This book, first published almost fifty years ago, was one product of the School Mathematics Study Group (SMSG) series on enrichment topics not incorporated into the standard high school syllabus. Pólya’s message, however, is as clear and striking today as it was when first written: mathematics is about problem solving,
It is my personal opinion that there is nothing of greater importance to be taught in mathematics to the high school student than the business of setting up equations… without getting to understand what a problem is about and what is relevant to it and (when appropriate) translating it from words in formulae, there is no mathematical education (p. 136).
And problem solve he does.
The book is organized into four content-rich chapters separated by one single-page chapter marking what Pólya terms the “tidal flow” between mathematics and physics. The chapters are long, subdivided into sections, and full of applications that follow the historical development of mathematics and physics. It is not an easy read — the reader has to do the mathematics along with Pólya — but his explanations are clear and his dry wit lightens the work along the way. Topics are arranged historically and one follows developments with a sense of a time traveler watching the concepts as they unfold.
The volume will appeal to a broad audience. Certainly the original addressees, high school teachers, could benefit from the plentiful selection of applications of mathematics. Not only are examples presented but solution strategies are reinforced. Pólya repeatedly asks the questions that we ask when modeling problem solving, “What is given? What is to be found? How many equations? (p. 147)” A second audience, not mutually exclusive from the first, is secondary and tertiary instructors who are interested in the history of mathematics. He flavors each section with a sense of the time and place in which the mathematician labored. Finally, the close link between physics and mathematics makes this text a tool that can be used by teachers in either discipline.
Katherine Safford-Ramus is Professor of Mathematics at Saint Peter’s College, the Jesuit College of New Jersey. She has been teaching mathematics at the tertiary level for 28 years. From October 2005 to October 2006, she served as the co-director of the Adult Numeracy Initiative, a project of the United States Office of Vocational and Adult Education, a division of the Department of Education. Safford is the author of Unlatching the Gate: Helping Adult Students Learn Mathematics. Her current research continues to focus on adults learning mathematics and, in particular, professional development of teachers as adult learners.