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Abstract
The purpose of this study was to understand how teachers make sense of their students' mathematical thinking. Specifically, learning trajectories and professional noticing were used as a way to examine how teachers understand and use students' mathematical thinking in their teaching practices. A qualitative research methodology was employed to address three research questions that focused on teachers' informal learning trajectories, what teachers notice about students' mathematical thinking during classroom interactions, and the ways that teachers respond to that thinking during classroom interactions. Two high school geometry teachers were observed and interviewed during one semester. In addition, the teachers attended biweekly working-group meetings to discuss students' mathematical thinking. I created two learning trajectories for the teachers' lessons to represent their thoughts about their students' mathematical thinking before and after lessons: the projected learning trajectory (PLT) and enacted learning trajectory (ELT). The two learning trajectories were compared to identify instances of teacher noticing. The PLT and ELT were similar in some instances but not others. The teachers described and interpreted what they noticed in terms of their uncertainties and surprises about students' mathematical thinking. Furthermore, the teachers described and interpreted what they noticed about students' mathematical thinking in terms of the mathematics tasks, their own mathematical knowledge, and their actions with students in the classroom. The teachers typically responded to students' mathematical thinking in five ways: posed a question, asked students to share, told the students something about the mathematics, posed another task, and used a pedagogical content tool. What the teachers noticed in the classroom interactions and the ways that they responded to students affected their development of PLTs and ELTs, and vice versa. When the students led the ELT, the teachers gave detailed descriptions for the learning trajectories and gave more details for what they noticed during classroom interactions. In contrast, when the teachers led the ELT the teachers struggled to describe learning trajectories and what they noticed during classroom interactions. The reporting and analysis of the data reveal implications for both research and teacher education.