It’s the truth. The techniques of algebra, geometry and calculus were hard to create, and they’re hard to learn. But saying so forthrightly doesn’t come naturally to a lot of teachers — or to commenters on education. “Math Is Not Hard: A Simple Method That Is Changing The World,” reads a headline in HuffPost, extolling an approach that aims to help ease kids into the subject. I embraced rhetoric like this when I was an apprentice college instructor. I was constantly telling students, at the outset of a computation, “Now this is pretty simple” — encouraging them, or so I thought. My mentor, the master teacher Robin Gottlieb, now a professor at Harvard, set me straight. When we say a lesson is “easy” or “simple,” and it manifestly isn’t, we are telling students that the difficulty isn’t with the mathematics, it’s with them. And they will believe us. They won’t think, “I’ve been lied to,” they’ll think, “I’m dumb and I should quit.”
This applies to parents, too. I’ve been teaching math for two decades, and I still find myself telling my kids that a math concept they’re struggling with is “not that hard.” That’s not encouragement — that’s evidence of my frustration with watching them struggle, and it’s not part of teaching.
One big problem is that math teachers mastered the concepts so long ago, we’ve forgotten their difficulty. A fellow mathematician once told me that high school calculus was as easy as following a recipe. And that’s exactly right, in one sense: Following a recipe is easy once you know how to cook. But recipes require tacit knowledge and substantial experience that novices just don’t have. How much salt is a dash? What’s a rolling boil? You learn to cook by cooking, in the presence of someone who knows how, and at first you flail; you make plenty of mistakes; you get results that are right in some ways but very wrong in others; and the outcome of all that work is that you become another person who thinks cooking is easy.
This isn’t just true of calculus, which most non-mathematicians accept is supposed to be hard. It goes for supposedly easier things, too, like fractions, a third-grade Common Core standard. When we first present fractions to children, we’re asking them to make a huge conceptual leap. For their whole life until that moment, the definition of a number was something that answers the question “how many?” A fraction is a totally different thing, not so much a number as an amount. And yet you are supposed to be able to add and subtract them, just as you can “regular numbers.” The popular economics blogger Noah Smith, a fervent advocate of math education, recently tweeted, “We don’t really start teaching math til junior high.” Not true! Even the concept of expressing a number as a string of digits is a deep, hard-won idea that takes time to grasp, a concept we shouldn’t treat as trivial just because it’s old hat in MMXXI.
The idea that math is supposed to be easy gets in the way of the most effective learning tool students have: asking questions. If math is easy, you should just get it. And so students are afraid to ask questions in class, because they’re afraid of looking stupid. The situation is even worse for students who by reason of gender or race or accent or household income have a justifiable fear that their classmates — or worse, their teacher — will jump to precisely that conclusion. If we were honest about how difficult and deep mathematics is — at every level — this would be less of a problem; we could move toward a classroom where asking a question meant not “looking stupid” but “looking like someone who came here to learn something.”
That switch would help all children, not just students who find themselves struggling. Yes, some kids have no trouble picking up the basic rules of algebraic manipulation or geometric constructions. They should still be asking questions, of their teachers and of themselves. For example: “I have done what the teacher asked, but what if I’d tried to do this other thing that the teacher didn’t ask of me? And for that matter, why did the teacher ask what they asked and not the other thing?” No matter how much math you’ve learned, there’s something adjacent to what you’ve mastered that you don’t know, and that’s where to point your eyes. If math class is easy, though, that curiosity just means you’re doing it wrong.
How do we build a classroom full of questions, in which students probe math’s daunting challenges? The past year of largely online teaching has given me some ideas. Teaching math on the screen has been, for everyone I know, a poor substitute for being together in the classroom. But there’s one aspect I’d love to somehow retain when we return to in-person teaching: the chat window.
Chats let students float questions that might not be fully worked out in their minds, without having to worry that they’re interrupting the whole class. Sometimes students talk out these questions together in side conversations, and sometimes I, with one eye on the chat, bring one of these questions forward and fold it into the lecture. It may be no coincidence that Art of Problem Solving, in my opinion the gold standard of online math instruction, has no video or audio component; the whole class is a chat window, and questions are plentiful. Lowering a barrier to question-asking means lowering a barrier to learning.
I get it: “Math is hard” can be discouraging. But “Math is easy” is just false, which is even worse. We can be truthful without being demoralizing. We can tell our students: Math is hard — and you can do it.