Doing well in your courses: Andrej's advice for success (2013)

This is fun to do during lectures but in my experience only about 5-10% of my learning happened in math class. The other 90% happened at home as I worked through the problem sets.

Essentially the lectures served as an inefficient way of delivering me a set of notes which I’d then reference during homework sessions. I could often predict what was coming next in the lecture but the really hard parts were the key parts in some technical lemmas that were necessary to complete the theorem. Learning how to figure out a key step like that had to come completely on my own (with no spoilers).

In a lot of ways, math lectures really started to turn into an experience similar to watching a Let’s Play of a favourite video game. Watching those can tell you exactly what you need to do to get past the part where you’re stuck but they don’t in general make you better at video games. For that you need to actually play them yourself.

The viewpoint of a lecture as an inefficient note delivery system is a pretty common and reductive view. Your "Let's Play" analogy was pretty apt though.

I think their (potential) value seems pretty clear when you look at language courses -- you can't possibly hope to develop fluency in a language by studying it in isolation from books -- forming your own sentences and hearing how other human beings do the same in real time is pretty decisive.

With math classes, YMMV, especially since they are rarely so interactive at the upper division and graduate level, but at the very least seeing an instructor talk about math and work through problems (and if you are lucky to have a particularly disorganized one, get stuck, and get themselves unstuck) can go a long way. But to be fair I regularly skipped math lectures in favor of reading too, heh

I rarely skipped math lectures in university (only when the prof was really bad; but then I watched video lectures taught by a different prof from a previous term).

The lectures in the hardest math classes I took did not feature any “working through problems.” They were 50 minute pedal-to-the-metal proof speedrun sessions that took me 2-3 hours of review and practice work to fully understand. I don’t know how anyone can see a lecture like that and not see it as an inefficient note delivery system.

I did have math classes where profs worked through problems but those were generally the much easier applied math classes. Those were the ones I least needed to attend lectures for because there you’re just following the steps of an algorithm rather than having to think hard about how to synthesize a proof.

For language learning it’s hard to beat full immersion. When we learn our first language (talking to our parents as children) we don’t learn it by theory (memorizing verb conjugations), we learn it by engaging the language centre in our brains. I think language classes are more useful if you want to learn to write and translate in that language, where you need a strong theoretical background. If your main goal for language learning is being able to speak with loved ones or being able to travel and speak fluently with locals, then sitting in a classroom listening to a lecture seems like a very difficult way to do that.

I had a math professor in college that would often say to our class, "You cannot be like Michael Jordan by just watching Michael Jordan. If you want to be better at basketball, you have to practice. Math is no different." No matter how you spin it, he was correct -- unless you are like Ramanujan and a Hindu god just reveals a solution to you.

Honestly though, I believe I learn better in a similar manner to what you described. I would rather just read the textbook and learn on my own. I find that to be a far more efficient learning style for me. However, I typically always went to class for a handful of reasons:

1. To signal that I cared about the subject to the professor (whether I honestly cared or not). Though I had some classes that actually penalized a lack of attendance.

2. There is comradery in group struggle. It was nice way to meet other students that had a common goal. I made many friends during my time. Some of which I still keep in touch with a decade later. In fact, I met my SO in one of my classes -- all because we studied together.

3. The main reason being, I paid for the class, and I wanted to get my money's worth out of it. While passing the course and learning the material was the goal. I'd hate knowing I just paid to teach myself everything. I could have done that for free, so I wanted something more out of the deal.

One of thing I should add is that I am poorly disciplined and have poor executive functioning, so I probably picked up more in class that I would admit -- I didn't have a control to compare against. Still to this date, I rely heavily on solutions to the problems. Not in a way that allows me to cheat, but I would likely be unable to be certain I was teaching myself correctly if I didn't have the answers or know of a method to verify my work. I am confident that I cannot be confident in my answers to nearly anything. I am prone to too many mistakes.

If one goes far enough in math, one will encounter solutions where there are not clear answers and one must use all of their knowledge and abilities to support their answers. And that my YN friends, is why I am not a mathematician despite my love for the subject.

I consider the value in math lectures to come from the speaker’s explanation of why to expect certain things. Is this an obvious fact in another context, rewritten for this application? Is this a surprise? What reasons besides the rigorous argument are there for believing the theory?

A lot of the theorems I learned in school weren’t particularly amenable to intuitive explanations like that.

For example, take Galois theory. The fact that a polynomial’s solvability by radicals depends on the solvability of its Galois group is surprising and not intuitive at all. The fundamental theorem of Galois theory is a very technical result utilizing purpose-built mathematical structures that were developed specifically to study the solutions of polynomial equations.

- I find that writing notes in class helped me learn just through the physical action of my hands. (I think there is some formal study of this as a phenomenon). I am poorly disciplined so at least getting that hour or so of writing notes is probably more than I would have managed alone.

- In class, sometimes the lecturer provides helpful intuition for something through informal speech or even intonation. For example I struggled with the concept of ergodicity from a textbook until I saw someone explain it to me like I'm 5. I find that often, textbooks are like man pages, in that they are almost afraid to provide informal/intuitive writing for fear of appearing unserious.

p.s. if ChatGPT existed 30 years ago I would have managed to learn so much more instead of spinning wheels on dry writing. ChatGPT is really good at being a "personalized manpage explainer"

I meant "problems" in a broad sense -- I loved disorganized professors who would pause and stare at their lecture notes in silence for a minute, realize their proof or example contained some flaw, and then have to correct it on the fly.

I found those moments really valuable if course-correcting was non-trivial -- the typical Definition-Theorem-Proof-Example format certainly is essential for organizing one's thinking and communicating new math in a way that's digestible to other mathematicians, but it is not how mathematicians actually think about math or solve novel problems

In the grad analysis sequence this "course correcting" mechanic was built into the course, since we were required to regularly solve a challenging problem and then present its proof to the class and withstand intense questioning from both the professor and peers. If you caught an error in someone's proof and could help the presenter arrive at a correct proof, you'd both earn points.

The thrill of surviving an incredulous "Wait a second..." from that particular professor (who later became my research advisor) was hard to beat

Anyway my intent was to analogize math lectures (whatever they might look like) with language courses or immersion in the sense that they are an opportunity to practice speaking and listening, and to immerse yourself in cultural norms. I think it goes a bit deeper than this, in that language is inextricably connected to most thought and vice versa -- we experience this in a very explicit way whenever we find our thinking clarified in the process of formulating a question, but it's always there

That said, pure immersion for language learning is actually easy to beat -- lots of research shows that immersion together with explicit grammar instruction has far better learning outcomes than immersion alone. Immersion alone misses lots of nuance -- and it relies on the speaker being acutely aware of the difference between their output and target forms.

With your verb conjugation example, lots of time can be saved by knowing that there's a thing called the subjunctive and that it is distinct from tense and it shows up in a myriad of places tending to concern hypotheticals

Similarly, I gain a lot from talking to mathematicians and attending conferences. But I also need to spend time alone consulting relevant theory, reading papers, and playing with examples. Both are important, but in math it seems you one get away with less immersion