Mastodawn

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

Terence Tao Jul 19, 2023

@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.

Patrick Honner Jul 19, 2023

@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!

Terence Tao Jul 19, 2023

@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.

Khanmigo Education AI Guide | Khan Academy

We make education free and accessible for all. By joining Khan Labs, you can help us develop new features that will empower hundreds of millions of learners around the world!

Khan Academy

Talia Ringer Jul 19, 2023

@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.

Patrick Honner Jul 19, 2023

@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.

Michael Pershan Jul 20, 2023

@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

Talia Ringer Jul 20, 2023

@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.

Michael Pershan Jul 20, 2023

@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.

Talia Ringer Jul 20, 2023

@mpershan I am pretty bad at calculation and I strongly enjoyed higher-level math courses and use their content for my research often

Talia Ringer Jul 20, 2023

@mpershan I don't feel like struggling at introductory calculus in any way impacted my ability to excel at, say, abstract algebra

Talia Ringer Jul 20, 2023

@mpershan and notably I found the Real Analysis sequence much easier than the introductory calculus sequence because it focused on concepts and not calculations

Talia Ringer Jul 20, 2023

@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

Michael Pershan Jul 20, 2023

@TaliaRinger I hear you! But in education it's dangerous to make generalizations from success stories -- there are a lot of kids out there.

Michael Pershan Jul 20, 2023

@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.

Dan Drake 🦆Jul 19, 2023

@TaliaRinger @tao @phonner this spring I took a computer science course on compilers, and found ChatGPT super helpful. I could ask it about details about a theorem and its large role in the associated theory and get a sensible, very helpful response.

It was also super helpful with Java. I know mostly C# as my "big, standard OOP" language, and in many cases I knew exactly what I needed to write in my Java code, but just needed to know the name of the API/class or the usual Java idiom.

Talia Ringer Jul 20, 2023

@ddrake @tao @phonner I'm curious how often it was correct, and how easily you were able to tell when it was incorrect, in the context of compilers. Also, what sorts of questions you asked it about compilers, if you have any examples. (I am teaching undergrad PL & compilers in the fall.)

Lîm Tsú-thuàn Jul 23, 2023

@TaliaRinger @ddrake I have same use for writing rust code, lookup rust document is too complicated, the organization of it is not what I familiar with.

Therefore, I use chatgpt and ask small piece of code like how to convert 4bytes to int64, it’s usually correct! Since task are deterministic, if it contains errors, I can fix them easily.

I guess that would be fun to see result of asking harder question like how a graph coloring algorithm good.

Dan Drake 🦆Jul 24, 2023

@dannypsnl @TaliaRinger "usually correct" Yes!

ChatGPT is like asking a colleague who's very knowledgeable, but doesn't have their head exactly in the context you're working. So you ask them for advice, trying to describe the problem, and even if they answer confidently, you know that because they haven't been working with *this* exact bug report or section of code, you need to take their answer and do a little thinking to be sure it'll work.

People are telling all these stories about copy and pasting code from ChatGPT, but the truth is, we do that from our colleagues all the time. "Hey Mike, when using the Whatever API, how do you do X?" Mike says "you just call whatever.do_the_thing with the first argument set to blah blah blah." You do that, because Mike is super smart, but then later learn that for your precise snippet of code, you need to do something slightly different.

You need to think about Mike's response; you need to think about ChatGPT's response.

Lîm Tsú-thuàn Jul 24, 2023

@ddrake @TaliaRinger Your colleague explain is great!

I was feeling it's also like Yellow Duck, since I have to stop and describe the question in natural language, that process helps me frame the problem, even solve the problem!

Talia Ringer Jul 24, 2023

@ddrake @dannypsnl Forgive if typos---eyes dilated and cannot see. I think the "we copy and paste code from people/StackOverflow too" thing is a bit of a worrying comparison, because these chatbots tend to mess up in ways that are quite different from how people mess up, and so require different training to properly evaluate

Dan Drake 🦆Jul 24, 2023

@TaliaRinger @tao @phonner this is my favorite interaction, on Chomsky normal form and why it only has two productions:

https://chat.openai.com/share/6ed8e1e7-ef7c-4311-b726-23d8ee8ac8fb

I would also often use ChatGPT to explain something I already thought I knew; I just wanted to get another explanation. Say the lecturer explained topic A. I think "okay, yes, I completely understand that, but I suspect there may be something idiosyncratic or unique about this explanation." So I would ask and see if, say, everyone else thinks about topic A in a slightly different way. (This is especially helpful when the teacher says "in this class..." or "for this assignment, you just need to know...")

