A student came to me today to discuss a conjecture that we made in class. First, I am impressed that my students are coming up with conjectures spontaneously. Second, I am impressed that some of the students are trying to prove them.
The conjecture was that for all elements g in a group G, the order of g divides the order of the group G. This student proved this by independently developing the notion of a coset, which I think is a difficult idea for students. We are going to learn about them next week in class, so I will find out then how difficult students find this idea.
Unfortunately, I think that this student had these skills before I ever had him as a student—I wish that I could take credit, but I cannot.