Grant Sanderson (of 3Blue1Brown fame) recently launched the 3b1b podcast as part of the SoME1 math exposition challenge. To celebrate, some friends and I decided to launch an experiment on this blog by annotating the first episode of the podcast:
In this episode, Grant Sanderson (GS) interviews Alex Kantorovich (AK) about all things academia, and what follows is some commentary by John Jasper (JJ), Hans Parshall (HP), and me (DM). Feel free to discuss further in the comments.
2:53 – HP: AK seems to share a belief among mathematicians: math is the best arena to train problem-solving skills that are widely applicable beyond math.
2:53 – DM: I’m stealing this joke.
4:21 – DM: The comparison to reading is funny. It’s wild to think of a society in which everyone supports the study of technical things like math. There’s certainly a push for STEM careers in the US today, but it hasn’t yet kept folks from bragging about how they were never good at math.
5:01 – HP: I am essentially ignorant of how mathematics is taught outside the US; I hope that GS digs deeper into these questions in a future podcast.
8:36 – HP: I strongly agree with AK here: “you need both”. There is a synergy effect: mechanical skills and conceptual understanding reinforce one another. “Intuition” is a hard thing to define (try it!), but it seems impossible to build mathematical intuition without rolling up sleeves and calculating.
8:54 – DM: I agree with the need for both understanding and drilling. Drilling is less attractive for obvious reasons, but it allows one to solve problems quickly. Does drilling have a place in higher-level math? For example, I can produce a proof of the spectral theorem given enough time, but I can also quickly recite the theorem from memory, and this is useful in research. Do we spend enough time drilling key theorem statements in class?
9:37 – JJ: My worry is that our teachers don’t have enough training to teach problem solving. Problem solving is much harder to teach than algorithmic stuff that avoids thinking.
11:08 – DM: The application of drilling here is familiar to me. But I’m also quick to code things up in a computer to help formulate conjectures. After discovering a conjecture, I tend verify small cases by hand to observe symbol manipulations that can prove the desired theorem.
15:32 – HP: AK’s point is really strong here. Mathematical “adults” can be frightened off by a vast literature surrounding an open problem, but there is still plenty of room for mathematical “kids” to play around (and teach the grown-ups).
15:36 – JJ: AK describes “racing” colleagues during research. Both GS and AK describe the excitement that kids sometimes have during math games on car rides, especially when they see excitement from adults. I think these are related. In particular, as a kid/student the positive feedback that we receive is a strong motivator. Indeed, it is an indication from people with more expertise (adults, profs, etc.) that we are doing something worth doing. During research we sometimes get the same validation from being the first to provide the proof of the claim we are all thinking about. This points to expert approval being an important motivator for learning. How can we best use this form of motivation for students? This “racing” makes research sound kind of intense (which it often is!). This may be a turnoff for some potential mathematicians.
20:01 – HP: I suspect that most mathematicians are too quick to discount their ability to engage in meaningful outreach. We should all care about making sure that people understand what we’re doing!
21:27 – HP: AK is being modest here; being editor-in-chief of a journal is both an honor and a responsibility. His description of Experimental Mathematics is appropriately sexy.
27:51 – HP: Now I want GS to interview a writer at Quanta about their background and process!
31:12 – JJ: Tweeting about your papers is a great exercise, if only to force one to write out something about the content of your paper in language designed for broader audience. AK suggests that people may have an easier time getting an idea of what is in his papers from reading a tweet. This sounds like the role that an abstract is supposed to play. Perhaps abstracts are only playing this role for experts in the field. Maybe what we need is a non-expert abstract in addition to the one written for experts. This could also be a remedy to a problem AK mentioned earlier, namely, that colloquium talks should be at the level of a 1st year graduate student. For colloquia, we could have speakers submit an abstract that is readable by a 1st year graduate student. Forcing colleagues to do this seems difficult, but forcing graduate students to do this for their job talks is very doable!
31:12 – DM: Oof. I think papers should be written in a way that maximizes understanding. If 140 characters elucidates a proof, then some version of those characters should appear in the paper. This is the opposite of a waste of space. We should remove the stigma of helpful explanation in papers much like how Experimental Mathematics removes the stigma of formulating conjectures and providing computational evidence for those conjectures. I have never received a review of one of my papers that asked to omit a helpful explanation.
