1 Characteristics of Good Problem Solvers Annie Selden TTU Mathematics Department Mathematics Education Seminar September 2, 1998 [Have the De Franco problems (p. 219) projected on the screen when folks enter.] 1. Thanks for coming. Since this was to be a seminar on solving problems, I thought I'd put up some problems to start. I'll use them later, but put them away for now. 2. Introduction. I want to back up a bit and say something about mathematics education research and the kinds of results you can reasonably expect from it (since that's where the results on problem-solving that I'm going to talk about fit in). 3. Background. Mathematics education research as a discipline in its own right (as distinct from psychology) is only about 30 years old. E.g., JRME published its 25th anniversary retrospective in 1994. Math ed research has as its aim to discover results about the teaching and learning of mathematics, with mathematics as the primary focus. This differs from psychologists from Thorndike (The Psychology of Arithmetic, Macmillan, 1922) on, who have used (and still use) mathematics problems as a means of studying how people think generally. For them, the mathematics problems are just nice structured problem (like anagrams, logic puzzles, and other things they use). One might consider math ed research as a discipline beginning some thirty years ago (approximately in the New Math Era) with Freudenthal (and the establishment of PME) and Ed Begle. At the time, there was a lot of interest in being scientific (meaning like the hard sciences -- some have even called this "physics envy"). It was also a time when the behaviorism of Skinner still dominated much of U.S. psychology. It seemed reasonable at the time to try to find out which teaching methods were more effective using experimental vs. control group studies (sometimes called the "agricultural model") or trying to identify "critical variables." (Cf. Critical Variables in Mathematics Education: Findings from a Survey of the Empirical Literature, NCTM & MAA, 1979.) Much of what was published in JRME in the early 70's was of this nature -- short (5-10 page) statistical studies reporting the (sometimes) better performance of the experimental group (often in terms of a p < .05) over the control group who were taught traditionally. The design of the experimental method of teaching was often guided by one's best intuition, much like calculus reform courses were designed in the last 10 years. The traditional teaching was assumed to be uniform, i.e., everyone did it the same way. Gradually math ed researchers got dissatisfied with this and looked to other methods of inquiry, often borrowing methods from cognitive psychology, cognitive science