Transcript No Slide Title

Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
1
The role of a visual language
in reconnecting a compartmentalized curriculum
Shannon Holland
Matthias Kawski
Ctr. for Innovation in Engin. Educ.
Arizona State University
Tempe, AZ 85287
[email protected]
http://ciee.eas.asu.edu/fc/microscope
Department of Mathematics
Arizona State University
Tempe, AZ 85287
[email protected]
http://math.la.asu.edu/~kawski
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)
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
2
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!
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
3
Disconnected Curriculum I
• Common occurrence, cycles…..
– Efficiency: establish standard syllabus with
well-delineated courses
– specialists perfect each syllabus
– while communication with original customers
fades away
– sudden uproar asks to re-evaluate objectives…
– courses adapt, or are replaced by new “courses”…..
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
4
Disconnected Curriculum II
• Concerns:
– Waste of resources, endless duplication
– Not taking advantage of structural reinforcement
through “cross-links” (c.f. A.Gleason, Samos 1998)
– Poor public image w/ all undesired consequences…
– Uninspired students, math is conceived as a collection
of unrelated facts, rules, algorithms,…..
– …...
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
5
A specific case
• VC and LA have often been combined….
• In 1995 ASU FC identified an integrated course
in VC - DE - Circuits as desirable from organizational point of view (registration,….).
Lots of colleagues/students wondered/asked:
Do VC / DE share, have anything big in common?
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
How badly even our knowledge
is compartmentalized
6
During presentation on vector calculus at professional meeting
with very good mathematicians in audience: Which of the pictured
vector fields is linear?……….
Our tenet: Can’t talk about differentiation w/o first understanding “linear”!
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
How badly even our knowledge
is compartmentalized
7
During presentation on vector calculus at professional meeting
with very good mathematicians in audience: Which of the pictured
vector fields is linear?……….
Our tenet: Can’t 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!
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
8
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 don’t trust these “pictures”…
WHY NOT?
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
9
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!
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
If zooming is so compelling in calc I 10
why 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 <=> 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 deriv’s
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
11
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-plane
the range is represented by arrows
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
12
Zooming for derivatives
Derivatives always involve a difference:
First 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
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
13
Solid knowledge of linearity is critical
Zooming for a derivative
of a linear object returns
the same object!
Recognize linearity!
1st subtract drift
L(cP)=c L(p)
L(p+q)=L(p)+L(q)
Then center the lens.
Linear objects appear
the same on any scale!
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
14
Decompositions of linear fields: Basic ideas
The easiest case:
 a b


  b a
 0 b


  b 0
Skew symmetric
“rotation”, “curl”
 a 0


 0 a
Multiple of the identity
“divergence”, “trace”
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
15
Interactively visualizing continuity/integrals
Zooming of zeroth kind
magnifies only domain.
Visual approach to
“continuity” =“local constancy”
needed for:
solutions to systems of DEs
(Euler, Runge Kutta), and
for Riemann integrability
(line/surface integrals).
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
16
Interactively visualizing curl/divergence
In complete analogy to
--- lines/slopes before calculus,
--- linear functional analysis
before convex analysis
develop 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).
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
17
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”,“limit”
Lens 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).
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski
Visual language / reconnect compartmentalized curriculum
RUMEC SouthBend 9/98
Interactively visualizing various flows
18
•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
Shannon Holland and Matthias Kawski, Arizona State University
http://ciee.eas.asu.edu/fc/microscope
http://math.la.asu.edu/~kawski