Interactive Visualization across the Curriculum Matthias Kawski Department of Mathematics Arizona State University Tempe, Arizona U.S.A. http://math.la.asu.edu/~kawski kawski@asu.edu

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 Applications DMS 00-72369 Algebra and Geometry of Nonlinear Control Systems EEC 98-02942 Engineering Foundation Coalition http://math.la.asu.edu/~kawski kawski@asu.edu

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

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

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

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

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

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

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

Curl & divergence  derivatives? http://math.la.asu.edu/~kawski kawski@asu.edu

Curl: Coherence. ) or fragmentation. ) and connections, i. e 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

Compartmentalization / Fragmentation ! Complex Analysis Linear Algebra Differential Equations http://math.la.asu.edu/~kawski kawski@asu.edu

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

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