A small anecdote in relation to a recent coffee conversation I had with @TaliaRinger (which they relate over at https://twitter.com/TaliaRinger/status/1681410191278080000 ): Yesterday I spoke with a children's book author who was interviewing me as part of a series she was writing on contemporary scientists. She freely admitted that she did not have great experiences with her math education at an under-resourced school and chose very early on to focus on writing instead. Nevertheless we had an excellent conversation about many mathematical topics that she was not previously familiar with, such as proof by contradiction, Cartesian coordinates, Mobius strips, or compressed sensing, all of which she found fascinating (and said she would read up on more of these topics herself after our interview). I posed to her the isoperimetric problem (using the classic story of Queen Dido from the Aeneid as the intro) and she correctly guessed the correct shape to maximize area enclosed by a loop (a circle), and instantly grasped the analogy between this problem and the familiar fact that inflated balloons are roughly spherical in shape. I am certain that had her path turned out differently, she could have attained far greater levels of mathematical education than she ended up receiving.

This is not to say that all humans have an identical capability for understanding mathematics, but I do strongly believe that that capability is often far higher than is actually manifested through one's education and development. Sometimes the key thing that is missing is a suitable cognitive framework that a given person needs to align mathematical concepts to their own particular mental strengths.

Talia Ringer on Twitter

“Terry Tao and I spoke over coffee for like two hours yesterday, in part about diversity in how people think about math. We both agreed that people who hit these walls early mostly don't learn the way of thinking about math that works for them. It's an educational failure”

Twitter
@TaliaRinger In this specific case, I guessed (correctly, as it turned out), that framing mathematical concepts and problems as narratives (ideally involving children) would be particularly effective in communicating mathematics to a writer of children's books. For instance, in addition to the Dido story, I could explain proof by contradiction through the story of young children challenging each other at recess to name the largest number, until one realized that because they could always add one to the number the other child proposed, that there was in fact no largest number. Compressed sensing I could explain due to the need to have a child sit still for several minutes in an MRI scan before the modern CS algorithms became implemented in the machines. Mobius strips I could explain via a proposed children's activity of cutting such strips to encourage mathematical exploration. These were handpicked examples, but in general I think a lot can be done with creatively reframing the way we present a given mathematical topic.
@tao @TaliaRinger Imagine trying to find a "suitable cognitive framework" for each of 175 individual students every day for every topic and you will begin to understand the challenge of being a K-12 classroom teacher!
@phonner @TaliaRinger I have hope that AI tools could partially alleviate this problem in the future. Already there are promising experiments such as Khan Academy's "Khanmigo" https://www.khanacademy.org/khan-labs or Upward Mobility Foundation's "uME" https://www.theumf.org/ (full disclosure: I am an advisor for the latter project). Of course, the majority of students will still benefit the most from personalized attention from expert human teachers, but as you say this is a limited resource.
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@tao @phonner My Uber driver back from the beach yesterday mentioned that he is preparing to go back for his masters, and is taking linear algebra online right now. He has had trouble understanding the way concepts are presented in class, but has been using ChatGPT to ask about concepts in a way that makes sense for him, as a supplement to the class. He said the professor is too busy to give him this kind of personal attention for the amount of time he wants it (about an hour weekly).

Like everything with AI, this both worries and excites me. We have to be careful that it doesn't become a replacement for more personalized attention when that could be available, and we have to be careful to build tools that are either correct or that help users calibrate to their untrustworthiness and think critically about automatically generated responses. If we can do both of those things, I think it can be great. I still worry there are not strong economic incentives to do those two things; often replacement with inferior automation is unfortunately profitable.

@TaliaRinger @tao I share your concerns and, to an extent, your optimism. (Though, maybe my optimism is just the pragmatism of a classroom teacher who knows that these tools are already being used and are here to stay!)

Having watched this space for many years, a separate concern I have is that the evidence offered as proof-of-concept of tools like these are often atypical. The Uber driver motivated enough to go back to school to study advanced math. The HS student driven enough to search out supplementary help videos online. The professional willing to commit to taking an online course. Sure, AI and Khan Academy and MOOCs can add value for these self-starters, but maybe being motivated is the real key to success. And of course not everyone is so motivated!

And as you point out, these things do tend to gravitate more toward providing profit than providing equity.

@phonner @TaliaRinger @tao While AI is certainly a distinct and promising technology, the exact same things have been said (less plausibly maybe) about every new media technology going back to radios.

I think everyone with an interest in this should read Tyack and Cuban's "Tinkering Toward Utopia."

If people want to read something shorter, they might read an essay I co-wrote for the blog Slate Star Codex a few years ago, particularly the last section. https://slatestarcodex.com/2018/09/04/acc-entry-does-the-education-system-adequately-serve-advanced-students/

[ACC Entry] Does The Education System Adequately Serve Advanced Students?

[This is an entry to the Adversarial Collaboration Contest by TracingWoodgrains and Michael Pershan (a k-12 math teacher), on advanced students in the education system] “What do America&#8217…

Slate Star Codex

@mpershan @phonner @tao Hmm, but technology really has opened up a lot to students who learn differently. My handwriting was so bad in elementary school that I needed occupational therapy for it, but I learned how to type extremely efficiently and that has been wildly amazing for me. And as someone who struggles with details, but does very well at high-level intuition, both calculators and computer-aided proof have done wonders for me.

Neural tools and especially language models gain in flexibility what they lose in guarantees. That combination is both more exciting and more worrying. A calculator program is very simple to implement and to fully understand; a formal proof assistant has a deliberately small logical proof-checking kernel meant to be understood by people. A language model is much less limited in what it can take as input, but correspondingly its outputs come with pretty much no guarantees by default, and the functionality of any given language model is obfuscated (I think that part is surmountable though).

Economically, I think there are few incentives to build correct software, because regulation of the tech industry is so lax. As software complexity grows, this becomes more and more of a problem. And software complexity is growing a lot, so this is a symptom of that.

@TaliaRinger @phonner @tao My sense as an algebra teacher is that scientific calculators have not radically opened up higher level mathematics for kids who struggled with calculation, because the higher-level skills often depend on lower-level fluency.

I'm not 100% sure that this is a useful analogy for LLMs or future AI, but I think it probably is. These tools are great -- and will probably be useful as teaching and learning tools -- but I don't expect this latest technology to radically open up education in big new ways.

@mpershan I am pretty bad at calculation and I strongly enjoyed higher-level math courses and use their content for my research often
@mpershan I don't feel like struggling at introductory calculus in any way impacted my ability to excel at, say, abstract algebra
@mpershan and notably I found the Real Analysis sequence much easier than the introductory calculus sequence because it focused on concepts and not calculations
@mpershan so insofar as higher-level math courses are closed off to folks who struggle with calculation by hand, I really think this is a massive failure of the educational system, and is not inherent to mathematics, especially tool-aided mathematics
@TaliaRinger I hear you! But in education it's dangerous to make generalizations from success stories -- there are a lot of kids out there.
@TaliaRinger And the point isn't that hand-calculation is a big obstacle itself, it's that to understand (say) compactness you typically need a pretty robust understanding of (say) fractions which itself requires fluency with (for example) multiplication. These skills really are in general truly foundational for students in a way that AI can't help with.