Proof Positive

By Lani Harac — February 18, 2005 13 min read
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The unadorned fourth-floor classroom in Harvard University’s Science Center was an appropriate foil for the theories being discussed within it. On a chilly November evening as gray as the room itself, a rounded older man with a bushy white mustache conducted a math lesson on the concept of infinity.

“What we’re trying to figure out is if there’s a one-to-one correspondence between points on a line and points on a plane,” he told the 17 students, dragging chalk along the blackboard to create a square with dotted lines.

The students had already learned that a line consists of an infinite number of points; pick any two numbers (0 and 1, for example), and there’s always another between them (say, 0.5). The same is true for a plane, except each point is represented by two numbers, commonly called x and y coordinates, that relate to two perpendicular edges of the plane.

Bob and Ellen Kaplan

The class was trying to determine whether a plane and a line contain the same number of points—it was either that or acknowledge that there are different sizes of infinity. They’d studied, in particular, the work of Georg Cantor, a 19th century German mathematician who’d come up with a rule to establish the correspondence, but they hadn’t been told how he did it. And now, after adding, dividing, and trying other functions with various numbers, they were stumped.

Still, there was time left. “We’ve spent almost an hour on this,” said the instructor, Bob Kaplan. “Of course, Cantor spent three years nonstop.”

It’s important to note that these students weren’t Harvard undergrads. They were 9- and 10-year-olds voluntarily participating in what Kaplan and his wife, Ellen, call the Math Circle, their version of a roughly century-old Eastern European practice.

Now looking at long decimal numbers, one boy suggested squaring the x and y coordinates. Another wanted to multiply them by the number pi. And a girl with a British accent proposed focusing on every other number after the decimal point.

“At this point,” Kaplan asked, “how many of you think there is a one-to-one matchup between the points in a square and the points on the line? One, two, three and a half. ... How many people think there’s not a one-to-one correspondence?” He paused, and almost all the hands went up. “So you think that Cantor, looking for a new size of infinity, has found one looking at the points on a plane?”

One student, suspecting a trick, asked: “Would you teach us something that was impossible?”

To anyone sitting in on the Kaplans’ classes for the first time, this question would seem appropriate. “Mathematics alone,” the French philosopher Simone Weil once wrote, “make us feel the limits of our intelligence.” But the Kaplans prefer the sentiment of Cantor himself: “The essence of mathematics is freedom.”

For all the talk about “math wars” in this country, most public school classes focus more on rote memorization than on theory. But the Kaplans, after 40-plus years as educators, believe that learning the mathematical process is an end in itself. The approach that they and their Math Circle instructors employ during these after-school sessions, which serve kids ages 5 to 18, is simple: Present an abstract concept as a puzzle, then let the students wrestle with it and come up with their own solutions. Not only is the process fun, the Kaplans believe, but it can be applied to other subjects, as well.

“Everything comes around again,” Ellen Kaplan says. “The ideas of mathematics are universal.”

Earlier that afternoon, Ellen Kaplan was leading a class of her own in a third-floor space identical to her husband’s. Her dozen or so students, 7 to 9 years old, were trying to come up with a formula for adding a string of numbers—such as 1, 2, 3, 4, and 5—without having to do so sequentially.

Starting with 1, Kaplan began writing odd numbers on the blackboard. She turned to the kids after 7 and asked, “How far up should I go?”

Some of the boys started shouting. “Nine thousand, nine hundred and ninety-nine!” “Nine million!”

“I hear ‘nine.’ I think that’s a good number,” Kaplan replied, writing “1 + 3 + 5 + 7 + 9.” “OK! If I wanted to do this, and I’m far too lazy to add all that up, ... what was Gauss’s clever and brilliant plan for avoiding work? Yasmin.”

Yasmin Siraj attends Math Circle

A little girl wearing a spangled figure skating costume recounted a method devised by Carl Friedrich Gauss, another 19th century German mathematician. The class had previously used “Gaussing” to add counting, or positive whole, numbers. But would it work for nonconsecutive ones?

Suddenly, one of the boys banged on his desk and blurted the next step. “You, um, under the 1,” he said, “you put nine-plus-seven-plus-five plus-three-plus-one.” Kaplan wrote it all down as the boy completed the process and arrived at the correct sum.

“Yes, well,” she responded, “the interesting thing is not what’s the answer, which we got, but how we get there.”

Each Math Circle class focuses on one topic per semester. Students are given a complex problem to solve and have to come up with ways to tackle it; the instructor simply facilitates the process. The classes take place once a week—either after school for an hour on weekdays or, for kids 11 to 18, three hours each Sunday. Tuition is $225 for 10 weeks ($450 for the Sunday session), although scholarships are available for families who need it. While some kids have been compelled by their parents to attend, most choose to be there, and close to 80 percent go to public schools.

