The problems of selecting problems

Abstract

We face several teaching problems where a set of exercises has to be selected based on their capability to make students discover typical misconceptions or their capability to evaluate the knowledge of the students. We consider four different optimization problems, developed from two basic decision problems. The first two optimization problems consist in selecting a set of exercises reaching some required levels of coverage for each topic. In the first problem we minimize the total time required to present the selected exercises, whereas the surplus coverage of topics is maximized in the second problem. The other two optimization problems consist in composing an exam in such a way that each student misconception reduces the overall mark of the exam to some specific required extent. In particular, we consider the problem of minimizing the size of the exam fulfilling these mark reduction constraints, and the problem of minimizing the differences between the required marks losses due to each misconception and the actual ones in the composed exam. For each optimization problem, we formally identify its approximation hardness and we heuristically solve it by using a genetic algorithm. We report experimental results for a case study based on a set of real exercises of Discrete Mathematics, a Computer Science degree subject.