Effective Teaching and Uses of Instructional Representations in Secondary Geometry: A Comparison of a Novice and an Experienced Mathematics Teacher

Abstract

The purpose of this qualitative case study is to investigate the uses of instructional representations of a novice secondary mathematics teacher and an experienced mathematics teacher in the content area of secondary geometry. Instructional representations are the tools teachers use to communicate their mathematical knowledge to students (Berenson & Nason, 2003). Classroom observations and semi-structured interviews were conducted and analyzed in order to find emergent patterns among the participants' uses of instructional representations. Patterns are reported from each participant's uses of instructional representations and from a cross-case analysis of both participants' uses of instructional representations.

The Pirie-Kieren theory (Pirie & Kieren, 1994b) for students' growth of mathematical understanding serves as the framework for the study. The Pirie-Kieren theory describes eight potential levels of student understanding based on the cognitive changes that occur during the processes of learning mathematics. These eight levels were adapted for the purposes of this study and used to describe how the participants' uses of instructional representations allowed and fostered opportunities for students to engage within those levels of mathematical understanding.

In order to facilitate students' growth of mathematical understanding, effective teachers have a well-developed knowledge base for teaching, including knowledge of multiple instructional representations and the connections between them (Lesh, Post, & Behr, 1987; Moseley & Brenner, 1997; NCTM, 2000; Rider, 2004; Wilson, Shulman, & Richert, 1987). The results of this study reveal the importance of examining how they use those representations in order to better understand how teacher knowledge contributes to effective teaching and student learning.