George Pólya's Influence on Mathematics Education

A century from now, George Polya will be -known through his books, papers, and a handful of videotapes and films of his teaching. No one then alive will have been one of his students or colleagues. What will his influence be on future generations of mathematics teachers? How will mathematics be taught differently because Polya considered how people learn and do mathematics, and then he spoke and wrote and taught? Currently, Polya's most obvious influence on mathematics education is through his ideas on how problems can be solved. "Problem solving" has become the watchword among trendy mathematics educators, and George Polya is their guru. The modest list of questions and suggestions for solving problems that he put forward in How to Solve It [2] has been transformed into a set of definitive "problem-solving strategies" that both describe human "information processing" for researchers and prescribe for teachers what students should be taught to do. The four phases of problem solving into which he grouped his observations-understanding the problem, devising a plan, carrying out the plan, and looking back-have metamorphosed into four stages through which problem solvers pass. The "Polya model" has become a ritual learned in teacher education courses and passed on to classes of baffled children, who wonder, for example, why anyone would want to look back at the banal problems with which they are beset. The theme of the 1980 yearbook of the National Council of Teachers of Mathematics [1] was problem solving, and when the advisory panel for the yearbook met to examine the more than 80 proposals for articles, they found references to Polya in almost every one. Polya has become the Marx and Lenin of mathematical problem solving; a few words of obeisance need to be offered in his name before an author can get down to the topic at hand. This spate of misreading and sanctification has, not surprisingly, led some iconoclasts to reject much of what is understood to be the message of Polya's writings. We are told that Polya's advice on heuristics is useless: Real mathematicians do not follow it, students cannot learn from it, researchers cannot get measurable results from it, and perhaps most appalling of all, computers cannot depend upon it in their attempts to simulate intelligent behavior. Polya is criticized for not having put his "strategies" into operational form, for neglecting in his "theory" the importance of conceptual knowledge, for overlooking the uses of mathematics in " real" situations, for failing to provide a complete and foolproof system. In other words, he was unable to make problem solving a science. The fact that Polya did not attempt to make problem solving a science but rather considered it, like teaching, a practical art often gets lost amid the homage and the disparagement. Polya drew upon his own experience as a teacher and as a problem solver in offering his advice. He knew better than anyone that it provided neither system nor theory. That is one reason he set up part of it as "A Short Dictionary of Heuristic" [2, pp. 37-232]. Dictionaries require the reader to employ them selectively, as they become intelligible and connected to what the reader is thinking about. They amplify the reader's thoughts; they do not dictate the form or arrangement of those thoughts. Polya saw his advice as providing some guidance to the baffled problem solver-some encouragement to stay with a problem, try to figure out what it is about, try to develop a reasonable plan for attacking it, and try to learn something useful from your experience with it. Polya considered the solution of problems to be the backbone of mathematics instruction. But, as he pointed out: