Teaching to the Mathematical Point: Knowing and Using Mathematics in Teaching.

Abstract

Teaching is purposeful work. Teachers are responsible for organizing time and resources so that what students do leads to learning. Knowing “the point” of instruction helps manage this work. This dissertation examines the central, yet underexplored, terrain of what it means to have a clear understanding of the goals of instruction and to use those goals to manage the work.

Through an iterative analysis of the literature and data from 17 preservice teachers’ mathematics lessons, this dissertation identifies and analyzes the practices and knowledge demands of determining the mathematical goals of an instructional activity and using those goals to design and steer instruction. The result is a conceptualization of “teaching to the mathematical point” as three interrelated types of work: (1) articulating the mathematical point; (2) orienting the instructional activity; and (3) steering the instruction toward the mathematical point. The first two constitute the work of “mathematical purposing” and, together, aim to help manage problems in steering instruction toward the mathematical point.

A main product of the dissertation is a conceptual framework that unpacks the work of mathematical purposing. Drawing on theories of distributed cognition, the framework takes a distributed perspective on the design of instruction. Thus, depending on the context, the work of mathematical purposing will be differently distributed between the teacher and various resources (e.g., curriculum materials, established instructional routines, longer-term planning, standards, etc.). The study also identifies problems in steering instruction toward the mathematical point, provides examples of strategies for managing these problems, and highlights issues that can arise. In addition, the study explores the relationship between mathematical purposing and mathematical knowledge for teaching (MKT). This analysis contributes to the ongoing development of the practice-based theory of MKT, in particular, by elaborating two domains of pedagogical content knowledge: knowledge of content and teaching (KCT) and knowledge of content and curriculum (KCC). Results from the study inform research on mathematics instruction, teacher knowledge, and teacher education, as well as the design of educative curriculum materials and tools for teacher education, such as protocols for lesson planning and reflection.