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Drawing on recent developments in complexity theory, ecology, and hermeneutics, Brent Davis and Dennis Sumara present an "enactivist" model of cognition and contrast it to popular notions of what it means to learn and think that pervade formal education. They illustrate their model by drawing from their experiences during a year-long study in a small, inner-city elementary school. According to this model, cognition does not occur in individual minds or brains, but in the possibility for shared action. An enactivist theory of cognition, the authors suggest, requires teachers and teacher educators to reconceive the practice of teaching by blurring the lines between knower and known, teacher and student, school and community.

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.

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