Teachers’ knowledge of subject matter draws an increasing attention from policy makers in recent years in all over the world, since more emphasis is given to highly qualified teachers (Crespo & Nicol, 2006; Hill, Rowan, & Ball, 2005). The National Council of Teachers of Mathematics (2000) reported that teachers who have strong subject matter knowledge give details in their lesson, link the topic to other topics, ask students many questions, and stray from the textbook. Grossman, Wilson, and Shulman (1989) defined subject matter knowledge as knowledge of the key facts, concepts, principles, and explanatory frameworks of a discipline. Subject matter knowledge of teachers has been an interest of educators; however, more recent time there has been a shift to pedagogical knowledge especially to pedagogical content knowledge (Johnston & Ahtee, 2006). Pedagogical content knowledge (PCK) differentiates expert teachers in a subject area from the subject area experts (Shulman, 1987). The PCK concerns with how teachers relate their subject matter knowledge to their pedagogical knowledge and how subject matter knowledge is related to the process of pedagogical reasoning (Shulman, 1987). Students in elementary school generally have difficulties in developing an understanding of certain mathematics concepts, which are abstract in nature (Ball, 1990; Mack, 1990). Operations with fractions are one of these topics. Although many students memorize the rote procedures needed to manipulate the symbols, they soon forget the procedures and thus find it difficult to learn fractions (Mack, 1990). Division of fractions, where conceptual understanding is critical, often considered the most mechanical and least understood topic in elementary school (Tirosh, 2000). Teachers’ understanding of a topic is important in helping students to comprehend those topics in depth. Thus, in this research study we aimed to investigate Preservice Teachers’ (PST) knowledge structures on division of fractions. Changes and improvements in teacher education programs are frequently recommended by many researchers, but it is critical that before making any changes in teacher education curricula, the details of the knowledge of PSTs should be analyzed. Since PSTs constitute the population of the tomorrow’s teachers, the conceptions that PSTs hold on fundamental operations should be the concern of teacher educators (Tirosh & Graeber, 1991; Tirosh, 2000). It is believed that exploration of teachers’ content and pedagogical content knowledge and relationship between these knowledge structures will give invaluable implications to the mathematics educators and policy makers in designing the course content of the teacher education programs in all over the world. It should be noted that teacher knowledge is critical issue in many European countries since it is the most important factor that influence students’ understanding and learning mathematics. Thus, this study aims to answer the following questions: 1. What is the nature of PSTs’ understanding of mathematical relationships related to division of fractions? 2. What kind of strategies do preservice elementary mathematics teachers prefer to use in explaining mathematical relationships related to division of fractions? 3. How does preservice elementary mathematics teachers’ subject matter knowledge influence the strategies they use to explain mathematical relationships?

To investigate the PSTs’ knowledge structures on division of fractions, 17 senior preservice elementary mathematics teachers in an undergraduate teacher education program in Ankara, Turkey were selected by using purposive sampling at the end of the spring semester of 2004-2005 academic year. The Qualitative case study design was used. To gather and triangulate information: Division of Fraction Questionnaire and follow up semi-structured interviews were used. The DFQ was developed based on instruments used by several other researchers (Ball, 1990; Ma, 1999; Schifter, 1998; Simon, 1993; Tirosh, 2000). The questionnaire focused on assessing PSTs’ understanding of mathematical relationships and strategies that they used to explain those relationships on division of fractions. After administering the DFQ (see appendix) to senior PSTs, semi-structured interviews were conducted to obtain a more complete picture of PSTs’ subject matter and pedagogical content knowledge. Constant comparative method of data analysis was utilized in order to analyse data.

To investigate the PSTs’ knowledge structures on division of fractions, 17 senior preservice elementary mathematics teachers in an undergraduate teacher education program in Ankara, Turkey were selected by using purposive sampling at the end of the spring semester of 2004-2005 academic year. The Qualitative case study design was used. To gather and triangulate information: Division of Fraction Questionnaire and follow up semi-structured interviews were used. The DFQ was developed based on instruments used by several other researchers (Ball, 1990; Ma, 1999; Schifter, 1998; Simon, 1993; Tirosh, 2000). The questionnaire focused on assessing PSTs’ understanding of mathematical relationships and strategies that they used to explain those relationships on division of fractions. After administering the DFQ (see appendix) to senior PSTs, semi-structured interviews were conducted to obtain a more complete picture of PSTs’ subject matter and pedagogical content knowledge. Constant comparative method of data analysis was utilized in order to analyse data.