>Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything.
This is so true. I was exactly like the author. I could not understand any math beyond algebra. Trigonometry was way beyond my ability.
But one day, I found an old trig book lying around when I went to visit my parents - it was one of my sibling's books when she took it.
I picked it up and had a thought - all those people who DID understand trig and higher math were not that much smarter than me. I mean, yeah, the super smart ones were, but I'm just talking about the regular students. So I got a bug to show myself that if they could do it, then I could do it.
My approach that I would do is to read the first section of the first chapter until I understood it completely and not move on until I did. I would not even move on from a sentence, or indeed a word or term, until I understood exactly what was happening. I would look all over the internet, watch youtube videos on that one single concept, or that one single word, until I understood, and I was absolutely adament about not moving on until I understood.
Word by word, sentence by sentence, section by section, I rolled on. I wanted to completely completely completely understand the very very basics of trig. It was slow going, as you might imagine. Super slow.
Well........all of a sudden, about 4 or 5 chapters into the trig text, my mind exploded. It was blown apart by a blinding light. In one single instant, I swear, one instant, I knew all of trigonometry. I got it. All of trig. One single instant. It was probably one of the most singular moments in my life.
So I then started paging through the entire rest of the trig book, and I knew it all. I swear. I would just glance over the pages and know what everything was about.
Now, I was NOT an expert, by any stretch of the imagination, I'm not claiming I became a savant. No. I'm just saying I "got" it.
As you said, I has the exponential results. So exponential, it was instantaneous.
>If the fundamentals are well understood, the applications feel trivial, intuitive, beautiful, clever - when the fundamentals are taken for granted, one word describes the relationship to the rest of the material: arbitrary.
This is it exactly. You describe it perfectly.
I did not continue on, because I actually had other things in my life that I had to move on to. But it was great because I knew then that math was easy. No doubt I could do the exact same with any math class, no doubt in my mind whatsoever.
For whomever wants to do great in almost any area of study, study HARD on those first 3-5 chapters. They set the tone for the rest of the course. If you study hard on them, they remainder of the course is trivial, with some things here and there that you need help with, but for the most part it is simple.