ChatGPT

A conversational AI system that listens, learns, and challenges

Dan Drake 🦆Jul 24, 2023

@TaliaRinger @tao @phonner here's another conversation on Julia and dataframes where it got things wrong: https://chat.openai.com/share/f0f4b11a-878f-4030-b6e0-5e5d19fe1b74

I'm not seeing any of the conversations where it just got things wrong, and I said so, and then apologized and gave it another go. Weird...

ChatGPT

A conversational AI system that listens, learns, and challenges

Patrick Honner Jul 19, 2023

@tao I am always excited about the opportunities new technologies create for teaching and learning! And I am always skeptical of those who sell us on revolution. (Khan Academy has been promising revolution in education for 20 years. Has it delivered?)

That being said, I think AI tools will have a big impact on teaching and learning, and I believe that impact can be positive. I have found your public musings on AI (how it can effectively be integrated into your workflow; how it is more like a probability kernel than a function) very helpful in seeing ahead. One of my goals for next year is to expose my (high school) students to more at the intersection of math, programming, and AI.

For me, the fundamental units of teaching exist within relationships, and so technology's impact will always be limited in scope. But as with calculators, and Desmos, and Scratch, and Geogebra, and Python, the impact can still be substantial!

Matthew Steffen Jul 19, 2023

@tao @TaliaRinger My favorite example of this phenomenon is one I learned several years ago. Consider two questions:

Matthew Steffen Jul 19, 2023

1. Four “Bicycle”-brand playing cards are laid out on a table. Some have red backs, some have blue backs; some are face-up, some are face-down. You would like to check the proposition “if the number on the front of the card is even, then the back of the card is red.” Which of the following cards do you need to flip over to determine if this proposition is true?

A. Face up, showing a five
B. Face down, showing a blue back
C. Face up, showing six
D. Face down, showing a red back

Matthew Steffen Jul 19, 2023

2. Four housemates share a car between them. The house rule is that anybody can use the car, but you have to fill up the tank with gas before you bring it back. If you know the following facts about the housemates, who would you need to talk to to ensure the rule is being followed?

Alice did not take out the car
Bill did not fill the tank
Claire took out the car
Dale filled the tank

Many people, even with some math background, find the second easier, even though it’s logically equivalent

robryk Jul 19, 2023

@tao I found it somewhat interesting that there's a very natural way to crotchet a Mobius strip (not by sewing it together from a rectangle), where if you keep crotcheting you will keep adding more width by going around its only edge. I wonder how many people have their first encounter with the concept by failing to correctly crotchet a tube (i.e. side surface of a cylinder).

Barry Goldman Jul 19, 2023

@tao @TaliaRinger 'suitible cognitive framework': over the decades, i've learned to do many things i never thot possible in my youth, such as dance, music, dating... this experience infuses my teaching.

yeah i've seen so many people shied off of math and science cuz really BAD teachers. a pity.

maybe in grade school kids should have MULTIPLE exposures to math and science even when they get off that track.

they get english every frikin year...

blake shaw 🇵🇸Jul 19, 2023

@tao @TaliaRinger i hated mathematics in K-12, but wound up discovering a love for it during an undergrad introduction to set theory, where I was introduced to mathematics as a practice of reasoning rather than application of rules introduced via rote memorization.

I was also diagnosed with ADHD as a kid, which I think is more of a disfunction with following instructions rather than an incapacity to maintain attention, because when left to my own devices to study a subject I can make great progress, but when asked to do something according to some prescription I often have trouble, and many friends I know who were ADHD as kids and then did great in university expressed similar experiences.

The problem is that most people's ADHD prevents them from going to a decent university because their grades are so badly effected. I was very lucky that my school district started a high school for self-learners that they sent to for high school, and graduated early with near perfect GPA

Talia Ringer Jul 19, 2023

@rml I also have ADHD! At some point I learned to embrace the way my brain works rather than fight it, and it made a huge difference for me. The flipside to impulsiveness is fearlessness; the flipside to distraction is exploration. And I don't know if you experience hyperfocus, but while it can be harmful if it's on a video game or something, it can be great when it's on research.

