matthias kawski. “ technology and doing mathematics” asu – fym seminar october, 2002 kawski...

http://math.la.asu.edu/~kawski [email protected]

Matthias Kawski. “ Technology and doing mathematics” ASU – FYM seminar October, 2002

Technology and “doing mathematics”Matthias Kawski

Arizona State University, Tempe, U.S.A.



Outline

• Brief intro– Visual, iconified language

– Doing math …. experimentation

• Demo: Divergence & Gauss’ theorem

• Questions and discusion



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 exclusivelyalgebraic-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).



“Richer”, iconifiedcf. 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



Efficient “professional” tools

“Cannot afford

wasting time with low-level manipulations!”



Doing mathematics ???

Axiom

Definition

Theorem

Lemma

Proof

Example

Exercise



“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 workin teams to solve complex problems.”

http://www.ehr.nsf.gov/EHR/DUE/documents/review/96139/summary.htm

“inquiry based learning’”

“problem solving”’



Doing mathematics ???

Axiom

Definition

Theorem

Lemma

Proof

Example

Exercise



Doing mathematics !!!

Question/problem (context!, application …)





Experiment ( Math is NOT a science, method of proof ! )






Observation (patterns!)







Conjecture







Conjecture

Theorem (formulation)







Conjecture


Definition (to make thm/pf elegant)







Conjecture


Definition (to make thm/pf elegant)

Proof (search for counter exa)




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



“across the curiculum” . . . for Fourier and complex analysis, differential geometry, linear algebra . . . see 2000 AMS-Scandinavian Congress http://math.la.asu.edu/~kawski



These formulas as “root of the concept image” ? ? ?



No words ( needed ) . . . cool! . . . but, . . . meaning?



The glorious highlight of the course . . . . . . but do the formulas have any meaning for the student?



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 ?



Alternative: Launch the vfa2 now

• Visual language– iconified– mouse input– rapid experiments

• Algebraic symbols– at end, if at all. Just as needed to interface with CAS



Recall: “linear” and slope

Divided differences,

rise over run

Linear <=> ratio is CONSTANT,INDEPENDENT of thechoice of points (xk,yk )

y y

x x2 1

2 1

y

x



Telescoping sumsRecall: For linear functions, the fundamental theorem is exact without limits, it is just a telescoping sum!

Want: Stokes’ theorem for linear fields FIRST!

F b F a

F x F x

F x F x

x xx

F x dx

k k

k k

k k

a

b

( ) ( )

( ( ) ( ))

( ) ( )

( )

1

1

1



Telescoping sums for linear Greens’ thm.

This extends formulas from line-integrals over rectangles / trianglesfirst to general polygonal curves (no limits yet!), then to smooth curves.

L Nds

L Nds

trL A

trL A

trL A

C

Ck

k k

kk

k

The picture new TELESCOPING SUMS matters (cancellations!)



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”)

matthias kawski. “ technology and doing mathematics” asu – fym seminar october, 2002 kawski...

Documents

mathematics asu fym

discusion slide

iconified language

mason language example

interactive visual language

americas smet faculty

algebraicsymbolic language

science facts