In this episode, Grant Sanderson (GS) interviews Steven Strogatz (SS) about research, pedagogy, and exposition, and what follows is some commentary by John Jasper (JJ), Emily King (EK), Clayton Shonkwiler (CS), and me (DM). Feel free to discuss further in the comments.
2:40 – DM: Seems like a fun problem. How hard could it be?
7:21 – CS: The “Steven has real talent” note is a nice story, but it also strikes me as extremely weird. Like, why not tell SS directly that he has real talent, rather than writing it in a note to the headmaster that he eventually gets a carbon copy of?
8:00 – EK: My intro discrete math prof gave us the Twin Prime Conjecture as a “bonus problem” but didn’t tell us that it was a famous open problem.
8:35 – CS: I wish I was better at generating problems that are extremely challenging but solvable for students at various levels.
8:35 – DM: We should definitely pose appropriate math puzzles to interested young folks. There’s great source material from Martin Gardner, Project Euler, and the IMO, for example. For younger kids, this might be a useful resource.
9:18 – EK: That is a fun problem. I’m going to be distracted thinking about it rather than doing the work I need to do.
10:37 – EK: THIS! I think far too many students these days look up proofs online without struggling sufficiently.
10:37 – JJ: Is it possible that the benefit of being able to readily look things up will outweigh the benefits earned by the struggle? I don’t think so, but I want to consider the possibility.
11:35 – DM: Even today, it seems that my best math ideas come when I’m procrastinating on an administrative task.
14:32 – CS: I can identify with this!
16:50 – JJ: I like the idea that the first part of any exposition (lecturing or writing) should be to get the students interested in the question. This is a nice way to structure the beginning of a lecture. Even more, I like the idea that motivation can simply come from asking an open-ended question like “what’s the distance between the sine and cosine functions?” In particular, the motivation need not come from a physics application or even an application to another area of math.
17:03 – CS: I love this formulation.
19:40 – EK: I enjoy questions where the first step is discussing what the actual meaning of the question is.
21:40 – DM: SS makes a great point here. I hate sitting through a talk in which I don’t care about the problem but I’m being dragged through the solution.
21:40 – CS: This hits a little close to home, in that a collaborator and I just finished a paper in which we answered a question that nobody had asked. But it also makes me reflect that my paper-reading is unbalanced: if I’m reading a paper, it’s almost always because it contains a result I want to use, and almost never just because I want to learn to love a new question.
25:54 – DM: His description of this proof gives me goosebumps. Pretty inspiring.
27:50 – JJ: This is a great take on a super important question: what motivates students? SS seems to agree with my take (from podcast #1 at 15:36) that approval from experts and competition are powerful sources of validation and motivation. SS adds that history, applications, and beauty are also sources of motivation for some students.
28:25 – JJ: The distinction between “beautiful” and “satisfying” strikes me as really useful language. I think I have used synonyms for beautiful when it would be more relatable (less exclusionary) to say satisfying. Indeed, some proofs are very satisfying in they way they come together, perhaps we can start a subreddit à la r/oddlysatisfying.
28:25 – CS: I like this as an alternative to the “math is beautiful” trope. In school I didn’t really love math, but I did like how it was a realm in which things usually fit together in a satisfying way. And now I’m not really coming up with beautiful theorems or even beautiful proofs, but the realization that this tool from over here is perfectly suited to solving that problem over there is really delightful.
28:55 – EK: “Beauty is exclusionary” is an interesting take.
30:01 – DM: It’s a good thought to be mindful of students who are motivated to do math as a way to change their socioeconomic status. Reminds me of how I would memorize vocabulary words to study for the SAT, since I was singularly motivated by being accepted into college. It also highlights the privilege wrapped up with insisting on being inspired in order to learn math.
32:47 – CS: What is the appropriate place for rigor in math? I mean, it seems clear that research papers should be pretty rigorous (which doesn’t mean they shouldn’t also communicate intuition), but probably most math talks, even seminar talks, would benefit from being less rigorous and more intuitive. And if most papers need to be fairly rigorous, that presumably means we need to train graduate students in being rigorous somewhere along the way. But what’s the appropriate venue for that? And how far down the scale does this really apply? And to what extent does mathematical rigor in a form different than you would find in a research paper come into this (I guess I have in mind engineering students, for whom rigor might mean something slightly different while still being important)?
