I was recently asked the question "How do you know when you've become a better mathematician/better at mathematics?," and I realized that at that moment I did not have a valid answer, since I have been using my performance on tests to make that judgement.
After putting some thought into it, I would say that one could at the very least use the following criteria:
- Learning material of similar difficulty at a faster pace with the same level of retention and understanding.
- Ability to bring together a larger number of theorems/lemmas/etc to use in showing a result (as opposed to doing problems that almost follow directly from the theorems).
While those might encompass a good amount of information that would indicate whether you have been improving, I know that at my level there is still a lot more for me to experience and learn.
So I would like to pose this question to everyone here, how do you know when you have become better at mathematics?