How to find the right balance between a challenging and an enjoyable mathematics exercise session? 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?
As a grad student now who took undergrad linear algebra with a professor I really liked, I will attempt to generalize what I saw that worked.

The best way for students to learn something that is as potentially abstract as linear algebra is to relate it to something concrete. My professor always liked to point out that once you leave R2 and R3, everyone visualizes geometry differently. As a result, he often tried to keep his visual explanations in that space.

However, for higher dimensional vectors, you can still relate the material to multi-dimensional concepts without drawing a picture. A great example (and pretty interesting exercise) is Markov chains. When we went over eigenvalues and eigenvectors, we motivated all of the math by walking through Markov chains. He used Markov chain gambling schemes (e.g. holding a bet steady when you win, doubling when you lose) to illustrate eigen-decomposition in a practical way. In that way you can help students develop intuition by talking about something that makes sense outside of the math-sphere.

In terms of practical exercises, I've found that progressive problems tend  to work best. That is, where each problem is working towards one ultimate problem solution. Generally, each step is a little more complex as well. For example, maybe the first step is finding the Eigenvalues and Eigenvectors, the second is diagonalization, the third is something tangential like inverting the matrix, and the last is applying what you've calculated to some practical problem like a Markov Chain.

In other words, ease students into the harder material by showing them the simpler ways to do it first. Once you can intuitively understand the general concept at basic level, it's easier to move on to more advanced views of the same and similar concepts. Plus, when students see complex material from the viewpoint of something simpler to understand (like maximizing gambling winnings), it provides a reference point for them to start from.