This book is a record of math circle activities with guidelines for their use with younger students. The authors begin by citing the breakthrough experience of Natasha Rozhkovskaya in 2009 when she pioneered a program for grades 1-3 at UC Berkeley. The program was a smashing success and led to follow-up programs for elementary age students at UC Berkeley, Stanford, Dominican University and the University of Oregon (Eugene). For those unfamiliar with the Math Circles programs (is this a non-empty set?) they often operate as enrichment opportunities in after school environments and feature a broad array of challenging mathematical activities and discussions for students. The great traditions of the Russian training system in mathematics lie behind much of its success.

So exactly what can such a program do for students who are so young that the idea of multiplication may be new to them? As one might expect, many of the topics involve visualizing patterns and developing arithmetic facts related to them. Indeed, this book begins with figurate numbers and geometric arrangements of “balls” or squares on graph paper. The starting points are simple (How many figures can be made with 3 balls? 4 balls?) and progresses to relations amongst the various figures (for a square arrangement of 9 balls how many must be added to two sides in order to make a larger square?). Somewhere along the way the students are led to discover that the sum of odd consecutive numbers is indeed a perfect square. Since all of this is not given in lecture format, the discussion among the students must drive the discovery process and one of the strengths of the Math Circles program is the nature of the follow-up questions that allow students to deepen their understanding along the way

How many odd numbers are in the sum 1 + 3 + 5 + 7 +…+ ? + ?? =11 X 11 ? Find the missing numbers ? and ??. Using the fact that 10 X 10 = 100, calculate the sum without multiplying 11 by 11 and without adding all the odd numbers.

Of course, this is only the beginning and the discussion continues with triangular and cubic tiles, and triangular and tetrahedral numbers and pyramids. Since the discussions seem to involve some understanding of multiplication of small numbers my suspicion is that the approach has a greater chance of success with 3rd or 4th graders.

Following Chapter One, are chapters introducing combinatorics, Fibonacci numbers, Pascal’s triangle and areas (which includes some discussion/derivation of the Pythagorean Theorem.) Coloring problems, nonsense languages and parking problems are all used to develop intuition about these concepts. So much lovely mathematics is contained therein without a single equation being exhibited! For instance, a count of the number of linear strip arrangements of length n using pink 2-space strips and blue 1-space strips leads directly to the nth Fibonacci number. Now the challenge becomes whether or not the various properties of the Fibonacci numbers can be worked out just using this tiling definition (to fans of Bijective Combinatorics this is a mathematical activity well worth pursuing). Additionally, a fair number of different counting scenarios (socks, ancestral trees, spirals and golden rectangles) are all treated in some depth so that students can develop the beginnings of a very important skill- recognizing common structures). The chapter on area also includes a number of problems that lead up to a development of Pick’s formula for regions bounded by lattice points and straight edges. All in all, this is a delightful selection of problems that are well-illustrated and will appeal to youngsters and oldsters alike.

Elementary teachers will find much of use here in cultivating mathematical curiosity among their students. I highly recommend it!

Jeff Ibbotson is the Smith Teaching Chair at Phillips Exeter Academy.