Why Does Exploding Dots Work?

A few months from now, if James Tanton and his Global Math Project co-conspirators have their way, ten million schoolchildren will take a huge mathematical step from the twenty-first century all the way back to the Bronze Age: instead of using a gadget with a state-of-the-art interface (say, a telepathic smartphone that tells you the answer to an arithmetic problem when you merely think the question), these kids will solve arithmetic problems by moving counters around on boards, the way people did thousands of years ago.¹

But if you think Tanton is a back-to-basics reactionary, you’ve got him all wrong: he’s a math-Ph.D.-turned-math-educator possessed by the conviction that math can be made understandable to, and exciting for, everyone. Tanton’s “Exploding Dots” approach to precollege math is designed to bring illumination and joy to a subject that students all too often associate with mystery and misery, and the Global Math Project’s aim is to carve out one week each year (“Global Math Week”) from the grade-K-through-12 academic calendar, in which every student gets a chance, if only for an hour, to experience that illumination and joy.

Time and Tesseracts

“The fourth dimension became a thing you talked about, without knowing what it meant.”

— Marcel Duchamp

“The fourth dimension!” The mere phrase makes some small part of my brain shiver. And it’s not just me that feels that there’s something awe-full, and maybe awful, about the fourth dimension. 16th-century scholar Gerolamo Cardano¹ said it was contrary to Nature. 17th-century thinker John Wallis² found the very idea of it monstrous. And now 21st century virtual-reality pioneers are about to bring it into your home, if you want it there.

One of the most famous monsters inhabiting the fourth dimension is the tesseract, which you might have heard mentioned ever-so-briefly in the recent film “A Wrinkle in Time”. The movie is based on the classic book by Madeleine L’Engle, which has introduced generations of children to the idea of dimensions beyond the three that we see around us.

Who Knows Two?

All the work they made them do was rigorous.

— Exodus 1:14

As I write this, it’s April 5, midway through the eight-day festival of Passover. During this holiday, we Jews air our grievances against the ancient Pharaoh who enslaved and oppressed us, and celebrate the feats of strength with which the Almighty delivered us from bondage — wait a minute, I think I’m mixing up Passover with Festivus. In any case, on the first two nights of Passover, Jews tell the story of the Exodus from Egypt before dinner, reading from individual copies of a book called the haggadah (“the telling”), and after dinner they sing a few extra songs, one of which is called “Ekhad Mi-Yodeah?”, or “Who Knows One?” It’s a cumulative song, with each verse containing one more line than the verse before, and in my family the game is to try to get through the increasingly long middle part of each verse without taking a breath. That’s tricky when you get to the last verse, whose English translation is

Who knows thirteen? I know thirteen!
Thirteen are the attributes of God;
Twelve are the tribes of Israel;
Eleven are the stars in Joseph’s dream;
Ten are the commandments;
Nine are the months to childbirth;
Eight are the days to circumcision;
Seven are the days of the week;
Six are the order of the Mishnah;
Five are the books of the Torah;
Four are the mothers of Israel;
Three are the fathers of Israel;
Two are the tablets of the covenant;
One is our God in heaven and earth.

But what does it mean to know a number?

The Genius Box

Mr. Jabez Wilson laughed heavily. “Well, I never!” said he. “I thought at first that you had done something clever, but I see that there was nothing in it after all.”

“I begin to think, Watson,” said Holmes, “that I make a mistake in explaining.”

— Arthur Conan Doyle, “The Red-Headed League”

When I was a kid living in the Long Island suburbs, I sometimes got called a math genius. I didn’t think the label was apt, but I didn’t mind it; being put in the genius box came with some pretty good perks. For one thing, the kids who thought I was a genius at math had lower expectations of me when it came to other things, so to the extent that I was endowed with a normal measure of certain traits (like a healthy sense of humor), people were impressed by how normal I was — and to the extent that I had deficits of athleticism or common sense, people tended to be forgiving: “What do you expect? He’s a genius!” (See Endnote #1, though.)

