Examining the work of constructing a representational context in elementary mathematics teaching.

Abstract

Despite wide support for teachers to use models, diagrams, metaphors, manipulative materials, and mathematical tools in mathematics teaching, little is known about what it takes to use such representations to support student learning. This omission in the literature leads to a gap for teacher education, as it is not clear what to teach teachers about the effective use of representations in elementary mathematics teaching. In the dissertation, I investigate the work of launching and preparing to use a representation such that a representation becomes a usable tool for mathematics learning. Across three teaching episodes, I study the teacher's work in launching and preparing to use four types of representations used in mathematics teaching: a concrete representation (square tiles); a visual and abstract representation (a multistory building); and a mathematical object (the number line). The method of inquiry involves using records of practice (e.g., videotaped lessons, student work, and teacher notes) to study the work of teaching in order to make more visible what teachers need to know and be able to do to use representations effectively. To this end, the analyses focus on what the teacher <italic>does</italic> and <italic>thinks</italic> about in the course of teaching. By teaching I mean both the interactive work within the classroom as well as the design work done in preparation. The study's findings comprise a set of resources for the work of launching and preparing to use a representation. I propose categories of work that a teacher educator might help teachers learn to do and identify three entailments of the work of constructing representational contexts in mathematics teaching: (1) the demand for mathematical knowledge, skill, and sensibilities; (2) the need to be judicious in how language and mathematical symbols get used; and (3) the importance of attending to both the affordances and constraints of drawing on students' prior knowledge and experience. The conclusion provides examples of how the findings might inform the content and pedagogy of a teacher education course on mathematics teaching. Connections are drawn between the study's focus and method of inquiry in relation to contemporary efforts to improve teacher education by situating teachers' professional learning in the actual work that teachers do.