Mathematics: A Fresh Perspective

A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.
Polya, 1944, preface to the first printing

Introduction

A common characteristic of high school mathematics instruction is the lack of motivation teachers provide for learning abstract concepts. I often hear students ask, "Why should I learn this concept?" "Where could I use it? Is it applicable in everyday life?"

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Another serious drawback of conventional mathematics instruction is that the students do not get the sense that mathematics is a process. The compulsory curriculum and traditional teaching methods used by many teachers fail to illustrate the way mathematicians actually do mathematics. Students do not realize that all the results in mathematics are obtained only after many attempts, after long struggles to solve certain fundamental problems. They view mathematics as a given set of rules and laws established many years ago that they are now obliged to use. They can see only the petrified structure, with no trace of the creative process that produced it. This fact raises another question for me: How can I help my students understand the road that mathematicians travel to obtain mathematical solutions?

Mathematicians know that they owe much to the work of their predecessors. Math teachers know that the mathematics we are teaching now is the result of centuries of mathematics research. We also know that much current mathematical theory has its beginnings in the work of 18th- and 19th-century mathematicians. But what about the students - do they have this historical perspective? Would a course on the history of mathematics help students understand how mathematics has developed?

As I have tried to find answers to these questions, it has become clear to me that mathematics can be richer and more meaningful if it is taught in a historical context, and in connection with other fields.

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