The Math Learning Experiment

by Qiaochu_Yuan 1y21st Mar 201831 comments

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Last Sunday I and a small group of volunteers ran an experiment. We got a bunch of CFAR alumni together and asked half of them to tutor the other half in math (in a broad sense, so including computer science) while paying close attention to what was happening in the minds of the tutees, focusing on issues like where the tutees were confused, where they were frustrated, where they were anxious or afraid, etc.

I had a few motivations in trying this. One was a sense that most rationalists could benefit from learning more math on the margin. Another was a sense of there being various obstacles standing in the way of that, for example an identity of "not being a math person" or various flavors of math anxiety (in addition to the fact that it's a hassle / expensive to find a math tutor, and that people have more urgent things to do), and wanting to see the extent to which those obstacles could be debugged at scale. A third was to explore running events at all.

The fourth and most important motivation for me came out of thoughts I've been having since reading Inadequate Equilibria. There's a cluster of people, including but not limited to Eliezer, Critch, and Nate, who (according to me) have what I internally call "trustworthy inside views," another name for which might be the ability to reliably generate useful gears models, and act based on them. This is the thing they do instead of using modest epistemology; it's the thing that allowed Eliezer to write HPMoR, among many other things. And what all of the people who seem to me to have this ability have in common is that they all have strong backgrounds in a technical subject like math, physics, or computer science (in addition to something else, this isn't sufficient). Probably there are other ways to acquire this ability, but acquiring it through learning technical subjects in a particular way seemed promising, and I wanted to see if I could help others make any progress in that direction at all, or at least understand the obstacles in the way of that.


I learned less about inside view than I was hoping. My plan was to run around observing all of the tutor-tutee pairs, but it was harder to do this productively than I expected: I kept feeling like I was missing important conversational context and not knowing how to model the tutee because of that. We tutored for 3 rounds of 30-40 minutes each, and that wasn't long enough; I think to really start approaching the inside view skill would require longer rounds and also particular skills on the part of the tutor that I didn't have the time to attempt to transfer to all of the tutors. I think if I want to learn more about the inside view skill in the future I should try to tutor someone myself, and for longer periods of time; I'll probably try this next.

An interesting point that came up after the 1st round is the unusual role that definitions play in mathematics, as well as the strange way mathematicians typically treat them in textbooks or similar. It's easy to have the experience of reading a new chapter of a textbook, absorbing all the definitions, understanding all the proofs, but still feeling like the wool was pulled over your eyes in some sense. (General topology can really strongly trigger this feeling.) Textbooks generally write down definitions as if there is nothing or very little that needs to be explained in doing so. In fact mathematicians have put a lot of work into writing down the "right" definitions (which is something like writing down the "right" ontology), but this work is basically invisible - I have never seen any textbook seriously discuss it, and it only comes up briefly in discussions of history of mathematics. People don't even talk about it in graduate school, despite the fact that graduate students are expected to be able to generate new mathematics as their job. At no point in a standard math education are students explicitly given the opportunity to practice approaching new mathematical territory with no definitions to help them orient towards it, and coming up with useful definitions on their own.

I consider it an important feature of my approach to mathematics, which feels related to the inside view skill, that I consistently get frustrated at definitions that I don't understand how to reinvent instead of taking them as given. A large part of my math blogging is about motivating definitions. Sometimes it would take me years between my first exposure to and frustration at a definition and the time that I finally had a satisfying motivation for it; for chain complexes it took something like 4 years, and the satisfying motivation is much more complicated to explain than the definition. (For an example that hasn't finished yet, I am still frustrated about entropy, even after writing this post, which clarified a lot of things for me.)


As far as the logistics of tutoring went, the tutor-tutee numbers worked out even on the first 2 rounds, which was remarkable, and on the 3rd round we had some extra tutees. I asked them to be "meta" for a tutor-tutee pair - keeping track of where they were in the discussion, slowing down the tutor if the tutee looks confused, etc. - and people reported that this was extremely helpful. This jives with activities we've been running at CFAR workshops around people talking in groups of 3 in this way (2 people having some sort of conversation and the 3rd person doing various kinds of meta for them) and I'd like to try doing things this way by default if we run something like this again (the next thing I might try is zeroing in on teaching people how to prove things).

I also had forgotten to plan for some things logistically (making sure I had enough hands to set up the space we were using easily, deciding in advance whether to order in lunch or ask people to go out), but they were magically taken care of by my volunteers, for whom I'm very grateful. In the future I'll try to set more time aside earlier in planning for Murphyjitsu.

Overall we at least ended up having a reasonably fun social time, and some people picked up some math in a way that was probably good. I expect the primary impact of this event on the tutees will be to help them feel more like learning math is a thing they can actually do, which is great. The tutors may have learned something about tutoring, which I'm also happy with.

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