This is the third in a series of discussions intended to provide an informative and sometimes provocative overview of the thinking of some key researchers in mathematics education and educational psychology.
Polya describes a four-stage approach to mathematical problem-solving. He bases his approach on common-sense questions that would naturally occur to an experienced problem-solver. Polya claims teachers should pose these questions to students in as natural and unobtrusive a way as possible, the goal being to encourage independence and internalization of this framework.
Polya, G. (1957). In the classroom. In Polya, G., How to solve it (pp. 1-32). Garden City, NY: Doubleday and Company, Inc.
o Effective problem-solving consists of four main phases: understanding the problem, devising a plan, carrying out the plan, and looking back. Understanding the problem includes labeling and identifying unknowns, condition(s), and data, and determining the solubility of the problem. Devising a plan means drawing on prior knowledge to frame an appropriate technique, restating the problem if necessary. The chosen technique is then carried out in the third stage. The final phase consists of checking the correctness of the solution and adding the problem to one's store of knowledge for use in solving future problems.
o The teacher prompts the students in applying these phases with the use of key suggestions and questions.
o An important characteristic of Polya's framework is generality: it can and should be applied to many different types of problems.
o Another key is practice. This framework should be applied often and in many different situations so that students can gain experience and confidence in problem-solving.
Direct Quotes (and some comments):
o "There are two aims which the teacher may have in view when addressing to his students a question or a suggestion...: First, to help the student to solve the problem at hand. Second, to develop the student's ability so that he may solve future problems by himself" (p. 3-4).
o "If the same question is repeatedly helpful, the student will scarcely fail to notice it and he will be induced to ask the question by himself in a similar situation. Asking the question repeatedly, he may succeed once in elicting the right idea. By such a success, he discovers the right way of using the question, and then he has really assimilated it" (p. 4).
o "[Appropriate questions and suggestions] have two common characteristics, common sense and generality. As they proceed from plain common sense they very often come naturally; they could have occurred to the student himself. As they are general, they help unobtrusively; they just indicate a general direction and leave plenty for the student to do" (p. 4).
o Regarding the inadvisability of such specific questions as "Could you use the Pythagorean Theorem?" -- a possible question when trying to find the length of the diagonal of a rectangular parallelepiped -- Polya says:
"The intention may be the best, but the question is about the worst.... [T]here is a long sequence of objections against that sort of help:
(1) If the student is near to the solution, he may understand the suggestion implied by the question; but if he is not, he quite possibly will not see at all the point at which the question is driving. Thus the question fails to help where help is most needed.
(2) If the suggestion is understood, it gives the whole secret away, very little remains for the student to do.
(3) The suggestion is of too special a nature. Even if the student can make use of it in solving the present problem, nothing is learned for future problems. The question is not instructive.
(4) Even if he understands the suggestion, the student can scarcely understand how the teacher came to the idea of putting such a question. And how could...the student...find such a question by himself? It appears as an unnatural surprise, as a rabbit pulled out of a hat; it is really not instructive" (p. 22).
o "Begin with a general question or suggestion..., and, if necessary, come down gradually to more specific and concrete questions or suggestions till you reach one which elicits a response in the student's mind" (p. 20).
o "A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution" (p. 15).
o "One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else....The teacher should encourage the students to imagine cases in which they could utilize again the procedure used, or apply the result obtained" (p. 15-16). [Polya uses the example of finding the measurement of the diagonal of a parallelepiped. From that problem he poses the related, real-life problem of determining the length of several wires needed to secure an erect flagpole].