The unadorned fourth-floor classroom in Harvard University鈥檚 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.
鈥淲hat we鈥檙e trying to figure out is if there鈥檚 a one-to-one correspondence between points on a line and points on a plane,鈥 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鈥檚 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.
The class was trying to determine whether a plane and a line contain the same number of points鈥攊t was either that or acknowledge that there are different sizes of infinity. They鈥檇 studied, in particular, the work of Georg Cantor, a 19th century German mathematician who鈥檇 come up with a rule to establish the correspondence, but they hadn鈥檛 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. 鈥淲e鈥檝e spent almost an hour on this,鈥 said the instructor, Bob Kaplan. 鈥淥f course, Cantor spent three years nonstop.鈥
It鈥檚 important to note that these students weren鈥檛 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.
鈥淎t this point,鈥 Kaplan asked, 鈥渉ow 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鈥檚 not a one-to-one correspondence?鈥 He paused, and almost all the hands went up. 鈥淪o you think that Cantor, looking for a new size of infinity, has found one looking at the points on a plane?鈥
One student, suspecting a trick, asked: 鈥淲ould you teach us something that was impossible?鈥
To anyone sitting in on the Kaplans鈥 classes for the first time, this question would seem appropriate. 鈥淢athematics alone,鈥 the French philosopher Simone Weil once wrote, 鈥渕ake us feel the limits of our intelligence.鈥 But the Kaplans prefer the sentiment of Cantor himself: 鈥淭he essence of mathematics is freedom.鈥
For all the talk about 鈥渕ath wars鈥 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.
鈥淓verything comes around again,鈥 Ellen Kaplan says. 鈥淭he ideas of mathematics are universal.鈥
Earlier that afternoon, Ellen Kaplan was leading a class of her own in a third-floor space identical to her husband鈥檚. Her dozen or so students, 7 to 9 years old, were trying to come up with a formula for adding a string of numbers鈥攕uch as 1, 2, 3, 4, and 5鈥攚ithout 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, 鈥淗ow far up should I go?鈥
Some of the boys started shouting. 鈥淣ine thousand, nine hundred and ninety-nine!鈥 鈥淣ine million!鈥
鈥淚 hear 鈥榥ine.鈥 I think that鈥檚 a good number,鈥 Kaplan replied, writing 鈥1 + 3 + 5 + 7 + 9.鈥 鈥淥K! If I wanted to do this, and I鈥檓 far too lazy to add all that up, ... what was Gauss鈥檚 clever and brilliant plan for avoiding work? Yasmin.鈥
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 鈥淕aussing鈥 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. 鈥淵ou, um, under the 1,鈥 he said, 鈥測ou put nine-plus-seven-plus-five plus-three-plus-one.鈥 Kaplan wrote it all down as the boy completed the process and arrived at the correct sum.
鈥淵es, well,鈥 she responded, 鈥渢he interesting thing is not what鈥檚 the answer, which we got, but how we get there.鈥
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鈥攅ither 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鈥攚ith help from a half-dozen other instructors鈥攁re 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 鈥済raduated鈥 in the past 10 years.
Yasmin Siraj is a sibling of a former student. 鈥淚 like learning about strategies, how to add numbers and subtract numbers really fast,鈥 the 8-year-old says of her Math Circle experience. 鈥淓very problem we鈥檝e tried, we鈥檝e figured out the answer.鈥
Yasmin engages in a multitude of extracurricular activities, including figure skating. But her father, Ra鈥檃d, says the alternative math lesson is worth fitting into the family鈥檚 busy schedule. 鈥淭he focus is not on getting all the arithmetic right before you can tackle the advanced concepts,鈥 he says. 鈥淭he traditional curriculums don鈥檛 seem to capture the fun of math.鈥
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鈥檇 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. 鈥淲e were sitting on our couch saying, 鈥業sn鈥檛 it baffling? Students hate math,鈥欌 Bob recalled.
