Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
1
Interactive visualization reconnects
a compartmentalized curriculum
Shannon Holland
Matthias Kawski
Department of Mathematics
Ctr. for Innovation in Engin. Educ.
Arizona State University
Arizona State University
Tempe, AZ 85287
Tempe, AZ 85287
[email protected]
[email protected]
http://ciee.eas.asu.edu/fc/microscope http://math.la.asu.edu/~kawski
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
2
Vector Calculus
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The sophomore level classes VC, LA, DE are an exciting places to fully
take advantage of computer algebra systems (on higher levels these are
more unstable, on earlier levels their prowess leads to endless fights about
CAS and the need to learn basic manipulatory skills).
3D-visualization and animations are so compelling that even the most diehard colleagues find it hard to resist. Here we find the beginnings of an
emerging visual language that at the least will complement the traditional
algebraic symbols, and likely displace a large amount of these.
Primarily anecdotal evidence suggests that we are doing a particularly lousy
job at teaching VC: It is full of un-enlightening formulas, is focused on very
narrow applications (E&M, fluids), and is completely disconnected from
the rest of the world. It is of little surprise that students not only remember
hardly anything of value, but also largely hate the traditional course.
Before students learn about derivatives in single variable they spend more
than a year with lines (ratios, slopes, inter/extrapolation).
Before we allow anyone into nonlinear functional analysis we require a year
of linear functional analysis.
But in the standard 3rd semester calculus course we do it all at once:
Introduce new (difficult) objects (vector fields), their derivatives (curl,
divergence), and the integral theorems -- no surprise that students don’t
understand these as more than strange combinations of partial derivatives!
Nowhere do we tell them how this kind of derivative, like ALL derivatives
is essentially (the characteristic property of) a linear approximation.
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
3
Disconnected knowledge
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The following is a most telling experience about how bad the compartmentalization of our curriculum, and both, students’ and faculty’s knowledge is:
Both locally and at professional conferences we showed images of vector
fields to various people, many of them excellent mathematicians.
Asked which of the images represent linear vector fields, most could not
tell. This perplexing result was easily explained by the fact that these
questions were always asked in a context of vector calculus (which most as
students only experience as a course full of formulas).
After suggesting:”Just think about DEs”, almost everyone immediately
made the right choices (the images of linear fields are very familiar from
the qualitative analysis about the equilibrium points of dynamical systems).
If faculty think of DE and VC as disconnected subjects, how shall students
make the connection?
Now add the visual images, the visual language:
Do the images above
belong into VC, LA or DE -- can’t tell? Isn’t that what we want!
Is the field/DE linear? What does this have to do with the curl and with the
period of a (linearized) pendulum being independent of the amplitude?
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
We asked one simple question:
If zooming is so compelling in calc I
why not zoom for curl, div in calc III?
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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 is the one concept that
connects ALL notions of derivative -- yet in the past students
often had trouble connecting calc 1, curl/div, Frechet derivatives
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
5
Interactively visualizing continuity/integrals
The naïve way of zooming on a vector field only magnifies the domain,
leaving the scale of the range unchanged. This form of “zooming” is
the visual approach to “continuity” (=“local constancy”) which is the
heart of any notion of solution curves to DEs (Euler, Runge Kutta) and
of Riemann integrability (line integrals).
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
6
Interactively visualizing curl/divergence
In complete analogy to lines/slopes before calculus, and linear functional analysis
before nonlinear functional analysis, one ought to develop curl & divergence first
in a linear setting -- this is almost linear algebra, and the images are compelling:
It is almost as easy to SEE the curl and the divergence of a linear field as the
slope of a line -- and as the lens is dragged, the curl and div change (if the field is
nonlinear, and they are constant in linear fields). Currently these images are found
only in the section on qualitative study of equilibria in DE-texts (source,sink,foci).
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
7
Irrotational is a local property
A critical test case for student understanding asks whether the pictured field is
“irrotational” - most take a global view and say “NO”, clearly exhibiting that they
do NOT understand that any kind of derivative only provides info about LOCAL
properties. The tactile experience of dragging a lens and changing the zoom-factor
dramatically convey “local”, “limit”, and show that the field is irrotational/closed
(a key property of the magnetic field about an infinitely long straight wire carrying
a constant current, or of the complex field 1/z, the origin of algebraic topology).
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
8
Interactively visualizing various flows
Individual integral curves are studied in DE (lots of software available).
The important application is to consider how small regions evolve under
the full nonlinear flow, under the linearized flow, or only components
of the linearized flow: trace (divergence!), symmetric part (chaos!), and
skew symmetric part (curl). Our utility will provide all these integrated
analogues of the lenses for zooming (differentiation).
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
9
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Technical background 1
The JAVAscope uses a powerful, fast parser written by David Wanqian Liu.
It includes hyperbolic functions, and floor/ceil which allow the “piecewise
defined” formulas. Selected fields (magnetic/gravitational,…) are
preprogrammed on a pull-down menu for faster access.
Standard window controls like grid size, window size etc are available.
The scales for arrows are explicitly displayed.
Various lenses display “magnified” vector fields and may be
dragged across the screen. The magnification factors may be changed
interactively (and soon also: independently).
