Ask HN: How to learn math from zero for adults?

I see students hit a wall in linear algebra, very good at "trained seal" but profoundly resentful that the time has come to go meta: Part of one's learning has to be investment in reflection and experiment on how one learns.

They generally believe that intelligence is innate, not deliberate gardening. There were thousands of people with Michael Jordan's body (alas, I'm not one) but he spent decades creating the athlete he became. Einstein found math difficult. It's like running: If you find running boring, you're not running hard enough. Math is a constant struggle, like mountaineering, and you shape, invent your mathematical mind. Those of us who love math (or mountaineering) are drawn to the struggle.

One syndrome I've seen often enough to classify: A student is uncomfortable with their calculus background. They want to pick the hardest calculus book they've heard of (often Apostol) and retreat to a desert island till they've mastered every word. I cringe in horror, and try to explain what's wrong with this approach.

It is widely observed that grad students learn four times faster than undergraduates. My first few years of grad school, it felt like I doubled what I knew every year. It was profoundly depressing when this subsided; I could have "been someone" with just a couple more doublings. What was going on?

Grad students learn what they need any given day, for goals they've set for themselves. Undergrads are taking on faith what authorities say is good for them. It's far easier to understand something when you see the point as you're learning it.

I know few mathematicians who read math comfortably, as if reading a novel. We mostly get angry and go think on our own, then return to realize that's just what the article said. Communication conventions in math are horrendous; one writes crappy machine code, then asks the reader to reverse engineer one's thoughts. Anyone in the tech sector knows how easily reading someone else's code can be "just kill me now" territory. As horrendous languages go, mathematical notation is a profound achievement. Nevertheless, beginners feel inadequate when faced with the inadequacies of mathematical notation to convey intuition.

If one wants to learn math, one needs to learn how to play, whatever that means. Find anything whatsoever in the text that leads to an example one can expand beyond the text. Does the text remind you of a pattern you've seen elsewhere? Perhaps the author just isn't saying. Play on your own, trying to decide if the ideas are in fact related.

Would it be possible to teach mathematics by theorem proving ab initio? I guess conventional algebra would be hard to digest for a first grader, but I maybe something like the Peano axioms can be thought of as rules for a game that the students can play, where subsequent arithmetic lessons will be about finding shortcuts to the tedious application of the rules in order to solve problems.

I'll absorb the same amount - during that first hour, but after that nothing further is getting in my head.

Also the computerized thing about a diagram vs words isn't a learning style difference - both of those are visual, just one is more effective (obviously a diagram is better).

His on-the-street test is also bad, someone with a different style will, for example, say the image name out loud, if they need it auditory.