Next semester, I will be leading exercise sessions in a first year course of mathematics. More precisely, the course is Linear Algebra and the audience consists of young, first year undergraduate mathematics-students. The course is taught at a university in continental Europe. Judging from my personal experience, I know that most first year students of mathematics have plenty of troubles with the new, abstract material that is taught at a university. Hence, it would be good to make the student's experience in my exercise sessions as "easy" and enjoyable as possible.
However, I am not a fan of just doing simple, easy and short exercises. My opinion is that you do not make things easier for the students, if you omitt doing the hard exercises and just do the easy ones. Of course, this does not mean that everything I want to do is hard, but every now and then, something difficult will have to be discussed in the exercise sessions. Often, the hardest exercises are the most enlightening ones.
Moreover, my experience tells me that most students have wrong expectations of an exercise class: they expect to understand everything without (only a tiny bit) personal contribution and blame the instructor (which would be me in that case), if they do not understand everything. This has happened more than once to me and if I tell them that they have to work, I am considered to be the 'strict teacher', whose subject is the most important one and who does not understand that the new material is hard. I would be totally fine with that image, but this demotivates students and let them feel safe because they have someone to blame (Once, I heard the sentence "I would have understood the new material, but the exercise class was too bad and on a too high level" - in my opinion, I discussed everything in detail and pointed out which things are the most important ones and focussing on these).
In short, I find myself in the following vicious circle (it seems like you can achieve two of the three things below at a time, but not all three together):
- Maximizing the students' enjoyment level (i.e., they should understand as much as possible and just have a good time during my exercise class)
- Doing an exercise class which is on a reasonable (not too high, but also not too low) level
- Keep the students motivated and tell them quite clearly, that they have to work to understand the new material
Or - in very short - the question is: Which aspects constitute a good exercise class (in mathematics) and which of these aspects are the most important ones?