33:58 – DM: Another way to motivate: You must complete this task in a short amount of time, so you need to learn and memorize enough to succeed at that task. Drilling has its benefits.
35:25 – EK: I recently saw a Tweet revealing a “life hack” that you could “flip” percentages, e.g., 4% of 75 is 75% of 4, with the latter being the easier calculation. A lot of people were shocked at the outcome, but maybe that’s because the “life hack” nature of commutativity (e.g., easier to add two 9’s than nine 2’s) was never explained to them.
35:35 – JJ: I really like teaching intuition before formalism. Two technical problems with this that I’m not 100% sure how to overcome. First, students will have trouble distinguishing between formal arguments and intuitive arguments. Calculus students are ONLY taught intuition, but then they get to real analysis we ask them to do formal proofs on the same material. Second, it’s difficult, but not impossible, to test their intuition.
36:06 – DM: The main reason I don’t post lectures on YouTube is that I fear exhibiting a level of perfectionism by blowing too much time on video editing. On the other hand, we consistently post talks from the CodEx seminar series on YouTube with very little editing. This is a benefit of separating the content creator from the person who posts on YouTube.
39:20 – CS: I agree that it in principle it would be cool to connect, say, animators and mathematicians, but it’s a major logistical challenge which no institution is really incentivized to solve. But if you’re a mathematician, you don’t actually need an animator (or designer, or whatever): you can make your own animations!
40:40 – DM: This reminds me of the teacher-student collaboration model that GS proposed when launching the SoME1 competition. A similar collaboration was already successful in this video. It’s an interesting thought that this sort of interaction might scale.
43:12 – DM: It’s very believable to me that Elias Stein’s Complex Analysis course played a pivotal role in SS’s education. I had the pleasure of sitting in on this class as a grad student, and I was floored by Stein’s teaching style. The experience had such an influence on me that it made an appearance in my teaching statement when I was on the job market.
44:20 – DM: Reminds me of my own experience in undergrad. I was naturally drawn to math, but my family’s experience in the Air Force biased my career trajectory. I ended up doing math as an officer in the Air Force for 13 years before joining the faculty at Ohio State.
47:05 – CS: This whole chapter is a really good advertisement for one of the major potential benefits of being an undergraduate at an R1 university. I went to a very small college where the faculty (the math faculty, at least) was just not very engaged in research, so working on major open problems with world experts in your senior thesis was just not a thing that was going to happen. (For drawbacks in going to an R1 as an undergrad as opposed to a place which values teaching more highly, see earlier in the video.)
55:05 – DM: This ribbon story is great! I love it when a physical model helps to clarify thinking on a math problem. Another noteworthy example is the space-time globe for understanding special relativity.
55:58 – EK: That’s a cool application and test of topological invariants.
58:54 – DM: Learning while teaching is a great (but sometimes stressful!) way to learn. When developing my topics course on data science, I selected various results from papers that I already had a working knowledge of, but I needed a much deeper understanding before I’d feel comfortable enough to teach those things. It ended up being a lot of work, but definitely worth it.
59:10 – CS: I’m going to have to steal this line.
59:21 – JJ: There’s some great stuff about learning here. SS talks about how he could learn the subject (group theory) now, but he’d have to teach it. This is in contrast to his experience in school where he failed to learn it. He points out two differences: There’d be a “gun to his head,” and he could do what he couldn’t while in school, which is look stuff up. What seem obvious here is that we should be listening to the experts on what is the best way to learn math. Student presentations can mimic this for my students, but I don’t know how we model the “gun to the head” part, which I think is also super important. The pressure of presenting is good, but many student presentations take the form of reading a prepared speech full of statements that they don’t fully understand. Should they also be required to field questions coherently?
1:01:06 – DM: I had no idea about the Latin derivative of “radical.” That’s a nice connection.
