1. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Technology and “doing mathematics” Matthias Kawski Arizona State University, Tempe, U.S.A.

2. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Outline Brief intro –Visual, iconified language –Doing math …. experimentation Demo: Divergence & Gauss’ theorem Questions and discusion

3. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Vision (M.K., Odense 2000) We are at the beginning of a new era in which an interactive visual language not only complements, but often supersedes the traditional, almost exclusively algebraic-symbolic language which for generations has often been confused with mathematics itself, (and which may be largely responsible for the isolation, poor public perception, and extremely difficult re-entry into mathematics due to the imposed vertical structure).

4. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu “Richer”, iconified cf. J.Mason language example from: Jerry C. Hamann, U Wyoming E.g. Rossler attractor: System of equations and MATLAB-SIMULINK screen Higher information content Intuitive, efficient interfaces for manipulating objects

5. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Efficient “professional” tools “Cannot afford wasting time with low-level manipulations!”

6. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics ??? Axiom Definition Theorem Lemma Proof Example Exercise

7. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu “Shaping the Future of SMET” (1997 NSF-report, Mel George) “ The goal – indeed, the imperative – deriving from our review is that: All students have access to supportive, excellent undergraduate education in science, mathematics, engineering, and technology, and all students learn these subjects by direct experience with the methods and processes of inquiry.” “America's undergraduates – all of them – must attain a higher level of competence in science, mathematics, engineering, and technology. America's institutions of higher education must expect all students to learn more SME&T, must no longer see study in these fields solely as narrow preparation for one specialized career, but must accept them as important to every student. America's SME&T faculty must actively engage those students preparing to become K- 12 teachers; technicians; professional scientists, mathematicians, or engineers; business or public leaders; and other types of "knowledge workers" and knowledgeable citizens. It is important to assist them to learn not only science facts but, just as important, the methods and processes of research, what scientists and engineers do, how to make informed judgments about technical matters, and how to communicate and work in teams to solve complex problems.” http://www.ehr.nsf.gov/EHR/DUE/documents/review/96139/summary.htm “inquiry based learning’” “problem solving”’

8. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics ??? Axiom Definition Theorem Lemma Proof Example Exercise

9. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …)

10. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …) Experiment ( Math is NOT a science, method of proof ! )

11. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …) Experiment ( Math is NOT a science, method of proof ! ) Observation (patterns!)

12. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …) Experiment ( Math is NOT a science, method of proof ! ) Observation (patterns!) Conjecture

13. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …) Experiment ( Math is NOT a science, method of proof ! ) Observation (patterns!) Conjecture Theorem (formulation)

14. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …) Experiment ( Math is NOT a science, method of proof ! ) Observation (patterns!) Conjecture Theorem (formulation) Definition (to make thm/pf elegant)

15. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …) Experiment ( Math is NOT a science, method of proof ! ) Observation (patterns!) Conjecture Theorem (formulation) Definition (to make thm/pf elegant) Proof (search for counter exa)

16. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …) Experiment ( Math is NOT a science, method of proof ! ) Observation (patterns!) Conjecture Theorem (formulation) Definition (to make thm/pf elegant) Proof (search for counter exa) Axiomatize

17. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu “across the curiculum”... for Fourier and complex analysis, differential geometry, linear algebra... see 2000 AMS-Scandinavian Congress http://math.la.asu.edu/~kawski

18. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu These formulas as “root of the concept image” ? ? ?

19. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu No words ( needed )... cool!... but,... meaning?

20. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu The glorious highlight of the course...... but do the formulas have any meaning for the student?

21. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Until the symbols have meaning...... what value do the formulas have ?... for how long will they be remembered ?... will they instill positive attitudes twds math ?

22. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Alternative: Launch the vfa2 nowLaunch the vfa2 now Visual language –iconified –mouse input –rapid experiments Algebraic symbols –at end, if at all. Just as needed to interface with CAS

23. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Recall: “linear” and slope Divided differences, rise over run Linear ratio is CONSTANT, INDEPENDENT of the choice of points (x k,y k ) yy xx

24. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Telescoping sums Recall: For linear functions, the fundamental theorem is exact without limits, it is just a telescoping sum! Want: Stokes’ theorem for linear fields FIRST!

25. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Telescoping sums for linear Greens’ thm. This extends formulas from line-integrals over rectangles / triangles first to general polygonal curves (no limits yet!), then to smooth curves. The picture new TELESCOPING SUMS matters (cancellations!)

26. Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Geometric definitions Here, the (reasonably nice), closed curves C “shrink” to the point p, and the denominator is the signed area of the region “inside” the curve. Interpretation: (Infinitesimal) rate of expansion (new out-flow per area), and (infinitesimal) rate of circulation (“distance” from being “gradient”) of divergence and of rotation (“scalar curl”)