There are FOUR different scales that ones needs to understand:
(assuming equal scaling along x and y-directions):
SCALE 1: domain of x and y
SCALE 2: range of the vector field (magnitude of arrows)
SCALE 3: magnified domain inside the lens Dx and Dy
SCALE 4: magnified range inside the lens (arrows/arrow-heads)
SCALE 1 (window size, grid size) is user determined.
SCALE 2 is calculated so that arrows don’t overlap
SCALE 3 = e * SCALE 1 magnification chosen by user
SCALE 4 = SCALE 2 when zooming for continuity/integrability
SCALE 4 = e * SCALE 5 when zooming for derivatives
The basic SCALE 5 for the “arrow-heads” in the lenses for differentiation
uses a numeric estimate of the C1-norm of the field (sampling the numeric
Jacobian at about 100 points inside the large window). Specifically, no
matter where the lens is moves, the arrows inside the lens shall not overlap.
Further magnification of domain and range is normally in lock-step.
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
10
Technical background 2
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The various scales are visible in the window (we may change the format
according to feedback we get). These scales NEED to be addressed in class
-- too many students do not understand that almost always the scales for the
domain (tickmarks on the axes) and for the range (arrow length) are not the
same but are simply chosen so as to maximize arrow length while avoiding
overlaps. Specifically, the images of F(x,y) and cF(x,y) look the same for
any positive constant c - distinguishing underlying space and tangent space
is critical to make sense out of applications, to provide basis for later work.
The nonlinear field is decomposed using the “projection”
 J ( F ( J 1 ( x)))  2 F ( x)  J 1 ( F ( J ( x)))
( ( F ))( x) 
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This is a reasonably geometric construction that is feasible in real time in
JAVA. It assures that the components of the nonlinear field converge to the
(skew) symmetric parts of the Jacobian linearization.
(Note: The “transpose”makes no sense for general nonlinear fields….)
The flow-module uses an adaptive scheme that adjusts the number of points
along the boundaries of the region that are integrated using standard RungeKutta algorithms, nothing fancy (some details are still under development).
Key features are that flows of linear fields preserve linear features,
nonlinear flows distort linear edges immediately, curl generates the
rotation, divergence is rate of expansion, symm part shows principal axes
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
11
Morale, lessons learnt
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Technology, primarily CAS and JAVA, open exciting new doors -- but lots
of creativity is asked for. We need to re-evaluate our approaches to subjects
at all levels, as the many basic assumptions may no longer be true.
Visual tools provide a powerful alternative to the traditional, almost
exclusively algebraic formalism. Not only can analytic concepts (existence
of a limit….) be rigorously formulated using the visual language, this
language also provides considerably more compelling connections between
traditionally separately presented topics. Here, no x’es can provide the
obvious connection between VC and DE as compellingly as the illustrations
of the vector fields, their derivatives, and their flows…..
This visual language suggests a new look the vertical prerequisite structure
of mathematics, at access for everyone, at reaching and committing
traditionally underrepresented groups..
Even for utilizing new technology at the introductory level there is a requirement for deep mathematical understanding. In our case typical questions
that arise are: How to split a nonlinear vector field in a geometrically sound
fashion (that is computationally feasible) into components that converge to
the symmetric and skew symmetric parts of the Jacobian? How to handle
DE applications where the states have non-commensurable dimensions (no
notion of angle, curl?, yet integrability & exterior derivatives make sense)?
It makes little sense to use calculators only to do yesterday’s exercises
faster -- most of these never had any intrinsic value: They were appropriate
for the then-technology to deepen the understanding. With new technology
we need new exercises to effectively learn a re-evaluated curriculum.
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)
Interactive visualization reconnects a compartmentalized curriculum
“Shaping the Future” of SMET education, Univ NE Lincoln , May 1998
12
To find out more
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The JAVAscope is freely available on the WWW
– The “differential version” (zoom lenses) works fine, except
for a few minor bugs, and the separate zoom controls……..
– The integrated version (flows) will be completed this summer.
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An accompanying technical preprint is available on-line.
- technical information about the program
- rationale and geometric background
- suggestions for exploratory exercises
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A “Book of Zooming”:
Limits = zooming : From sequences to Stokes theorem is under preparation.
Almost a coffee-table picture book with background math, about 50 kinds
of zooming and related topics, accompanied by an interactive MAPLE-CD
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Calculus of Vector Fields
Under a new 1998-99 NSF grant we are working on a radically new
approach to vector calculus, which develops all key concepts in a linear
setting first (all the way to Stokes’ theorem), which uses state-of-the-art
visualization throughout, and which firmly integrates traditionally separate
points of view: VC, LA and DE.
By affirmatively integrating DE and VC, this approach provides much
richer applications - largely inspired by geometric control -- of the integral
theorems than the usual narrow focus on E&M and fluid dynamics.
A short paper-text, MAPLE-worksheets & JAVA-applets on CD, and
extended projects shall appear throughout the next 2 years.
Shannon Holland and Matthias Kawski,
http://ciee.eas.asu.edu/fc/microscope
Arizona State University
http://math.la.asu.edu/~kawski
This work was partially supported by the National Science Foundation:
DUE 97-52453 (Vector Calculus via Linearization: Visualization …)
DUE 94-53610 (ACEPT), and EEC 92-21460 (Foundation Coalition)