I think a certain amount of blame has to go to the teachers though as well. My personal anecdote, I did extremely well in Calc AB in high school, aced the AP exam, aced first semester Calc III in college, then had a professor in linear algebra who, in hindsight many years later, was a terrible teacher. He zoomed through everything, didn't explain, and just presented rather than taught. I still remember his comment to help us understand - "if you're having trouble picturing 11 dimensions, picture a 3D picture, but in 11 dimensions." Thanks! My last math course, Partial Diffs I did well again. To a certain degree I feel "math is math" but how it's taught is different from prof to prof.
My intro physics class was kind of like that. There was a departmental standard that all the profs were instructed to use, and it apparently just involved showing us equations and how to derive them (generally after we had already had to derive them on online coursework).
The tests were actually applying those equations, being able to pull apart a problem into what bits of information you had, what bits you needed, and being able to combine and substitute out formulas to get the missing bit you were asked for.
In hindsight, I can vaguely see how the instruction might have helped on the test (I'd have gotten used to substituting out and deriving new equations from the old), and I can see how I might have done better on the tests (write out what equations I remembered. Write out what data I was given. Write out what data I was missing. Start substituting things out until I found a way to calculate the thing they asked for), but at the time, the only thing that helped me pass was actual instruction and practice outside of class on solving problems.
Halfway through Calc III it was interesting to watch the visual students, who normally would do very well, have a rough time, while other students, who just treated it as an abstract system had more overhead/trouble/were slower when learning initially, but it paid off when getting to un-visualizable systems.
In my experience, virtually all of Calc III is "visualizable" (even somewhat esoteric stuff like Lagrange multipliers[1]), because it's mostly about vectors (which have a natural geometric interpretation). My claim would be that to be really good at math, you need to be skilled at both visualization (and other intuitions) and abstract systems. They complement each other well.
> I still remember his comment to help us understand - "if you're having trouble picturing 11 dimensions, picture a 3D picture, but in 11 dimensions." Thanks!
To be fair to the prof, you cannot actually picture 11 dimensions. That looks like a joke to me (hey, I laughed).
I had an excellent math teacher in High School who would intersperse questions in his lectures, turning around and waiting long enough for a few hands to go up and then choosing someone to answer.
I was never able to adjust to the recitation-style math lectures in University and ended up having to, as you say, teach myself by reading the book and posing questions to myself as I went along.
I was and still am angry that quality of education I was receiving in exchange for tens of thousands in tuition was considerably poorer than what I was used to receiving for free in public High School.
I feel your pain. But part of college is learning how to learn things on your own. It varies a bit by field, but in theory by the time you end your undergraduate work you are almost out of work that other people have done before, and now it's up to you to figure out new things.
Probably one reason (among many) I didn't bother trying for a PhD.
Learning on your own is certainly core to college - but lectures that just where the TA just takes an outline from the book and regurgitates it are a waste of time. I certainly never attended those classes. I imagine they are useful for auditory learners but I’m not one of those.
Even at the college level there are plenty of class formats where either teaching was done, or at the very least was done via discussion.
The "here’s are the course materials read to you by a very bored smart person who has better things to do", isn’t a class format - it’s a series of exams.
Geoffrey Hinton uses this same "picture a 3D picture, but in 11 dimensions" quote in his Neural Networks Coursera class. Honestly I think it's the only way to imagine >3 dimensional objects.
