Kaplans Teach Students 'The Art of the Infinite'

During one session of The Math Circle—a Cambridge-based extracurricular program that aims to teach kids math “the right way”—a 12-year-old girl demanded that Robert Kaplan tell her the answer to a confusing question. The exchange found its way into The Art of the Infinite: The Joys of Mathematics, a recent book which Kaplan, the program’s co-founder, wrote with his wife, Ellen F. Kaplan ’57.

“Our approach is very peculiar,” Robert says. “We don’t tell them anything. We just nudge.”

A mathematical problem is posed at the beginning of each class, and students then lead the discussion wherever they wish. The Kaplans say they insist on a collegial environment above all.

The Math Circle is a fairly recent phenomenon—but people from across the country are already taking note.

“There’s no competition in the Math Circle,” Robert says. “Math is too hard for us to get on one another’s nerves.”

They also try as much as possible to ensure that only students who are truly interested come, as opposed to those prodded by parents.

“The Math Circle doesn’t work when parents force kids to come,” he says.

The Kaplans partially accomplish this objective by scheduling classes at awkward times. Those for younger children meet weekdays around 5:00 p.m., after a long school day.

Adolescents meet early Sunday morning, Robert says with a grin, “the only day of the week they get to sleep late.”

In 1999, Robert published his first book, The Nothing That Is: A Natural History of Zero. The book was a bestseller, praised by The Times of London and The Wall Street Journal and translated into eight languages.

It was then that Robert realized that he had stumbled onto something.

Although he attributes his success to the fact that “there are a lot of loony people to whom zero is important,” he says that “there is a tremendous hunger for certainty.”

“Mathematics offers that, but there’s a cloak around it,” he says.

Similarly, The Art of the Infinite aims to make mathematics more accessible to the general public. One tactic the Kaplans use is to present math along with the history behind it.

“Through history, anecdote, illustration and biography, we make the approach so enticing…that people become engrossed,” Robert says.

The book contains three “interludes”—short chapters that depart entirely from math. Robert explains their presence by drawing a parallel to a Math Circle class.

“There’s a moment when the eyes glaze, and you have to stop,” he says.

“It has to do with the pace at which one reads,” Ellen adds. “Math has to be read at the speed of lyric poetry because it’s that condensed. It provides a smoothness so that you can glide across a philosophical conversation.”

The Kaplans say they understand that mathematics fills many people with dread.

“The written language of it is off-putting, clumsy, medieval,” complains Robert. “You have to be able to see through that to the beauty of the structure, like in music.”

To this end, their book’s treatment of technical notation is very wary, according to Ellen.

“We start in conversational language, speaking qualitatively,” she says. “Then it becomes more precise. You have to make notation you own. It is barbed wire in your face until then.”

The book also addresses the terror of mathematical notation in its appearance. All the diagrams are hand-written by Ellen, who also illustrated The Nothing That Is, and each chapter begins with headings in a whimsical typeface.

That way, explains Ellen, “it doesn’t seem that it was given from above.”

The Kaplans sit in the airy living room of their house off Central Square, sipping tea and nibbling on shortbread. Their conversation ranges from depictions of the crucifixion from the Renaissance to hitch-hiking through Ireland. Ellen’s paintings line the room’s walls. One, a reproduction of a painting she saw in Florence, was the result of the museum not selling a postcard of it, she admits.

Their novel approach to mathematics began on “a Tuesday evening” in 1994, Robert recalls.

Ellen had been teaching at the time, and was sick of “that kind of authoritarianism” so common in math classes.

“Why is math taught so badly,” they lamented at the time, “especially in Cambridge?”

The couple decided to take matters into their own hands. That very Saturday, they gathered about 30 students in the basement of a nearby church.

“They were as randomly chosen as you could get,” Ellen says.

But the Kaplans realized that they had touched a nerve, and enrollment grew—purely through word of mouth. They now teach nearly 200 people every semester.

The Kaplans have never repeated a curriculum.

“Each semester starts from scratch, so veterans and novices move together,” Robert says.

They try to find topics which are “mysterious and accessible at the same time,” Ellen says.

The results are often so creative that last year one class had its findings published in FOCUS, the Mathematical Institution of America’s most widely distributed periodical.

But the Kaplans never considered themselves math whizzes in their younger days.

“I am the poster child for the mathematics failure,” Ellen says.

She describes her elementary school as a progressive school “where you only learned what you wanted.” At age 10, she could add numbers under 20, subtract without borrowing, but knew no division.

“And my mother was a mathematician,” she says with a laugh.

Ellen’s parents had her switch schools, but she never developed an interest in math and never took a math class in high school beyond Algebra 2.

After high school she attended Radcliffe Collge, where she studied classical archeology and steered clear of math classes.

But several years later, after she was hired at a local high school teaching biology, she found out that she would be teaching math as advanced as calculus.

It was then that, with the help of summer lunch hours with colleague Barry Mazur she learned “everything from long division on.”

Robert also admits experiencing the “standard terrible teaching” as a student, learning math from the football coach. He says he rediscovered mathematics by way of philosophy, which he studied as a graduate student at Harvard.

“They are intimately related,” he explains. “If you want to know how things are and what they are, in my belief you’re asking a question of structure. Mathematics is not only the art of the infinite, but also the science of structure.”

He says his primary interest remains philosophy, which is probably why the subject comes up so often in the new book.

The Kaplans say they have high hopes for the future of the Math Circle. They’d love to expand the program and predict that their current book tour will increase interest.

Their main obstacle remains finding teachers. Some Harvard graduate students participate, but there is still a shortage because the type of teacher they want is unique.

“You need someone who knows their stuff but doesn’t say so,” Robert says. “At this point, we’re thinking of reshaping humanists.”

The Kaplans say they receive numerous e-mails about starting Math Circles in other cities. While they say they support the idea, they add that they’re hesitant to give too much advice apart from the ground rules about environment. However, those seeking more wisdom from the Math Circle will have a new Kaplan tome to peruse in 2004, entitled Out of the Labyrinth: Mathematics Set Free.

After speaking to a crowd of about 50 earlier this month at a Wordsworth Books appearance in support of The Art of the Infinite, the Kaplans embarked on a national book tour this week.

During their tour, the Math Circle will be under the direction of James Tanton, who teaches several courses with the Kaplans at the Harvard Extension School.

In the future, Robert says that the Math Circle’s courses may be put under the authority of the Extension School. While he says he’s somewhat wary of making them so “official,” he admits “you have to know when to hold ’em and when to fold ’em.”

The Kaplans say they plan to continue their mission of opening mathematics up to ever expanding audiences.

“The pleasures are so enormous and no one knows about it,” says Robert. It’s our lost native language…it’s our connection to truth.”