Studying Graduate Level Mathematics Outside Lecture Hours

Question

I am not exactly sure if this should be posted on math.se or academia.se, but I think the question is more mathematical in nature.

I am a first year graduate student in mathematics, and I would like to think that I am a diligent student at that. My study habit involves reproducing almost all the theorems that we have gone through in class by myself. This has worked really well for me when I was taking undergraduate level mathematics classes. But when I took grad level courses, I realized that reproducing and studying a 1.5 hours worth of lecture is taking me almost a whole day. This kind of freaked me out at first. But I thought I'm just not working hard enough and I am still in the undergraduate to graduate transition. Anyway, I asked my classmates in the courses I'm taking in if they even bother to reprove what we covered in the lectures. To my surprise, most of them told me that they do not. They just use the theorems as they are to answer the exercises/problem sets. Now, I'm thinking that it might actually be a more efficient way of passing the course if you can pull this off. You might be able to 'study' a lot more since you can save time; you'll be able to solve more exercises from the text book, and have more time for problem sets if you do this. But I feel very uncomfortable using theorems that I did not even try to prove for myself (and this takes away the fun of doing mathematics!). Also, dissecting the important proofs that one sees in the lectures tend to give you an idea of the recurring techniques that are likely to appear later.

This has gone too long; my question is that, should I compromise on reproducing lectures, maybe cut the time I'm spending in reproving the theorems we have already done in class in favor of doing end of the chapter exercises / problem sets?

Answer 1

You need to be able to do all of it really, both the problems and understanding the proofs from classes. You'll just need to prioritize as you go.

I always went over lecture notes after the class to make sure I understood the proofs and formulations, however I didn't re-write it all as I was doing this, I would just add some extra explanation at parts where I felt I needed it. What I would do was to make sure I understood the lecture notes and then do the problems, then once the whole course has finished I would go right back through and re-write the proofs and notes, but in a condensed form if I could.

Just to make sure I am being absolutely clear you definitely DO need to make sure you understand everything that happens in lectures, often with solving problems in maths we need a thorough understanding of the subject. I have known plenty of people who just try and apply theorems without understanding the content of them and it never ends well.