In 1998, a room of approximately thirty teenagers listened to Brian Conrey give a lecture on elementary number theory lecture. First came a crash course in terminology and notation, including recursive definition and the definition and notation for "a divides b". After introducing the Fibonacci numbers, Conrey invited the students to try to prove that the 5th Fibonacci number divides every 5th Fibonacci number that comes afterwards. After a few minutes, a quiet voice said, "I think it can be generalized. I think Fn divides Fmn."
It was thirteen-year-old Tienkai Liu. As the story goes, Conrey began stalling for time, asking for ideas on how to prove it, and for examples. "But I can prove it!" replied Tienkai, and he marched up to the board and gave a beautiful proof.
Tienkai Liu was one of the six high school students on the 2002 US International Mathematics Olympiad team. Count Down is the story of those six members and the problems they solved on that Olympiad. The book is as fascinating as the individuals themselves, who, in addition their passion for mathematics and problem solving, enjoy word games, music, politics, creative writing, and sports.
One of author Steve Olsen's aims is to convey a sense of the playfulness of the creative mind and of the deep enjoyment mathematical ideas can bring, and he does this exceptionally well. The core of the book consists of six chapters entitled Insight, Competitiveness, Talent, Creativity, Breadth and A Sense of Wonder. Each chapter is devoted to an exploration of one of these attributes as embodied by a particular team member. For example, the Breadth chapter is devoted to Oaz Nir, who improvised brilliantly on a television talk show episode devoted to the IMO team. In High School, Oaz played water polo player, competed for the debate team and, in college, chose a double major in Math and English. After reading Count Down it is clear that, among team members, such breadth is the rule rather than the exception.
Each student's story is told via a problem he solved on the 2002 competition. This allows Olsen to talk meaningfully about creativity in mathematics by describing the break-through each competitor had. Ian Le was simply returning from a bathroom break when he suddenly realized that an obscure inequality, named "Jensen's Inequality" for the man who discovered it in 1906, would unlock the problem. Oaz Nir simply "dumbassed" his problem, a strategic move that, while inelegant, saved him time for other problems. Reid Barton and Gabriel Carroll drew on properties of the complex number field.
By describing the contestants themselves, their interests and enthusiasms, by offering samples of their mathematical humor, and even the word and card games with which they passed the time, Olsen reveals these teenagers as both exceptional and typical. Despite his or her talent, it is clear how hard an Olympiad team member must work to reach the necessary level. Many begin by taking part in MATHCOUNTS, a mathematics enrichment, coaching and competition program for middle school students. Some, like Tienkai Liu, are lucky enough to take part in Mathematical Circles, a grass roots effort by mathematicians to enrich US mathematics education for the gifted. Team members are the top scorers in a whole sequence of mathematics competitions, first on the AMC12 (formerly AHSME), then on the AIME, then on the United States of America Mathematical Olympiad (USAMO). The top twelve USAMO scorers attend a summer camp, and a final test determines which six are chosen for the US team.
How do students discover that they are good at mathematics, or even that they like mathematics, particularly when so much of school mathematics is flat? Sometimes the discovery is accidental. Melanie Wood is the only girl to have even been a member of the US team. In 7th grade, Melanie was asked to fill a vacant spot on her school's MATHCOUNTS team. To her surprise, she won the competition. Before this experience, she had never thought of herself as particularly good in Math. The fact that, worldwide, fewer women than men compete in the IMO prompts a discussion of the theories behind gender differences which is intelligent and complete.
Woven throughout the book is material of interest to mathematicians and teachers of mathematics, and more generally to anyone who has experienced the pleasure of solving problems. Having said that, the book is clearly written for a very general audience. While Olsen describes the problems and their solutions, he does so in a very gentle way, and relegates technical details to an appendix.
Students who plan to become teachers should read this book. The CBMS-MAA report on The Mathematical Education of Teachers recommends that future teachers take a capstone course linking college and high school mathematics, and Count Down would be a wonderful supplemental text for such a course. Although many students would need additional information and hints from the instructor, it would be possible to ask students to read the account of each solution, fill in the missing details, and comment on the creativity and insight involved.
More importantly, the book challenges the mission of so many mathematics educators to "break the problem down" so that every student can understand it. My students think this is the essence of good teaching, but my own children find the step-by-step process unrelentingly dull. I loved reading that in Romania, according to Titu Andreescu, US IMO coach, 90% of the population love math, and it is typical for a taxi driver to tell a mathematician, with enthusiasm, "I was really good at math!" (Compare that to the typical US response, "I could never balance my checkbook!")
What is responsible for such different reactions? Olsen describes how teaching culture in the US differs from teaching culture in Europe and Asia. He cites the well-known statistics, the divergent attitudes towards multi-step problem solving, the watered down content and the low expectations in the US. It's provocative material, and Olsen presents it well. In fact, the book provokes thought on so many fronts, including the nature versus nurture debate, the gender question, and the nature of creativity, all while being thoroughly entertaining, that I think you'll enjoy it tremendously, and that you'll enthusiastically recommend it to students, friends, and family!
Kathryn Weld (Kathryn.firstname.lastname@example.org) is Associate Professor of Mathematics and Computer Science at Manhattan College.