Amazon.com: Mathematical Circles: Russian Experience (Mathematical World, Vol. 7): 9780821804308: Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg: Books

Russia perennially places among the top three performers in the International Mathematical Olympiad, the world's most prestigious mathematical competition for high-school students. The "mathematical circle" is undoubtedly one element of the mathematical culture that has contributed to Russia's success in that competition.

A Russian mathematical circle is not a geometrical shape, but rather a group of mathematically motivated students guided by a university-level mathematician who helps the students enlighten themselves about simple, yet beautiful and powerful, mathematical concepts. Fomin's Mathematical Circles is a strikingly elegant, practical tool for enabling American high-school teachers and math coaches to replicate the Russian mathematical circle here.

Mathematical Circles has two parts, each intended to be taught over one year. The first part has sections covering parity, combinatorics, divisibility and remainders, the pigeon-hole principle, graphs, the triangle inequality, and games. The second part has sections covering more advanced topics in divisibility, combinatorics, and graphs, as well as sections on invariants, number bases, geometry, and inequalities.

Each section begins with a short introduction addressed to the teacher and then proceeds to a series of problems periodically interspersed with concise explanations about new concepts being introduced through the problems and pedagogical advice related to those concepts. In any given section, the first problem is generally extraordinarily simple. The first problem in the parity section is:

Problem 1. Eleven gears are placed on a plane, arranged in a chain as shown [in a diagram with eleven gears interlocking in a circular arrangement]. Can all the gears rotate simultaneously?

What a beautiful first problem this is for illustrating the utility of the parity concept as a mathematical tool! The parity-based argument not only leads rapidly to a solution, but also disposes of the whole class of problems of this sort, regardless of the number of gears.

In the inequality section, Fomin introduces the triangle inequality, the most important inequality in elementary geometry. When this inequality is proved in conventional American geometry textbooks, the proof generally involves the construction of an altitude and comprises multiple lines of statements and reasons. Fomin's proof is algebraic and comprises all of one line.

Mathematical Circles overflows with such poetry. The problems are well conceived, well composed, well sequenced, and outright interesting. For those teachers interested in deepening their own understanding of mathematics, in search of material to enhance the traditional curriculum, or coaching math clubs or teams, Mathematical Circles is an invaluable tool. I recommend it without reservation.