Interactive Visualization across the Curriculum
description
Transcript of Interactive Visualization across the Curriculum
http://math.la.asu.edu/~kawski [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
Matthias Kawski Interactive Visualization across Curriculum AMS/MAA mtg NewOrleans Jan 2001
Curl & divergence derivatives?
?
http://math.la.asu.edu/~kawski [email protected]
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 [email protected]
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 [email protected]
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 [email protected]
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