The role of problem solving and discovery in the undergraduate curriculum

Matthias Kawski Tempe, AZ Oct 2, 2010

Matthias KawskiAssociate Director for Undergraduate Programs

School of Mathematical & Statistical SciencesArizona State University

Outline

• Problem solving ?• What do we actually teach• Genuine problem solving

-------------------• Outreach activities:• March 2010: Julia Robinson Math Festival• Math Circle at ASU Tempe• Visits (ugrads!) to schools (math clubs ?)

Your experience ?

• When you were an undergraduate student?

• How much problem solving?

• What do/did you mean by problem solving?

• How much was problem solving valued?How much problem solving was on exams?

(good indicator: average number of minutes per question?)

What does“problemsolving”mean?

Why does /would anyone want it ?

Polya:

A flowchartfor problem

solving?

in mathematics

Alan Schoenfeld

Problems ?

Problems ?What do we ask on exams?

Nomenclature has changed:problems vs

exercises

At least awareness by publishers

Lip-service by profs?

Exercises

aptly named

just as will be on test

Problems?

a start

how many of these

on the test ?

But what do we ask on exams?

• Problems solving?

• Drill exercises?

• Why? Afraid of the truth?

MAT 300, day 1any better?

MAT 300

MAT 300: discrete math examples I

On the test: new problems ?

Retrospection undergrad math• In many of our classes we see lots of exercises,

applying/practicing a-priori known techniques• Pockets of problem solving:

Putnam, olympics, discrete, math club, REU, …• Tests all too often cover specific sections

“need to study for the test”• Rare: “every exam is comprehensive K-2-now”• Still a lot of:

axiom-defn-lemma-proof-thm-proof-example• Much too little:

problem-example-conjecture-proof-thm-defn-axiom

Math circles

• 100 years plus Bulgaria• Russia, Kolmogorov school• 1990 immigrants in US, both coasts• recently: NMAC, MSRI, NSF, “circle on the road”

• after school tradition: ballet, swim, piano, soccerwhat about “math club” ?

• no TV crew for JRMF in March 2010 at ASU Tempeno TV, newspaper picked up news releases, stories

Math Circle at ASU Tempe• the first of, hopefully, many math circles

catering to diverse clients/participants

• highly motivated high-school age

students

• connect w/ research mathematicians

• 8 to 11 weekly meetings/semester

• focus on problem solving, primarily

discrete math, algebra, elementary

number theory, geometry, topology

• plan funding by corporate sponsors

http://math.asu.edu/~mathcircle

Tuesdays 5:40-7:10 pm ASU Tempe

warm-up: treasure huntA man has a treasure map. It shows an island with a gallows, an elm tree and an oak tree. The map has instructions. They say go to the gallows and walk to the elm tree, counting your paces. Then turn 90 degrees right and walk the same number of paces as you had just counted. Mark that spot. Go back to the gallows. Now walk to the oak tree, counting your paces. Turn 90 degrees left and walk the same number of paces as you had just counted. Mark that spot. The treasure is buried halfway between the two marked spots.When he got to the island, he could find the trees, but not the gallows. How can he find the treasure?

well known problem, this copy from http://trickofmind.com/2009/08/treasure-hunt.html

Math circle: discrete math exampleN students stand in a circle, each wearing a colored hat from a set of N known colors. Some colors may be repeated, or not assigned at all.None of the students can see the color of their own hat, but each can see the colors of all the other. They are not allowed to communicate once the teacher has placed the hats on their heads. However, they knew about the problem set-up beforehand and were allowed to make a plan BEFORE the hats were placed on their hats.

The objective is that at least one student correctly guesses the color of her hat, no matter how the teacher assigns the colors.

Well known problem, often stated in terms of prisoners, executioner, numbers on forehead, see e.g.: http://sbjoshi.wordpress.com/2009/09/26/puzzle-6-prisoner-and-the-matter-of-life-and-death/

Two popular geometry problems

google solutions? example in class?

CSUMS undergrad student research samples http://math.asu.edu/CSUMS/

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ASU undergrads EAGER to visit HS classes / math-clubs to tell their story. ASK us to come!

• we recognize problem solving when we see it• we know (and believe in) its superior value (data? proof?)• we all know how young kids love problems (keep ‘em kids)• success in teaching problem solving is difficult to assess,

(and we may be disappointed by outcomes)

• we do not live alone and need to compromise, but we should

not “over-compromise” • let us work together, send a coherent message to media, and push

real math in classrooms, find alternate models were needed.• train the students to be problem solvers – we want them!

The role of problem solving and discovery in the undergraduate curriculum

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