What the Kaplans—with help from a half-dozen other instructors—are not running is a tutoring or test prep program. Theirs is a nonprofit organization working in space donated by both Harvard and Northeastern universities, and many students are repeat participants. This past fall, 125 children were enrolled in the Math Circle, but attendance has been as high as 200, and thousands have “graduated” in the past 10 years.

Yasmin Siraj is a sibling of a former student. “I like learning about strategies, how to add numbers and subtract numbers really fast,” the 8-year-old says of her Math Circle experience. “Every problem we’ve tried, we’ve figured out the answer.”

Yasmin engages in a multitude of extracurricular activities, including figure skating. But her father, Ra’ad, says the alternative math lesson is worth fitting into the family’s busy schedule. “The focus is not on getting all the arithmetic right before you can tackle the advanced concepts,” he says. “The traditional curriculums don’t seem to capture the fun of math.”

This notion of fun is what got the ball rolling in 1994, when Ellen was teaching at the private Commonwealth School in Boston. Bob, who’d taught there for 34 years, had a taken a post elsewhere the year before, and they both continued to encounter kids resistant to learning mathematics. “We were sitting on our couch saying, ‘Isn’t it baffling? Students hate math,’” Bob recalled.

“And they don’t even know what it is,” Ellen interjected. “It’s not fair.”

The Kaplans, who met when Ellen was a freshman in college and who have been married close to five decades, recounted this story as they ate lunch in a Harvard cafeteria. Like many older couples, they finish each other’s sentences and piggyback onto each other’s ideas. They decided 10 years ago to call some friends and invite them to a Saturday morning get-together to simply “talk about math,” Bob said. “And that was the first Math Circle. Twenty-nine people came.”

The math circle idea orginiated in Eastern Europe more than a century ago as an extracurricular way for educators to share with younger people the adventure of mathematical problem-solving.

The math circle idea isn’t new. It originated in Eastern Europe more than a century ago as an extracurricular way for educators and professionals to share with younger people the adventure of mathematical problem-solving. Although many circles have popped up in the United States as prep groups for math olympiads, some hew to the original intent, and the Boston group specifically eschews competition, grades, and homework. Classes are open to kids of any skill level, although there is a certain amount of self-selection: More than half of the Kaplans’ alumni have gone on to study mathematics in college.

The proliferation of math circles in the United States is perhaps no surprise. In the 1990s, the U.S. Department of Education took part in two video studies, during which 8th grade lessons in several countries were recorded. The second study, conducted in 1999, showed that of the seven countries surveyed, the United States—ranked lowest in terms of test scores—was the only one that neglected to teach its students conceptual skills. Meanwhile, in its statement of beliefs, the National Council of Teachers of Mathematics explicitly notes the positive role of mathematical reasoning; council president Cathy Seeley says that math circles like the Kaplans’ are not only educationally sound but also “likely to help students in their mathematical thinking.”

The irony is that when they were kids, the Kaplans hated math. Bob described his first encounter with the subject as “badly taught in a Quaker school in upper New York state.” He always failed math classes, “and I was rather proud of that,” he remembered. “Here was this nonsense that other people thought was worth a lot, ... and I was damned if I was going to put a lick of work into it.”

He did, however, put effort into learning Greek, Sanskrit, and any other topic that would further his philosophy studies, his first love. But later, as a high schooler, “I came to realize [that] to attack the problems I was interested in, I’d have to understand some mathematics and also to approach the problems through mathematics,” he recalled. “And then the sheer beauty of it just hooked me.”

Ellen’s experience, in New York City, began with a disastrous high school class, and she avoided the subject in college, studying archaeology and biology. She and Bob met in 1953, when she was at Radcliffe, and married several years later. Shortly thereafter, they planned to join a dig in Turkey, but after Ellen discovered she was pregnant (the Kaplans have one child, a son), she looked for a teaching job instead. Commonwealth hired her to teach biology.

Weeks before the first term began, the school revised her assignment to one period of biology and four of mathematics. Ellen, then 23, knew nothing about math—but she and Bob did have a mathematician friend. “And mathematicians are like Jesuits,” she said. “I mean, if you say, ‘I don’t understand math,’ they say, ‘Oh, you just didn’t learn it right. Come with me, my dear, I’ll teach it to you.’ So we would meet every day at lunch time.”

That approach has informed the way she’s taught math ever since.“When you’re in charge of it, as opposed to being the passive victim,” Ellen added, “you can make some sense of things, and you go over it until you can explain it to somebody else. I fortunately realized, ‘Now that I can explain it to somebody else, I won’t explain it to somebody else! I’ll let them have the same experience.’ ”

Ellen stayed at Commonwealth for close to 40 years, founding the Math Circle with Bob three years before leaving the school for good.From the start, the couple opted to run the program independently so they could retain control of class content and managerial decisions. They’ve since written two critically well-received books, which attempt to do in print what they’ve done in the classroom: Bob’s The Nothing That Is: A Natural History of Zero, published in 1999; and their co-authored The Art of the Infinite: The Pleasures of Mathematics, in 2003.

