In high school, I held up a pretty-decent-level calculus class because I was confused about something. Specifically that thing where you rotate a curve around some spatial axis (like sculpting pottery) and calculate the volume of the resulting enclosed 3D shape.
I kept being confused, and the teacher (who was super nice and knowledgeable and good-at-teaching [1])... her explanations kept not-getting-through to my brain.
"How do we know y=x^2's 'vase' volume? Wouldn't it be infinite since it's open at the top?" --> [explanation involving rotating around the Z-axis so it's like y=sqrt(x), or something idk] --> "But that doesn't seem very principled! What's the rule law for how to turn the shapes?"
Then some other student in class said like 1-2 sentences, but the only key info I needed was the phrase "domain and range".
Then I was like "oh, I get it completely now, thanks!" And then the class laughed/sighed/was somewhat exasperated.
I developed a maybe-seemingly-trivial hypothesis, that if someone receives explanation E_1 of a concept C, and they're paying attention, and they still don't intuitively grok C, then they need at least one more different explanation E_2.
An idea immediately came to mind: Could you teach someone any advanced math concept, by throwing every explanation at once at them? Could this work on anybody without more-straightforward mental disabilities? [2]
So I've long had a back-of-my-mind idea, which I labeled "Mathopedia". This is not to be confused with any other existing math website that someone would find useful, including MathWorld, Khan Academy, MathOverflow, Wikipedia, Mathematics Stack Exchange, YouTube, Arbital, the OEIS, Metamath, Tricki, ProofWiki, nLab, Hypertextbook, and... uh... at least one literally called Mathopedia. Might need a new name then...
The idea was simple: a math-learning tool that explains advanced uni/graduate/research-level mathematical concepts by gathering a huge number of explanations per concept, and putting them together in an extremely-multimodal (bordering on seizure-inducing) format.
This led me to a few more trains of thought:
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A core "Mathopedia" website, a wiki where each concept gets a page. A page's subsections would go from more-intuitive/motivational/extensional-definition/multimedia/seizure explanations, to the more technical ones, ending with a ton of examples. In my head, this could involve a strong community of contributors.
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A few desktop-software ideas that, if useful, seem (to me) too-powerful to give to non-alignment-researchers. I am probably wildly overestimating the utility of relatively-simple non-ML-based desktop software that hasn't already been invented. Still, being careful.
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[Reading the Arbital Postmortem while shaking my head so
other people know thatI understand what went wrong there and how my "Mathopedia"" would do better.] -
[Reading Paul Lockhart while alternately nodding and shaking my head so I agree with his emphasis on open-ended learning but dislike how mathematics is taught in US K-12 schools (as elaborated in Lockhart's colorful examples/analogies).]
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[Just cross-referencing 3 textbooks, Googling, and asking Discord, like every other mathematician since the days of Pythagoras. If the resources work for everyone else, shouldn't they work for me?]
I'm still not sure of whether a real "Mathopedia" is worth the effort to build, in some kinds of short-AI-timelines. (Here I'm wanting such a website/tool to mainly be of use for technical AI alignment research, though if it worked it would aid many causes). Then again, when some people entering the field still lack linear algebra on arrival, maybe it is worth it.
Despite the clear emotional/self-serving/imposter-syndrome biases at play, I'm still legitimately unsure as to whether "make advanced maths easier to grok" is secretly the same activity as "stop filtering for the intelligence/conscientiousness needed to wade through terse jargon-heavy not-always-standardly-written-or-correct explanations quickly, in a way that would kneecap any sub/field that actually did make it easy to metaphorically inject concepts into one's brain without wading through terse jargon-heavy not-always-standardly-written-or-correct explanations quickly".
How does my original hypothesis look? What, if any, marginal value is there in this sort of project? Does "making math understandable quicker" make things worse? And, of course, can any of this be tested and/or used within a decade or less?
She also encouraged "free play" in maths, which I didn't really grok the importance of until much later. ↩︎
Especially if they don't already share my ADHD, which wasn't diagnosed until college. One person's "flow" is another person's "overstimulation". ↩︎
That sounds about right to me. I'm not an expert on pedagogy, but from my own experience self-learning as an adult (a couple of languages to vary levels of fluency, math concepts that I never learned in school, a moderate amount of various fields) your hypothesis is correct. Being exposed to a concept from multiple angles or in multiple forms reduces the chance that someone won't be able to understand it.
I'd be willing to bet money that people who actually study the psychology of learning have a name for this concept, but I don't know what it is.
Yeah, we'd hope there's a good bit of existing pedagogy that applies to this. Not much stood out to me, but maybe I haven't looked hard enough at the field.
Why just math? This may seem like an obvious question, but your hypothesis applies to any type of learning: how to be a better parent, how to self-regulate emotions, moral philosophy, systems thinking or statistical thinking, physics, and so on.
Maybe! I'm most interested in math because of its utility for AI alignment and because math (especially advanced math) is notoriously considered "hard" or "impenetrable" by many people (even people who otherwise consider themselves smart/competent). Part of that is probably lack of good math-intuitions (grokking-by-playing-with-concept, maths-is-about-abstract-objects, law-thinking, etc.).