I used to take math really seriously. As a student, I worried about getting the “right” answer or figuring out the “right” way to think about things. While there’s nothing wrong with trying to be correct, these feelings really impeded my ability to do good math. Anxiety makes it hard to learn new things or see things from a different perspective, which is exactly what you need to do good math. Instead, I started seeing math as a creative endeavor, and I figured out that I do my best work when I’m having fun.
You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine.
- D. Hilbert
I used to see the same thing happen when I tried to write songs or essays. If you try to write the perfect song on your first try, you’ll never be able to write a single word. You’ll freeze up.
In writing classes, a lot of people are taught to do what I call “vomitting on paper.” You just write whatever’s on your mind, stream of conscious style. Get a bunch of stuff down and then go back and revise. Move things around, maybe organize stuff into bullet points. Delete some stuff. Rinse and repeat.
There is no such thing as good writing. There is only good rewriting.
- Harry Shaw, Errors in English and How to Correct Them
This is what I do with a lot of my blog posts. If I really want to polish, I’ll iterate a lot. For other posts (like this one) maybe just once is enough.
Similarly with song writing, what I like to do is just sit down and turn on the recorder on my phone. Always, 100% of the time. Even if I have no idea what I’m going to play, or if I’m just fooling around. That’s because my best creative work happens when I’m in the moment, just playing around, not worried about sounding dumb or playing the wrong thing. And if the recorder’s on, I can always go back and remember those moments, taking the best ones and stitching them together into something coherent.
So you can kind of apply this to math, right? Most people wouldn’t say that math is like writing an essay or a song. But really it is. And it’s just as creative.
This is the problem with math homework and tests. The kids are worried about getting bad grades, so they don’t take risks; their imagination just shuts down. But really, to see why an answer might be true, you have to throw a bunch of stuff at the wall and see what sticks. A lot of math problems can’t be solved by just applying formulas and shuffling around symbols. You really have to see the analogies and know where you’re going, so each step you take is in the right direction.
One should not try to prove anything that is not almost obvious. - Alexandre Grothendieck
Really, writing a proof is just something you do after you’ve basically solved the problem. It’s for the benefit of other people (and yourself), so that you can prove to them (and yourself) that something is true. But it doesn’t really reflect how you got there.
We often hear that mathematics consists mainly of “proving theorems.” Is a writer’s job mainly that of “writing sentences?” - Gian-Carlo Rota
So my process for doing math is more or less the same as writing songs and essays. I get a copious amount of paper, write down my problem, and then I write down anything that comes to mind, as it comes to mind. If I need to draw pictures, I draw pictures. If I need to write down an equation and annotate what each part really means, I do that.
This helps me really “go down the rabbit hole” so to speak. If I try to solve a math problem in my head without writing anything down, I can only keep track of 1 or 2 possibilities, and follow those ideas a few steps each. But if I write everything down, it’s like I have inifinite brain space. And then I’m not afraid to try crazy ideas.
But the key thing, again, is playing around. If I ever feel like doing the math is a chore, I know I’m going to have a bad time, and that’s the only time when math ever really feels “hard.”