Seeing the Problem: An Explanation From Pólya

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Abstract
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 Pólya. We use the analysis of the lesson to test the applicability of what is known about instructional explanations in elementary mathematics to higher mathematics. The most salient characteristic of Pólya's lesson is the profusion of models and representations used to develop a sense of the problem and to support the instructional explanation. Pólya used analogical models to transform the complex FFT to a simpler one, and he used representations to extend the perspective on the problem. Introducing such models and representations requires keeping track of the links between them and the original problem. Pólya excelled in this endeavor, and his passage to each new representation or model was generally justified in terms of the goals of the explanation. Another very important feature of an instructional explanation is problem identification, a fragile goal state that needs to be constantly maintained when the problem being explained is complex. For this lesson, we examine how Pólya established and maintained that goal. Finally, we offer some insight on instructional explanations in a situation in which the teacher needs to fulfill two goals at the same time. Here, the first goal was to teach students both how to solve FPP and 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.

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