I think it happens to a lot of students that (at least it feels like) really understanding the material does not really contribute to having a decent grade. It seems to me that quite often exams are more a test of general intelligence than a test whether a student really mastered the material. Of course its usually the smarter students that also master the covered material more deeply than the rest, but still. It seems to miss the point of a final exam, which is to test whether the students mastered the material. How do you guys handle the problem of designing exams that avoid this pitfall? Moreover do you think that testing mastery of the material is the right goal to aim for?
EDIT: One example I could give are proofs that only require knowledge of a few basic operations that one is allowed to perform, but demand that one sees some sort of trick. Almost no (deep) understanding of the material is required and almost no advantage is gained from understanding the ideas behind the concepts. Someone smart without deep knowledge will easily outperform someone less talented who spent a lot of time understanding the concepts. But not only does it not reward understanding, but tricks that are really hard to see are heavily dependent on luck. I might see the trick one day but I won`t on another. Of course both factors are impossible to rule out completely but the might well be mitigated if the teacher would try. If exams are designed like the one I complain about, they put too much focus on the performence of students under exam situations while a lot of other qualities are neglected.
Take the following question:
Determine the radius of convergence for the following infinite sum:
One day I might do the right transformations and on another day I might not. But I am not sure whether questions like this test deep understanding of mathematical concepts.
On the other hand I think that there are also exam questions that are too easy. For example if one wants to test whether the students understood the jordan canonical form, I think it is a very poor test to simply require the students to calculate the matrix and cernels etc. . This can be done by any student who memorizes the required algorithm without understanding what it actually means.