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Individual and collective analyses of the genesis of student reasoning regarding the invertible matrix theorem in linear algebra


In this study, I considered the development of mathematical meaning related to the Invertible Matrix Theorem (IMT) for both a classroom community and an individual student over time. In this particular linear algebra course, the IMT was a core theorem in that it connected many concepts fundamental to linear algebra through the notion of equivalency. As the semester progressed, the IMT took form and developed meaning as students came to reason about the ways in which key ideas involved were connected. As such, the two research questions that guided my dissertation work were : 1. How did the collective classroom community reason about the Invertible Matrix Theorem over time? 2. How did an individual student, Abraham, reason about the Invertible Matrix Theorem over time? Data for this study came from the third iteration of a semester-long classroom teaching experiment (Cobb, 2000) in an inquiry-oriented introductory linear algebra course. Data sources were video and transcript of whole class and small group discussion. To address the second research question, data from two individual semi-structured interviews, as well as written work, were also analyzed. The overarching analytical structure of my methodology was influenced by a framework of genetic analysis through the notion of cultural change, using two interrelated strands of microgenesis and ontogenesis (Saxe, 2002). I utilized two analytical tools, adjacency matrices and Toulmin's Model of argumentation, to analyze the structure of explanations related to the IMT both in isolation and as they shift over time. In addition to an in-depth analysis of the complex ways in which Abraham and the classroom community reasoned about the IMT throughout the semester, this dissertation presents methodological contributions regarding the two analytical tools. First, it necessitates the definition of four new argumentation schemes that are expanded versions of Toulmin's model. Second, it further adapts and refines the use of adjacency matrices within mathematics education research to analyze student thinking, both at static moments and over time. Finally, the present study lays a strong foundation for a 2-fold analytical coordination. The first coordinates results from adjacency matrix analysis with those from Toulmin's Model to demonstrate they were often compatible, with the tools varying in their respective strengths and limitations. Second, the present study lays a foundation for coordinating the mathematical development of both the individual and the collective units of analysis

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