Definitions are important: the case of linear algebra

In this paper we describe an experiment in a linear algebra course. The aim of the experiment was to promote the students' understanding of the studied concepts focusing on their definitions. It seems to be a given that students should understand concepts’ definitions before working substantially with them. Unfortunately, in many cases they do not. Influenced by their high school experience they concentrate in learning patterns for problem solving without deep understanding of the relevant concepts. In order to educate the students to appreciate the importance of understanding definitions, we changed the homework problems and the didactical contract with the students. We discuss the positive outcomes of these changes from theoretical and educational points of view. We chose to try our approach in a linear algebra course since it is probably the most abstract course for first semester students. We hope that our approach can be useful in teaching other undergraduate courses.