This book, by Kaplan and Kaplan, a husband and wife team, discusses the authors’ experience running “The Math Circle”. Given my own experience setting up and running a math circle with my wife, I was very interested in digging into this.

The introductory chapters make the authors’ perspective clear: mathematics is something for kids to enjoy *and create* independently, together, with guides but not with instructors. The following quote gets across their view on the difference between this approach and their perception of “school maths”:

Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult.

What a perverse fate for one of our kind’s greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you’re an adult you’ll never have to listen to music again.

I find the authors’ perspective on mathematics education, and their anti-competitive emphasis, appealing. Later in the book, when discussing competition, Math Olympiads, etc., they note two standard arguments in favour of competition: that mathematics provides an outlet for adolescent competitive instinct and – more perniciously – that mathematics is not enjoyable, but since competition is enjoyable, competition is a way to gain a hearing for mathematics. Both perspectives are roundly rejected by the authors, and in any case are very far removed from the reality of mathematics research. I find the latter of the two perspectives arises sometimes in primary school education in England, and I find it equally distressing. There is a third argument, though, which is that some children who don’t naturally excel at competitive sports do excel in mathematics, and competitions provide a route for them to be winners. There appears to be a tension here which is not really explored in the book; my inclination would be that mathematics as competition diminishes mathematics, and that should competition for be needed for self-esteem, one can always find some competitive strategy game where mathematical thought processes can be used to good effect. However, exogenous reward structures, I am told by my teacher friends, can sometimes be a prelude to endogenous rewards in less mature pupils. This is an area of psychology that interests me, and I’d be very keen to read any papers readers could suggest on the topic.

The first part of the book offers the reader a detailed (and sometimes repetitive) view of the authors’ understanding of what it means to do mathematics and to learn mathematics, peppered with very useful and interesting anecdotes from their math circle. The authors take the reader through the process of doing mathematics: analysing a problem, breaking it down, generalising, insight, and describe the process of mathematics hidden behind theorems on a page. They are insistent that the only requirement to be a mathematician is to be human, and that by developing analytical thinking skills, anyone can tackle mathematical problems, a mathematics for The Growth Mindset if you will. In the math circles run by the authors, children create and use their own definitions and theorems – you can see some examples of this from my math circle here, and from their math circles here.

I can’t say I share the authors’ view of the lack of beauty of common mathematical notation, explored in Chapter 5. As a child, I fell in love with the square root symbol, and later with the integral, as extremely elegant forms of notation – I can even remember practising them so they looked particularly beautiful. This is clearly not a view held by the authors! However, the main point they were making: that notation invented by the children, will be owned and understood by the children, is a point well made. One anecdote made me laugh out loud: a child who invented the symbol “w” to stand for the unknown in an equation because the letter ‘w’ looks like a person shrugging, as if to say “I don’t know!”

In Chapter 6, the authors begin to explore the very different ways that mathematics has been taught in schools: ‘learning stuff’ versus ‘doing stuff’, emphasis on theorem or emphasis on proof, math circles in the Soviet Union, competitive versus collaborative, etc. In England, in my view the Government has been slowly shifting the emphasis of primary school mathematics towards ‘learning stuff,’ which cuts against the approach taken by the authors. The recent announcement by the Government on times tables is a case in point. To quote the authors, “in math, the need to memorize testifies to not understanding.”

Chapter 7 is devoted to trying to understand how mathematicians think, with the idea that everyone should be exposed to this interesting thought process. An understanding of how mathematicians think (generally accepted to be quite different to the way they write) is a very interesting topic. Unfortunately, I found the language overblown here, for example:

Instead of evoking an “unconscious,” with its inaccessible turnings, this explanation calls up a layered consciousness, the old arena of thought made into a stable locale that the newer one surrounds with a relational, dynamic context – which in its turn will contract and be netted into higher-order relations.

I think this is simply arguing for mathematical epistemology as akin to – in programming terms – summarizing functions by their pre and post conditions. I think. Though I can’t be sure what a “stable locale” or a “static” context would be, what “contraction” means, or how “higher order relations” might differ from “first order” ones in this context. Despite the writing not being to my taste, interesting questions are still raised regarding the nature of mathematical thought and how the human mind makes deductive discoveries. This is often contrasted in the text to ‘mechanical’ approaches, without ever exploring the question of either artificial intelligence or automated theorem proving, which would seem to naturally arise in this context. But maybe I am just demonstrating the computing lens through which I tend to see the world.

The authors write best when discussing the functioning of their math circle, and their passion clearly comes across.

The authors provide, in Chapter 8, a fascinating discussion of the ages at which they have seen various forms of abstract mathematical reasoning emerge: generalisation of when one can move through a 5×5 grid, one step at a time, visiting each square only once, at age 5 but not 4; proof by induction at age 9 but not age 8. (The topics, again, are a far cry from England’s primary national curriculum). I have recently become interested in the question of child development in mathematics, especially with regard to number and the emergence of place value understanding, and I’d be very interested to follow up on whether there is a difference between this between the US, where the authors work, and the UK, what kind of spread is observed in both places, and how age-of-readiness for various abstractions correlates with other aspects of a child’s life experience.

Other very valuable information includes their experience on the ideal size of a math circle: 5 minimum, 10 maximum, as they expect children to end up taking on various roles “doubter, conjecturer, exemplifier, prover, and critic.” If I run a math circle again, I would like to try exploring a single topic in greater depth (the authors use ten one hour sessions) rather than a topic per session as I did last time, in order to let the children explore the mathematics at their own rate.

The final chapter of the book summarises some ideas for math circle style courses, broken down by age range. Those the authors believe can appeal to any age include Cantorian set theory and knots, while those they put off until 9-13 years old include complex numbers, solution of polynomials by radicals, and convexity – heady but exciting stuff for a nine year old!

I found this book to be a frustrating read. And yet it still provided me with inspiration and a desire to restart the math circle I was running last academic year. Whatever my beef with the way the authors present their ideas, their core love – allowing children to explore and create mathematics by themselves, in their own space and time – resonates with me deeply. It turns out that the authors run a Summer School for teachers to learn their approach, practising on kids of all ages. I think this must be some of the best maths CPD going.