Exploring middle grade teachers' knowledge of partitive and quotitive fraction divisions
Lee, Soo Jin
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The purpose of the present qualitative study was to investigate middle grades (Grade 5-7) mathematics teachers’ knowledge of partitive and quotitive fraction division. Existing research has documented extensively that preservice and inservice teachers lack adequate preparation in the mathematics they teach (e.g., Ball, 1990, 1993). Especially, research on teachers’ understanding of fraction division (e.g., Ball, 1990; Borko, 1992; Ma, 1999; Simon, 1993) has demonstrated that one or more pieces of an ideal knowledge package (Ma, 1999) for fraction division is missing. Although previous studies (e.g., Ball, 1990; Borko, 1992; Ma, 1999; Simon, 1993; Tirosh & Graeber, 1989) have stressed errors and constraints on teachers’ knowledge of fraction division, few studies have been conducted to explore teachers’ knowledge of fraction division at a fine-grained level (Izsák, 2008). Thus, I concentrated on teachers’ operations and flexibilities with conceptual units in partitive and quotitive fraction division situations. Specifically, I attempted to develop a model of teachers’ ways of knowing fraction division by observing their performance through a sequence of division problems in which the mathematical relationship between the dividend and the divisor became increasingly complex. This is my first step toward building a learning trajectory of teachers’ ways of thinking, which can be extremely useful for thinking about how to build an effective professional development program and a teacher education program. The theoretical frame that I developed for this study emerged through analyses of teachers’ participation in the professional development program where they were encouraged to reason with/attend to quantitative units using various drawings such as length and area models. As part of the larger research project, I observed all sessions of the professional development project, which met 14 times for 42 hours in a large, urban, Southern school district. The data collected for this qualitative study included videotaped lessons, reflections, and lesson graphs of the five relevant instructional meetings, and pre-, post-, and delayed-post assessment interviews and interview graphs for eight teachers.