These are my experiences about learning, written from the perspective of a math student. I do think that my advice is applicable to other theory intense subjects such as physics and economics.
So I used to be the smart guy in high school. Doing well in math competitions, not listening in class and all that stuff. Basically I was a smart jerk. Needless to say, when I got to Harvard, I wasn’t the smartest in my class anymore. My freshman year, I enrolled in math 25, the honors class for math majors. I did pretty well in that class, but had to work really hard. Now, looking back at that experience, I see a couple of things that, given the chance, I would have done differently.
One could say that I was good at math, but not at learning math, if that makes sense. In high school, the emphasize for me had been at problem solving, while math studies in college are much more theory oriented.
How do you efficiently learn so much theory?
1) Don’t assume everything will work out
The first mistake I made is an obvious one. When I did not understand a concept, I thought I could get away with it anyway (for me, this concept was differential forms). “It will probably make sense a week from now,” I thought to myself. Wrong. A week from then, I was completely lost. Never (never!) think that something that confuses you know will magically be clearer a week from now.
In essence, all my other mistakes can be derived from this mistake.
2) Be prepared to spend extra time
When you don’t understand a concept, spend extra time trying to understand it right away. Believe me, in the long run, this will save you a lot of time.
3) Go to office hours
When you don’t understand something, go to office hours. Just go. Most people don’t go to the professor’s office hours, so chances are you will be sitting with only a few other students together with the professor. Then you can ask the question you thought was to embarrassing to ask during class.
4) Get an intuitive understanding of definitions and theorems
When you first learn a new concept, chances are it will seem weird and arbitrary. If you want to study math or any other theory intense field, you have to be able to routinely take something weird and arbitrary and make sense of it. How do you do this?
- Don’t focus too much on the equations. Ask yourself, what does this equation mean, and try to translate it into plain English (this should probably apply to softer fields such as economics as well). Ask yourself what something really means. Is continuity really “epsilon delta blah blah blah” or is it simply a way of writing down mathematically that points near each other end up near each other after we apply the function? Answer: Both! But one of them is easier to understand, guess which one?
- Draw a picture. Yes, your high school geometry teacher was right after all. Draw a picture, make up an example, try to understand what is really going on.
5) Know when to give up
When you have drawn all the pictures you can think of, tried to translate what you are doing into plain English and are still not getting anywhere, take a break. Do you core reading, eat a banana, tickle your roommate or go to sleep. Then go to office hours.
6) Have fun
Learning math should be fun. If it is not, you should probably reconsider taking those math classes. Life is to short to waste your time learning useless facts about differential forms, vector spaces and compact sets if you don’t even enjoy it.