Studying Math

Grad school has been very difficult. I'm constantly learning how to "be" a mathematician.

I'll be frank, undergraduate classes were simple. Most of the work required was just getting the hang of whatever you were doing. This worked fabulously back at the University of Minnesota, Duluth. Especially in the philosophy program. I'd skim/read the material, doing my best to get the general impression. Then, I'd attend class and the professor would cement the general ideas in my mind. The details mattered only every once in a while. In fact, this process even worked backwards: I could attend class, get the general idea from the professor, then read the book and I'd be ready for an exam.

It doesn't work that way now. No matter how many times I read through a chapter, even if I use different books to get many different angles on a subject, it isn't enough. It's still difficult to follow lectures. I still have trouble completing the homework.

Today, I finally had the courage to ask one of my professors for advice. He said you've got to read a sentence, set the book aside, then play with it.

For example, here's the first sentence from Set Theory (Jech 2002).

Intuitively, a set is a collection of all elements that satisfy a certain given property.

Like most definitions in mathematics, I'd basically try to memorize it, and that's not necessarily wrong. But according to my professor, you've got to start testing the definition out. "What kind of property would work?" you might ask. "What about the property 'numbers less than 5'?" So, that means a set would be the numbers 0, 1, 2, 3, and 4. "Is that okay? Are numbers 'elements'?" I'm supposed to ask. "What if the property was not being equal to itself?" No number (or anything in the universe, for that matter) isn't equal to itself, so that would be an empty collection.

Of course this is the obvious thing to do. But it's hard in practice to do it. It's painful to stop reading a book every sentence, put the book aside and actually think. It reminds me of the saying "You can send them to college, but you can't make them think."

I suppose studying math is like exercising. Watching, listening and reading other people do mathematics is like watching someone on a treadmill. You don't gain anything unless you're on the treadmill yourself.

I sure hope I get the hang of it. I'm very proud of where I am and I want follow the path in front of me. It's just strange having to admit that I still need to improve my studying habits.