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In this article we offer a theoretical discussion of teachers' mathematicsfor-teaching, using complexity science as a framework for interpretation. We illustrate the discussion with some teachers' interactions around mathematics that arose in the context of an in-service session. We use the events from that session to illustrate four intertwining aspects of teachers' mathematics-for-teaching. We label these aspects "mathematical objects," "curriculum structures," "classroom collectivity," and "subjective understanding". Drawing on complexity science, we argue that these phenomena are nested in one another and that they obey similar dynamics, albeit on very different time scales. We conjecture (1) that a particular fluency with these four aspects is important for mathematics teaching and (2) that these aspects might serve as appropriate emphases for courses in mathematics intended for teachers.

Regional differences in performance in mathematics across Canada prompted us to conduct a comparative study of middle-school mathematics pedagogy in four regions. We built on the work of Tobin, using a theoretical framework derived from the work of Maturana. In this paper, we describe the use of video as part of the methodology used. We used videos of teaching activities as prompts for discussions among teachers and the video recordings of such discussions became the data sources for our comparative research. Our use of video revealed a number of advantages and disadvantages which influenced the research.

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