I am a teaching assistant in mathematics at a university in Continental Europe. The course for which I am an assistant is a third year course, so usually the students are expected to know the basic notions from calculus and linear algebra. However, I often notice that students are not as familiar with the basic material as I would wish them to be$^1$. So, the natural question to ask is:
What should be my teaching strategy?
There are several possibilities. Each of them has its own pros and cons, which might contradict each other; but in fact, they don't, because you can regard it as a pro or a con, depending on your general attitude towards teaching.
1. Go very slowly and repeat everything.
- They can actually learn a lot from the exercise class.
- Everything I do in the exercise class is self-contained; they find everything relevant to a certain topic concentrated on a few sheets of paper. Whence:
- Learning the material is made easier for the students (at least in a certain way).
- People who already know the basic material get bored easily.
- I have trouble finishing; I lose plenty of time repeating material that should be known.
- When learning the material, they lose focus on what's important and what's not. This goes hand in hand with:
- Due to me repeating everything, the density "content per time" seems to be very high for the students, although they should already know a lot of things I say.
- A funny aspect which I encountered is the following: If I repeat basic material, students do not seem to understand the material, but if I skip the repetition and just use the results (e.g., I say "follows from linear algebra"), they think that they understand. But in fact, they don't, because they do not really understand what happens at a particular step in an exercise.
2. Just assume that they know the material.
- I can actually finish in time, so they get solutions for every exercise.
- The students who know the material can follow and don't get bored.
- The students should look up the material on their own; in fact, one goal of the university is to make people work independently and scientifically.$^2$
- They don't really understand what happens during a particular step of an exercise.
- As a TA, I somehow feel bad when using this method, because I know that there are a lot of students who don't understand the material. Of course, I want them to learn as much as possible.
- At the latest they should be able to apply the material in the exam. If they don't learn it fast, they never will, and they will most likely fail the exam. (See footnote 2: With this method, I don't encourage them as much as with method 1 to really learn the material.)
3. A mixture of 1. and 2.
Somehow combines a few pros and cons of each 1. and 2.
Which method do I apply?
Usually I go by method 1., because my aim is to make the students (all students! - even if this is utopian) fully understand the material. However, I am a bit sceptical because of the cons.
Should I even care about students as in footnote 1?
Assume that the students would be interested in the material but forgot a lot of basic notions. Does this change the answer to the main question?
Would it be a good idea to go by method 2. and give a sheet of paper with the used basic theorems to the students?
$^1$ This is actually an euphemism. Unfortunately, the students I talk about do not care about grades or even understanding the material. They just want to pass the exam.
$^2$ ... but this does not work very well in practice. The kind of students mentioned in the footnote above just don't care and blame the TA for giving bad exercise classes. They don't understand that at a university not everything is served up on a plate so that they only have to eat the meal - they also have to cook. But it is a matter of the TA's attitude if one should prepare the ingriedients for them (like in a cooking class) or just give them the full recipe.