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 )... 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? 
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:
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.
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.
[Reading the Arbital Postmortem while shaking my head so other people know that I 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).]
[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". ↩︎
These days, when I try to learn something, I often download multiple textbooks on the same topic, or watch YouTube videos from multiple authors. Because sometimes different sources better explain different parts of the puzzle.
I wonder how this applies to schools? The "half-empty" perspective is that having one teacher per subject sucks; the "half-full" perspective is that at least we have two sources: the textbook and the teacher. (Or the teacher and the recommended literature, at the university level.)
The cost of reading or watching multiple explanations of the same thing is that it takes more time. Perhaps a specialized website could enforce time limits (like 20 minutes for a TED talk). For example, you choose a topic, and then you can watch three 10-minute videos from three different authors that try to provide an introduction to the topic. If the topic is too complicated to be explained in 10 minutes, you only get the most general explanation, and all complicated parts become separate topics. (But would that require to make a "tech tree" of the topics? And is that a good or bad idea?) -- This is different from what you propose, and possibly has a greater barrier to contribute.
I think a good start would be to choose a topic and make an example page. Examples sometimes communicate the idea better than descriptions. (Skip the overhead of making a wiki, and just post the example page as a LW article; with a short explanation on the top.)
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.
There are teaching styles that explicitly encourage discussion among students, precisely because different students express things differently, so there is more opportunity to "click" for the students who didn't get it.
First students get a problem that is only slightly different from what they already learned, then they are given some time to figure it out individually, and then the ones who got it explain their solutions to their classmates, and other students ask them questions.
Very good points, yeah!
I actually attempted making an example-page in a Wikipedia sandbox, but did not have the energy/deeper-requisite-knowledge for the topic I chose (Godels Incompleteness Theorems ;-;), so I didn't finish it. But I do agree that, if I launched this, I'd need at least one good example-page.
Another part of the problem, which Arbital especially failed at, was getting others to contribute. Reddit and StackOverflow solve this by basically giving people literal "status points" for writing helpful effort-signaling posts. So I'd want some kind of MediaWiki/other plugin that says "Hey, new contributor! Here's a list of subsections of different articles, where we want X type of examples for concept Y. Mind adding more stuff there?" I even had a harebrained idea for a multi-authorship-explicit-credit plugin of some sort.
You chose one of the most difficult examples. It would probably be better to choose something that is difficult for others but simple for you.
Another part of the problem, which Arbital especially failed at, was getting others to contribute.
Too late to cry over spilled milk, but one of the reasons I didn't contribute was that Arbital seemed too serious; I wasn't sure whether less serious topics were even allowed. Which might also become a problem of the website where Goedel's Theorems are the canonical example. Probably it would be better to use something simpler, for example prime numbers, or quadratic equations, just to encourage more people to contribute.
Hey, new contributor! Here's a list of subsections of different articles, where we want X type of examples for concept Y.
Yes, this would definitely work for me.
One problem with "using a simpler example", is that there's a lower bound. Prime numbers are not-too-hard to explain, at some levels of thoroughness.
Like, some part of my subconscious basically thinks (despite evidence to the contrary): "There is Easy Math and Hard Math. All intuitive explanations have been done only about Easy Math. Hard Math is literally impossible to explain if you don't already understand it."
Part of the point of Mathopedia, is to explicitly go after hard, advanced, graduate-level and research-level mathematics. To make them intelligible enough that someone can learn them just from browsing the site and maybe doing a few exercises.
Even if they need to go down a TVTropes-style rabbit-hole (still within the site) to find all the background knowledge they're missing.
Even if we add increasingly-unrealistic constraints like "any non-mentally-disabled teen should be able to do this".
Even if it requires laborious features like "there should be a toggle switch / separate page-subsection that replaces all the jargon in a page with [parentheses of (increasingly recursive (definitions)]], so the whole page is full of run-on sentences while also in-principle being explainable to an elementary schooler".
Even if we have to use some incredibly hokey diagrams.
I agree that too easy example does not make a good demo for "how to explain difficult things".
Maybe Complex Numbers would be a better topic, because you can start from really simple (an 8 years old kid should be able to understand C as a weird way of writing 2D coordinates) and progress towards complicated (exponentiation). Plus there is a great opportunity to use colors for C-to-C functions.
That said, "easy" is relative to the audience. As a challenge, you could take a smart 8 years old kid and try explaining as much about prime numbers as you can, in the time limit of 10 minutes. Do the same for a 10 years old, etc. (This is my pet peeve: There are many simple explanations of simple things which could be further simplified, but no one bothers to do that, because from the perspective of an adult, they seem already easy enough. Or because at some moment, too simple explanations just feel low-status. We need more and better distillation of all human knowledge. People say "you can't be a polymath anymore, because we already know too much". Yeah, but an average person could probably know 10x more than they do now, if our educational methods didn't suck, because we stop at "good enough".)
choose something that is difficult for others but simple for you.
choose something that is difficult for others but simple for you.
Yep, a broader life lesson I'm still learning haha.
IIRC Paul Graham recommended such a tactic, framing it as "easier gains from moving around in problem-space than solution-space".
And your other recommendations definitely make sense here. In my giant bookmarks folder about the "mathopedia" idea, this post and the comments are bookmarked.
Before building a whole website, just try this technique on some students. Whether with just paper or a quickly built web page for a few specific concepts.
Reminds me of the general teaching method of trying many different approaches of explaining something until one of them resonates. Mostly illustrated by different modalities or media, such as visual, textual, or emotional, physical. It is just more common in math where you need something that let's you build the abstractions on top of.
It seems like ChatGPT should do an excellent job of simply giving you a bunch of explanations for a mathematical concept and also responding to questions if something is unclear for you.
Forgot to mention this in the post proper, but: Pages would be organized in a multi-examples-per-subsection way, where each subsection corresponds to something like a part of an extended "ADEPT Method".
It may be worth doing, but I expect this dynamic to apply.
Yep! My main hope is that it works in a niche of people who needed specifically-it (or who find it more "intrinsically fun" to read and/or contribute to than the other options).
I don't think "throw every explanation possible" is the right takeaway from your experience. To me, it seems like the teacher was failing to model what you were getting stuck on, and so the takeaway would be something more like "try to model the learner better, so as to produce better (not more!) explanations".
"Throw every explanation possible" might still be learning-complete in some sense, so might be worth exploring.
Forgot to mention in-post, but the site BetterExplained is also quite good.
It sounds like an excellent foundation.
Ideas for improvement:
These ideas seem promising!
How do you distinguish feeling of epiphany and grokking?
How do you distinguish feeling of epiphany and grokking?
Good point, I haven't really done that here. We could differentiate by e.g. having practice-problems, and people can login to track their progress. Similar to the multi-explanations/teaching-methods setup, there could be a broad variety of example problems --> less likely someone gets lots of them right without actually understanding the concept.