1:01:51 – EK: Huh. I didn’t realize that convergence issues of Fourier series and not calculus motivated real analysis.
1:10:00 – CS: My sense is that most mathematicians think they know more about the history of math than they actually do. History is a difficult and complicated undertaking which is a long way from vaguely remembering a couple of paragraphs (which were probably apocryphal to begin with) about Hamilton inventing the quaternions or whatever. (Okay, GS and SS basically get into this a couple of minutes later.)
1:12:14 – CS: Somewhat distinct from history of mathematics, do people study mathematical folklore in the sense being discussed here? I guess Alberto Martínez’s book addresses some of these, but (not having read it), the book description seems to pitch it more as mythbusting than as studying why certain stories become popular and what purpose those stories might serve even if they’re not factually accurate.
1:15:59 – DM: Never let the truth get in the way of a good story! We embellish the truth even when describing events from the previous night. Stories are fun. In the context of math, stories get us to care about the material. Does it really matter whether a soldier actually disturbed Aristotle’s circles?
1:24:17 – DM: It seems that GS agrees with me.
1:25:00 – CS: This somehow reminds me of how my advisor Herman Gluck always described visualizing higher dimensions: “Visualizing higher dimensions is an exercise in creative lying”. Of course, the trick to it is knowing what it’s okay to lie about.
1:27:59 – DM: I really like this joke idea that Fourier wanted to solve the heat equation so that he could figure out why he needs to wear a heavy coat all the time. And I’m not concerned that this is historically inaccurate (in all likelihood).
1:29:09 – DM: For some reason, I have to fight myself to provide examples before discussing the more general underlying theory. I know it’s the right way to explain a thing, but for some reason, I want to hurry up and share the cool thing that’s calling the shots.
1:32:30 – CS: This is something that so many mathematicians say and think, and I find myself agreeing with it as well, which makes me almost want to push back on it. Like, if everybody thinks this is bad, but it doesn’t change, why is that? Is it actually good? Or is it just that everybody says they want a clear narrative and lots of motivation and examples, but then when they have to referee a 100 page paper they say “There’s only 20 pages of actual math in here, cut it way back!”?
1:33:00 – DM: I first learned of this one-sentence proof in this (otherwise very interesting) MathOverflow thread.
1:35:00 – DM: SS has a way with words.
1:38:14 – JJ: SS is talking about the zeta function, in particular, the equality of the infinite sum and infinite product forms. He explains what he calls a “very understandable proof.” This is really a byproduct of the notation for infinite sums and infinite products. We use this notation because it gives us useful intuition that we already have from finite sums and finite products. Just working with infinite series in this way is “considered not kosher” because many such manipulations would lead to false statements. I do think that this is the right intuition here, and, going back to an earlier point by SS, this is the right way to start the discussion about this fact. However, this is not the right way to PROVE this fact. This section is pointing to an important fact that a lot of math education is focused on producing the rigorous proofs, and much more should be focused on figuring out what is true, at least intuitively. Indeed, research time is mostly spent figuring out what is true.
1:38:48 – CS: I wonder how many results in math really have nice intuition and a nice story behind how the person figured out what was true. A lot of times for me, the answer is either “I computed a bunch of examples on my computer and this was always the result” or “I just knew it had to be true for reasons I can’t even entirely articulate”, neither of which is actually that interesting or informative to read. But maybe this is an argument there should be a lot fewer math papers in the world?
1:39:58 – EK: “is” vs. “ought” — I like that distinction as applied to mathematics.
1:48:42 – DM: I like this idea of people giving talks about work that they didn’t do. SS mentions that some seminars that follow this model, but I haven’t heard of it before (outside of presentations in my topics course).
1:50:35 – CS: GS is basically describing the Current Events Bulletin at the Joint Meetings, which I think basically everybody enjoys, so it seems like there should be more venues for this kind of thing (other than writing textbooks, that is).
1:53:15 – DM: Here’s a link to the previous collaboration between GS and SS. In hindsight, this early 3Blue1Brown video follows GS’s suggested SoME1 model of having an elder collaborator provide the material and having a younger collaborator develop the animation.