The Story of Six Prodigies and How We Can Learn to Emulate ThemBy Katherine S. Ryan
Count Down: Six Kids Vie for Glory at the World’s Toughest Math Competition
By Steve Olson
256 pages, Houghton Mifflin Co, $24.00
Steve Olson's new book, “Count Down,” explores a phenomenon that few people know exists: the International Mathematical Olympiad. Here, in the 2001 competition, 473 high school students spent nine hours over two days, sitting at tiny desks dotted across a basketball court, pondering and scribbling their way through six problems of deceiving simplicity, that utilized only concepts from basic math courses.
An example: Prove that
a/(a2 + 8bc).5 + b/(b2+8ac).5 + c/(c2 + 8ab).5
is greater than or equal to 1 for any positive numbers a, b, and c. The solution involves only appealing to Jensen's inequality and carrying out trivial algebraic manipulations but, as Olson contends, many math professors would cringe at having to complete the problem in the time allotted the Olympians.
Like its cousin “Spellbound” -- the movie that introduced us to the brilliant contenders for the top prize at the National Spelling Bee -- Count Down acquaints us with the young mathematical whizzes of the American Math Olympiad team. Unlike “Spellbound” -- which explored individual personalities and American cultural values in charming detail -- Olson’s book does not fixate much on the particulars of the students. In broad strokes, of course, he outlines who these team members are, but there is a sense that any smart mathematicians could have filled their seats. These six students are simply a doorway into the larger themes that Olson wants to explore. One theme is the dearth of women in mathematics. Another is the beauty of following mathematical arguments from start to finish. What looms largest, though, is the question of genius.
Olson recognizes that these six students have something that majority of people do not. They possess a rare capacity to attack mathematical problems at a young age; most people would agree that they are in fact prodigies. But Olson wants to debunk this recalcitrant idea that there is some innate, magical capacity in kids like these. After all, they are just good at math. They must have learned it somewhere. By surveying a wealth of data and observing the team members in practice, he decides that this thing, genius, can in fact be distilled.
It involves, most importantly, dedication. No one, he argues, got to this math competition without hours of grinding away at symbolic conundrums. The hours, though, were well spent; it took a directed kind of training to get to the top. These champions emerged from years of work not so much adept at spouting out mathematical formulas as excellent at visualizing problems, making connections between distinct areas in the field, and driving themselves forward with the promise of competition. As Olson winds through study after study, finding out, for example, that Mozart was good but also had put in 3,500 hours of practice by the tender age of six, his argument seems clear: anyone who is personally dedicated to the task of achievement can be great. Genetics, shmetics.
From this, a far more subtle argument emerges. If anyone can become brilliant in mathematics, then why is the American school system producing hordes of students who wholeheartedly detest the subject? The answer, he believes, lies not in the lack of enthusiasm or effort, but in the way the material is introduced and tested. Unlike in Romania -- where students are taught through problem-solving sessions and where taxi drivers will brag about their own mathematical skills -- most students in the US learn by repetition and memorization. Instead of proving the Pythagorean theorem, most will instead be given problem after problem on how to use the formula -- problems that ask in a tedium of modifications what the value of a right triangle’s hypotenuse might be. This is no way to inspire a love of the subject.
Olson is suggesting that we re-examine our methods, looking to the training of the Math Olympians as an example. The students on the team learned through a program developed by their coach Titu Andreescu, who has his own ideas on what American school students might need. They also used a text called “The Art and Craft of Problem Solving,” which is considered to be excellent for the task and has now been on the market for five years. He believes that many of the resources put toward a small number of students isolated in talent searches -- students who, incidentally, are not later significantly more successful than their peers -- could better be used to train math teachers to engage students in the skills of difficult problem solving. Perhaps, with these strategies, it will be possible to reform Americans into eager math-o-philes.
Now, then, can someone please explain to me the proof of Jensen’s inequality?