"How To Solve It"

Polya

My interest in geological-mathematical problem solving is the downstream consequence of "discovering" George Polya in a local bookstore. The champion of problem solving, George Polya was a legendary mathematics educator. His book, How to Solve It (2nd ed., Princeton University Press, 1957, 253 pp.) is still available in popular bookstores nearly a half-century after its first publication.

Polya was interested in the "stimulating questions" that mathematics teachers ask to help their students solve problems. According to him,

"... the teacher is led to ask the same questions and to indicate the same steps again and again. Thus, in countless problems, we have to ask the question: What is the unknown? We may vary the words, and ask the same thing in many different ways.... Sometimes, we obtain the same effect ... with a suggestion: Look at the unknown! [Such questions and suggestions] aim at the same effect: they tend to provoke the same mental operation" (p.1-2).

Steps to Problem Solving

In How to Solve It, Polya classified the questions and suggestions that are helpful in discussing problem solving into four sequential categories. This classification constitutes "Polya's Four Steps", which are reproduced in varying forms in numerous mathematics-education texts. The four steps are:

1. understanding the problem
2. devising a plan
3. carrying out the plan, and
4. looking back

Polya expanded his views in a two-volume work, Mathematical Discovery: On Understanding Learning, and Teaching Problem Solving (Wiley, v. 1, 1962, 216 pp; v. 2, 1965, 191 pp.). Following are three concepts that you might find interesting on the importance of problem solving in general, and word problems in particular. (All quotes are from v. 1.)

1. Know-how is the important thing.

   "Our knowledge about any subject consists of information and of know-how. If you have genuine bona fide experience of mathematical work on any level, elementary or advanced, there will be no doubt in your mind that, in mathematics, know-how is much more important than mere possession of information."

   "What is know-how in mathematics? The ability to solve problems -- not merely routine problems but problems requiring some degree of independence, judgment, originality, creativity." (p. vii-viii)

2. Problems, by definition, contain obstacles.

   "Solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim which was not immediately attainable." (p. v)

   "A problem is a 'great' problem if it is very difficult, it is just a 'little' problem if it is just a little difficult. Yet some degree of difficulty belongs to the very notion of a problem: where there is no difficulty, there is no problem." (p. 117)

3. Do many problems.

   "Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.... (I)f you wish to learn swimming you have to go into the water, and if you wish to become a problem solver you have to solve problems." (p. v)