An activity theory analysis of linear algebra teaching within university mathematics

The focus of my research was to explore the teaching of linear algebra to a large group of mathematics undergraduates (> 200). With this thesis I present a characterisation of a university mathematics teaching practice in the context of linear algebra. The study took place over a twelve week period, one academic semester, at a UK university with a strong tradition in engineering and design technology. Two researchers working closely with a mathematician, the lecturer of linear algebra, collected data in interviews with the lecturer, and in observations of his lectures (which were audio-recorded). Students' views were sought via two questionnaires and focus group interviews. Data analysis was largely qualitative. Linear algebra is an introductory module in most standard first year undergraduate degree courses in mathematics. Research shows that students nd the highly conceptual nature of linear algebra very difficult and challenging. The lecturer, a research mathematician, had re-designed the linear algebra module based on his own experience of students' dfficulties with the topic in the previous year. He followed an inductive approach to teaching instead of a more traditional DTP (definition-theorem-proof) style. He based his teaching on informal reasoning about examples that were designed to engage students conceptually with the material. Through this research I gained insight into the lecturer's motivation, intentions and strategies in relation to his teaching. In applying an activity theory analysis alongside a traditional grounded theory approach to my research, I conceptualised the lecturer's teaching practice and presented a model of the teaching process. This takes account of the lecturer's didactical thinking in planning and delivering the linear algebra teaching. Findings from the study give insight into the educational practice of a mathematician in his role as a teacher of university mathematics. I present some of the outcomes of the study in terms of mathematics (three linear algebra topics - subspace, linear independence and eigenvectors), in terms of the didactics of mathematics and in terms of the theoretical basis of Mathematics Education as a discipline