How Mathematical Knowledge for Teaching Intersects with Teaching Practices: The Knowledge and Reasoning Entailed in Selecting Examples and Giving Explanations in Secondary Mathematics.

Abstract

Teaching requires both knowing and doing. This dissertation helps bridge between research on teacher knowledge and research on teaching practice by conceptualizing the ways in which teachers draw on their knowledge when enacting specific teaching practices. Recent research on mathematical knowledge for teaching has specified different domains of teachers’ knowledge, but has focused less on how teachers use what they know in teaching. Similarly, several teaching practices have been found to matter for student learning, yet researchers have not adequately delved into the knowledge and work entailed in carrying out these practices. To investigate teachers’ knowledge use in practice, this study of eight Algebra II teachers focused on the content of rational expressions and equations and two foundational practices in mathematics teaching, selecting examples and explaining. Data were collected through classroom observations and interviews simulating the two teaching practices. Analyses probed the knowledge used in enacting each practice. One finding of this study is a description of components of the practice of selecting examples and the knowledge teachers draw on in enacting them. For instance, teachers sequence examples and draw on nuanced understandings of differences across a set of mathematical examples. Second, different categories of explanations were seen and teachers’ knowledge use varied by explanation type. Third, across both practices, the knowledge teachers drew on when enacting the practices was associated with differences in how teachers enacted the practice. For example, teachers drew on a broader range of knowledge types when giving mathematical reasoning explanations than when giving procedural explanations. The research also shows the complexity of knowledge use when enacting teaching practices. For example, during a single explanation, one teacher drew on common and specialized content knowledge, knowledge of content and students, and pedagogical knowledge. The findings contribute to theoretical understandings of how teachers use knowledge in teaching by conceptualizing the ways in which teachers draw on their mathematical knowledge for teaching when enacting specific teaching practices. For instance, they better specify types of specialized content knowledge used in practice. This work also has implications for mathematics teacher education and the methods used to study teacher knowledge in practice.