One of the things some folks may misunderstand is the difference between mathematics and arithmetic. For a lot of people, their exposure to mathematics is just doing calculations — filling in their tax forms, balancing their checkbook. That is not mathematics. The analogy I like to give is that spelling is not literature. But you need to know how to spell in order to be a writer. Calculations are part of mathematics, but it’s not really what it’s about. It’s a broader thing than that. Unfortunately, people are not exposed to that much in K–12.
I’ve often asked some of my colleagues, if there was one idea from mathematics that you want an educated high school student to know. From a literature point of view, if I said I’m going to see Romeo and Juliet tonight at the theater, and you say, what’s that? And I say, the play by William Shakespeare, and you say, who’s that? That’s really a problem. I don’t expect you to know all the plays of Shakespeare and all the sonnets, but Romeo and Juliet and Shakespeare — yeah, you should know what that is. So what’s the analogous thing in mathematics?
In my opinion, it’s the fact that there are infinitely many prime numbers — and how to prove it. It’s not super difficult, but if you’ve never thought about the question, you would really struggle to understand how you can possibly know there are infinitely many. How do you even begin to do this? But it can be taught, and it opens up a whole other way to think about an idea from mathematics that’s really cool. It was known to the Greeks. Euclid knew this. It’s not very current, but it’s still very interesting, and I’d want every high school student to know this.
I don’t care if they know how to solve quadratic equations. Boring. Not really interesting. But the idea that you can use mathematics to answer questions like that — that’s pretty amazing.
And yes, I think it’s important that you don’t struggle to calculate. It’s just like if I were to sit down and write a story that I want to get published, and I have no idea how to spell any words, or I have no idea about grammar — I’m stuck. This basic arithmetic is a foundation on which mathematics sits, in the same way that spelling and grammar is something on which literature sits. You need to know some of that.
And by the way, with some of the spelling and grammar, if you’re going to break the rules, sometimes you can do that for great effect. But you have to first be grounded in the rules, know when to intentionally break them, and then sometimes that’s creative and really kind of cool.
I was taught my multiplication tables, and that was good. I was in grade school in the 60s — it was quote-unquote the new math. There was this panic because the Soviets launched Sputnik, and we’re so far behind. There were certain things that were really kind of interesting. We saw sets in elementary school. We created operations besides plus and times — innovative operations — and figured out what properties they have. That was part of fourth- and fifth-grade education, so that was a hint that there’s some creativity in there.
If it were up to me, no student would touch a calculator until about ninth grade. When you’re doing your science and so forth, yeah, you really don’t want to be dividing 387 by 942 on paper. I’m old enough that we were doing all that with slide rules when I was in high school. There’s a value to that also, because you’re not just blindly punching numbers. You have no decimal point on your slide rule, so you have to keep track of what you’re doing and always ask yourself: does this make sense? I think you lose that with the calculator. I’m old school — no calculators until about ninth grade. I think they’re just a detriment.
When I learned geometry, we were doing proofs. I loved it. Geometry’s wonderful, and proofs are wonderful. Math without proofs is like science but never doing an experiment. Well, how do you know it’s right? That’s what proofs are. That is what mathematics is. It’s more than proofs, but that is the core of it, and it’s gone from K–12 education, as far as I can tell.
The creativity and enthusiasm of a teacher is very influential. Very influential. If your teacher hates math, you will hate math. I was very fortunate that I had several teachers who were very enthusiastic about math and very supportive. Even my fifth-grade math teacher — she was wonderful. I just remember how much enthusiasm and creativity was brought to the subject, even that far back, and I’m sure that was a big influence on me.
