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Abstract
Teacher knowledge of students’ mathematical thinking has been a growing research area in mathematics education. Although researchers have made substantial progress in understanding students’ mathematical thinking and created research programs and curricula tailored to the research findings, teacher education programs have, by and large, not been grounded in similarly extensive research on the cognitive nature of teachers’ knowledge and its development. Moreover, teacher knowledge and student-teacher interaction are often treated as disparate research areas, and research on their interplay is sparse.
In this dissertation study, I addressed these research gaps by characterizing the mental processes by which teachers develop knowledge of students’ mathematical thinking as they interact with students and reflect on students’ mathematical thinking. Specifically, I focused on how teachers construct knowledge of students’ mathematical thinking including how this construction, in turn, supports the teachers’ personal growth in mathematical knowledge.
Adopting and integrating radical constructivism, Piagetian learning theories, and communication theories, I conceptualized the cognitive mechanisms by which teachers construct knowledge of students’ mathematics through social interactions with students. My theorization combined the notions of first- and second-order models, decentering, assimilation, perturbation, accommodation. I expanded the constructivist teaching experiment methodology to a methodology called teaching experiment^2. I included two high school teachers and supported each of them in engaging in cycles of interactions with individual or paired student(s) and reflection on those students’ thinking. Each cycle included (a) a planning session, (b) a teaching session, and (c) a reflecting session in the form of video-stimulated recall interview. One teacher worked with her students in the context of linear programming, and another teacher worked in the context of geometric dilation.
My analysis identified seven key mental processes involved in the teachers’ constructions of students’ mathematics: re-presenting, contraindicating, explaining, inferring, modifying, comparing, and restructuring. The specification of these mental processes contributes to the current theorization of teacher knowledge by operationalizing an anti-deficit, constructive, and dynamic perspective on teacher knowledge and knowing. The findings also inform teacher educators about how to advance teachers’ sensitivity to student thinking and supporting teachers in continually learning from their students in their own teaching.
In this dissertation study, I addressed these research gaps by characterizing the mental processes by which teachers develop knowledge of students’ mathematical thinking as they interact with students and reflect on students’ mathematical thinking. Specifically, I focused on how teachers construct knowledge of students’ mathematical thinking including how this construction, in turn, supports the teachers’ personal growth in mathematical knowledge.
Adopting and integrating radical constructivism, Piagetian learning theories, and communication theories, I conceptualized the cognitive mechanisms by which teachers construct knowledge of students’ mathematics through social interactions with students. My theorization combined the notions of first- and second-order models, decentering, assimilation, perturbation, accommodation. I expanded the constructivist teaching experiment methodology to a methodology called teaching experiment^2. I included two high school teachers and supported each of them in engaging in cycles of interactions with individual or paired student(s) and reflection on those students’ thinking. Each cycle included (a) a planning session, (b) a teaching session, and (c) a reflecting session in the form of video-stimulated recall interview. One teacher worked with her students in the context of linear programming, and another teacher worked in the context of geometric dilation.
My analysis identified seven key mental processes involved in the teachers’ constructions of students’ mathematics: re-presenting, contraindicating, explaining, inferring, modifying, comparing, and restructuring. The specification of these mental processes contributes to the current theorization of teacher knowledge by operationalizing an anti-deficit, constructive, and dynamic perspective on teacher knowledge and knowing. The findings also inform teacher educators about how to advance teachers’ sensitivity to student thinking and supporting teachers in continually learning from their students in their own teaching.