Spring 1993

Book Review: How to Solve It: A New Aspect of
Mathematical Method
G. Polya. Princeton, NJ: Princeton University Press, 1957, Second Edition.
Reviewed by Jennifer Norton, Graduate Student Associate, TRC

Polya's How to Solve It details the motives and procedures that lead to solutions in mathematical problem solving and shows teachers how to help their students learn how to solve problems. The interactive approach illustrated in this text is designed to help students with their problem- solving skills, while making sure they perform a reasonable amount of the work. Teachers use questions to guide students effectively and unobtrusively, and to enhance their problem-solving skills through imitation and practice. The book is divided into four sections:

In the Classroom: This section begins with a concise table that carries the reader through the four phases of problem solving: 1) understanding the problem, 2) devising a plan and recognizing the connection of parts of the problem, 3) carrying out the plan, and 4) looking back: reexamining, discussing, and checking the results in order to aid future problem solving. This section then details these aspects of Polya's approach and walks the reader through several examples.

How to Solve It: An imaginary dialogue between a student and teacher illustrates Polya's approach with respect to a particular mathematical problem.

Short Dictionary Heuristic: The dictionary provides references for particular aspects of problem solving, including such topics as the following: using analogies to aid problem solving, introducing auxiliary elements to aid problem solving, checking the result and deriving it differently, using the results of earlier problems to solve new problems, decomposing and recombining problems, thinking inductively, using notation, setting up equations, varying problems, and recognizing signs of progress.

Problems, Hints, Solutions: This section provides many sample problems to let readers test their knowledge and understanding of the approach introduced in this book. By encouraging the teacher/readers to participate in the learning process from a student's perspective, Polya helps readers internalize the approach and integrate it with their teaching skills.

Although the first edition dates from 1945 and the author is writing to teachers of mathematics, How to Solve It offers insights and practical solutions for the difficult task of teaching students to solve problems in several disciplines. If you find yourself solving problems for your students because they can't do it themselves, or frustrated that you can't get them to understand, try Polya's approach!