[weehoo]

The great grandparent to your post (the one who initially brought up the necessity of rote learning) is a quant at a market maker.

Anecdotally, my favorite professor from undergrad (born in China, PhD from the best department in his field [my personal opinion]) said he thought the reason for Russian/Chinese dominance in certain areas of math was due to how those areas benefitted very much from rote practice. He advised all of us (American undergrads) to drill and kill certain techniques in order to build up our pattern matching.

I don’t think they’re advocating doing hundreds of worksheets on the power rule or trig substitutions or memorizing line by line proofs. Our brains do follow formal rules when doing math, but the insight necessary to find a way to solve a problem that isn’t straightforward isn’t through application of rules, it’s through a tacit intuition that you build up by doing lots of math. There is no other way.

It’s like how everyone feels like they understand physics to a PhD level while watching the Feynman lectures, but if you were to hand them any of the problems afterwards, what seemed like such a natural stream of thought is just simply out of reach. It’s much easier to go over something and declare “this makes sense” than it is to come up with that something in the first place.

[whoisburbansky]

+1 on the drilling specific techniques. Any skill is reinforced by consistent targeted practice, and to think that math is an exception where you can break through with pure genius is just deluding yourself.

The only way I got through my undergrad math was by doing problem set after problem set until the concepts were second nature, and the courses that didn't have a sufficient breadth of exercises to drive home fundamental concepts ended up being the ones I struggled with the most.

[centimeter]

> break through with pure genius

Applying derivations that someone else invented doesn’t require “pure genius”; it only requires you to be able to follow the person who discovered it, which is a lot easier than finding it yourself.

> The only way I got through my undergrad math was by doing problem set after problem set

It sounds like you didn’t really understand what you were doing then. It’s hard to phrase this without just sounding like I’m bragging, but I never had to practice doing anything I actually understood. If I felt like I needed practice, that was a sure sign I didn’t get it, which I always tried to fix with careful thinking instead of repetitive memorization

[centimeter]

> It’s much easier to go over something and declare “this makes sense” than it is to come up with that something in the first place.

Obviously, but that’s not what we were talking about. We were comparing memorization to understanding, not inventing to learning.

If you’re good at math, you should be able to re-derive any formula or procedure quickly (up to, say, constant factors) without having to memorize it (after the derivation has been explained to you).

If you run into a problem that you can’t solve because you didn’t drill the steps hard enough, you don’t actually understand the problem. This isn’t necessarily your fault - many math courses teach by symbolic manipulation without the conceptual grounding required to actually re-derive the symbolic procedures yourself. Few students will seek that understanding on their own outside of class, in which case they’re stuck with memorization.

> the reason for Russian/Chinese dominance in certain areas of math was due to how those areas benefitted very much from rote practice.

I think this supports my point - these “certain areas” are small. There is a relative paucity of mathematical/physical innovation from China (especially per capita!). The west still dominates mathematical invention.