How Not to Be Wrong

Stephen Colbert Thinks “Number Sentences” Are Silly. They’re Not.

Stephen Colbert
Stephen Colbert: wrong about number sentences.

Photo by Stephen Lovekin/Getty Images for Robert F. Kennedy Center For Justice And Human Rights

People who teach math, like me, hate it when students ask us, “When am I going to use this?” We don’t hate it because it’s a bad question. We hate it because it’s a really good question, and one that our curriculum isn’t really set up to answer. And that’s a problem.

In my day job I’m a pure mathematician, specializing in the most abstract parts of number theory and geometry. But I’ve also been writing here at Slate for more than a decade about the connections between the mathematical world and the things we think about every day. The foreignness of mathematical language, from arithmetic to calculus, can create among outsiders the misapprehension that these spheres of thought are totally alien and contrary to how most people navigate the world around them.

That’s exactly wrong. Math is built on our natural ability to reason. Despite the power of mathematics, and despite its sometimes forbidding notation and abstraction, the actual mental work involved doesn’t differ much at all from the way we think about more down-to-earth problems. Rather, mathematics is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.

In my new book, How Not to Be Wrong, I write about the many ways math is woven into our thinking, covering everything from lottery schemes to the obesity apocalypse to the Supreme Court’s view of crime and punishment to the existence of God. It’s not the kind of book where math is a big floating blob to be admired (or feared) from afar. We get right up next to it and get our hands dirty. Because that’s what math is about. 

This week, I’ll be math-blogging at Slate—sometimes about stuff from the book, other times about math that’s in the news right now. Let’s get started right now, because I have something to get off my chest about mathematics education.

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We are angry about the way math is taught. We have always been angry about the way math is taught. One day Pythagoras started talking up his theorem, and the next day, people were saying, When I was a kid we just measured the hypotenuse. Easy peasy. Why do you have to make things so complicated, Pythagoras?

Nobody’s expressed this anger more precisely or more hilariously than Tom Lehrer did in his song “New Math” (from the early 1960s, in case you’re under the mistaken impression the math wars are a contemporary development). “But in the new approach, as you know,” Lehrer says, “the important thing is to understand what you’re doing, rather than to get the right answer.”

At the moment, the anger is centered on the Common Core, a fairly mild-mannered suite of goals and standards that’s become a stand-in in the popular mind for everything from overreliance on standardized tests (if you’re a Democrat) to jackbooted federal intrusion into local culture (if you’re a Republican). A Florida legislator said it would turn our kids gay. Louis C.K. said it made his kid cry.

And then there’s the “number sentence.” OK, number sentences aren’t actually mentioned anywhere in the Common Core, and they’ve been part of the math curriculum for decades. They sound weird and unfamiliar, though, which makes them fair game for schools-these-days tsk-tsk-ery. A New York principal argued in the Washington Post that the number sentence concept was too sophisticated for young children. And Stephen Colbert mocked the phrase, suggesting that students should also have the option of using “a word equation or formula paragraph.”

The phrase “number sentence” was new to me, too, when my 8-year-old son brought it home from school this year. But I didn’t make fun of it. I cheered it! When we call an equation like

2 + 3 = 5

a “number sentence” we’re not dumbing down. We’re telling it like it is. That equation is a sentence. It’s a sequence of symbols that makes an assertion about the world. “Dave” is not a sentence; it’s a proper noun. “Vsdsgs” is not a sentence; it’s a meaningless glurg. But “Dave is my brother” is a sentence. It has a subject (“Dave”), a verb, (“is”), and an object (“my brother”). Just like 2 + 3 = 5: the subject is “2+3,” the verb is “=,” and the object is “5.”

What’s wrong with calling 2 + 3 = 5 a good old “equation”? First of all, not all number sentences are equations. A number sentence is a sentence about numbers, which could be an equation, like 2 + 3 = 5, but also an inequality, like 5 > 3.

Worse, “equation” has been around so long that it’s lost contact with its literal meaning, “something that equates.” Students in my courses routinely refer to “5 > 3” as an equation. The same with a mathematical expression like “x^2 + y^2,” which is no more an equation than “Dave” is a sentence.

When we call 2 + 3 = 5 a “sentence” we engage in the radical act of insisting that mathematics has meaning. That shouldn’t be a radical act. But, too often, we teach our students that “doing mathematics” means “manipulating clusters of digits according to rules presented to us by the teacher.”

That’s not math. And when we teach our students to do that, and only that, we are training them to be slow, buggy versions of Excel. What’s the point?

If a student doesn’t truly grasp that “2 + 3 = 5” is a sentence, a statement about the world that might be true or false, it’s hard to see how, when algebra comes around, they can grasp that

x^2 + 3 = 5

is a sentence, too, one which is true for precisely two values of x (namely, the positive and negative square roots of 2) and false for all the rest.  

One solution, of course, is to double down, addressing algebra, too, in a purely algorithmic way. You have an equation involving x, certain modifications of the equation are allowed (because the teacher says they’re allowed) and when you get to something that has just “x =” on the other side, you’ve won the game. If you can do this, you can get an A on a typical algebra test. But can you do algebra? I’m not so sure.  

My kids really liked a tablet game called DragonBox, which, according to its promotional material, “secretly teaches algebra to your children.” Kind of like the way you can secretly feed kale to your children by grinding it up and hiding it in a meatball. But what DragonBox actually teaches are the rules of algebra: that you’re allowed to add the same symbol to both sides, that multiplying a sum by a symbol requires you to multiply each summand by the same symbol, and so on. Getting these rules in muscle memory is what you need if you want to get fluent in algebraic computations.

There’s just one thing missing, but it’s a big thing: the fact that algebra is made of sentences, that it means something, that it refers to something outside itself. An algebraic statement isn’t just a string of symbols with an x stuck in there somewhere. It’s an assertion about a relationship between quantities (or, when you get to more advanced algebra, between functions, or operations, or even other assertions.) Without that animating idea, algebra is a dead and empty exercise.

To be fair, Jean-Baptiste Huynh, who created DragonBox, gets this. He says in an interview that “DragonBox does 50 percent of the job. We need to teach the rest.” Fifty percent is not such a small number, and it sounds about right to me. Computation is important! We lose just as badly if we generate students who have some wispy sense of mathematical meaning but who can’t work examples swiftly and correctly. A math teacher’s least favorite thing to hear from a student is “I get the concept, but I couldn’t do the problems.” Though the student doesn’t know it, this is shorthand for “I don’t get the concept.” The ideas of mathematics can sound abstract, but they make sense only in reference to concrete computations. William Carlos Williams put it crisply: no ideas but in things.

Because math is about things. Not, despite how it sometimes looks, about itself, and not about getting a good score on the test. It’s about which things—which sentences—are right, and which are wrong, and how to tell the difference.

Update, June 12, 2014: I should have said that my thoughts about DragonBox, formalism, and algebra were touched off by this 2013 John Holbo post at Crooked Timber and the associated comment thread, whose existence I’d forgotten about when I was writing the piece.