A study of prospective mathematics teachers' knowledge development and beliefs changes for teaching fraction division

The purpose of this study is to examine prospective mathematics teachers' knowledge development and beliefs changes for teaching fraction division through the undergraduate mathematics method course to the field practice. Further, it reveals the correlation between the knowledge development and beliefs changes. Therefore, this study uses a qualitative methodology. I analyze the data from two time periods using three steps. In the method course period, interviews concerning knowledge and beliefs are triangulated with the tests, surveys, concept mapping and the writing assignment. There are two steps in this time period. First, I focus on a total of 27 prospective teachers' subject matter knowledge (SMK), including common content knowledge (CCK) and special content knowledge (SCK), and its development. Further, I examine their beliefs changes towards fraction division and mathematics teaching and learning during the method course. Next, I choose six participants from the total 27, based on different mathematics achievement. I do this to identify (1) whether CCK differences impact SCK development and (2) whether SCK development influence beliefs changes in the method course. In the field practice period, classroom observation of fraction division is triangulated with the interviews. I follow up one prospective teacher in his field practice and focus on the way his beliefs influence his teaching behavior and the development of the pedagogical content knowledge (PCK), through the teaching.;The results indicate that the prospective teachers developed both CCK and SCK in their method course. Their beliefs towards to teaching and learning fraction division progress from procedural-oriented to conceptual-oriented. The knowledge development and beliefs changes derived from the different learning experiences from their past school experiences and method course. Moreover, prospective teachers who had high CCK developed his/her SCK significantly. Thus, his/her beliefs changes became more significant. Further, the prospective teacher's beliefs changes in the method course influenced the way of teaching behavior in the field practice and SCK impacts PCK in teaching. On the other hand, field practice changed prospective teacher's beliefs and the development of PCK. Therefore, further attention is called for in the prospective teachers' knowledge transition and beliefs changes from a student to a future teacher.