1. Interactive Visualization across the Curriculum
Matthias Kawski
Department of Mathematics
Arizona State University
Tempe, Arizona U.S.A.

2. Thanks for generous support by
Department of Mathematics
Center for Research in Education of Science,
Mathematics, Engineering, and Technology
Arizona State University
INTEL Corporation through grant 98-34
National Science Foundation through the grants
DUE Vector Calculus via Linearization:
Visualization and Modern Applications
DMS Algebra and Geometry of Nonlinear Control Systems
EEC Engineering Foundation Coalition

3. “Across” the Curriculum
Moved practically all courses aggressively into predominantly
electronic format, emphasizing interactive visualization, exploration
and experimentation – but also proving via programming or via CAS.
Last 5 years:
Vector Calculus
Differential Equations
Linear Algebra
Calculus I, II, III
Adv. Engineering Math
[[Advanced Calculus]]
[How to write a proof.]
Differential Geometry
Complex Analysis
Partial Diff. Equations
Techn. in Classroom
Advanced Math via CAS

4. Everything is on the WWW. . . . . . the key is to use it intelligently!
Reconsider purpose/objectives of each course...
Reconsider selection of content / specific topics
Reconsider choice of delivery methods
“Inquiry-based learning” Doing math = experiment, make observations, conjecture, further test, formulate theorem, prove  definition, axiomatize….
Concentrate on core topics, and study these in depth There are very few fundamental concepts, emphasize “coherence”, Build rich “rooted” concept (“procept”) images (often w/ visual core!!!) and remember them for life (as opposed to: memorize lots of formulas for next exam only)
Efficiency and effectiveness Intense, highly interactive sessions via modern software tools. Continuous assessment & evaluation, permitting and relying on

5. E.g. choice between CAS and JAVA
Selected parameters
preprogrammed
freely available WWW, JAVA 2
mouse-clicks,
“kinesthetic”
completely flexible
start-up costs both time & $$$
command line
Coming next: Selected examples from
Differential geometry
Complex Analysis
Vector Calculus

6. JAVA 2 or CAS? Complex analysis
Convergence of Laurent series
Zooming on essential singularities
Winding numbers & branch cuts
Conformal mappings

7. Differential Geometry, Gauss curvature
After proving (!) e.g. the Theorema Egregium with MAPLE ((-- is that really doing mathematics?? ask Thurston!)), the real exciting math is just beginning!

8. JAVA 2 or CAS? Differential Geometry
Geodesic spheres, distance function
Conjugate points, focusing/curvature
Visualizing tensors

9. Vector calculus: Curl & divergence
The central object of study in vector calculus.
A horrible formula
that few students remember beyond the next exam.
Traditionally: almost exclusive use of algebraic symbols
little insight (one-sided, or fragmented, concept image)
major hurdle for re-entry students
invitation to further study higher math?

10. Curl & divergence  derivatives?

11. Curl: Coherence. ) or fragmentation. ) and connections, i. e
Curl: Coherence*) or fragmentation? *) and connections, i.e. hyperlinked architecture in brain just as on WWW

12. Compartmentalization / Fragmentation !
Complex Analysis
Linear Algebra
Differential Equations

13. JAVA - Vector field analyzer
Interactively zoom (move the lens!) to see derivatives of vector fields, curl and divergence, .....
Switch to flows – connect VC w/ DEs
Coming soon: Line / flux integrals

14. This was just a “teaser”!
Also p.65: Friday 1:00 – 3:00 p.m. NSF – DUE CCLI poster session
Sheraton, Pontchartrin A 3rd floor
Course materials
JAVA applets
CAS worksheets
past conference presentations (mostly .ppt )
(p)reprints of published articles
are, of course, all on the WWW just use your favorite search engine, and look for “kawski”