Self learning math

How should you be learning math? Even when I was a grad student, I did so much math, and I tried to do a certain number of problems every day. I remember I had a goal of—I think it was 20 proofs a day. It was very hard to keep up. It's very hard to do 20 mathematical proofs every single day, especially when you're in grad school.

So what I would do is go back and do easy proofs, proofs that perhaps I've already done, just to hit that magic number of 20. You can do stuff like that, and it's good, but you're going to burn out.

I think the right answer to the question of how much math should you do every day has to start from a different place. It has to start from a place of discipline, and that discipline is built upon doing math every day. The real question should be, "How can I get myself to do math every day?"

It's like that old saying: You can't learn to run until you learn to walk. The same thing with mathematics, right? You're not going to be able to sit down and do five hours of mathematics every day if you're not even doing math.

And I think that it comes down to decision paralysis. That is the thing that affects most people in all areas of life. I have a lot of friends who have great ideas, and they don't do them. I have ideas, and I don't do them. But I try, and I'm aware of it, and at least I do math every single day. Every day, no matter what, I do at least one problem. It keeps my mind sharp, and it's worth it.

How can you get yourself to do math every day? Well, I think the best way—and this is the way I do it, which is why I think it's the best way—is to tell yourself you're going to do at least one problem every single day.

Because when you tell yourself you're going to do at least one problem every single day, chances are you're going to do it. If you tell yourself, "Oh, I'm only going to do 30 minutes," that already is creating perhaps some negativity in your mind. Because maybe you wake up today and you don't want to do any math at all, right? Zero. So looking forward to the prospect of having to do 30 minutes is a very negative thing.

It's all mental. It's all up here. If you can control your mind, everything else is going to be great. You just have to be able to have a strong mind, and math is one way to do that. I think mathematics and physical activity—both of those things are extremely important, and they'll help your mind.

Tell yourself you're going to do at least one problem a day. I think that's a good way to do it. Just do it. Don't overthink it. Don't fall into the decision paralysis trap of, "Oh, what type of math should I do? What math book should I use?"

And what's going to happen when you do that is, on those days when you don't want to work, on those days when you don't want to do anything, and you do that one problem, sometimes those days will end up being productive. Sometimes you just have to start. And once you start, it's like a ball of ice rolling down a hill, like in the cartoons—you know, it just gets bigger and bigger and bigger. The snowball just grows the farther it goes.

A lot of times, that's how it happens with mathematics. Sometimes you'll do one problem, and you'll be like, "Oh, I'm done. I can't think today. My mind is in a fog. I need to go do something else." And then you go do something else. But at least you did one problem.

You want to look back at the end of every single day and say, "Did I do enough? Did I do enough today to progress my life?" Did you do enough math problems? Did you train? Whatever it is you're doing in your life, whatever you're trying to do—did you do enough to progress?

If you're in school, it's pretty easy—well, easier—because you have this external motivation. The specter of grades is hovering over you. It's like you have to succeed. You have grades to pressure you. That doesn't make it fun, but it forces discipline upon you.

But for others, self-study is the way. And in order to self-study, you have to have discipline. It makes you stronger. And I always think that the morning is the best time.

If you do a problem right when you wake up, after you have coffee or whatever, just do a problem. Do an integral, whatever—just do some math. It's going to stay with you for the rest of the day. You'll be able to look back later in the day and say, "Hey, oh yeah, I remember that integration problem I was doing earlier in the day. That was a really interesting substitution," or "Oh yeah, I remember using partial fractions to solve that," or maybe you did a proof—"Oh, that was a really cool proof. We took the maximum of a finite set to create an upper bound on the sequence. “

You know, whatever those things those ideas that you have in the math they stay with you and you think about them and it gives you time it lets your mind progress. It's much better than cramming that's why they say cramming doesn't work. I mean it does work to an extent I've crammed for so many tests.

I think what holds people back is they set expectations that are too high so don't set any expectations. Just set the expectation of I'm going to do at least one math problem. Once you can do that a couple times then you can start having higher expectations. But be careful because you don't want to set them too high and you don't want to break your routine. You want to make sure that you do it every single day, it makes such a difference.