This year, the graduate class format changed dramatically for me. I went from a normal class, filled with students, to classes with at the most 4 students and a professor. Actually, my Complex Analysis II class has only another student enrolled. As such, the format has changed. The professors no longer give 3h-lectures, the students do, each in turn.
Basically, each graduate student will prepare a 3h-lecture¹. In one class, that means that I lecture every 4 weeks. In another, it’s every other week². Preparing the lecture involves going over the textbook and the proofs. Depending on how detailed the proofs are, you’ll need to flesh them out further, and make them understandable, citing the right theorems, propositions, etc. Depending on what book/resources you are using, this might take quite a few hours. It also depends on the overall complexity of the class and the overall sparseness of the authors of the book. Atiyah’s books is very sparse. The proofs are sometimes quite short and they need to be expanded significantly.
It takes me between 6 and 12 hours to prepare for the lecture. This involves going over the proofs multiple times, researching other proofs via the Internet, and ensuring that I know what I am talking about, because the professors will quiz you while you are giving the lecture to make sure that you understand and that the other students understand. At the same time, if any other theorems or propositions are being cited, you need to familiarize yourself with them.
The professors usually take some time every other week to present solutions to some exercises. The students will also present solutions as well. This doesn’t stop the professor from explaining a concept further than you did, while you are presenting. In my case, since I present in English and the other students are more or less fluent in that language, the professor explains to them in Mandarin. Students will also ask questions on topics that they don’t understand fully.
I’ve noticed that while preparing and right before starting the lecture can be extremely stressful³, you do grasp the subject matter very well. While I’m presenting, it’s not really stressful anymore because I have been a teacher for years. However, you also need to keep on your toes, because if you made a mistake in your proof, the prof will catch it and expect you to correct it on the fly. Since you have spent hours preparing, it’s not that difficult, but it can be nerve-wracking if you’re not used to being criticized by a professor.
Initially, I didn’t see the benefit of this format, but I’ve come to agree that students do learn a lot more and quicker this way because when you present, you know the subject matter well. It’s doubly difficult for me since I didn’t do Commutative Algebra I last semester, so I’ve had to catch up with the class this semester.
Between my two classes⁴, Commutative Algebra II is the hardest. The other three students are graduate students in pure algebra. I’m a graduate student in analysis. On top of that, the three graduate students are the professor’s graduate students, so when I took the class, I sort of disturbed their little clique. Needless to say it was made apparent to me in the beginning, and the classes were extremely uncomfortable. But, grudgingly, the prof has accepted my presence and I’ve come to like the subject.
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[¹]: I haven’t frequented that many maths departments in my time, but I think that some math departments use this format. It all comes down to each individual professor. I seem to remember that undergrads despised coming to the front of the class and presenting anything, especially exercises. I remember that when one graduate student from France expected the undergrads to do so, there was an uproar.
[²]: Technically speaking, in Complex Analysis II, we will shortly begin the lecture. Until now, the professor has been giving lectures. There are only 2 students in this class.
[³]: When I’m stressed, I go to the bathroom often. This combined with coffee isn’t a good mix. I also tend to wait to eat something after the class, because I’m too nervous beforehand.
[⁴]: I actually have three classes, but the last one is a colloquium class, so I don’t have to prepare, just be present.