1. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Outline Personal background –ABET EC 2000 –“Shaping the future”, experimentation Demo: Divergence & Gauss’ theorem Questions and discusion

3. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu ABET 2000 http://www.abet.org Criteria for accrediting programs in engineering in the US http://www.abet.org Criterion 3. Program Outcomes and Assessment Engineering programs must demonstrate that their graduates have (a) an ability to apply knowledge of mathematics, science, and engineering (b) an ability to design and conduct experiments, as well as to analyze and interpret data (c) an ability to design a system, component, or process to meet desired needs (d) an ability to function on multi-disciplinary teams (e) an ability to identify, formulate, and solve engineering problems (f) an understanding of professional and ethical responsibility (g) an ability to communicate effectively (h) the broad education necessary to understand the impact of engineering solutions in a global and societal context (i) a recognition of the need for, and an ability to engage in life-long learning (j) a knowledge of contemporary issues (k) an ability to use the techniques, skills, and modern engineering tools necessary for engineering practice. Criterion 4. Professional Component The Professional Component requirements specify subject areas appropriate to engineering but do not prescribe specific courses. The engineering faculty …….. The professional component must include (a) one year of a combination of college level mathematics and basic sciences (some with experimental experience) appropriate to the discipline

4. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu ENGINEERING CRITERIA 2000 PROGRAM CRITERIA PROGRAM CRITERIA FOR ELECTRICAL, COMPUTER, AND SIMILARLY NAMED ENGINEERING PROGRAMS Submitted by The Institute of Electrical and Electronics Engineers, Inc These program criteria apply to engineering programs which include electrical, electronic, computer, or similar modifiers in their titles. 1. Curriculum The structure of the curriculum must provide both breadth and depth across the range of engineering topics implied by the title of the program. Graduates must have demonstrated knowledge of probability and statistics, including applications appropriate to the program name and objectives; knowledge of mathematics through differential and integral calculus, basic sciences, and engineering sciences necessary to analyze and design complex devices and systems containing hardware and software components, as appropriate to program objectives. Graduates of programs containing the modifier "electrical" in the title must also have demonstrated a knowledge of advanced mathematics, typically including differential equations, linear algebra, complex variables, and discrete mathematics. Graduates of programs containing the modifier "computer" in the title must have demonstrated a knowledge of discrete mathematics.

5. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu The Foundation Coalition The NSF, ABET, …. are dead serious. The FC alone is to receive 30 Mill. $ from the NSF over twice 5 years. And the FC is only one of 6 original coalitions, most refunded for second 5 year term. (Compare calc reform?)

6. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Foundation Coalition Courses at ASU Team-based learning Technology intensive Integrated Curriculum 1st Year: Intro to Engineering, Calculus 1+2, Physics, Chemistry, English composition 2nd Year: Vector Calculus, Diff Eqns, Mechanics (previously with: Electric Circuits, Linear Algebra, Intro Macroeconomics)

7. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Technology: why? which when? Amount of knowledge in math, sciences, … is growing ever faster … Available time for instruction is shrinking! Need to continuously re-evaluate every curriculum item – identify, and concentrate on core & current items. Abolish the others. ( Recall the square root algorithm! ) Need to employ more efficient, and more effective instructional tools / learning tools

8. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Efficient “professional” tools “Cannot afford wasting time with low-level manipulations!”

9. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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

10. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics ??? Axiom Definition Theorem Lemma Proof Example Exercise

11. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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”’

12. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics ??? Axiom Definition Theorem Lemma Proof Example Exercise

13. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Doing mathematics !!! Question/problem (context!, application …)

14. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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 ! )

15. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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!)

16. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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

17. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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)

18. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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)

19. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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)

20. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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

21. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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

22. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu These formulas as “root of the concept image” ? ? ?

23. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu No words ( needed )... cool!... but,... meaning?

24. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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?

25. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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 ?

26. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002 http://math.la.asu.edu/~kawski kawski@asu.edu Alternative Visual language –iconified –mouse input –rapid experiments Algebraic symbols –at end, if at all. Just as needed to interface with CAS

27. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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

28. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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!

29. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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!)

30. Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 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”)