# Creating exercises: Get students to do it

One interesting technique I’ve tried is to ask students to create an exercise and its solution. The advantages of this are many. One, creating exercises is hard and I can use all the help I can get. The best exercises I can use in the following year or for revision purposes. Two, the students get practice in calculating when they are creating the solutions. Three, students have to think deeper. Sometimes a lot deeper. This gives them much more insight into the technique(s) being practised.

For example, when teaching vector calculus to engineers, I asked the students to produce an exercise on the calculation of a surface integral. Constraints were important as a shamelessly flippant student could set the exercise of integrating the constant function over a square. Hence, the surface should not be flat and the function should not be constant or consisting of only linear terms.

This worked quite well with most of the students being able to get a good exercise from it. Admittedly, many did not stray far from the type of question they had already seen and essentially just changed some coefficients. Nonetheless, they got a sense of achievement from this. The task was frustrating for some students though. This was particularly true if during working out the answer to their question they discovered that they have created something too hard and had to simplify. Their assessment was that the previous work is somehow wasted.

Interestingly, when questions were created to give to others, some students tried to set the hardest possible question that they themselves could answer in an attempt to torture their peers.

I also tried this to generate problems involving eigenvalues and eigenvectors for $2\times 2$ and $3\times 3$ determinants. I’m sure that there are many other possibilities but I would say that this task is limited to routine calculation exercises. It is hard to create a question asking for a proof that is in the sweet spot of being just hard enough to be non-trivial yet doable. However, if anyone has ever managed to get their class to do this I would be interested in seeing the results.