The Math Learning Experiment

Cool experiment, I hope you run more things like this!

Re: definitions, this is a serious struggle I have with the way math is presented. The definitions and notations we have now have been simplified, condensed, and iterated upon for hundreds of years, and what sticks is a matter of practical utility for mathematicians. To present the product of this process as a fait accompli without explanation or motivation is really misleading.

I mean, look how long it took to come up with the modern definition of the group! Even after it was named, for the longest time, mathematicians only considered permutation groups.

I would lay some of the blame at the feet of Platonism. Mathematical Platonism, the idea that "mathematics is out there in idea-space and we just discover it" is at best a useful metaphor or fake framework and has been taken way too far. It implies that definitions are somehow floating in thought-space and the mental move to discover them is to look really hard. In practice, the construction of definitions is more like engineering: there's some weird thing that's there and we'll try to build a vessel that captures its shape as closely as possible.

Note: I tried verifying the received wisdom that "most mathematicians are Platonists" and found the answer to be much murkier than I expected. I would still say that mathematics, the way it is presented, suggests that things like chain complexes and schemes are "things out there in reality" instead of "definitions progressively engineered by people to fit reality."

ARGH! OK, this seems like a really great idea, and I'm thrilled you did it. On the other hand, I feel like LessWrongers fail to optimize tons of effort by not collecting results when they do this class of experiment, so the people involved, and everyone else, fail to gain as much as they could. Obviously you won't get a large sample, and the evidence is weak, but it's REALLY valuable for you and for others considering running similar events.

If I were to specify what types of data I'd hope could be collected:

• Ask mentees whether they feel they learned something interesting or worthwhile in each session. What was it, specifically? (Both regarding the topic, AND what they gained.) How useful was the session to them, on a scale from 1-10, where 1 is not worth their time, 3 is would consider doing it again, 7 is would certainly attend if it was done again, and 10 is would be willing to pay for similar events. Ask what they would want to change if it were done again.
• Ask mentors whether they thought their mentee learned something interesting or worthwhile. Ask them to predict what their mentee would say on the above scale. Ask whether the mentors gained anything, and if so, what. Ask what they would want to change if it were done again.

These are all things you could do via a google survey, and you could tell the participants the results would be made public. You don't need to have the datasets matched to figure out who mentored whom, but it might be kind of nice.

Maybe you could still do this, via an emailed survey? If you'd like me to set up the google form, I'd be thrilled to do so!

Awesome idea! I'm curious about some of the details of the event if you'd be willing to share, partly because I've thought of trying something similar for programming stuff:

• What math topics did you cover and at what level of detail did you cover them?
• Was the instruction purely informational (one person explaining to another) or were there also problems involved (tutor posing a problem and giving the tutee time to think it through and try to solve it)?
• Were there people who consider themselves "bad at math" or "not math people" included? Did their views change at all after the event?

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).

What makes you think this is the result of the technical background rather than a selection effect (where the kind of people who are good at thinking chooses to read technical subjects)?

> for chain complexes it took something like 4 years, and the satisfying motivation is much more complicated to explain than the definition.

You can't just say that and not tell us what the motivation is! Is it on your blog somewhere?

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 feel lucky in that I got some experience with this, though none of it came as part of my "standard" education.

I remember when starting analysis I felt really lost on the basics of calculus like I didn't know why we were doing anything. So I worked through Serge Lang's "A First Course in Calculus" and it fixed me right up, but only because it was a book that was very careful to work through the intuitions behind why you do things. When you get the intuitions down you can't get confused the way you can if it's all just symbol manipulation.

I got more of this when I did two semesters of directed, independent topology study. The prof and I just sat down and worked through proofs until I could explain them forwards and backwards and work out examples easily because I really groked the basics. Same sort of thing when I switched into graph theory and was doing original research.

By contrast I'm still confused by a lot of stuff in measure theory and PDE and other things because I didn't get a great grounding on the intuitions from classes and didn't have time to go back and get it.

I will say that when I was in computer science the courses did a much better job of training intuitions about the constructs you were working with. My best guess is that there was something of an engineering mindset in that department that was missing from the math department.

Yay rationality experiments and reports!