Tracing the Development of Understanding Rate of Change: A Case Study of Changes in a Pre-Service Teacher's Pedagogical Content Knowledge

Abstract

This investigation adopts an interpretive approach to study the development of understanding rate of change through a pre-service high school teacher's emerging pedagogical content knowledge over a two-year period of time. Shulman's (1986) definition of pedagogical content knowledge and Ma's (1999) definition of profound understanding of fundamental mathematics provides the conceptual framework for the study. The three categories of pedagogical content knowledge identified by Shulman (1986) assert first that knowledge of students' ideas consist of the teacher knowing the areas of student difficulty and the requirements for student learning. Secondly, knowledge of representations includes the symbols, words, graphics/pictures and other representations used by the teacher. And thirdly, knowledge of instructional activities is having a repertoire of activities to use in the classroom. Ma's (1999) profound understanding of fundamental mathematics is also fed into pedagogical content knowledge with subcategories of teachers' having basic mathematical ideas, connectedness and longitudinal coherence. This conceptualization of pedagogical content knowledge was used to analyze the pre-service teacher's construction of knowledge about teaching mathematics.

Lesson plan study, classroom observations, journal reflections and interviews were coordinated to understand what changes took place in order to trace the pre-service teacher's development of pedagogical content knowledge. By assigning each component to a single construct, pedagogical content knowledge, the components functions as parts of a whole. As a result, lack of coherence between components can be problematic in developing and using pedagogical content knowledge, and increased knowledge of a single component may not be sufficient to effect change in practice.