There’s more to mathematics than grades and exams and methods

When you have mastered numbers, you will in fact no longer be reading numbers, any more than you read words when reading books. You will be reading meanings.
(Harold Geneen, “Managing”)

When learning mathematics as an undergraduate student, there is often a heavy emphasis on grade averages, and on exams which often emphasize memorisation of techniques and theory than on actual conceptual understanding, or on either intellectual or intuitive thought. There are good reasons for this; there is a certain amount of theory and technique that must be practiced before one can really get anywhere in mathematics (much as there is a certain amount of drill required before one can play a musical instrument well). It doesn’t matter how much innate mathematical talent and intuition you have; if you are unable to, say, compute a multidimensional integral, manipulate matrix equations, understand abstract definitions, or correctly set up a proof by induction, then it is unlikely that you will be able to work effectively with higher mathematics.

However, as you transition to graduate school you will see that there is a higher level of learning (and more importantly, doing) mathematics, which requires more of your intellectual faculties than merely the ability to memorise and study, or to copy an existing argument or worked example. This often necessitates that one discards (or at least revises) many undergraduate study habits; there is a much greater need for self-motivated study and experimentation to advance your own understanding, than to simply focus on artificial benchmarks such as examinations.

It is also worth noting that even one’s own personal benchmarks, such as the number of theorems and proofs from you have memorised, or how quickly one can solve qualifying exam problems, should also not be overemphasised in one’s personal study at the expense of actually learning the underlying mathematics, lest one fall prey to Goodhart’s law. Such metrics can be useful as a rough assessment of your understanding of a subject, but they should not become the primary goal of one’s study.

Whereas at the undergraduate level and below one is mostly taught highly developed and polished theories of mathematics, which were mostly worked out decades or even centuries ago, at the graduate level you will begin to see the cutting-edge, “live” stuff – and it may be significantly different (and more fun) to what you are used to as an undergraduate! (But you can’t skip the undergraduate step – you have to learn to walk before attempting to fly.)

See also “there’s more to mathematics than rigour and proofs”.

I also recommend Keith Devlin’s opinion piece “In Math You Have to Remember; In Other Subjects You Can Think About it”. (Note: the title of the piece is actually the opposite of Devlin’s (and my) opinion; read the article for the explanation.)