Difficulty of learning mathematics

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This page is about the difficulty of learning mathematics. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?

Differences in difficulty between pre-rigorous mathematics and rigorous mathematics


NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.

Satvik Beri makes the case that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.

Vipul Naik: "While linear algebra lacks any single compelling visual tool, it requires either considerable visuo-spatial skill or considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."

Brian Bi: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."

Christopher Olah: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."

https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related are https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH and https://www.greaterwrong.com/posts/de3xjFaACCAk6imzv/towards-a-new-decision-theory/comment/DRQciHjC8GMonBCFe

Michael Nielsen on Simpson's paradox: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."



Bill Thurston: https://mathoverflow.net/questions/38639/thinking-and-explaining

Tim Gowers: "Mathematics becomes hard. Every mathematician will be able to tell you rather precisely when it was that they found that mathematics had stopped being an easy subject that they could understand with very little effort and became a difficult subject that they had to struggle with if they wanted to get anywhere. It isn’t necessarily an advantage if this happens to you later rather than sooner. For example, some Cambridge students find the course difficult right from the start, whereas others largely coast through the first year and then find that they can’t coast through the second year. The people who found it hard in the first year may by this time have developed good study habits that the people who found it easy in the first year do not have." [1]

https://news.ycombinator.com/item?id=7331775 "Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can."

Nate Soares: https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things

See also