The role of the teacher is one of the essential elements in improving instruction and students’ understanding in mathematics classrooms (Hiebert et al, 1997). Effective teachers should know the concepts and topics that students often have difficulty with, and the ways to clarify students’ common misunderstandings (National Council of Teachers of Mathematics [NCTM], 2000). Most of the studies that concentrate on teacher’s role in increasing the effectiveness of instruction emphasized that teachers’ content and pedagogical content knowledge are major determinants of mathematics instruction and learning (Ball, 1990; Crespo & Nicol, 2006; Fennema & Franke, 1992). This study attempts to illuminate the role of mathematics teaching method courses on pre-service teachers’ content and pedagogical content knowledge. Mathematics teachers who have deficiencies in subject matter knowledge are likely to pass their misconceptions and misunderstandings on to those they teach (Even, 1990). On the other hand, teachers who have solid mathematical knowledge for teaching is more capable of helping his/her students achieve a meaningful understanding of the subject matter and students’ understanding of the topic goes beyond procedures (Even, 1990; Tirosh, 2000). Teachers who have conceptual understanding can answer students’ questions about the meaning behind their symbolic manipulations. However, subject matter knowledge is not enough to achieve this goal. Teachers should also transform the content into representations that help students develop understanding (Shulman, 1986). Shulman (1987) stated that knowledge of subject matter for teaching consists of two overlapping domains: content knowledge and pedagogical content knowledge. His conceptualization has been accepted as a framework for most of the researchers on teacher knowledge (Ball, 1990; Ball & McDiarmid, 1990; Borko et al., 1992; Shulman & Grossman, 1988). Shulman (1987) mentioned that pedagogical content knowledge differentiated expert teachers in a subject area from the subject area experts. The pedagogical content knowledge concerns how teachers related their subject matter knowledge (what they know about what they teach) to their pedagogical knowledge (what they know about teaching) and how subject matter knowledge is related to the process of pedagogical reasoning (Shulman, 1987). The importance of teachers’ knowledge of students’ learning has encouraged many researchers to examine pre-service and inservice teachers’ content and pedagogical content knowledge in various topics in mathematics (Ball, Lubienski & Mewborn, 2001). Operations with fractions are one of the interests since fractions are considered the most complex mathematical domain in elementary school curriculum (Mack, 1990). 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). However, studies revealed that pre-service and inservice teachers’ knowledge on operations with fractions is largely procedural (Ball, 1990; Graeber, Tirosh, & Glover, 1989; Ma, 1999; McDiarmid & Wilson, 1991; Tirosh, 2000; Tirosh & Graeber, 1991, Simon, 1993). In line, researchers emphasized the importance of teaching mathematics method classes and changes in the education programs in order to increase content and pedagogical content knowledge of pre-service teachers (Kinach, 2002; Robinson & Adkins, 2002, Tirosh, 2000). In this study, we are interested in the role of method courses in developing pre-service teachers’ understanding about elementary students' misconceptions on operations with fraction, and the role of method courses in developing pre-service teachers’ performances in transforming a mathematical expression related to operations with fractions into a real world problem. That is, this study aims to answer following research questions: 1) Do teaching methods courses have any role in developing PSTs' understanding about elementary students' misconceptions on operations with fractions? 2) Do teaching methods courses have any role in developing PSTs' performances in transforming a mathematical expression related to operations with fractions into a real world problem?

The study was conducted in the school of education of a large public university in Turkey. The participants were in a four-year regular teacher education program. Twenty-seven pre-service mathematics teachers participated in a two-semester research study which is based on the data collected from two teaching methods courses; one general teaching methods course, and the other one was mathematics teaching methods course. The participants were given a two-item test three times at different times to measure their pedagogical content knowledge for mathematics teaching. In addition, six of the participants were interviewed to obtain an in-depth understanding of their responses. While the first question in the two-item test was adapted from Van de Walle (2005), the second question was adapted from Ma (1999). Both questions are on fractions.

In the first question, we asked PSTs to explain how they would handle a sixth grade student’s difficulty related to operations with fractions. The results indicated that a majority of PSTs (25 out of 27), at the beginning of the first course, could detect the 6th grade students’ misconceptions but explained them by only the Standard Algorithm (SA). They could not provide a reasoning and justification of the SA. On the other hand, they used wider range of strategies to explain the student’s misconception in the second assessment. The content of the first teaching methods course might be the reason for the increased number of strategies used by PSTs. A review of the interviews confirms this assumption as one of the participants stated, “I used the discovery method in my solution; I think it is the influence of what I learned in the methods course.” The responses to the first question changed dramatically in the third assessment. To be specific, 14 PSTs emphasized the part-whole relationship. Another remarkable finding is that there was no response without any reasoning. Thus, their responses were more conceptual at the end of the second methods course. In the second question, the results indicated that while 9 PSTs in the first assessment and 12 of them in the second assessment could not write a real-world problem, all of them wrote a real-world problem representing a division with fractions in the third assessment. Most of the responses (24 out of 27) presented a measurement model of division whereas two of the responses presented a partitive model of division. These particular findings indicated that participants’ level of understanding went up after the second methods course. An overview of interviews supports this claim. The participants stated that they improved their level of mathematics and the knowledge for mathematics teaching in the second course. To sum up, the analysis of the participants’ responses revealed that methods courses have an important role on PSTs’ PCK on the topic of fractions. Since teaching methods courses play a critical role in developing PSTs’ PCK, it is important to design systematic research studies investigating to what extent these courses help improve PSTs’ conceptual understanding of teaching mathematics. It is hoped that the results will contribute to goals of the Life Long Learning Program of the EU as the study makes suggestions to improve the education of teachers.