http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

Interactive Visualizationacross the CurriculumMatthias Kawski

Department of MathematicsArizona State UniversityTempe, Arizona U.S.A.

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

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 97-52453 Vector Calculus via Linearization: Visualization and Modern ApplicationsDMS 00-72369 Algebra and Geometry of Nonlinear Control Systems

EEC 98-02942 Engineering Foundation Coalition

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

“Across” the Curriculum

• 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

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:

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

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 WWW...

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

E.g. choice between CAS and JAVA

preprogrammedfreely available WWW, JAVA 2

mouse-clicks,

“kinesthetic”

completely flexiblestart-up costs

both time & $$$command line

Selected parameters

Coming next: Selected examples from • Differential geometry• Complex Analysis• Vector Calculus

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

JAVA 2 or CAS? Complex analysis

• Convergence of Laurent series• Zooming on essential singularities• Winding numbers & branch cuts• Conformal mappings

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

Differential Geometry, Gauss curvature

After proving (!) e.g. the Theorema Egregium withMAPLE ((-- is that really doing mathematics?? askThurston!)), the real exciting math is just beginning!

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

JAVA 2 or CAS? Differential Geometry

• Geodesic spheres, distance function

• Conjugate points, focusing/curvature

• Visualizing tensors

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

Vector calculus: Curl & divergenceThe 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?

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

Curl & divergence derivatives?

?

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

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

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

Compartmentalization / Fragmentation !

Linear Algebra

Complex Analysis

Differential Equations

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

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

http://math.la.asu.edu/~kawski kawski@asu.edu

Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001

This was just a “teaser”!

• 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”

Also p.65: Friday 1:00 – 3:00 p.m. NSF – DUE CCLI poster session Sheraton, Pontchartrin A 3rd floor

http://math.la.asu.edu/~kawski