First, some context: I'm a PhD student, and this term, I'm running a bunch of seminar groups for first years (I believe these are what would be called "discussion sections" in the US - essentially, each group is 20-ish students out of a much larger course, and I'm doing stuff to support the lectures. I'm marking, but not setting, the assignments). My previous teaching experience has all been with what I believe would be "upper division" courses in the US, and thus far I've tried to approach this in essentially the same way, but there are some pretty significant differences that lead me to think that I'm not using the time well. Specifically:
- In previous teaching, I've been able to rely on students (at least after a little prior prompting) turning up with questions that we can discuss, being able to tell me which parts of the course they're struggling with, etc. and dealing with these issues has taken up a large part of the available time (essentially, because dealing with the things that they're definitely struggling with seems more reliable than me trying to guess what they're struggling with). This doesn't seem to be the case with these students.
- Previous students have been able and willing to get involved in discussions, either in small groups or all together, and have even started such discussions. The default expectation from these students seems to be that they can sit there and do very little and I'll somehow pour mathematics into their brains.
- Previous courses have all been heavily proof-based, and I've spent a good part of the available time helping with details of proofs that students have not been comfortable with, providing alternate proofs, and trying to drill down to the underlying concepts. This course, thus far, consists entirely of matrix algebra, and the expectation seems to be something far more computational.
As far as what I've done so far, besides the usual start-of-course administrative stuff, is briefly summarise the content of the course thus far, provide a couple of worked examples, and get them to work on some practice questions (theoretically in groups, but I haven't managed to overcome their unwillingness to talk to each other very successfully yet), while I walk around and provide support. I also experimented with asking a question involving slightly more thought ("so we've seen that it is not always the case that AB = BA, but we've also seen that this does hold when either A or B is the identity or zero matrix. Can we say any more about when AB = BA does or doesn't hold?"), which they found significantly more difficult than I had anticipated, and needed a lot of help to even start.
In general, this doesn't seem like that effective a use of the available time: in particular, there's very little that I've done that couldn't have been done by handing them a textbook with plenty of practice questions. I also haven't gained much information about concepts that they struggle with - most of the errors that I've found, both in their assignments and in the class, are purely failures of mental arithmetic, rather than any conceptual issues.
Essentially, I'm looking for ideas of how I can adjust my teaching to make it more productive. Ideas for encouraging more active participation would also be much appreciated.