Reflection of The Art of Problem Posing: What-If-Not

The What-If-Not strategy impresses on me as a way to verify a given concept. As stated in the text, the act of inducing an always true conclusion is far more difficult than attempting to find an exception to the case. In this way, the what-if-not process is strangely familiar as the question of “what if this is not true?“ is regularly employed as the precursor or the first step in a Proof by Contradiction or the less satisfying Proof by Counterexample.

Not only does the question what-if-not help to verify new concepts, as in the case of the Pythagorus example, it allows a teacher to clarify the concept and by showing where the hypothesis fails, a student can more readily accept a definition or the set of conditions whereby the hypothesis succeeds. Furthermore, by exemplifying a what-if-not strategy, one is more likely to recognize when and when not to use a given formula or concept.

This strategic line of questioning is one that I have used while attempting to prove something to myself in order to internalize new ideas and the more I explore a concept, the more likely I will be able to use that concept as a tool for further examination of other ideas. This strategy as a teaching tool may highlight to the student, where a formula may hold, but also why or why not it always holds true. Without having been consciously of aware of doing so, asking what if or what if not is a very natural line of questioning in our everyday lives. I find it is at the heart of imagination and eventually invention. Using such a method in mathematics is, by any means, too much of a stretch.