| Although the idea of pedagogical content knowledge (PCK) has been elaborated in numerous studies, there has been little clarification of what constitutes it or research into its development. Furthermore, studies that have investigated PCK, or mathematical knowledge for teaching (MKT) as introduced by Thompson and Thompson, have historically focused on pre-service teachers at the elementary level. This study contributes to filling these voids by investigating in-service secondary school teachers' ways of thinking that supported or constrained their capacity to reflect on their practice as they engaged in activities designed to promote powerful mathematical knowledge for teaching as proposed by Silverman and Thompson.;The study's participating teachers expressed understandings and ways of thinking that constrained their capacity to reflect productively. There were very few occasions where teachers' behaviors could be characterized as reflective. Teachers kept their personal meanings private, operated with tacit assumptions unquestioningly and unawarely, operated freely with underlying incoherent meanings, and were unperturbed when their meanings were misaligned with their students or with their colleagues. Teachers did not question their own or their colleagues' meanings, did not question their assumptions, did not question the coherence of their meanings, and did not question whether their meanings were aligned or even compatible with those of their students or colleagues. I propose an orientation toward learning and teaching of mathematics (called an empirical orientation) as a potential way of thinking that accounts for teachers' ways of operating and helps to explain teachers' reticence to reflect. An empirical orientation hinders productive reflection because it constrains teachers' capacity to take their own actions and thoughts as objects of thought, and it constrains teachers' ability to take the points of view of their students and of their colleagues. |
