TITLE: What remains in a student's mind
QUESTION [124 upvotes]: 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.

REPLY [2 votes]: I am a late-comer to this question but I hope I have a good answer to this question.
I am a HS math teacher, inevitably after some times into a semester, an overwhelmed student would come up to me and asks if he has to remember all of these formulas when going to college. Generally, this is my answer: No, you don't, you need to remember only the most basic concepts, but what I expect to remain in a your mind is the mysterious thing called  math maturity.
Math maturity is the chemistry that clicks in a student's mind when he is tackling a tough question in SAT, it is also the lubricant that makes his brain runs faster when in college.
Perhaps we can extrapolate this to learning higher math during college, as asked by the OP? Thank you.