As a high-schooler I did a summer course at Columbia about "graphics programming". In the morning, the instructor would cover the very basics of the theory of what we were trying to model, from the physics of it to the math required to represent the physics. The point of it was that, starting from scratch, we would build our own little ray-tracer in C++.
The course lasted 4 weeks (and was intensive, we were in class 5hrs a day if memory serves me right). It may have been the most unique and interesting educational experience of my life. There was so much motivation to LEARN the math and physics, and I vividly remember going home and looking at objects, and thinking about the angles that the light would bounce off of objects, etc. Plus, there was something truly empowering with starting from a completely blank sheet of code and building something like this, fully understanding what each part of the code was doing.
It was this course that introduced me to vectors and vector operations, as well as matrices, before I fully dove into them in school. It was absolutely perfect. Ever since I've had a soft spot for linear algebra, and am often disappointed at how frequently I've encountered it completely in the abstract, when it may be one of the topics in math that you can most readily connect to interesting, non-trivial real world problems.
This is the only way I've ever learned anything mathematical, at least beyond school-level maths. I wish it were more common to find courses - even online courses - that 'motivate'[0] maths with engaging projects, which for most of us here would mean coding projects. (The same goes for mathsy areas of programming: information theory, the prereqs of compression or computer graphics, modular arithmetic, linear algebra, set theory, topology, etc.)
And it's not just us, either. So many people I speak to have had the same exact experience. I have mathematician friends who agree with this, based on their experience of teaching maths. I think mathematical pedagogy is just broken, with teachers getting high on the perceived difficulty of their special wisdom.
[0] ...to use the semi-slang my university lecturers used. In other words, give you a reason to want to learn it.
I share a similar experience. I have a course on Numerical Optimization, and I feel that I have learned more linear algebra, and matrix calculus in 2 weeks than I did my whole undergraduate course. I also delved a bit into differential geometry to argue the correctness of the implementation of my gradient functions using the finite difference method.
Honestly, this has been the most fun I have had in a course in a long time. It's difficult at times, and there are many things one needs to learn, but this is how learning should be done imho.
That's really cool. If you have any pointers to any resources along those lines, I'd be really interested.
And I absolutely agree. It's a tragedy that maths is taught in such a dry, unengaging way. I think far more people would enjoy it and excel at it if only it were taught in a way that motivated them. Your experience is one of a million demonstrations of that.
For what it's worth, there are a couple of Wikipedia pages which I found to be great jumping-off points - effectively 'sitemaps of mathematics' - to explore mathematics and what parts you may not know yet:
They also present 'ways of looking' at mathematics which I find can be eye-opening with any subject. I don't know if you're a lumper or a splitter (https://en.wikipedia.org/wiki/Lumpers_and_splitters), but I'm a lumper, so it helps to see connections and unities between things deep down. (Anyone who's ever read the annoyingly separate study areas of automata vs state machines, and the replicated abstractions and principles – despite them being literally the fucking same thing – will probably see what I mean.)
The course lasted 4 weeks (and was intensive, we were in class 5hrs a day if memory serves me right). It may have been the most unique and interesting educational experience of my life. There was so much motivation to LEARN the math and physics, and I vividly remember going home and looking at objects, and thinking about the angles that the light would bounce off of objects, etc. Plus, there was something truly empowering with starting from a completely blank sheet of code and building something like this, fully understanding what each part of the code was doing.
It was this course that introduced me to vectors and vector operations, as well as matrices, before I fully dove into them in school. It was absolutely perfect. Ever since I've had a soft spot for linear algebra, and am often disappointed at how frequently I've encountered it completely in the abstract, when it may be one of the topics in math that you can most readily connect to interesting, non-trivial real world problems.