K-8 Teacher Candidates' Use of Mathematical Representations and the Development of Their Pedagogical Content Knowledge as Exhibited in Their Lesson Planning
Ward, Robin A., Anhalt, Cynthia O., Vinson, Kevin D., Focus on Learning Problems in Mathematics
A study was carried out involving thirty-one K-8 teacher candidates enrolled in an elementary mathematics methods course to investigate and document their thinking as they plan for mathematics instruction. The teacher candidates submitted lesson plans at three intervals during a semester-long methods course which were coded based on the planned uses of mathematical representations. Analysis of the data revealed trends in the choices of representations. Recommendations are presented highlighting the potential benefits of incorporating the knowledge base on mathematical representations into a mathematics methods course and a discussion ensues on the development of these teacher candidates' pedagogical content knowledge through their choices of mathematical representations in their lesson planning.
A primary goal of a methods course is to prepare future teachers to teach the subject matter in effective and engaging ways such that student understanding is maximized. In other words, the authors posit that methods courses provide opportunities for teacher candidates to develop and further their pedagogical content knowledge. In an attempt to document and explore how K-8 teacher candidates represent mathematical ideas in ways that are understandable to students, a study was carried out in which K-8 teacher candidates submitted lesson plans at three intervals during a semester. Serving as a lens during the coding and analyses of these lesson plans were the five representations defined by Lesh, Post, & Behr (1987). Of interest to the authors were the representations these teacher candidates might choose to use as they plan for teaching a mathematics topic in the most effective way as possible; that is, so that student understanding is maximized. Also of interest was the potential emergence of trends in these K-8 teacher candidates' choices of representations in their lesson planning as they progressed through their semester-long elementary mathematics methods course. After analyzing the lesson plans of these teacher candidates at three intervals, the authors were able to document the planned choices of and trends in both teacher and student use of various representations within their lesson plans. These choices and trends will be described and a discussion will follow presenting the potential benefits of incorporating the knowledge base on mathematical representations into a mathematics methods course to contribute to the development of K-8 teacher candidates' pedagogical content knowledge.
Research on pedagogical content knowledge
Many researchers have described the various types of knowledge that are needed by teachers. In particular, Shulman (1986) defines pedagogical content knowledge as the ability to represent ideas in ways that are understandable to students. Shulman argues that teachers need to know things such as what topics children find interesting or difficult and what the representations are that are most useful for teaching a specific content idea. Shulman (1987) further elaborates on pedagogical content knowledge as the capacity "to transform the content knowledge he or she possesses into forms that are pedagogically powerful and yet adaptive to the variations in ability and background presented by the students" (p. 15).
For Carter (1990), pedagogical content knowledge represents an attempt to determine what teachers know about their subject matter and how they translate that knowledge into classroom curricular events. Doyle (1992) contents that this ability distinguishes a teacher from a non-teaching specialist; for example, "Knowing biology is necessary, but certainly not sufficient, to know how to represent biological content to students in a teaching situation" (p. 498).
Kennedy (1998) offers that recitational knowledge; that is, the traditional mode of defining facts and terminology is not sufficient for teaching. …