There were other kids I heard about, most of them living in New York City and attending Stuyvesant High School and the Bronx High School of Science, whose mathematical achievements were described in awed tones on the high school math grapevine, and I envied their endowments and their successes, but I didn’t think they were geniuses either. They knew lots of things that I didn’t know — things I wanted to know — but they didn’t know things in a different way than I did. The things those kids knew weren’t facts, but rather habits of thought: the kind of habits that won you top honors in math competitions and science fairs.

I got acquainted with some of those kids in my junior and senior years of high school, and they were awesomely good at solving math problems, but there was no uncanny aspect of their performance that seemed worth explaining away by appealing to some mystical inborn attribute like “genius”.

Roasting a Dodo and Biking on Mars: The Magic of Dimensional Analysis

The first time I encountered dimensional analysis, I didn’t recognize it for the magic wand that it is; it seemed to be just a form of mathematical hygiene — useful for avoiding mistakes, but not much else. To get a sense of what I mean by hygiene, consider a question that came up in my own household (arising from the incident described in Endnote 1): how do we convert 6 feet per second to units of miles per hour? We know that there are 60 × 60 = 3600 seconds in an hour and 5280 feet in a mile, but suppose we aren’t sure what to do with those numbers to get our answer; we suspect we should multiply 6 by 3600 / 5280 or by 5280 / 3600, but we’re not sure which. To keep ourselves on the right path, we use something called the factor-label method: we attach units to those numbers and we multiply three fractions that carry those units along in their numerators and denominators, with the three fractions representing the assertions “The speed is 6 feet per second”, “3600 seconds equals 1 hour”, and “1 mile equals 5280 feet”:

This gives an answer with units of miles per hour, after we’ve cancelled sec with sec and ft with ft. If we had tried to combine the fractions the wrong way, multiplying where we should have divided or vice versa, the appearance of unfamiliar units in the answer (like square-foot-hours per square mile per second) would have hopefully alerted us to our mistake.

But dimensional analysis can do much more for us. In many cases, it gives us a way to figure out the answer to a physics problem without actually applying the laws of physics. Bertrand Russell once wrote: “The method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil.” In many ways, dimensional analysis gives us the same bargain: it offers us a way to cheat the universe at its own game.

On Size, Death, and Dinosaurs

Here’s how Roald Dahl describes what happened one morning when the Queen of England invited a large and unexpected guest (“The BFG” ) to have breakfast with her:

There was a frantic scurry among the Palace servants when orders were received from the Queen that a twenty-four-foot giant must be seated at breakfast with Her Majesty in the Great Ballroom within the next half hour. The butler, an imposing personage named Mr. Tibbs, was in supreme command of all the Palace servants and he did the best he could in the short time available. A man does not rise to become the Queen’s butler unless he is gifted with extraordinary ingenuity, adaptability, versatility, dexterity, cunning, sophistication, sagacity, discretion, and a host of other talents that neither you nor I possess. Mr. Tibbs had them all.

The redoubtable Tibbs may have possessed dexterity and sagacity, but his grasp of allometry was sadly deficient. Allometry is the part of biology that studies how the size of a creature relates to other aspects of the creature’s life. Mr. Tibbs, noting that the Big Friendly Giant was four times the height of an ordinary man, decided that the BFG needed a quadruple-sized meal:

Everything, Mr. Tibbs told himself, must be multiplied by four. Two breakfast eggs must become eight. Four rashers of bacon must become sixteen. Three pieces of toast must become twelve, and so on. These calculations about food were immediately passed on to Monsieur Papillion, the royal chef.

To Tibbs’ consternation, eight eggs weren’t enough to put a dent in the giant’s appetite; even seventy-two eggs weren’t enough to satisfy him.

What was Tibbs’ mistake? And how many eggs should he have offered the BFG?

The Roots of Unity

Two primal pleasures from my years of childhood (and maybe my years of infancy too, though how would I know?) are feelings I’ll call coziness and spaciousness. The first is the feeling I’d get at night pulling the blankets up over my head; the second is the feeling I’d get standing on a beach staring out at the ocean. These pleasures seem like opposites, but sometimes you can have both at once — for instance, when beneath those blankets you have a flashlight and a book full of magic and adventure. I’ve learned that the adventure called mathematics has its own ways of combining the love of the large with the delight of the small, and I’m going to tell you about one of my favorite combinations of these two opposite pleasures: the roots of unity.