When I was taking linear algebra and calculus in university, I found that there was a lot of focus on deriving formulas from underlying principles, with the notion that this constituted a "fundamental" understanding of the mathematic concept. I got quite good at being able to derive formulas for anything, and did well enough to scrape by on my exams. However the concepts didn't really stick with me. Going through Khalid's site I quickly discovered I had a terrible intuitive understanding of mathematical concepts, almost embarrassingly so. Somehow, derivation from first principles doesn't quite capture intuitive insights for me, especially once I start worked at higher levels of abstraction removed from easily understood foundations (i.e. multi-dimensional vectors).
The two things that I found most helpful in relearning math is (1) building up a foundation for mathematical concepts through betterexplained's intuitive method and (2) turning it into code as soon as possible. For the latter, I have a side project that is a sort of platform to test all my various ideas, from city performance modeling, to procedural form generation, where I am constantly trying to rework or tweak with new math formulas. It's amazing how much more efficient and useful this is as a learning method.
Kalid from BetterExplained here, it's awesome to hear the approach is working for you. I had very similar reservations, I could derive many results (like a robot) but had little intuition behind e, i, pi, radians, etc. (let alone how they all came together in something like Euler's Formula).
My litmus test became: If I can't intuitively describe i^i (an imaginary number to an imaginary power) I don't understand it. I don't care if I can derive the equation 15 different ways. If I couldn't spit out some properties of i^i (positive or negative? Real or imaginary? Big or small?) after a glance then I knew I didn't know it. (Why can I spit out properties of 2^3 or 3^(-4) in a few seconds, but not i^i?)
Code is an excellent way to practice these ideas; the bugs in your logic correspond to the bugs in your thinking, and you see (very explicitly) where to correct them.
Kalid: you should really check out geometric algebra (a.k.a. Clifford algebra). It will give you a much deeper understanding of what i is and what the exponential function is, and how they generalize to higher dimensions and more complicated models, and it will help stitch together the weird inconsistent little fragments of understanding provided by imaginary numbers, quaternions, matrix algebra, Lie theory, differential forms, etc. into a more unified/cohesive model.
This is the model all high school and college students will be taught in 100 years, or perhaps even in 50 years, and it will prevent an enormous amount of confusion and misunderstanding. It’s already becoming the practical tool of choice in many geometric computing problems, and among niche groups of physicists.
(Or books might be better sources for going in depth. Search for New Foundations for Classical Mechanics, Geometric Algebra for Computer Science, Geometric Algebra for Physicists)
Unfortunately, university lecturers in many subjects, including mathematics, are often doing that job only because they have to as part of their main research position, not because they actually want to or have any aptitude for it. There are some great lecturers who are happy exceptions, but they are few and far between in my experience.
The sooner traditional universities and their ineffective, anachronistic teaching practices die out as the main form of tertiary academic education, the better. They could be replaced tomorrow by some combination of recorded lectures given by presenters who are actually good and in-person tuition given by people who can actually teach and care about their students' education, and I suspect absolutely nothing of value would be lost. The students would be both better educated and probably much better off financially, while the academics with a talent for research but not for teaching could make better use of their time and skills.
Problem: Who wants to spend a lot of time learning to teach when you don't really see this as more than a requirement to get that job? Which I completely understand, I don't blame the professors. Having tried a little bit of teaching myself, it's just very, very time-consuming to prepare yourself for doing it really well.
At university in the 1990s I preferred learning from (good) textbooks over most lectures, and today there's the Internet - I've actually gone back to learning with >50 courses on Coursera and edX over the last few years. So I think the problem has shrunk significantly since you can choose to learn from great teachers more easily than ever. But I'm a "pull" person when it comes to learning, meaning I don't even want someone to tell me what I'm supposed to know, I prefer to go out and look and select and get it myself.
It sounds like we're of a similar generation, but my experience was that unfortunately finding good textbooks for mathematics was also very hard. The average textbook was a dry, boring regurgitation of the standard definitions and results in a field and some associated proofs, with no added insight or motivation or context or example applications whatsoever. Many of them were written by the same academics whose lecturers were similarly bland and uninspiring.
Most of those academics could have done with learning about the ADEPT method we're discussing here and similar ideas. Unfortunately, they just had no interest in doing so. As you say, teaching well takes time, but it also takes a willingness to try to teach well.
I remember a particular meeting at the end of an academic year where the teaching representatives of the faculty at my university were seeking feedback from the undergraduate students. When one of the students boldly (but entirely fairly) asked why the presentation skills of most of the lecturers were so bad and why they weren't required to undertake training to improve when the university's teaching so fundamentally depended on them, the reply was essentially "We know and we agree, but they wouldn't accept it." In almost any other profession, the response to substandard performance of a key job function and refusing to undertake measures to improve would be getting fired.
These days, with students here in the UK paying thousands of pounds in fees every year on top of what we used to have, I can't imagine that official response would go down any better than it did for us, but as you say, these days there are more promising alternatives. This is why I think universities need to stop being the main tertiary academic education, at least in anything like their current form.
My problem is that I hate learning from video lectures. Do you know any MooC sites that don't require videos i.e. that present the material in written form (either chiefly or to duplicate lectures)?
Your first paragraph perfectly captures how I felt when I attempted to work through Linear Algebra Done Right, by Sheldon Axler. It's an awesome book, and the derivations and exercises show the beauty of maths.
But I had no mental model on which to hang the various results, and struggled to keep it all in my head, particularly after a break of a few weeks. IIRC, I stopped around the point when eigenvectors are introduced.
Oh, and please share examples of your coding+learning approach :)
The two things that I found most helpful in relearning math is (1) building up a foundation for mathematical concepts through betterexplained's intuitive method and (2) turning it into code as soon as possible. For the latter, I have a side project that is a sort of platform to test all my various ideas, from city performance modeling, to procedural form generation, where I am constantly trying to rework or tweak with new math formulas. It's amazing how much more efficient and useful this is as a learning method.