The arrival of Mathematical Circle Diaries evoked a memory of the last scene in a 2004 science fiction movie The Day After Tomorrow. During a cataclysmic storm, with temperatures dropping to sub-freezing, a small group led by Jake Gyllenhaal survives by burning the vast book collection in the New York Public Library. The movie clearly leads one to believe that this may be the only remnant of the city population. However, when the worst is over, and the survivors are being flown away to the less affected Mexico, they are surprised to see numerous groups of people emerging on rooftops and waiting for transportation. (One may of course wonder what kept them warm but this is beside the point.) The scene is a forceful tribute to human adaptability, the ability to survive even in the face of unimaginable adversity.
What does the movie have to do with Anna Burago’s book?
The book is a compelling testimony to our ability to overcome the alleged ills of the education system. Emergence of math circles is a relatively new phenomenon in the US. The first ones were “aspiring to prepare our best young minds for their future roles as mathematics leaders.” These were associated with colleges and universities and run by college instructors. What we observe now is the spread of grass-root initiatives, some of which have developed into Sunday and even full fledged schools. More importantly, they involve children of a much earlier age and, in principle, do not expect participants to have exceptional aptitude for mathematics. In the book under review the author shares her experience of running such math circle for middle school kids — for about a decade.
The first part of the book contains 29 lessons, so that a group that meets once a week has enough material to occupy themselves for a whole school year. The lessons are very detailed: each starts with stating the goal of the lesson and the required supplies (most quite modest). Then come warm-up exercises, which are followed by a set of solved problems, often in the form of teacher/student interaction, and ending up with a list of problems to be solved independently. There are also multiple inserts with advice for teachers.
The second part of the book contains descriptions of contests and games that can be used to entertain circle participants. Several chapters serve as practical examples for such activities: (Mathematical Auction — Chapters 4, 15, 24; Math Hockey — Chapters 9, 18; Mathematical Olympiad — Chapters 12, 21). The third part gives constructive advice on lesson planning and discusses the principles of running a math circle. Solutions to all the problems in the book have been collected in the fourth part of the book. There is also a short bibliography consisting mostly of Russian books and books from the MSRI/AMS Mathematical Circles Library.
The meticulous presentation of problems, their context and solutions should make it possible for a math teacher with no circle experience to start one and make it interesting to the students. The book also aims at parents who might want to try following the author’s footsteps. I may suggest doing this a step at a time — the book is very much suitable also for that purpose; many of the problems could be resolved conversationally during a drive to a movie or to some extracurricular activity. For example, no supplies are needed to discuss the following (Problem 3.10):
The country of FarAwaynia is composed of several states and also has several political parties. Once, a group of FarAwaynian politicians got together for a dinner. It is known that the group contained people from at least two different states and from at least two different parties. Prove that there were two politicians at the dinner that represented different states and belonged to different parties.
To give an insight into the level of the problems, this one is starred to indicate the difficulty above average. Still, if there is a difficulty, the parent may ask: “Let’s just pick out a couple of politicians. Are we lucky? And if not why? What can be said about the pair?” (For any of these questions the parent may choose to wait for a reply.)
Is this actually mathematics? As every mathematician will confirm, it is and very much so. As the author explained in the circle opening speech, The most unusual thing about our circle is that we are not going to study mathematics. I mean, we are not going to study the same kind of mathematics that you learn at school. (This is actually not quite so, because, e.g., chapters 5 and 6 are concerned with word problems. The methods employed, though, are different — more visual and intuitive.)
I do not agree with the thesis that due to the logical structure of mathematics its study fosters in students the ability for logical thinking. My impression, however, is that the manner in which the problems have been selected, grouped, solved and discussed in the book may prove otherwise. As the author mentions, The book is intended for teaching students in grades 5 through 7. It is an age of curiosity and of openness to learning: the discoveries students make in class and the triumphs of having a problem solved make them proud of their achievements. The learning skills and the thinking habits acquired by the students at this time stay with them for life and will come handy for future learning.
More than an educational reform, our society (and politicians, of course) needs a change in perspective of what education of young minds is about. I do believe that Anna Burago’s book spotlights the first steps in the right directions.
Alex Bogomolny is a former associate professor of mathematics at University of Iowa. He lives in New Jersey, maintains a popular site Interactive Mathematics Miscellany and Puzzles, with a server somewhere in Michigan, and blogs at CTK Insights. In an extreme need you can tweet him at @CutTheKnotMath.