How I Rewired My Brain to Become Fluent in Math

Understanding is "facile" [superficial], but rote learning is the real way....?

Seems reversed. An explanation is she's in Electrical Engineering - my ugrad experience of EE was a emphasis on using tools (formula etc) as opposed to understanding them (that's for Science). Engineers, after all, are paid to get stuff done, not just sit there and grok it. "Fluency" with tools works well for EE.

OTOH, all discplines have "tools" - even pure maths has algebraic manipulation. If you're not fluent, it will slow you down (fortunately, you'll have years of algebraic practice from school).

OAH, most professional mathematicians don't think symbolically (a survey found, IIRC, about 70% visual, 25% kinesthetic, 5% linguistic) - notation not so much a tool of thought as a serial representation (serialization/data format), for recording/communicating. So practice in thinking is what's helpful.

I think fluency with "standard" modes of thought is a double-edged sword. Yes, you become expert with those tools, fast and able. But that very expertise biases your perception and reasoning in terms of them. Thus, it's hard for you to see another way; you'll tend to build on top of them instead. Fortunately, since our tools are pretty good, this approach works well. It's just that you're less likely to see fundamentally new approaches (though to be fair, that's pretty damn unlikely anyway).

That is really humbling and gratifying to hear, thank you.

I think that it is strange that she is saying that there is too much focus on understanding at school. From what I could see, most people learned maths by just memorising the process. Except for a few who invested an exceptional amount of effort, they scored poorly. In contrast, everyone who I knew who focused on gaining a deep understanding was able to pick up maths quickly.

There is one area, though, where there is probably too much focus on understanding at the cost of memorisation and that is in the Olympiads. I remember that many of us held the rather uncharitable attitude that the people who tried to succeed in maths by memorising it were 'stupid'. But clearly, understanding combined with targeted memorisation of the key building blocks will lead to the most success.

I believe what is described is repetition, but not rote learning. That, to me, means memorizing facts without connecting the dots. Rote memorization is learning for an exam by memorizing the previous years' problems, or learning history by memorizing dates and events. If I focus on knowing the previous years problems by heart, I may accidentally make the connections to be able to solve more varied problems, but thats rather inefficient. Similarly, knowing historical dates doesn't imply an understanding of why and how things happened.

Learning different conjugations of verbs by using them in sentences isn't rote learning, either. It's rather the opposite, trying to connect the new verb into the existing knowledge of the language.

Rote learning alone is useless for most students. Repetition is certainly important in retaining information, no doubt about it, but as with the math examples, you need to apply basic arithmetic in the context of solving equations, for example, to really get a "feel" for it, a sort of automatism where you not only know how to do 1+1, but also WHEN to use this knowledge.

As far as complex algorithms and such go, I do believe they can be broken down into pieces, but you really do have to know the pieces and their relations and implications innately, and with complex algorithms, it's really hard to know all the pieces of the puzzle, and sometimes even finding out what is missing is difficult. You can drill something ad infinitum without gaining anything if you're missing something.

I believe this to be the secret skill that the article omits. In order to learn effectively, you have to recognize where your knowledge is lacking, find out what it is you should know, and drill those parts. A teacher should help students find these gaps and help fill them. The rest is just a little bit of willpower and discipline.

Do you end up writing perfect code when you learn from examples?

Just like you, I learn from examples rather than by reading documentation... and I suspect that just like me, you don't end up spending 5 minutes writing perfect code. If true, to me, that means your friend's method is superior.