Do you have an example of a "drawing" of a theorem, in this context? (I've seen these for fairly trivial theorems but not for more complex ones, so I'm curious.)
As someone who very much relates to the GP’s anecdote, I might suggest determinants as a good example.
As an undergraduate studying maths, I encountered a standard theorem in one of my first courses, which says that about 947 different conditions are equivalent to a matrix having a determinant of zero. I dutifully memorised these. I also dutifully memorised the algorithm for how to calculate a determinant. I might even have remembered some verbatim proofs of some of the equivalences.
However, I developed absolutely no intuition about what a determinant is. I had book knowledge, but no insight. It was a long time ago now, but I’m fairly sure that when I graduated I still did not truly understand even this very basic (by undergraduate standards) subject. I think it was probably a few years later, when I came across some of the same theory but in a much more practical context at work, that most of the connections in that equivalence theorem first “clicked”.
Meanwhile, here is what a gifted presenter with the right illustrations can do in about ten minutes:
The 2,000 or so substantially identical comments below that video are very telling.
Given the understanding you’d get with that quality of presentation, the equivalences I mentioned above would have been obvious and constructing the proofs from first principles would have been straightforward.
In the process of learning about a family of algorithms in machine learning I also gained some physical intuition of determinants (same diagram as 3Blue1Brown, but applied in a different context of "squashing and stretching" probability mass): https://blog.evjang.com/2018/01/nf1.html
Ahh I did a bit of googling but couldn't find anything nice — sorry! Most of the time, the complex stuff is broken down into smaller "lemmas" with their own manageable proofs, and then the proof of the whole theorem will be something like "Follows from Lemma 2.1, Lemma 2.2, and a basic application of Theorem 1.4"
As an undergraduate studying maths, I encountered a standard theorem in one of my first courses, which says that about 947 different conditions are equivalent to a matrix having a determinant of zero. I dutifully memorised these. I also dutifully memorised the algorithm for how to calculate a determinant. I might even have remembered some verbatim proofs of some of the equivalences.
However, I developed absolutely no intuition about what a determinant is. I had book knowledge, but no insight. It was a long time ago now, but I’m fairly sure that when I graduated I still did not truly understand even this very basic (by undergraduate standards) subject. I think it was probably a few years later, when I came across some of the same theory but in a much more practical context at work, that most of the connections in that equivalence theorem first “clicked”.
Meanwhile, here is what a gifted presenter with the right illustrations can do in about ten minutes:
https://www.youtube.com/watch?v=Ip3X9LOh2dk
The 2,000 or so substantially identical comments below that video are very telling.
Given the understanding you’d get with that quality of presentation, the equivalences I mentioned above would have been obvious and constructing the proofs from first principles would have been straightforward.