A study was conducted at the Technical University Berlin involving students who twice failed the written examination in the first semester course Linear Algebra for Engineers in order to better understand the reasons behind their failure. The study considered student understanding in terms of Bloom's taxonomy and the ways in which students approached problem solving. The results indicate that students rely on lower-order thinking processes and these processes are linked to solution approaches. Thus, by investigating solution strategies in homework sets and in classwork, an instructor can easily identify students at risk of not understanding at the appropriate level. In this contribution, the study is related to the framework set forth by the European Society for Engineering Education (SEFI) Mathematics Working Group.