But the passion that prevents them from retiring is teaching. During those November classes, Ellen, in particular, was dealing with a fairly chaotic bunch of 7- to 9-year-olds. Still, she kept the class on track.

“Let’s see if we can find a method that works if we’re not starting with 1,” she said, resuming her lesson on Gaussing. “Sarah, tell me a number, any number, but let’s say bigger than five and less than 50. ... There’s no trick to it, you can pick any one you want.”

“Thirty-eight?” the girl answered tentatively.

“That’s gonna be our first number,” Kaplan responded. “Now, we want to do something that we add regularly on. We’ve done adding by ones, we’ve done adding by twos. Yasmin, what would you like to do?”

By the end of the hour, the kids had moved into the abstract, using letters instead of numbers. Because they control the process, arriving at a solution means they’ve discovered something new on their own. The Kaplans argue that this method can be applied in other subjects—to studying the rhyming structure of sonnets, for example, or observing how elemental gases respond to heat.

But the couple feels there aren’t enough educators who see things the way they do. Seeley of the NCTM characterizes U.S. math education as “sit and git”: The teacher talks, the students listen and then practice repetitive exercises. So the Kaplans have begun propagating their approach. In December, they demonstrated for educators and mathematicians at a conference in California, and they’ve already established circles in Great Britain. Their plans for 2005 include training sessions in Massachusetts, Indiana, and New York City and publication of their book about the Math Circle, Out of the Labyrinth: Mathematics Set Free.

They’ve also recruited internally: Sam Lichtenstein, an 18-year-old who joined the Math Circle shortly after its inception is now an instructor. A high school senior, he still attends the three-hour Sunday session. “We all work together to solve the problem,” he says. “In school, sometimes competition can detract from the overall experience.”

Lichtenstein, who’s considering a math-related major in college, takes classes at Harvard in place of those offered at his public school—a decision he attributes to his years with the Kaplans. The first thing he worked on at Harvard? Cantor’s theories about infinity. “That’s the seminal Math Circle class,” he says. “That’s what got me hooked.”

Lichtenstein’s first teacher was Bob Kaplan, who by the end of that November class on infinity had hands as white as his hair from chalk dust. He hedged after being asked whether he’d given the students an impossible task with the line-and-plane problem. “There’s a third possibility,” Kaplan responded, “that this is beyond human capability.”

Indeed, the kids seemed to have reached their limit. They’d been working with a pair of lengthy decimal numbers beginning with 0.99 and 0.84 that, respectively, were the x and y coordinates for a point on the plane. But they hadn’t yet created a rule to collapse them into a single number on the line—the one-to-one correspondence.

Kaplan didn’t want to keep the students dangling, however. He was about to give a gentle nudge when Sophy Tuck, the girl with the British accent, haltingly revisited a way of combining x and y.

“Sophy’s suggestion is this: Skip every other one,” Kaplan repeated.

A student writes on the chalkboard.

Even Sophy wasn’t sure what she meant, but suddenly the class had direction. As they continued to struggle, the phrase became a mantra. Kaplan repeated it again and again, interlacing his fingers for emphasis. The kids knew they were close.

Sophy would say later that her 6th grade math class, at a private school, is still studying fractions and decimals, though not the way the Math Circle does. “I like how you work out the solution to that kind of thing,” the 11-year-old said. “[The Kaplans] push us enough, but then they let us work it out ourselves. It’s really satisfying.”

It was Sophy herself who recognized the solution in class. She gasped, eyes wide, as a classmate spoke, and Kaplan turned to her and uttered, “Sophy, say it.”

“OK. We take the 9, and then we take the 8—”

“Ah! Watch this. ... ”

Suddenly, the rest of the class understood: Mesh the two decimals together by alternating digits from both numbers. The pair 0.99 and 0.84, then, becomes 0.9894—a single point on the line. Any pair of decimals combined in this way would create a unique number. They’d found their rule.

Recalling Cantor, Kaplan said, “Three years, he believed there was no one-to-one correspondence, and he was going mad, as you were going mad. And then he thought, Why don’t I just dovetail, like that? ... There are exactly as many points in this square as there are on the line.”

There’s also a reason each class tackles only one topic: Every proof leads to more questions. Sophy and her classmates had successfully compared a line and a plane, but they still had three more weeks of Cantor.

“Fantastic, everybody!” Kaplan announced. “So, next week, on to new infinities. If there are any. ... ”


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