Ars Technica: Do you have a favorite moment where your students had a significant breakthrough in their understanding of math?

Ben Orlin: They're magic when they happen, and they're usually small. The way understanding is built is not like there's a single click. It's not, "Oh, now I know all the differential equations." There's one moment I remember because it was so incongruous to me. I was teaching ninth grade in the UK, and I took a problem up with the board that had some fractions within fractions. I just multiplied through by a denominator to clear it so it wasn't a fraction within a fraction. It was now just a simple fraction. I never would've expected the reaction: a bunch of ninth graders cheering and whooping and hollering. They were so excited because they hadn't thought of that way of manipulating it. They didn't learn any new idea in that moment, but just the mastery of the mathematics was exciting to them.

Usually that's not what gets students excited. What gets them excited is really understanding things. More recently I was teaching an introduction to statistics. P-values are the MacGuffin of statistics. They come up everywhere. Students don't really know what they mean. But one day I threw a dozen simulated pretend studies at them and had them judge whether the null hypothesis was true or the alternate hypothesis was true. By the end of that, they were talking like statisticians. Even if they weren't applying the definition of a P-value, they just had a gut sense that, "Okay, P = 0.00001, yeah, okay, that's definitely a real effect. P = 0.32, no way that's a real effect. P= 0.06 or P = 0.03, that's borderline. Hard to tell. Got to look at the context a little bit." They didn't see themselves as doing anything magical, but it was exciting to me because at the beginning of the day, they didn't know how to interpret this number, and by the end of the day, they did.

I'm not sure it felt to them like a crisp mathematical understanding. But real-world uses of mathematics often are a bit fuzzy and negotiable. Part of the challenge of mathematics is that you have the world where everything is crisp and clean and perfectly defined—that's the great land of abstractions—but then you've got to bring those abstractions here into reality where the mapping is always a little messy.

Ars Technica: Speaking of messy mapping, you're very upfront in the book about how your analogies for the language of math—numbers are nouns, operations are verbs, grammar is syntax—are imperfect, in that they work for basic arithmetic but less so when you get to algebra and calculus. Why is that?

Ben Orlin: Nouns as numbers, I liked that because nouns are the simplest element of language. They're usually the first things you learn. Just naming things, literally. Numbers are not people or places, but they're the basic things of mathematics. There's this wonky philosophical point I like to make: prepare students for the kinds of abstraction leaps that they'll need to make as they learn mathematics, to give them the encouragement that they've already made one. That's just learning to count. There's a few ways to frame it, but one is that numbers in everyday speech are really adjectives. You can have seven staplers, and you can have seven cups of coffee. But seven is not a thing in reality. You can't touch seven. You can't sense seven. There's no direct material experience of the concept seven.

Already, just in learning to count, which my 5-year-old is pretty good at, they're already making a real philosophical leap. Later, you have to go from this particular addition process to what happens when you add X plus Y, or this particular calculation process to an arbitrary function where we can't even write down the calculation that you're performing. When you make those leaps, I hope people can draw a little encouragement from the fact that they've made such a leap already just by treating a number as a noun rather than an adjective. That part of the mapping felt pretty clear to me: to enumerate is to name the world. It's to assign names to all these quantities that we run into.

The place where the linguistic mapping is a little rockier is with operations and verbs. When you first meet them, the operations really do feel like verbs. Adding and subtracting are things you do to numbers. Adding is pushing the piles together, and subtracting is taking some away from the pile. Multiplying is making an array or a rectangle. Dividing is literally dividing, so they really do feel like verbs. But within the language of mathematics, they aren't. They're more like prepositions or conjunctions. I view that switch in perspective as really pivotal for mathematical learning, particularly when I've taught middle school and seen the students who are able to thrive in algebra.

A big part of that is being able to accept operations as just structure, the juxtaposition of two numbers. When you see A over B, no one's asking you to actually divide. You don't need to carry out any calculation process. It's just a ratio, A to B, and they can just sit there being that ratio, and that's all there is to it. That transition is tricky, and the fact that the language metaphor breaks down there is not a coincidence. There's everyday English where operations are verbs, and then there's algebraic language where operations are not verbs. Operations are prepositions, and the only verbs we really use are "equals" and "greater than" and "less than."

Ars Technica:  Then there's grammar, i.e., the rules that govern how you do these things.

Ben Orlin: Grammar combines a few different kinds of rules, especially mathematical grammar, which is this weird composite, because some of the rules of mathematical grammar are really ways of encoding deep truths about number. The distributive property is the classic one. You can see it as a fundamental truth about combining piles. If you have A piles of things and B piles of things, then you have A plus B piles, which sounds pretty simple until you start doing some algebra, when how to apply the distributive property becomes quite subtle.

So, some of the grammatical rules of math are about encoding deep truths. Some of them are almost stylistic. We tend to write AB and not BA for no special reason. It's just that when you have lots of things you're multiplying together, it would be hard to keep track of them all. If you had ABC and BAC and ACB, there are six different versions of that, so you just alphabetize your variables for the sake of making it easier to keep track of.

Equations are a very limited type of sentence. They're only "to be" sentences. You only ever say in an equation, this thing and that thing are the same, which wouldn't make for very good literature. No one wants to read that play or that poem. But in math, it's great, because that's the kind of information math gives us: different ways of representing the same object. So you can get really rich forms of mathematical discourse, which are just paraphrasing. This thing is the same as that thing, and that thing is the same as that thing. So the grammar of algebra is built to let us have those weird conversations.

Ars Technica: I especially enjoyed the mathematical vocabulary guide you included at the end as a kind of very long appendix. 

Ben Orlin: It was really fun to write. It's a complement to the other parts of the book, which are really going after the essence of mathematics that we try to teach in school. But I wanted just a tour of fun mathematical vocabulary terms of the sort that you could use in everyday life. I liked the idea of trying to plant more seeds of mathematical terms that could blossom into everyday speech. I like the idea of building bridges between mathematics and other parts of our intellectual lives.

Ars Technica: What was the biggest challenge while you were writing this book?

Ben Orlin: The challenge was the glut of possible things to say. When I sat down to write a chapter about addition, left to my own devices, I could write 40,000 words about addition. But I didn't want to write 40,000 words. I wanted to write 800 words. So, what's the right lesson to teach in this moment for the reader? That was hard. The book involved less research than my earlier books, and precisely for that reason it was harder to write because I had all these different ideas from 10–15 years of teaching knocking around in my head. How do I distill the one little tidy fun lesson for this chapter? So I would sit down and write a chapter and think, "Okay, if the book was going to end in another five pages, what would be the thing I absolutely need to tell you about math before I can let you go?"

I would write that chapter, and then the next day I'd sit down and think, "Okay, we've gotten this far, but if the book was going to end in five more pages, what would be the next thing I'd really have to tell you about math?" It was not the way I'd written any other book. That's a very strange way to write. I'm much more of a top-down planner. But for this book, that process felt right. I wanted every lesson to have that sense of urgency.