Experience, of course

There is a pattern here, the pattern of auxiliary figures, which has some promise and which we can describe as follows: Try to discover some part of the figure or some closely RELATED FIGURE which you can construct and which you can use as a stepping stone in constructing the original figure. 

This pattern is very general. In fact, the pattern of similar figures formulated in sect. 1.5 is just a particular case: a figure similar to the required figure is related to it in a particular manner and can serve as a particularly handy auxiliary figure.

Unavoidably, its greater generality renders the pattern of auxiliary figures less concrete, less tangible: it gives no specific advice about what kind of figure we should seek. Experience, of course, can give us some directives (although no hard and fast rules): we should look for figures which are easy to “carve out” from the desired figure, for “simple” figures (as triangles), for “extreme cases,” and so on. We may learn procedures, such as the variation of the data, or the use of analogy, which, in certain cases, may indicate an appropriate auxiliary figure. (Polya, Mathematical Discovery p.15)

Good for Polya for understanding that advice such as “discovery a related figure” is too general to serve as specific advice. As such, it would seem unlikely to be helpful advice for a student struggling on a problem.

What fascinates me is how little interest Polya seems to have in helping the reader grasp the intangible. “Experience, of course, can give us some directives” but he seems uninterested in truly specifying them beyond the bare tantalizing crumbs that he offers here.

What, then, is Polya’s interest? The book is titled “On Understanding, Learning and Teaching Problem Solving.” Why does he think that specifying things such as his “auxiliary figure” pattern is helpful for the reader, whereas “find simple figures” or “check extreme cases” can only be grasped via messy, pick-it-up-as-you-go-along experience?

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