenet Cant talk about differentiation w/o
first understanding linear!
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How badly even our knowledgeis compartmentalized
During presentation on vector calculus at
professional meeting with very good
mathematicians in audience Which of the
pictured vector fields is linear?.
Our tenet Cant talk about differentiation w/o
first understanding linear!
No answers -- until audience is prompted to think
in terms of DEs -- there the pictures are
familiar, everyone immediately answers!
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Connections between VC and DE

• Not much in terms of algebraic symbols(aside
  from the ubiquitous x and d/dx)
• Vector fields (arrows) clearly are an obvious
  tie.But students/faculty dont trust these
  picturesWHY NOT?

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Connections between VC and DE

• Vector fields (arrows) clearly are an obvious
  tie.
• But much more is true! Curl and divergence are
  very meaningful in DEs.Only via DEs do they
  really acquire meaning!HOW? -- Via interactive
  pictures, not via formulas!

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If zooming is so compelling in calc Iwhy not
zoom for curl, div in calc III?
In the pre-calculator days limits meant factoring
and canceling rational expressions and
secant lines disappeared to a point to reemerge
as tangent lines...
Today every graphing calculator has a zoom
button.The connection Derivative ltgt local
linearity is inescapable Local approximability by
linear objects underlies ALL notions of
derivative -- yet in the past students often had
trouble connecting calc 1, curl/div, Frechet
derivs
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Distinguish zooming for integrals / for
derivatives
For catalogue see fourth-coming book
Zooming and Limits From Sequences to Stokes
theorem
Zooming in the domain only is appropriate for
integrals and continuity Here the domain is the
xy-planethe range is represented by arrows
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Zooming for derivatives
Derivatives always involve a differenceFirst
step is to subtract the drift at point of interest
Then magnify domain (xy-plane)and range (arrows)
at equal rates to observe convergence to linear
part
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Solid knowledge of linearity is critical
Zooming for a derivative of a linear object
returns the same object!
Recognize linearity!
1st subtract drift
Then center the lens. Linear objects appear the
same on any scale!
L(cP)c L(p) L(pq)L(p)L(q)
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Decompositions of linear fields Basic ideas
The easiest case
Skew symmetricrotation, curl
Multiple of the identitydivergence, trace
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Interactively visualizing continuity/integrals
Zooming of zeroth kind magnifies only
domain. Visual approach to continuity local
constancyneeded for solutions to systems of
DEs (Euler, Runge Kutta), and for Riemann
integrability (line/surface integrals).
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Interactively visualizing curl/divergence
In complete analogy to --- lines/slopes before
calculus, --- linear functional analysis
before convex analysisdevelop curl divergence
first in a linear setting -- almost linear
algebra, images are compelling It is as easy to
SEE the curl and the divergence of a linear field
as the slope of a line.As lens is dragged, curl
and div change (if the field is nonlinear), are
constant (if field is linear).
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Irrotational is a local property
Test case for understanding Is pictured field
irrotational ? Many students take a global
view, say NO, i.e. do NOT understand that any
derivative provides info about LOCAL properties.
Tactile experience of dragging lens and
changing the zoom-factor dramatically convey
local,limitLens shows that field
irrotational (key property of magnetic field
about straight wire w/ constant current, or of
complex field 1/z, the origin of algebraic
topology).
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• Interactively visualizing various flows

• Individual integral curves
• Regions evolving under various flows
• Full nonlinear flow
• Linearized flow
• Components of lin. Flow -- trace
  (divergence!) -- symmetric part (chaos!) --
  skew symm. part (curl)
• User draws polygonal region and chooses the
  flow -- each corresponds to a magn.lens