I really strongly believe that ADHD brains are beautiful. I still struggle a lot with paperwork, chores, and so on, but that's OK, my neurotypical partner helps out. I struggle with details in research, and my collaborators help out, or I outsource those details to trustworthy computation. 😄

Talia Ringer Jul 19, 2023

@rml With ADHD I've also found the impact of interest on how well I do on something to be strongly amplified. My ADHD Coach called it an "interest-based nervous system" and I really like that. For me this often meant that I could not motivate myself enough to care about introductory courses and would find them very hard, but I'd get obsessed with later more "advanced" topics and I'd excel at them.

My intro CS lecturer in college told everyone that if you get a B in intro CS, you'll likely get a C in the second class and then fail the third, so you should just consider dropping the major. I had a B in intro CS and when I heard that, it almost stopped me from pursuing it further. But I excelled so much once I started to have freedom in what particular parts of CS I could explore. Like cryptology, programming languages, and complexity theory. It just kept getting more fun. CS research feels like the epitome of that. I can work on the things that captivate my interest at any given point, and somehow that's an actual job I can get paid for. Cool!!

blake shaw 🇵🇸Jul 19, 2023

@TaliaRinger exactly! I think ADHD is much more about hyperfocus than it is about being unable to focus; its simply diagnosed as a deficit because it manifests as a behavioural disorder in the class room.

everyone I've known who was diagnosed with ADHD as a kid and has wound up doing well shares a propensity towards becoming obsessively involved in their work, focusing on it all hours of the working day, and those I know from growing up who went to community college etc (because society failed them) have wound up with drug & alcohol problems. if you give people with ADHD something we can sink our teeth into, we honestly don't need adderall. and I think that goes to show that diagnosing it as an "attention deficit" is part of the problem.

Andrew Perfors Jul 19, 2023

@rml @TaliaRinger As a fellow ADHDer, I completely agree with this entire conversation :)

Paolo G. Giarrusso Jul 21, 2023

@rml @tao @TaliaRinger Modern science agrees, ADHD just hinders *controlling* attention, and doctors *should* know — the experts do.

Talia Ringer Jul 21, 2023

@Blaisorblade @rml @tao ADHD brains are beautiful 🦋

𝙺𝚊𝚛𝚕 DEFEND DEMOCRACY 𝙱𝚞𝚗𝚍𝚊𝚢 🗽Jul 19, 2023

@tao @TaliaRinger Professor Tao, as a longtime admirer of your writings and as a mathematics teacher to young pupils who seek advanced mathematics lessons outside school hours, I strongly agree with what you write here. Many American pupils have received such bad mathematics instruction in elementary school that they end up with correct understanding of mathematics being replaced with nonsense they hear in class, so that the first task for many outside teachers is to correct misunderstandings. That's regrettable.

I suppose you are familiar with the work of Liping Ma on how primary mathematics education is done in United States schools and how it could be better done, but I'll link to the latest edition of her book Knowing and Teaching Elementary Mathematics here for interested onlookers. Her book was life-changing for me and for many other readers.

https://www.taylorfrancis.com/books/mono/10.4324/9781003009443/knowing-teaching-elementary-mathematics-liping-ma

Knowing and Teaching Elementary Mathematics | Teachers' Understanding

The 20th anniversary edition of this groundbreaking and bestselling volume offers powerful examples of the mathematics that can develop the thinking of

Taylor & Francis

Boarders Jul 19, 2023

@tao @TaliaRinger I think of the preface to Penrose’s Road to Reality where he tries to explain to a mathematics-phobe what a fraction is and in doing so comes to realize there really is some essential subtlety to the notion of equivalence classes and perhaps it is he who hadn’t appreciated the concept in its fullness.

One thing that leads us astray (as Stephen Jay Gould repeatedly said in his writings against IQ determinism) is the thought that there is some unitary notion of intelligence both inside and outside mathematics. Imagine one is confused about: what is allowed in Euclidean geometry proofs, what is an inertial frame, what it means to assign probability to an event that happens once, why can we use Fourier series to find the solution to an equation etc. One can easily imagine someone prescribing respectively: a course in formal logic, a course in the philosophy of physics/probability, a course in rigorous real analysis etc.

Such a prescription would not necessarily allow everyone to get past their confusions, but at present we seem to assume that such confusions, up to (and sometimes including) undergraduate level education, point to a fundamental flaw in the individual and a sign that a subject as vast and multifaceted as mathematics is “not for them”.

It is a great tragedy that the free and full development of our capacities and personalities is left to the mercy of such pernicious notions as the inherent hierarchy of people.

∯ Suzza Jul 19, 2023

@tao @TaliaRinger I posted a similar comment to Talia when I saw this thread on Twitter. In my own experience, I found that I was able to learn math by treating it more like a language which is one of my personal strengths.

