matthias kawski. “ technology and doing mathematics” university of kansas september, 2002 kawski...

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

Matthias Kawski. “ Technology and doing mathematics” University of Kansas September, 2002

Technology and “doing mathematics”Matthias Kawski

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



Outline

• Personal background– ABET EC 2000

– “Shaping the future”, experimentation

• Demo: Divergence & Gauss’ theorem

• Questions and discusion



ABET 2000 http://www.abet.orgCriteria for accrediting programs in engineering in the US

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



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?)



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)



Efficient “professional” tools

“Cannot afford

wasting time with low-level manipulations!”



“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



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

• 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” university of kansas september, 2002 kawski...

Documents

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