Exciting Math Labs Inspiring Proofs

1. “The most exciting phrase to hear in science, the one that heralds the most discoveries, is not Eureka! (I found it!) but ‘That’s funny…’” Isaac Asimov Art Duval and Helmut Knaust Department of Mathematical Sciences The University of Texas at El Paso March 6, 2004

2. The Problem: • Math majors take 4-5 courses geared for science and engineering majors, heavy on computation, then show up for their first classes for math majors, and encounter proofs for the first time.  They do not know what a proof is, why you'd want one, or how to construct one.

3. The Problem (cont’d): • Others, unsure whether to major in math or something else, base their decision on those same 4-5 more applied courses, never discovering that math is primarily about proofs. • An existing course focuses on how to write a proof, once you have an idea of what you want to prove, but not on how to find the idea of the proof.  "Nuts and bolts" of proof-writing (sets, functions, relation), not enough time for big ideas and exploration, not very motivating.

4. The Intervention: • New course "Introduction to Higher Mathematics" (Math 2325) • Sophomore level • Only co-requisite: Calculus I • 6-7 “labs” on a variety of mathematical topics (two weeks each) • Goal: See why you’d want or need a proof, and how you might discover it.

5. devise experiment conduct experiment refine conjecture formulate conjecture The Laboratories: test conjecture

6. The Laboratories (cont’d): • Intriguing open-ended problems • Short exposition by instructor • Prove conjecture • Write report (students can resubmit for a new grade once per lab) • Instructor provides guidance and feedback all along

7. Group work: • Students work in teams of 2-3 on the experimentation part, while • writing up reports individually. • Ways in which this interaction takes place: • One student at computer, other thinks of experiments to try • One student experiments, while other looks for patterns • Both students look for patterns on computer • Students work separately, but check in with each other every so often

8. Sample Laboratory: Iteration of Linear Functions Convergence never xn = a xn-1 +b ; initial value xo Convergence only if x0=b/(1-a) • What happens to this sequence as n gets large? • How does the answer depend on a, b and xo? Convergence always Convergence only if b=0

9. “Great course; I’m very happy it was here this semester. I recommend it to all … students who are unsure of majoring in Math. The best part (personally) is that I can actually read a Math book and understand.” • Student comments* • * Course Evaluation, Math 2325, Spring 2003 “Challenging but very effective; not overwhelming. Instructor’s excitement for math is contagious.” “Cognitive thinking is better. Mathematical interest is at a peak level for me. Comprehension is strengthened and reinforced. Mathematical inspiration!!” “This class helped to seal the holes in my mathematical knowledge base! I was exposed to many forms of math which I had not been aware of; additionally, terminology and abstract concepts were clarified.”

10. Student comments* • * Course Evaluation, Math 2325, Fall 2003 “I have been very pleased all semester. ... Overall very informative and rewarding. I have enjoyed this class immensely.” “I enjoyed this class. I learned a lot of new material, which was quite interesting.” “I liked the way we explore new mathematical concepts. I like the way the instructor points out the mistakes and encouraged us to turn in a revision.” “The textbook is really good and correlates exactly to the class.”

11. All Questions Answered, All Answers Questioned* * Borrowed from Donald Knuth • □ Art Duval artduval@math.utep.edu • □ Helmut Knaust helmut@math.utep.edu