Almost all exams in mathematics courses up through the calculus sequence are composed of a series of problems for students to work. These questions are generally much less problems than exercises. The idea is for our students to demonstrate that they have acquired a certain set of skills and knowledge and that they also understand the mathematical concepts. But these two objectives have been conflated in mathematical instruction and students study for tests by essentially drilling on the problems they find on practice tests, review sets of problems and the homework. In fact the preponderance of homework problems are of the drill type. Math texts have gotten thicker and thicker not because of additional subject matter (There is some of that.) but because students demand to see more problem solutions modeled. The problems that once required students to use mathematical concepts in a new (to them) way have been turned into exercises.
It can be argued that doing such problems requires understanding of the underlying concepts – at least doing them easily and remembering how to do them. When I explain a problem, I write down the steps and verbally (mostly) model the thinking process – what concepts I use. But the students learn the steps. Of course they do. That is how they will be tested.
As I prepare for Precalculus II next term, I hope to separate the problems into skill problems and concept problems – drill and memory versus understanding. The tests should really be made up of the second type of problem but this would cause a student uprising.
By the way, making up good unique problems that test for concepts is difficult and as soon as the problems are used on test they will make their way into test files and thus will lose their purpose if reused on a test. The existence of the internet just accelerates a good problem’s obsolescence . They will circle the globe at the speed of light never again to have the impact their first use was designed for. This is like the difference between watching a movie for the first time and then watching it over and over again. No surprises and less intellectual challenge.