Terence Tao

So there's many, many different ways to access math.

But when you teach a class of 30, 40 kids, you can only teach one way.

And inevitably, many of the students in that class will not click with them, the style.

If there's some way to have multiple pathways to learn the same material, maybe outside the classroom, some enrichment activities, I think that would help.

Yeah.

So yeah, if the teacher doesn't care, the students won't either.

But good teachers are so precious and so rare, hard to find.

Yeah.

So it's really important in math to not just prove positive results, but negative results.

So results where you know you cannot prove the thing you want with the hypotheses you have, because you can find some counterexamples which satisfy all your hypotheses, but don't satisfy your conclusion.

Now, these hypotheses may not be true.

correspond to the real-world problem that you were working with.

But by comparing that counterexample with the real-world problem, you can see what you're missing.

So a lot of math is actually exploring the negative space of what doesn't work and what you know doesn't work.

And it's only after you sort of map out all the negative space can you see kind of the very narrow path which dodges all the pitfalls and gets you to your goal.

Sometimes, yeah, you can just, what's the modern term, like raw dog it or something?

Raw dog, oh.

That's what the young kids say.

Right.

Yeah, so it's good to not be emotionally invested in one outcome, that this has to be true or this has to be false, but to actually actively work to prove both, one conclusion or its opposite.