Prospective middle school mathematics teachers' processes of constructing student thinking models on the concept of ratio

This qualitative case study aims to examine how prospective middle school mathematics teachers participating in a research-based student-centered pre-service training program (a) develop models of student thinking regarding the concept of ratio and (b) utilize their reconstructed understanding of student thinking in their instructional decisions. For an in-depth analysis, two prospective teachers in their final year of the elementary mathematics education program at a state university were selected. During the first six weeks of the ten-week training program, each prospective teacher worked on understanding students' thinking processes related to ratio tasks. During the last four weeks, they developed and implemented original ratio tasks with students. Data were collected through the prospective teachers' solution papers for ratio tasks, video-recorded group discussions, individual interviews, and field notes taken by the researcher. The collected data were analyzed using content analysis. The analysis used a framework that defines prospective teachers' models of student thinking through four processes (identification, comparison, inference, and reconstruction). According to the findings from within-case and cross-case analyses, while explaining student thinking, prospective teachers repeated student statements, emphasized key points in student solutions, explained student solution methods in detail, or focused on general solution characteristics. During the comparison process, prospective teachers compared student solutions with their solutions and with the solutions of other students who responded to the same task. The inference process involved interpreting evidence from student work. Prospective teachers were able to predict student thinking using the models they created. Furthermore, in individual interviews, they evaluated student thinking based on research-based knowledge and participated in the reconstruction process by generating questions or problems. However, in cases where the reconstruction process was not observed, prospective teachers tended to guide/focus and teach/explain.