What did I learn today?
That in order to understand mathematics, you have to prove a lot of things. You can’t spectate, and you can’t peek at other solutions. It doesn’t help you. The homework problems are necessary for understanding what’s going on. If your mathematics is weak: exercise it by doing exercises.
I haven’t been doing enough exercises, although I read material nonstop. Also, I don’t communicate my ideas enough, and part of me is afraid to, but I’m going to continue on. Today revealed that I’m still not spending enough productive time on the material, as I read through the book several times, and had most of the definitions and theorems memorized.
Ah, another thing: memory. The first step is to memorize everything you see, even if you don’t understand it. I used to scoff at rote memorization, until I realized that’s the first step. You have to be able to push symbols without knowing exactly what’s going on… at first. It is a blind navigation and it requires faith, and I didn’t have faith in the process but today during the exam the symbol manipulation was fluid, and flowed right out of me, because I would continuously write down the equations and inequalities although I didn’t know what was going on.
To reiterate: memorize the theorems and definitions. You should be able to recite them without thinking, and all conditions need to be precisely stated. This is what makes Joon a good analyst, is that he knows the definitions cold. Those are the tools you need, before you even begin to construct your arguments.
Second: read your class notes. I spend too much time reading the book, instead of reading what it is the professor wanted the class to focus on. Class notes are there because the professor deems the material relevant, so pay attention to that. If nothing else, read the notes.
Third: Read the book sections before attending the class. Lectures should be supplementary, not the main point. They are the highlights of the book. I sort of did this in manifolds, but seemed to have forgotten this over time, and it’s biting me in the ass.
Fourth: talk to anyone about what you’re thinking, but especially the professor and TA. Communicating with Joon helps me to see where I’m faulty in explanation, clarification or interpretation and his guidance is invaluable. His precision is great, too.
I’m still writing a bit too much in this personal blog, and should move my expositions to the problems I am currently working on. I’m getting to it, I just hate tex-ing up things sometimes. But whatever, time to build a new habit.
I will pass prelims easily, I can already see it.
What did I learn today?