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PROBLEM SOLVING: WHAT I HAVE LEARNED FROM MY STUDENTS 1. Background

Source: www.mathnet.or.kr
Topic: Problem Solving

Sort Desciption: of our students are not adept at either problem solving or at constructing simple. proofs. I will first describe what I mean by problem solving and how that ...

Content Inside: Proceedings of KAIST International Symposium on Enhancing University Mathematics Teaching May 1216 2005 Daejeon Korea PROBLEM SOLVING: WHAT I HAVE LEARNED FROM MY STUDENTS JAMES SANDEFUR Abstract. Most students have difficulty transitioning from mathematics courses which are taught in a more algorithmic manner such as algebra and calculus often are into more theoretical courses which require students to comprehend and construct proofs. To understand why students have this problem and how they can overcome it for 4 years I have been videotaping my college math students attempting to construct proofs of statements of varying degrees of difficulty. These videos have given me insight into where students have problems why they become stuck and what support helps them make progress. In this talk we will preview some video-clips and discuss their implications for teaching proof and problem solving. I will also share how my teaching has changed and what impact it has had on Georgetown University math students as they transition into our upper level courses. 1. Background Most of our students can quickly learn the techniques we teach and can apply them to problems that are similar to the ones we have worked in class. On the other hand these same students often have difficulty generalizing techniques both to multi-step problems and to problems that are somewhat different from what they have seen. They often exclaim You havent shown us how to work this. In other words most of our students are not adept at either problem solving or at constructing simple proofs. I will first describe what I mean by problem solving and how that differs from constructing proofs. Consider a two-person game in which there is a pile of n beads and on each turn a player can pick up one two or three beads. The winner is the person who picks up the last bead. If each person plays optimally who wins the game the person who goes first or second and what is the optimal strategy? Problem so ...

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