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The role of problem solving and discovery in the undergraduate curriculum. Matthias Kawski Associate Director for Undergraduate Programs School of Mathematical & Statistical Sciences Arizona State University. Outline. Problem solving ? What do we actually teach
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The role of problem solving and discovery in the undergraduate curriculum Matthias Kawski Associate Director for Undergraduate Programs School of Mathematical & Statistical Sciences Arizona 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 ?
But what do we ask on exams? Problems solving? Drill exercises? Why? Afraid of the truth?
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/~mathcircleTuesdays 5:40-7:10 pm ASU Tempe
warm-up: treasure hunt A 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 example N 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/
CSUMS undergrad student research samples http://math.asu.edu/CSUMS/ Weather Modeling. Dawn Curtis, Lee Denison, Marcos Valdez, Andrew Brandon & Diana Gonzalez Advisors: Dr. Eric Kostelich & Dr. Alex Mahalov Cancer Modeling. Mary Alice Cameron, Audrey Whitmer & Taylor Hines. Advisors: Dr. Eric Kostelich and Steffen Eikenberry Modeling the Lagrangian Trajectories and Dynamics of Particles Released within the Terrain-Induced Wind Rotors found in Owens Valley, CA. This is a typical image produced by the model, depicting a tumor several months after diagnosis. 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!
