Someone who is really good at math (and did algebraic geometry in high school, 3x Putnam Fellow, 2x IMO Gold and currently doing world-class research in algebraic geometry) wrote (while he was still in high school !) this in regarding to learning math somewhere
Virtually all of undergraduate math can be learned by people with no actual ability to understand anything, hence why there are quite a few graduate students who have no real understanding of what they learned as an undergrad.
EDIT: To show what I mean by this:
For example, the proof of most theorems in later real analysis/basic functional analysis can be looked at in two ways:
- This proof is a bunch of lines such that each one proves the one after it!
- This proof is a set of ideas, relying on this hypothesis here and this hypothesis here, that cannot be generalized without overcoming _________, that has importance because of it's relevance in ________, is motivated because __________, and shows the power and limitations of ____________.
A lot of undergraduate education is focused on "rigor", so you end up with students who can see 1 but not 2.
I am reading basic group theory (from Herstein) and Analysis (from Abbott), and it's clear to me that I'm doing only 1. What to do so that I can learn to do 2 ?
Also, here's another thing he wrote (regarding contest math, viz IMO preparation)
You're thinking too much about PREPARATION.
You learn these ideas by just exploring.... if you see a probabilistic method with 1/2 for the first time, you should be asking 50 thousand questions, and one of them should be "what if p is a value that isn't 1/2?"
So many people spend 5 hours a day preparing and then lose to people who prepare 0 hours a day, because those people who prepare 0 hours a day, even though they don't do any formal practice, they just explore and find stuff.
How do you learn to explore and find stuff ?
PS: Please don't reveal his name here if you recognize him from his description.