Using dynamic geometry to explore linear algebra concepts: the emergence of mobile, visual thinking

This dissertation sheds lights on aspects of students’ thinking as they interacted with a dynamic geometric diagram of the concepts of eigenvector and eigenvalue. Given that the phenomenon of thinking is not directly observable, I attend to their use of the dragging tool, shifts in their attention and emerging ways of communicating the concepts through gestures and speech. I present the transcripts of one-on-one videotaped interviews with five university students and analyze isolated episodes. My analytic frame is informed by the theories of shifts of attention and instrumental genesis. The latter reveals evidence of the transformation of tool into an instrument of semiotic mediation by the process of internalization while the former highlights the significant role of attention and awareness in learning and understanding mathematics. The complementary use of the theories enables me to analyze the cognitive development of a student in a digital technology environment, because the student’s use of different dragging modalities can provide easily-visible evidence of shifts in her structure of attention and consequently can reveal her understanding of the concepts. Moreover, the dynamic geometric diagram stimulated the formation of kinaesthetic and dynamic imagery, as evidenced by the students’ ways of communicating. I thus incorporate aspects of embodied cognition into my analysis in order to account for the important role played by the body in students’ exploring and communication. My analysis suggests that the students mostly used a synthetic-geometric mode of thinking, but more importantly, their thinking involved facilities of process and time and vision, spatial sense, kinesthetic (motion) sense. These facilities enabled them to communicate dynamic and kinesthetic imagery using embodied expressions and gestures. I thus argue that dynamic geometric representations of eigenvectors enabled the students to develop dynamic-synthetic-geometric thinking. I also discuss the role of dynamic geometric diagram of the concepts in enabling students to experiment with the behaviour of eigenvectors. This is in opposition to static diagrams that can be found in textbooks. I conclude this dissertation with some pedagogical suggestions in terms of the use of dynamic geometric diagrams of the concepts of eigenvector and eigenvalue.