When I walked into the Harvard classroom, about eight students were already there, as I’d expected. But people passing by were surprised to see them—not because there was anything unusual in early arrivals at the beginning of a semester but because of what these students looked like: sitting in their tablet armchairs, their little legs sticking straight out in front of them. They were all about five years old.
This was the first class for these new members of The Math Circle. The parents, ranged at the back, may have been nervous, but their children weren’t, because it was simply the next thing to happen in a still unpredictable life.
“Hello,” I said. “Are there numbers between numbers?”
“I don’t know what you’re talking about,” said Dora, in the front row.
“Oh—well, you’re right—I’m not too sure myself.” I drew a long line on the blackboard and put a 0 at one end, a 1 near the other. “Is there anything in there?”
Sam jumped up and down. “No, there’s nothing there at all,” he exclaimed, “except of course for one half.” This wasn’t as surprising a remark as you might think, since Sam had just turned an important five and a half.
“Right,” I said, and made a mark very close to the 0 and carefully labeled it 1/2.
“It doesn’t go there!” said Sonya, sitting next to Dora.
“Really? Why not?”
“It goes in the middle.”
“Why?”
“Because that’s what one half means!”
I erased my mark and, with a show of reluctance, moved 1/2 to the middle, acting now as no more than Sonya’s secretary.
“Well,” I said, “is there anything else in there?”
Silence … five seconds of silence, which in a classroom can seem like five minutes. Ten seconds. Tom, at the back, got up and began to put on his jacket, because clearly the course was over: we’d found all there was to find between 0 and 1.
“I guess it’s just a desert,” I said at last, “with maybe a camel or a palm tree or two,” and began sketching in a palm tree on the number line.
“Now that’s ridiculous!” said Dora, “there can’t be any palm trees there!”
“Why not?”
“Because it isn’t that sort of thing!” This is an insight many a philosopher has struggled over.
Obediently I began erasing my palm tree, when I felt a hand grabbing the chalk from mine. It was Eric, who hadn’t said anything up to this point. He started making marks all over the number line—most of them between 0 and 1 but a fair number beyond each.
“There are kazillions of numbers in here!” he shouted.
“Is kazillion a number?” Sonya asked—and we were fully launched.
In the course of the semester’s ten one-hour classes, these children invented fractions (and their own notation for them), figured out how to compare them—as well as adding, subtracting, multiplying, and dividing them; and with a few leading questions from me, turned them into decimals. In the next-to-last class they discovered that the decimal form of a fraction always repeats—and in the last class of all came up with a decimal that had no fractional equivalent, since the way they made it guaranteed that it wouldn’t repeat (0.12345678910111213 …).
The conversations involved everyone, even a parent or two, who had to be restrained. Boasting and put-downs were quickly turned aside (this wasn’t that sort of thing), and an intense and delightful back-and-forth took their place.
A sample: after they had found all sorts of fractions in the second class (most but not all with numerator 1), and located them (often incorrectly) on the number line where they thought they should be, I began the third class by saying I was troubled by having 1/3 to the left of 1/4 (in that part of the desert between 0 and 1/2), as they had wanted.
“You may be right,” I said, “but I need convincing.” A long debate followed, which divided the class into those who would (for no very good reason other than reading my doubt) put 1/4 before 1/3, and those who argued for leaving things as they stood, on the grounds that since 4 was greater than 3, 1/4 had to be greater than 1/3.
At the start of the next class, Anne put up her hand. Would I please draw two pies on the board? I groaned inwardly: she must have had a conversation with her parents during the week; but I drew two large, equal circles for her.
“Now make the Mercedes sign in one in one and a cross in the other,” she said. I did. We contemplated the drawings—the message seemed clear. But Anne wasn’t finished.
“There, you see? If it’s a pie you like, a third is bigger than a fourth! But if it’s rhubarb, then a fourth is more than a third!” This wasn’t an argument I had encountered before.
You probably think this was a class of young geniuses, hand-picked for special training. In fact it was like any other Math Circle class: we take whoever comes—math lovers and math loathers; we don’t advertise but rely on word of mouth. True, this is in the Boston area, with its high percentage of academic and professional families, but we’ve had the same sort of classes with inner-city kids, suburban teenagers, college students, and retirees—from coast to coast, in London and Edinburgh, and (in German!) in Zurich and Berlin. The human potential for devising math, with pleasure, is as great as it is for creative play with one’s native language: because mathematics is our other native language.
We came up with the idea of having such sessions one warm September evening in 1994, while having dinner with our friend Tomás. The academic year was beginning, and what had long been bothering each of us inevitably came up: how could the teaching of math be in such bad shape—especially in a city known for mathematical research? All that our students, whether college freshmen or ninth graders, knew about math was that they had to take it and that they hated it.
We had chicken with cream and complaints, but the chocolate mousse that followed was too light for such gloom, and one of us said: “Why don’t we simply start some classes ourselves?”
Tomás thought he could rent the basement of a nearby church on Saturday mornings. We wrote up a list of friends with children and telephoned around: “We’re going to have some informal conversations about math this Saturday morning. Would your children be interested?”
One of those we’d spoken to had heard of Nina Goldmakher in Brookline, who ran a Russian literary circle for children, and called her up. She was wonderfully enthusiastic and sent all of her students to us. We began that first morning with twenty-nine teenagers.
Word spread. We outgrew the church basement by the end of the semester, and Professor Andrei Zelevinsky, at Northeastern University, offered us free rooms there. Parents of younger and younger students called us up; Professor Daniel Goroff at Harvard found rooms for them on weekdays. To keep classes small, we trained graduate students as teachers. We incorporated as a nonprofit in 1996, still rely only on word of mouth, averaging 120 students each semester, meeting in twelve different sections. We’ve kept the style informal, the bureaucracy minimal—and couldn’t have more fun.
What has stood in the way of enjoying and mastering math as one does music, how we have removed these barriers in The Math Circle, and how such circles for young and old may now spread, is the subject of this book.