Most of your math courses to date probably have been ones where the instructor lectured and you took down whatever he or she wrote at the board, then studied these notes before exams, and when you did homework problems, it was not too har dto decide what examples from class were the ones related to what you were being asked to do. The class lectures followed a "definitiontheoremproof" style, which is the standard for mathematics classes everywhere. No doubt you have become accustomed to this apprach, and have developed study habits that work well with it.
This style is effective from a pedagogical standpoint, but it has little to do with how mathematics works in practice. As you begin to develop new mathematical ideas, coming up with the right definitions and determining what the good theorems are is at least half the battle, and you don't get a lot of practice with that in a definitiontheoremproof lecture.
Since this course is not a prerequisite for anything else (and so doesn't have a specific set of ideas that we have to cover), we are free to be somewhat less rigid in the structure of the course. My plan is to have us work on developing some of the ideas together in a way that more closely mimics how mathematics is done by actual mathematicians. With a small class of experienced students like this, I think this will be a rewarding approach.
In practice, this means that you will be asked to develop some of the key ideas yourselves (with guidance from me, of course), and I will ask you to work out the details of some of the processes we will learn based on an outline we work out in class. As is often the case with mathematics, this may be an iterative process, where you get part way there on your own, then we come back to class and take those ideas a bit further, so you can work on them yourself again the next night. In particular, if you can't figure something out completely on the first try, that's OK; most interesting problems are not ones you can solve all at once.

