I finished Lockhart’s Lament, the 25-page version, and I have some feedback about it.
I regard proof as being the defining activity of math. Really the trio of axiom-theorem-proof. However, proof is the real heart of the matter.
I used to have a sort of mystical admiration for the idea of proof. Then Tyler taught me a little bit about Coq, the theorem prover, using Software Foundations. I learned that proof is always intended for an audience, and that the most objective audience would obviously be a computer, a program that is going to make the same determinations every time, thus formal proof is the best kind of proof. Tyler didn’t say this, but it was one of my takeaways anyway.
Now taking Basic Concepts of Mathematics, I’m seeing proof as a story. The audience may vary—I’m a weaker audience than my professor, so I need more hand-holding. But an informal proof can still be rigorous. And I think this is the point that Lockhart is making about teaching kids to prove: we should not hold them to the same standards of rigor, but we should nonetheless expect them to make a mathematical argument.
This reminds me of the “black box game” which I learned about, as a way to teach your kids something about functions. The way you play the game is they give you a number and you give them another number, and they have to figure out how you came up with it. For instance, if you give me 3, I might give you 3, and if you give me 4, I might give you 5. If they know about odd numbers, they may say, “oh you’re giving me an odd number, either my number or the next odd number.” This is a way of helping kids think about functions without being too limiting about how they are constructed. (Another key takeaway for me from this class was that a function is not a bunch of algebra, it is defined if there is a way to find the unique value associated with the argument, whether that is a bunch of algebra or not.)
On Math and Science
Tyler provides an interesting definition here: math is about deduction and science is about induction. In science, we chip away at irrelevant details of reality to try and generalize to everywhere. In math, we can never depart the realm of pure reasoning and make contact with actual reality. I find this thought quite beautiful. It’s almost like a limit, defined from the left and the right, but there’s no reaching the limit other than by assuming it. A beautiful thought, I guess, to me, apropos of nothing in particular.
Lockhart returns repeatedly to a metaphor of a music teacher who is unable to play an instrument or has never heard music but is simply able to manipulate the notation. There’s an allure to this metaphor so I can’t completely begrudge him but I do have some feedback here.
First, school can and often does have negative effects in every subject. I actually think you can point to music as an example of this: Benn Jordan recently did a video about why there are so few women in the music industry and one (of many) reasons is that we typically make girls learn symphonic instruments like violin and cello, whereas we allow boys to learn instruments like guitar and drums—which are actually used in the modern music industry. The folks with bumper stickers bemoaning funding for band class in school are of course not imagining that we’re going to fill the room with 808 clones and electric guitars with distortion pedals and allowing real noise to be made. They’re picturing 30 kids playing a chamber music piece written sometime between the invention of the printing press and the electric lightbulb to a room full of well-dressed parents calmly golf-clapping. Every subject is ruined by school, not just math. Hence the joke about those who can and those who teach. My apologies to a certain reader who absolutely can is about to start to teach.
That said, I think the curriculum is changing, albeit not to the extent that Lockhart would like. But this essay is 20 years old now—some sympathizers have had time to get power. My kids love their math classes—they’re in 4th and 1st grade—and one of them said her entire class asked for more math homework. They’re worksheets, sure, but they like now to show you a half-dozen ways of solving similar problems. My son clearly has multiplication tables memorized, but at no point did the school send him home with flash cards and tables to fill out. I don’t know exactly how he learned it, but he knows the material. Things are not the same as when I was a kid.
You can actually see the effect of this in places like National Review. We have here a great example of people demanding that things be different, and then being upset when they are not the same (which is exactly what you expect from National Review). But in fact, I recognize some of these problems and several of them are great—if you’ve been taught the material. But many of these problems are using new approaches and if you weren’t taught them and aren’t inventive enough to figure them out, well, you see what happens.
I had a powerful experience taking this class, and reading “A Mathematician’s Lament” while taking it helped me acknowledge that it is a powerful experience. Lockhart wants my kids to have that experience too. This isn’t a coherent vision of how to restructure math education. It’s a plea for people to experience math the way they experience art. I’m sad that I was 39 before this happened to me.
I want my kids to experience this too. Preferably before they turn 39.