33:11 – DM: Yeah, I appreciate thinking of the first-year graduate student as the intended audience of a colloquium. I wish more colloquium speakers applied this model of the audience.
33:56 – HP: AK makes an important point here: definitions are hard to parse without intuition, and intuition takes time to build. Different branches of math seem to rely on different intuition; I would also be upset to attend a talk that opened with an affine scheme over a totally real whatever.
36:41 – HP: Solid take-home advice from AK: to give better talks, give more talks and pay attention to the audience.
38:25 – DM: I’ve found that the chat and polling features in Zoom remove barriers to classroom discussion. On the other hand, Zoom can make it difficult for me to throttle the rate at which I cover various points in lecture, since Zoom can obscure nonverbal cues from students. I’m looking forward to returning to in-person lectures in the fall. I wonder if I should use an integrated approach. Live streams? Clickers? I’m certainly less afraid of technology now that the pandemic forced me to work with it.
42:39 – HP: AK mentions trying to collaborate with the math ed department. I wonder how well-known it is that math and math ed are two very distinct fields, and often (but not always) they are separate departments!
43:30 – DM: Oof, I don’t think mathematicians are the best problem solvers in all cases. By definition, they are the best solvers of math problems, but there are many types of problems in the world. George Pólya’s “How to Solve It” doesn’t explain how to solve the various social problems in the US. Also, I don’t think the NSF should mandate 10% time on non-math problems, but they can choose to fund more cross-disciplinary proposals, thereby incentivizing such research. Ohio State provides several opportunities for cross-pollination, e.g., the STEAM Factory and the Translational Data Analytics Institute.
46:17 – HP: Yes, please: more university-wide colloquia!
48:33 – DM: Heh. Lots of data science is about separating signal from noise, and there’s a lot of beautiful math that makes this work, but in many cases, industry can just purchase more signal. I hear this happens in big tech all the time. Pretty rough.
56:44 – DM: This reminds me of how I can use duality theory in convex analysis to systematically search for proofs of a certain form. I wonder what the learning curve is like for Lean.
58:56 – HP: GS asks a really important question here, wondering about the danger of asking an oracle.
1:01:05 – DM: Is Lean sufficiently user friendly to bring into the classroom?
1:02:21 – JJ: I can see huge value in using Lean to separate the instructor from the evaluator in proof-based courses. As AK points out, students (including himself) often feel that they are losing points because the evaluator is “nitpicking.” I’m concerned that the students might feel that subjecting their proofs to a computer checker is the height of nitpicking. Moreover, it is not even the standard for proofs in mathematics. Meanwhile, it’s super easy to see how programming in MATLAB is not a contrived exercise since folks use it in the wild. This may become moot once Lean becomes more commonplace in the math community.
1:02:21 – HP: I really appreciate GS justifying the position that learning programming helps one learn to think mathematically. My own opinion: it is more important to learn programming than it is to learn any one particular piece of software. I would caution students that LEAN is under rapid development, and from their FAQ: “Documentation is not the main priority right now.”
1:02:21 – DM: When grading proofs, it can be difficult to convey to the student why points are being taken away. For example, in early proofs courses, “your logic is not clear” can be interpreted as “I just don’t like your writing style.” I like the idea of pitting the student against the compiler so that the teacher can be a coach more than an adversary. I’ve seen this sort of social move used in other aspects of the classroom, for example, separating the evaluator from the instructor in coordinated courses or qualifying exams.
1:04:52 – HP: Thanks to GS for extracting a prediction! I would buy AK’s prediction that at least one math journal requires a “formalized” proof in 20 years time (are there any now?), and I would be flabbergasted if this was the norm. The quest for formalization of mathematics is old and controversial; see this.
1:09:17 – HP: I don’t know the reference to “Tarski and somebody”, but it seems plausible that AK is referring to Russell and Whitehead’s Principia Mathematica, which proves that 1 + 1 = 2 on page 83 of Volume II. They were self-aware: it is immediately followed by the comment “The above proposition is occasionally useful.” This should give us some appreciation for how difficult formalizing mathematics can be.
1:10:04 – HP: I agree with AK: papers should tell stories! I need to remember this advice.
1:10:50 – DM: It seems that AK agrees that papers should include helpful explanations after all. 🙂