I’m a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an intense study, I forget inexorably most things that I have learned. For example if I study algebraic geometry, commutative algebra or differential geometry, In my minds remain only the main ideas at the end. Viceversa when I deal with arguments such as linear algebra, real analysis, abstract algebra or topology, so more simple subjects that I studied at first or at the second year I’m comfortable.
So my question is: what should remain in the mind of a student after a one semester course? What is to learn and understand many demostrations if then one forgets them all?
I’m sorry for my poor english.
Let’s say you can’t remember what Theorem 42 says, from a course you took nine semesters ago. Then 20 years later someone mentions a term that sounds vaguely familiar to you—you’re not sure why. It’s in a discussion of a topic you’re curious about. You look in book indexes for terms sounding vaguely related, and google those terms, and after pursuing vague memories, it turns out to lead you to Theorem 42, which you’d forgotten, and that’s just what you need to answer the question that was on your mind. So all was not lost.
However, it is also often useful to actually remember things. Two things that help are (1) you find surprising connections among seemingly disparate things, and it impresses you, especially if one of them was something you were interested in; and (2) You teach a course that includes the statement and proof of Theorem 42. You teach that course six or seven times, and grade students’ answers to exercises in which they must use Theorem 42, or prove it by another method that is roughly sketched for them, or prove another result by the same method, and you answer lots of students’ questions about all this, and help them through difficulties they have with it.
When I took topology as an undergraduate I remember the instructor putting on the blackboard a huge list of various spaces and said this one has this property and this property but not this property, and we were supposed to learn to identify an example given those properties. They were quite different from each other: manifolds pasted together, spaces of sequences of real numbers, things put together out of transfinite ordinals, and I wondered: How can one remember all of this? But then some years later I found much of it fresh in my mind whenever a question arises that such examples answer. So another part of the answer is: just keep going.