SEEING THE PROBLEM - AN EXPLANATION FROM POLYA

In this article, we focus on the instructional explanation of guessing as a heuristic for solving the Five Planes Problem (FPP), given in a lesson by George Polya. We use the analysis of the lesson to test the applicability of what is known about instructional explanations in ele mentary mathematics to higher mathematics. The most salient characteri stic of Polya's lesson is the profusion of models and representations used to develop a sense of the problem and to support the instructiona l explanation. Polya used analogical models to transform the complex P PP to a simpler one, and he used representations to extend the perspec tive on the problem. Introducing such models and representations requi res keeping track of the links between them and the original problem. Polya excelled in this endeavor, and his passage to each new represent ation or model was generally justified in terms of the goals of the ex planation. Another very important feature of an instructional explanat ion is problem identification, a fragile goal state that needs to be c onstantly maintained when the problem being explained is complex. For this lesson, we examine how Polya established and maintained that goat . Finally, we offer some insight on instructional explanations in a si tuation in which the teacher needs to fulfill two goals at the same ti me. Here, the first goal was to teach students both how to solve FPP a nd how to use guessing as a problem-solving heuristic or strategy; the second goal was to show us how to teach guessing. Keeping these goals in mind, we offer some suggestions on how to teach metaskills through mathematical problems in accordance with the current understanding of constructivist approaches to learning.