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Out Of the Labyrinth: An Excerpt


Robert and Ellen Kaplan have taught courses ranging from Sanskrit to Gödel’s Theorem, to students ranging in age from four to seventy. In 1994, the Kaplans, along with Tomás Guillermo, began The Math Circle an esteemed, popular mathematics learning program. On their website they state that, “We are careful to choose topics which are unlikely to be in the school curriculum – we see our role as widening and deepening the river, rather than accelerating its flow between narrow banks.” In keeping with their mission, to supplement math education, they have written a book, Out of the Labyrinth: Setting Mathematics Free that reveals the secrets behind the Math Circle’s success. Below we have excerpted from the first chapter of Out of the Labyrinth. This piece illustrates just how much fun math can be for children. Be sure to check out DailyKos on Thursday when Peter Flom will explore math circles himself, in a new feature titled, Education Uprising, an effort to reform education from the ground up.

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.”


“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 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…

…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 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, 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.

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