This study examined how preservice secondary mathematics teachers developed mathematical knowledge for teaching (MKT) around representations and big ideas through mathematics and mathematics education courses. The importance of big ideas and representations in mathematics has been emphasized in national standards as well as in literature. Yet, many studies suggested that teachers' conceptions or understandings of representations and big ideas are problematic. Nonetheless, there is a lack of attention from the mathematics education community regarding how and in what ways teachers develop knowledge in these areas.

This is a longitudinal study focusing on three preservice secondary teachers who had taken mathematics and mathematics education courses for two years, one year as undergraduates and the other year as graduates in a teacher education program. The courses emphasized representations and big ideas and were designed to help preservice teachers develop MKT through problem solving and analysis of student thinking. As such, it was expected that preservice teachers would develop beliefs/conceptions and knowledge that were oriented around representations and big ideas.

This study used qualitative methods to analyze classroom data, two individual interviews, and teacher assessment portfolios from student teaching. Three areas of knowledge were examined: ability in problem solving, knowledge of tasks, and knowledge of student thinking. The results showed that the participants improved knowledge and beliefs/conceptions with regard to representations and big ideas within the courses. However, they had difficulties transferring their knowledge to their practice. In the area of problem solving, they were able to successfully solve problems at the grade 5--8 level with which they were familiar through the courses, when they were prompted to use non-algebraic approaches. Yet, they had difficulties solving many high school level Cartesian representation-related problems even with prompts. They also showed tendencies to use algebraic approaches in problem solving for both grade 5--8 and high school level problems when they were not prompted. In the areas of knowledge of task and knowledge of student thinking, they were able to design tasks and analyze student thinking with an eye on representations and big ideas, when they were familiar with them through the courses. However, they had an eye mostly on procedural skills and students' errors rather than on big ideas and students difficulties with big ideas, when they had to design their own tasks and analyze their students' mathematical thinking during student teaching. Their lack of profound content knowledge in the topics they taught was one of the major contributing factors for those phenomena. It is advisable that secondary teacher educators pay extra attention to developing teachers' content knowledge as well as knowledge of student thinking. Without knowledge in these areas, it is unlikely that secondary teachers could help students understand critical mathematical ideas.