Evan O’Dorney leaps from his seat and snatches the chalk from his teacher’s hand. “I think it goes like this,” the eighth-grader exclaims. He has come up with a formula for the number of ways to divide a set of objects into three subsets, and he explains it to the group.
“What do you think?” the teacher asks, turning to the group of teenagers. “Are you persuaded?”
“I am!” one student calls out. Others begin nodding their heads in agreement.
The students plunge onward, and together, they manage to come up with a general formula for dividing a set of n items into k subsets. The teacher raises questions to help lead the way but provides no answers. Before the hour is over, the students have worked out the problem on their own.
It is a typical session at the Berkeley Math Circle in California. The students have chosen to come here on a Tuesday night for their own enjoyment, not for a school assignment.
The math circle phenomenon is not confined to Berkeley. Across the country, mathematicians who are frustrated with the state of math education are taking matters into their own hands. Many of them have started ambitious extracurricular math programs.
Different groups have different philosophies and approaches, but they all introduce students to deep mathematical ideas that are not normally covered in classrooms, and they encourage students to tackle tough mathematical questions for themselves.
The idea of math circles began in Eastern Europe as an effort to train students for math competitions in which they have a few hours to tackle a small number of very hard problems. The competitions culminate in the selection of six students from each country to compete in the International Mathematical Olympiad.
Zvezdelina Stankova of Mills College in Oakland, California participated in math circles as a high school student in Bulgaria and went to the International Mathematical Olympiad twice, in 1987 and 1988. After immigrating to the United States, she was shocked to find that most American students don’t encounter a single proof in the course of their math education. In 1998, she started a math circle in Berkeley, California, fashioned after the club she had enjoyed in Bulgaria. “It’s world-class mathematics for kids,” she says.
At the weekly meetings of the Berkeley math circle, different teachers give talks on a wide range of subjects, covering geometry, number theory, topology, probability, game theory, and more.
“We talk about the beauty of math in topics that are not usually covered [at school],” Stankova says. “This program is really for talented, bright kids who want to be challenged and learn the depths of mathematics.”
Many kids from the Berkeley Math Circle have gone on to win prizes in various math contests. Several have made it to the International Math Olympiad, for which only six U.S. students are chosen each year from about half a million contestants. Two Berkeley Math Circle participants have won gold medals in the international competition.
Stankova says the competitions provide a focus for students, but the real point is to expose them to beautiful mathematics, train them to think mathematically, and encourage them to pursue math-related careers. “We want to make doing math as popular as being on the football team,” Stankova says.
Some other math circles take a very different approach. Since 1994, Robert and Ellen Kaplan have started a number of math circles in the Boston area. Their math circles now serve more than 125 students each year, from age 4 through high school. The Kaplans welcome students with widely varying degrees of mathematical know-how and comfort, and they strongly de-emphasize competition. Their classes meander in whatever directions the participants’ ideas take them.
Recently, Bob Kaplan was meeting with a new class of four- to six-year-olds for the first time. He walked into the room and said, “My name is Bob. What’s the area of a circle?”
The children rapidly ran into two difficulties: They weren’t precisely sure what a circle is, and they had no idea how to measure area.
By their fifth meeting, the kids had come up with a few ideas. They put a grid over a “3 step circle,” as they called it, which means a circle of radius 3. They counted 16 squares inside the circle, “plus junk,” and 36 squares outside the circle, “with some extra junk.” The area of the circle, they decided, was between 16 and 32.
Next, they wondered what to do about the junk. The children’s first idea was to collect it up somehow and put it together into squares, but they weren’t sure how to deal with the curvy sides. A girl named Emily had another idea: drop the squares and “do things with triangles.” In other words, inscribe a hexagon.
“But now we have a whole new problem, because we don’t know how to calculate the areas of triangles,” said Kaplan. He paused, cogitating, and then burst out, “It’s so exciting!”
In following the children’s lead, the Kaplans are even willing to travel down blind alleys. “A key part of our approach is that we go along with false conjectures,” Bob explains. “Whatever they want to suggest, we say ‘fine.’ They’ll find out.”
Ellen chimes in, “The math will show them!” She explains, “We frequently part in sadness and desperation, which is like real math. Then we come back together, and someone says, ‘I’ve been thinking about this, and it seems to me…'”
In their new book, Out of the Labyrinth, the Kaplans explain that they organize each of their courses around an alluring, age-appropriate problem. When they ask young children, “Are there numbers between numbers?” the question may inspire them to invent fractions, discover how to manipulate them, and develop decimals. Some might even stumble upon the disturbing existence of irrational numbers.
A ten-year-old once demanded of Ellen, “You have to tell me what ii is, because there is a kid at school who knows, and he won’t tell!” Ellen seized on the question to create a new course that explores trigonometry and Taylor polynomials to develop tools to answer the ten-year-old’s question. It follows the children’s ideas all the way.
It might seem that only exceptionally bright and motivated kids would be capable of working out such challenging mathematics for themselves. However, the Kaplans say their classes have attracted some students who were not very interested in math but had heard that the classes were fun. The Kaplans find that almost all of the kids who try the class stick with it, with the possible exception of a few who get dragged there by an overzealous parent.
Over time, even the most timid students develop strong mathematical abilities, the Kaplans say. In fact, the Kaplans have become great disbelievers in the notion of mathematical talent. “Anyone can learn to think like a mathematician and will, in the process, come to find pleasure in learning and creating math,” they say in their book.
The Kaplans don’t train their students for math competitions, although a few students enter contests through math clubs at their schools. “My first objection to competition is that one person wins, everybody else loses,” Ellen says. “If you’ve ever watched something like the football playoffs, the team that comes in second is an incredible team that’s done incredibly well. And they’re sitting there in tears, saying their whole season is ruined. To take something like mathematics that takes your whole brain, your whole passion, and reduce it to that!” She shudders.
In their classes, the Kaplans say that the lack of competition helps kids to become willing to say, “I didn’t get that. Could someone explain it to me?” They develop an admiration for one another’s abilities and learn to rely on one another’s strengths.
No matter what the approach, the kids in these clubs are discovering some wonderful mathematics, learning to think for themselves, and enjoying themselves at the same time.
Try this puzzle from the most recent Bay Area Math Olympiad:
The points of the plane are colored in black and white so that whenever three vertices of a
parallelogram are the same color, the fourth vertex is that color, too.
Prove that all the points
of the plane are the same color.
Solution:
Suppose not. Let A be a white point and B a black point.
Their midpoint C is one
of the two colors; without loss of generality suppose C is black. Now pick any point D not
collinear with A, B, C, and construct E so that CADE is a parallelogram. If D, E are both white,
then CADE has three white vertices and one black vertex, impossible; if they are both black,
then CADE has three black vertices and one white vertex, impossible. So D and E are opposite
colors.
But BCDE is also a parallelogram, since BC = AC = DE and lines BC, DE are parallel. However, BCDE it has three black vertices and one white vertex. Thus we have a contradiction.
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