I hit a wall in fourth grade mathematics unable to understand multiplication and had to repeat. When I treated multiplication more like a sentence, I could see the grammar. Recognizing what multiply meant as an operation.

Since then I've gone up as high as differential equations, It seemed unlikely as a kid that I could handle more advanced mathematics. I think I have managed to do well for someone who struggled for a bit.

Righteous Hazard Jul 19, 2023

@tao @TaliaRinger In addition to class and racial structural barriers, math education commits murder on wonder and curiosity in students.

A suitable cognitive framework is helpful (I was fortunate in that I encountered a math instructor early on who showed me a way think mathematically), but it is not enough.

Perhaps a story from athletics would help. I spent time coaching 8-13 year olds in cross country running, and was fortunate to attend a coaching clinic taught by a successful high school coach, who told me something important:

“All coaches want to push their athletes to maximize their performance, but the truth is that people don’t reach their peak as runners until their late 20s or early thirties. Yes, I could push my young athletes harder and get a few more meet victories, but if I damage their love of running before they reach their peak, I have failed as a coach.”

Math is a subject filled with enough wonder and astonishment to last a lifetime. Mathematics exists only in minds, it needs curious minds in order to survive. If we damage young people’s ability to apprehend that wonder and feel that astonishment, we fail mathematics itself.

Andrew Perfors Jul 19, 2023

@tao @TaliaRinger I completely agree with this! I teach introductory statistics to a huge cohort of psychology undergrads (over 1000 this year) and they consistently surprise everyone, especially themselves, with how much they end up understanding. One of my key tools is that I embed the entire thing in a long narrative. And I explain everything three ways: (a) verbally, with analogies; (b) visually, with pictures; and (c) mathematically, with notation.

My 8yo and 10yo kids can understand huge amounts through these techniques as well. I'm convinced that a lot of people think math is hard just because it's hard to TEACH.

Jason Murphy Jul 20, 2023

@tao @TaliaRinger
Culture tells women they can't do maths. We all find maths hard at some point! But at that struggle point society gladly lets women out of the field while telling men to put their heads down and push.
Take my beloved as an example: dropped maths in high school. But she loves doing the sudoku. I asked her about that one day, and she said, and I quote: "its not maths; it's just logic with numbers." !!
I'm pretty sure she would have dominated in a quantitative field if she'd pursued it.

Talia Ringer Jul 20, 2023

@tao Also, this has strong "Emily Riehl explains infinity" vibes to me. I loved these videos, they gave me warm and fuzzy feelings: https://youtu.be/Vp570S6Plt8

Mathematician Explains Infinity in 5 Levels of Difficulty | WIRED

YouTube

Esme Ciredutemps Jul 20, 2023

@tao are you advocating for massive investement in state éducation ?
Because i don't see how what you would like could happen without it 🤔

Sharon Machlis Jul 20, 2023

@tao @TaliaRinger @seanfobbe And other times what's missing is encouraging female students, students of color, and those from lower income households that they, too, can excel in math. Subtle and not-so-subtle social signals that “People like you don't do math well” are very real and have effects.

Talia Ringer Jul 20, 2023

@smach @tao @seanfobbe Very real. When my mom was in high school, her teacher told her, "you are really good at math! If you were a man, you could be an engineer." She liked math a lot, but majored in psychology in college, and never took math beyond that. I was really grateful nobody ever told me that, though I did have a college lecturer who tried to talk me out of majoring in computer science.

Sharon Machlis Jul 20, 2023

@TaliaRinger @tao @seanfobbe Terrible story. ☹️

Gopiandcode Jul 20, 2023

Proper representation in media and pop culture can also have a big impact as well! Looking back in retrospect, a major thing that got me interested in science and mathematics as a child was watching "Short Circuit 2" with the Indian scientist main character Ben, which was the first time I saw someone who looked like me being represented as cool and something to look up to (though in hindsight, maybe not the best example of representation, seeing as it was actually just a white guy in brown face :( ...). My sister has a lot of self doubt about her skills and capabilities, and I can't help but wonder if this might also have been a factor... :(

Colin the Mathmo Jul 24, 2023

Calling @Chartodon LR ...

CC: @tao @TaliaRinger

𝑃𝒓𝒂𝒋𝒂𝒚 Jul 25, 2023

@tao @TaliaRinger When God created the world it was dark and night then God said ,let Newton be it was light.