We all know that there is a difference between a person who is good at mathematics and a person who loves mathematics. Furthermore, even without delving too deeply into the so-called Math Wars, I do not think it would be controversial to say that a number of people who would have made excellent additions to the mathematical community never stray towards our side of campus because they did not have somebody ignite their spark of interest in mathematics when they were in high school or junior high. I know from my personal experience that I may never have become a professional mathematician if it were not for Steve Sigur, a teacher at the K-12 independent school I attended in Atlanta, who "discovered" me when I was in the 8th grade and who helped nurture my excitement in mathematics by feeding me a steady stream of fascinating problems, interesting math projects and competitions, and exposure to people who did mathematics and loved it as much as he did. (At this point, I hope the editor will indulge me and allow me to plug the fact that Mr. Sigur is giving an MAA Invited Address on triangle geometry at this summer's Mathfest in Providence, and also has a book on the topic due out with John Conway in the near future. But I digress...)
I imagine that most of us who are professional mathematicians had a similar experience — whether it was a teacher or a parent or someone else who nurtured that enthusiasm about mathematics. In particular, I would wager that most of us were already excited about mathematics when we headed off to college, and in many instances it was despite the courses we had taken rather than because of them. Unfortunately, there are not enough of these teachers to go around, and this brings us back to where we started — with a large number of 16-18 year olds who could someday become great mathematicians but whose interest is never tapped.
That is the situation that Tatiana Shubin and Peter Ross saw in 1998 and it inspired them to start the Bay Area Mathematical Adventures, or BAMA. The goal of BAMA was to bring research mathematicians in to speak to high school audiences about topics that they would find interesting and which would get them interested in mathematics in a way that normal coursework — and even the mathematics competitions which are prevalent at that level — could never do. They launched the series with a talk by Ron Graham on juggling, and have brought a number of excellent speakers in over the years (see here for a list) While this is useful to the few thousand high school students in the Bay Area, it does not seem to be of much use to students in Atlanta, or Philadelphia, or Des Moines. Until now, that is. Because Shubin, along with co-editor David Hayes, has put together Mathematical Adventures for Students and Amateurs, a book which collects nineteen of these talks.
The book opens with a paper by Carl Pomerance entitled "Prime Numbers and the Search for Extraterrestrial Intelligence", closes with Robin Wilson's quite funny "Alice In Wonderland: an Informal Dramatic Presentation in Eight Fits", and is just as varied in between as those titles suggest. There is a wonderful article by Joe Gallian about how different states use mathematics to determine driver's license numbers, a topic which introduces the idea of error correction quite nicely (and a paper which reads just as animatedly as Gallian would speak on the subject). There is an expanded version of Graham's talk on juggling patterns, which is cowritten by Joe Buhler. There is an article by Richard Scott discussing the concept of curvature and geodesics. Arthur Benjamin and Jennifer Quinn write an article on Fibonacci Numbers, which regular readers of Read This! will guess (correctly) that I found well-written and interesting. And there is much more (a full table of contents can be found here.
The essays differ not only in their topics, but also in their level of complexity and in their styles of exposition: chatty, terse, first-person narratives, dialogue format, with pictures, without pictures. The only running thread through them all is their enthusiasm and their mathematical quality.
A highlight to me — showing my personal mathematical biases, I admit — were the two articles that introduce the topic of elliptic curves in two very different ways. Karl Rubin introduces the topic by thinking about right triangles of a given area while Edward Schaefer asks when a number can be written as the product of both two consecutive integers and three consecutive integers, leading one to consider the equation y(y + 1) = x(x + 1)(x + 2). Both articles seem to do an excellent job of presenting some of the interesting properties of elliptic curves and why they are of deep interest to mathematicians, while never losing sight of the fact that their audience has yet to take calculus, let alone group theory or algebraic geometry.
As the title explicitly says, the essays in this collection are written for students and amateurs — in other words for people with little to no formal mathematical background. While some of the essays include proofs and rigorously developed ideas, many pay attention to the big ideas and gloss over the details. As a mathematician, this bothered me on occasion, even though as a teacher I completely agree with the decision. The goal of most of the essays in the book seems to be not to teach mathematics but to expose the reader to ideas from mathematics, hopefully whetting their appetite for further exploration.
The bottom line of any book review should be to answer the question "should I buy this book?" So, should you? I do not think this book would be particularly useful to any of us for our own sakes — the ideas are clever but very few of them present much that I did not know before (seeing as how I am neither a student nor an amateur), and those that do don't go into the kind of depth or rigor that I would desire from a book I was reading on my own. And as a teacher in a university, I find it hard to imagine many situations where I would use this book either — some of the essays would be appropriate to give to my students, but most of them would be too elementary for most students past the Freshman level.
I am not trying to say that this book is without use. Far from it, in fact. It seems to me that anyone who is interested in mathematics and is exposed to high school age kids, whether they are teachers themselves, parents, neighbors, or even as university professors getting involved in our communities through the high schools, would do well to read this book to see some excellent examples of how we can present real mathematics in ways that students can understand. Furthermore, we have all experienced that feeling at a party when someone finds out we are a mathematician and asks what we do. We try — and often fail — to figure out something to say that conveys what mathematics is really like yet does not require an intimate knowledge of Ahlfors or Hartshorne. And while I wouldn't necessarily suggest carrying this book around to hand to people in that situation, if you did so they would find contained within it nineteen well-written expositions of mathematics which anyone can understand with a little bit of work and thought. And that might be all it takes to turn someone who is good at math into someone who also loves mathematics.
Darren Glass is Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at firstname.lastname@example.org.