Book Review by David Palmer-Stone

The Art of Problem Posing
1983 by Stephen I. Brown and Marion I. Walter
The Franklin Institute Press

People tell horror stories about math teachers, and about math in general. Math dysfunctions have new, popular labels -- "math anxiety" and "innumeracy" (from the books by Sheila Tobias and John Allen Paulos). The standard curriculum has failed, according to this view, because of its authoritarian emphasis on the "right" way to get the answers, and its demoralizing judgments of those who make mistakes.

Adequate mathematicians can solve problems. But for some, math is intrinsically rewarding, like art. They don't just tolerate it: they like it, and choose to do it.

This book's authors suggest that good mathematicians have unconsciously developed strategies for fostering mathematical insight (in spite of the school curriculum), and that we can help students learn to consciously adopt these strategies.

This "non-standard curriculum" focuses on process and questioning, not on getting the right answer. With the threat of "judgment" and "failure" removed, self-esteem is no longer on the line, and the student becomes interested in exploring math through the following four interrelated stages:

1. List attributes of the problem, statement, phenomenon, etc., that you are starting with. This list must be as highly detailed as possible. (This first stage of intense observation is essential to any creative or critical-thinking activity).
   
2. Ask "What-if-not?" for the attributes. This is really the core of this process. You are asking, "What if this attribute were different?" There is a virtual infinity of ways that something can be different from what it is. Different people will notice different attributes, and replace them differently, so this and the previous step open the door for individual ideas.
   
3. Explore, ask questions, come up with examples, in the form of data, answers, ideas, etc.
   
4. After the problem is solved, ask "What does this mean or imply?; What have I actually done?; How far can it be extended?; Why did it work?" and so on. Analyze the problem for its significance.

Good mathematicians will look for regularities, generalities, limits and exceptions to rules, new viewpoints, implications, etc. The above steps provide a fertile hunting ground for these.

The authors suggest broad implications of this work:

1. Listing attributes – and then challenging them – can show us our unconscious assumptions that limit us -- our beliefs, our "givens";
   
2. Generating problems is vital for understanding and creating. You only understand something as you change it; just solving problems does not change the "given" that the problem provides. One should question the question itself.
   
3. Observing one's own learning ("metacognition") helps one learn how to learn. The exercises in the book provide an opportunity to watch and explore your own learning and thinking strategies, and to experiment with new ones.

I highly recommend this book. It has interesting ways to get started, and examples, including student work. The exercises (most are aimed at a high-school level) are generally easy enough to get into, yet hard enough to be challenging. The technique is readily transferrable to other, including creative and even practical, activities. It is good-natured, playful, fun, and at the same time profound. And I agree with its philosophy.