Mathematical Circles

(Russian Experience)

Dmitri Fomin, Sergey Genkin, Ilia Itenberg

Foreword

This is not a textbook. It is not a contest booklet. It is not a set of lessons for classroom instruction. It does not give a series of projects for students, nor does it offer a development of parts of mathematics for self-instruction.

So what kind of book is this? It is a book produced by a remarkable cultural circumstance, which fostered the creation of groups of students, teachers, and mathematicians, called mathematical circles, in the former Soviet Union. It is predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport, without necessarily being competitive.

Thus it is more like a book of mathematical recreations-except that it is more serious. Written by research mathematicians holding university appointments, it is the result of these same mathematicians' years of experience with groups of high school students. The sequences of problems are structured so that virtually any student can tackle the first few examples. Yet the same principles of problem solving developed in the early stages make possible the solution of extremely challenging problems later on. In between, there are problems for every level of interest or ability.

The mathematical circles of the former Soviet Union, and particularly of Leningrad (now St. Petersburg, where these problems were developed) are quite different from most math clubs in the United States. Typically, they were run not by teachers, but by graduate students or faculty members at a university, who considered it part of their professional duty to show younger students the joys of mathematics. Students often met far into the night, and went on weekend trips or summer retreats together, achieving a closeness and mutual support usually reserved in our country for members of athletic teams.

We are fortunate to be living in a time when Russians and Americans can easily communicate and share their cultures. The development of mathematics education is an aspect of Russian culture from which we have much to learn. It is still very rare to find research mathematicians in America willing to devote time, energy, and thought to the development of materials for high school students.

So we must borrow from our Russian colleagues. The present book is the result of such borrowings. Some chapters, such as the one on the triangle inequality, can be used directly in American classrooms, to supplement the development in the usual textbooks. Others, such as the discussion of graph theory, stretch the curriculum with gems of mathematics which are not usually touched on in the classroom. Still others, such as the chapter on games, offer a rich source of extra-curricular materials with more structure and meaning than many.

Each chapter gives examples of mathematical methods in some of their barest forms. A game of Nim, which can be enjoyed and even analyzed by a third grader, turns out to be the same as a game played with a single pawn on a chessboard. This becomes a lesson for seventh graders in restating problems, then offers an introduction to the nature of isomorphism for the high school student. The Pigeon Hole Principle, among the simplest yet most profound mathematics has to offer, becomes a tool for proof in number theory and geometry.

Yet the tone of the work remains light. The chapter on combinatorics does not require an understanding of generating functions or mathematical induction. The problems in graph theory, too, remain on the surface of this important branch of mathematics. The approach to each topic lends itself to mind play, not weighty reflection. And yet the work manages to strike some deep notes.

It is this quality of the work which the mathematicians of the former Soviet Union developed to a high art. The exposition of mathematics, and not just its development, became a part of the Russian mathematician's work. This book is thus part of a literary genre which remains largely undeveloped in the English language.

Mark Saul, Ph.D.
Bronxville Schools
Bronxville, New York

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