鈥淎nd they don鈥檛 even know what it is,鈥 Ellen interjected. 鈥淚t鈥檚 not fair.鈥
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鈥檚 sentences and piggyback onto each other鈥檚 ideas. They decided 10 years ago to call some friends and invite them to a Saturday morning get-together to simply 鈥渢alk about math,鈥 Bob said. 鈥淎nd that was the first Math Circle. Twenty-nine people came.鈥
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鈥檛 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鈥 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鈥攔anked lowest in terms of test scores鈥攚as 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鈥 are not only educationally sound but also 鈥渓ikely to help students in their mathematical thinking.鈥
The irony is that when they were kids, the Kaplans hated math. Bob described his first encounter with the subject as 鈥渂adly taught in a Quaker school in upper New York state.鈥 He always failed math classes, 鈥渁nd I was rather proud of that,鈥 he remembered. 鈥淗ere 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.鈥
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, 鈥淚 came to realize [that] to attack the problems I was interested in, I鈥檇 have to understand some mathematics and also to approach the problems through mathematics,鈥 he recalled. 鈥淎nd then the sheer beauty of it just hooked me.鈥
Ellen鈥檚 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鈥攂ut she and Bob did have a mathematician friend. 鈥淎nd mathematicians are like Jesuits,鈥 she said. 鈥淚 mean, if you say, 鈥業 don鈥檛 understand math,鈥 they say, 鈥極h, you just didn鈥檛 learn it right. Come with me, my dear, I鈥檒l teach it to you.鈥 So we would meet every day at lunch time.鈥
That approach has informed the way she鈥檚 taught math ever since.鈥淲hen you鈥檙e in charge of it, as opposed to being the passive victim,鈥 Ellen added, 鈥測ou can make some sense of things, and you go over it until you can explain it to somebody else. I fortunately realized, 鈥楴ow that I can explain it to somebody else, I 飞辞苍鈥檛 explain it to somebody else! I鈥檒l let them have the same experience.鈥 鈥
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鈥檝e since written two critically well-received books, which attempt to do in print what they鈥檝e done in the classroom: Bob鈥檚 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.
鈥淟et鈥檚 see if we can find a method that works if we鈥檙e not starting with 1,鈥 she said, resuming her lesson on Gaussing. 鈥淪arah, tell me a number, any number, but let鈥檚 say bigger than five and less than 50. ... There鈥檚 no trick to it, you can pick any one you want.鈥
鈥淭hirty-eight?鈥 the girl answered tentatively.
鈥淭hat鈥檚 gonna be our first number,鈥 Kaplan responded. 鈥淣ow, we want to do something that we add regularly on. We鈥檝e done adding by ones, we鈥檝e done adding by twos. Yasmin, what would you like to do?鈥
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鈥檝e discovered something new on their own. The Kaplans argue that this method can be applied in other subjects鈥攖o studying the rhyming structure of sonnets, for example, or observing how elemental gases respond to heat.
But the couple feels there aren鈥檛 enough educators who see things the way they do. Seeley of the NCTM characterizes U.S. math education as 鈥渟it and git鈥: 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鈥檝e 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鈥檝e 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. 鈥淲e all work together to solve the problem,鈥 he says. 鈥淚n school, sometimes competition can detract from the overall experience.鈥
Lichtenstein, who鈥檚 considering a math-related major in college, takes classes at Harvard in place of those offered at his public school鈥攁 decision he attributes to his years with the Kaplans. The first thing he worked on at Harvard? Cantor鈥檚 theories about infinity. 鈥淭hat鈥檚 the seminal Math Circle class,鈥 he says. 鈥淭hat鈥檚 what got me hooked.鈥
Lichtenstein鈥檚 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鈥檇 given the students an impossible task with the line-and-plane problem. 鈥淭here鈥檚 a third possibility,鈥 Kaplan responded, 鈥渢hat this is beyond human capability.鈥
Indeed, the kids seemed to have reached their limit. They鈥檇 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鈥檛 yet created a rule to collapse them into a single number on the line鈥攖he one-to-one correspondence.
Kaplan didn鈥檛 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.
鈥淪ophy鈥檚 suggestion is this: Skip every other one,鈥 Kaplan repeated.
Even Sophy wasn鈥檛 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. 鈥淚 like how you work out the solution to that kind of thing,鈥 the 11-year-old said. 鈥淸The Kaplans] push us enough, but then they let us work it out ourselves. It鈥檚 really satisfying.鈥
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, 鈥淪ophy, say it.鈥
鈥淥K. We take the 9, and then we take the 8鈥斺
鈥淎h! Watch this. ... 鈥
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鈥攁 single point on the line. Any pair of decimals combined in this way would create a unique number. They鈥檇 found their rule.
Recalling Cantor, Kaplan said, 鈥淭hree 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鈥檛 I just dovetail, like that? ... There are exactly as many points in this square as there are on the line.鈥
There鈥檚 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.
鈥淔antastic, everybody!鈥 Kaplan announced. 鈥淪o, next week, on to new infinities. If there are any. ... 鈥
