Pólya, Problem Solving, and Education

Nineteen forty-five was both a banner year and a year of great chaos for problem solving. It was the year that Max Wertheimer's Productive Thinking, a classic study of problem solving, first appeared in English. Jacques Hadamard's Essay on the Psychology of Invention in the Mathematical Field appeared as well. So did Karl Duncker's monograph On Problem Solving. At the same time mathematics instruction in the schools was mostly drill-and-practice, since it was based on the Associationists' of bonds (see [16]). Drill-and-practice was under attack, however. With Wertheimer leading the charge, the Gestaltists claimed that such instruction completely missed the substance of mathematical thinking. The Gestaltists argued that rote instruction was of little value, and that when students memorized without understanding they missed the underlying essence of the mathematics they studied. Unfortunately, the Gestaltists had no theory of learning or instruction, since they believed all the real action takes place in the subconscious. Hence they had few practical classroom suggestions. Moreover, they were counterattacked by the behaviorists, who claimed that mind and were useless constructs, and that all behavior (mathematical and otherwise) could be explained by stimulusresponse chains. It was against that background of psychological and pedagogical confusion that the most important problem-solving work of the time appeared. As indicated in the next section, How to Solve It was hardly Polya's first foray into the world of problem solving. It was, however, an absolutely critical one. How to Solve It marked a turning point both for its author and for problem solving. For Polya it was the first of a series of major volumes on the nature of mathematical thinking, the topic that became the focus of his work in his later years. How to Solve It was followed in 1954 by the two volumes of Mathematics and Plausible Reasoning, and in 1962 and 1965 by the two volumes of Mathematical Discovery. For mathematics education and for the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Polya. Since then Polya's influence both on the study of mathematical thinking and on the study of productive thinking in general has been enormous. One major purpose of this note is to trace out the main ideas in Polya's work. Another is to trace the influence of those ideas on subsequent research into the nature of mathematical thinking.