Review of "What-If-Not" strategy from "The Art of Problem Posing"

This review is in reference to pages 33-65 of the "The Art of Problem Posing" by Stephen I. Brown and comments on the strategy of "What-If-Not".

The term "What-If-Not", refers to a creative thinking process of examining mathematical/problem elements with a fresh perspective. With each concept, we identify key ideas & problems and then we try to expand on them in new directions, incorporating our own perspectives and thinking processes.

I really like this strategy for problem posing/solving. It's all about asking the right questions. Taking a theorem and changing a variable or condition to create a new situation and then examining it to see where it has taken us. This can be used extensively in the classroom, as we have begun to discover in our other classes. The students learn through inquiry and compound ideas and ultimately will have a more concrete understanding of the concept using this strategy.

We first explored this in SCED 316a with the aptly named 'UnDemo' in which the students were in control of the experimental process and merely guided by the teacher. In this way, the students were allowed to ask "what-if" and then examine "what-if-not". If students can understand what something 'is not', then they have a better understanding of what 'it is'. We next explored this in our Principles of Teaching lecture in which we talked about what it means to define a chair. And then from that, looking at a plethora of objects that were 'chair-like', but not chairs (i.e. thrones, stools, lounges, couches, etc.). In this way, the pythagorean theorem is a chair, but the other equations that were plucked from it, where 'pythagorean-like' and equally valid for examination. And by understanding what the pythagorean theorem wasn't, a student would have a better understanding of where it fit in their theoretical framework.

The only limitation to this method of inquiry, however, is that fact that this type of exploration can often be off 'topic' or time-consuming and may even lead to confusion. We cannot use this strategy like a blunt tool, or it will damage the process. Instead, we must use it as a fine instrument that requires skill and precision in utilizing for best results.