Product Description

"This is a sample of rich Russian mathematical culture written by professional mathematicians with great experience in working with high school students ... Problems are on very simple levels, but building to more complex and advanced work ... [contains] solutions to almost all problems; methodological notes for the teacher ... developed for a peculiarly Russian institution (the mathematical circle), but easily adapted to American teachers' needs, both inside and outside the classroom."

--from the Translator's notes

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

This book is intended for both students and teachers who love mathematics and want to study its various branches beyond the limits of school curriculum. It is also a book of mathematical recreations and, at the same time, a book containing vast theoretical and problem material in main areas of what authors consider to be "extracurricular mathematics". The book is based on a unique experience gained by several generations of Russian educators and scholars.

Product Details

• Amazon Sales Rank: #353334 in Books
• Published on: 1996-07
• Number of items: 1
• Binding: Paperback
• 272 pages

Language Notes
Text: English (translation)
Original Language: Russian

Customer Reviews

A superb collection of problems and mathematical insights.
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.

A great book for young students.
I bought this book to help me learn how to solve problems. However, when it arrived, I realised it was destined as a book for 12 to 14 year old students. Still, I gave it a try ( I am 19 years old). The problems are well stated, easy to do, and methodologicaly sound. I found the problems too easy, but my little brother ( 9 years old ) had trouble. It's great for some young students who would like to learn the basics of problem solving.

The Russians do Math Right
In sharp contrast to standard US math education, which
is generally a death march from algebra to calculus, this
book suggests a wonderful new way to organize the ideas
of elementary mathematics. The organizational principle
here is around fundamental ideas that underlie
every mathematical proof ever conceived: parity, the
pigeonhole principle, induction, counting (combinatorics),
etc. Each section starts off with easy problems that anyone
can get, and leads you through to more and more challenging
illustrations of that section's principle; the last problems
of each section are often quite sophisticated and rewarding.
Do the problems in this book, and you can't help but just
be smarter for it.

When I was a kid, I was mystified by puzzle problems that I
had no idea how to tackle, and intimidated by kids who could
solve those types of problems. Had this book been available
back then, it would have de-mystified those problems for me,
and I would have acquired the kinds of skills and insights
that make a real mathematician. Whatever your age, if you
are interested in developing your core competencies in math,
I can't think of a better endeavor than to do all the problems
in this book. If I were the US Secretary of Education, I would
make solving all the problems in this book a mandatory
requirement for all math teachers, and all graduating high
school students. Even a partial implementation of such a
policy would make this country mathematically literate in a
way that we can't even conceive of today. It would de-mistify
mathematical "genius" on a global scale.