Learning Math is Learning Words

On connections across disciplines

Here’s a little story about math and language learning.

This past Saturday, I was reading a book to my almost-four-year old son. The book was about trains. Actually I think it was called Train. Anyway, he quickly got bored (for whatever reason on the page about commuter rail). Not knowing what else to do, I asked him some counting questions.

“Let’s say you have two apples,” I said. “Then I give you one more. How many do you have now?”

Three, he said.

“Now you have three apples. I give you one more. How many do you have now?”

Four, he said, yawning.

“Now you have four apples. I give you one more. How many do you have now?”

Eight, he shouted.

“Say you have five apples. I give you one more. How many now?”

Fourteen, he shouted louder.

This was when I started wondering, not for the first time, what exactly was going on in our kid’s head. He’s a pretty curious guy. Always playing with his own shadows. Always doing weird things when he walks. Loves jumping off of things. Speaks with increasing precision and sophistication in effortful, sludgy sentences.

My feeling was, I think he could figure this out.

I said to the boy, let’s count together. One, two, three, four, five.

Now I went back to apples: “You have four, and I give you one more. How many now?”

This time, he said: five.

I bumped up the quantities, each time reciting the number sequence together. He answered correctly up to ten. And this learning has lasted over the past few days. Now he doesn’t need anyone to prompt him with the sequence (though he still gets the teens wrong).

He seems to have made a connection and it seems to be thanks to his knowledge of the number sequence, one, two, three… a bit of memorized language that turns out to be at the very foundation of all mathematical thinking.

one two three four five NUMBER BLOCKS

For over fifty years, but probably much longer, researchers have noted that students who struggle to read often also struggle with arithmetic. It was only more recently, in the last twenty years or so, that we’ve learned what’s going on more precisely.

To be absolutely clear—these two things don’t always go together! Plenty of kids can struggle with reading but not math. The other way around too. Besides, there are a lot of general cognitive issues that could impact both math and reading. Someone is going to say that I didn’t say this, but I’m saying it, alright? We’re cool?

OKAY. That said, there seems to be a distinctive thing that kids who have trouble in both areas are sometimes dealing with. And that is phonological processing, our ability to represent, remember, and retrieve the sounds of language. (Think: the thing phonics might help with.)

So, for instance, in a 2022 meta-anlysis the researchers found a persistent association between phonological processing skills and certain mathematical skills. “Early phonological processing skills are likely to promote math,” they conclude. And it’s not just word problems or things like that—there’s an association between these skills and pure number computation, e.g. 5 + 2.

Without getting completely in the weeds, phonological processing is typically measured in three different ways. These are then posited to reflect three distinct capabilities. One component is called RAN, which stands for rapid automatized naming. A task that assesses RAN is trying to name colors that have been randomly sequenced on a page. What we’re measuring, it’s thought, is a kid’s ability to easily retrieve words (stored as sounds) from memory.

Students who struggle to do this quickly have a nontrivial chance of struggling with computation. (See this 2017 meta-anlysis.)

Isn’t that fascinating? Why would aptitude at color-naming be associated with early computation ability?

But look back at the conversation I had with my son: it was a language-intensive situation. His phonological processing skills were extremely engaged. He had to remember and retrieve words that he has only a weak understanding of, like “seven.” He had to remember it in a number sequence (essentially a short nonsense poem). Then, in a bit or rewiring/learning, he realized that he can use that poem to advance one further, the way he already knew how to do with numbers below four.

There’s a lot more to all this but basically: if you have trouble pulling words, stored as sounds, from memory, it’s going to impact your ability to use the counting sequence to solve problems.

(Down the line it’ll also get in the way of retrieving and using memorized math facts.)

Obviously, this has implications for teachers and parents of young children who are having a hard time with counting or arithmetic. But it’s also worth dwelling on what this says about math itself.

Math—the discipline of adding, subtracting, multiplying, dividing, conjecturing, estimating, reasoning, solving—depends on learning about fifteen words by heart. At first they don’t have to make sense, but they do have to be in order. If you have trouble with this, it may make it hard to do any of those other things, at least at first. There is an arbitrary-ish poem at the start of mathematics.

And all this is also to say that there isn’t some single “math part of the brain.” Learning math is a dance between language, experience, and our natural-born number sense. These things are engaged together and at all times, and at all levels. It’s possible to see this in the way students and teachers talk about algebra, fractions, derivatives, but it’s maybe easiest to see it at the very start, when we’re just talking apples and numbers.