Interactive Visualization in Advanced Courses

Abstract

We demonstrate how modern technology may completely change the structure of even advanced classes in pure mathematics that until most recently appeared rather immune against such innovations. This talk will focus on differential geometry and complex analysis, demonstrating model uses of computer algebra systems and custom- designed JAVA applets. The main structural innovation is the use of these tools not simply as an add-on, e.g. for additional examples after a traditional treatment of axiom-definition-lemma-proof-theorem-proof-example. Instead, the modern computer tools suggest a complete reversal of this traditional routine even in these abstract, pure math courses: Starting with experimentation, observations lead to conjectures, followed by more tests, formulation of desired theorems, and attempts of proofs, that finally motivate clean definitions (and only in the very last step axiomatization). The class dynamics change dramatically as now the classes are driven by students' questions, where in the past, all too often, classes just supplied strings of "answers to questions that had never been asked". Selected examples from complex analysis and differential geometry clearly show how modern technology with mature implementations of computer algebra, graphical interfaces, and high-speed calculations and instantaneous animations are both a virtual prerequisite for these radical changes, as they also seem to inevitably force such changes.