Abstract In this article, we present a re-framing of tea- cher development that derives from our convictions regarding the enactive approach to cognition and the bio- logical basis of being. We firstly set out our enactivist stance and then distinguish our approach to teacher development from others in the mathematics education literature. We show how a way of working that develops expertise runs through all mathematics education courses at the University of Bristol, and distil key principles for running collaborative groups of teachers. We exemplify these principles further through analysis of one group that met over 2 years as part of a research project focused on the work of Gattegno. We provide evidence for the effec- tiveness of the group in terms of teacher development. We conclude by arguing that the way of working in this group cannot be separated from the history of interaction of participants.

Key words: mathematics teacher development, awareness, purposes, collaborative group, enactivism, expertise.

We illustrate and exemplify how the idea of reflection is framed by the enactive concept of “deliberate analysis.” In keeping with this frame, we do not attempt to define reflection but rather work on the question of “how do we do reflecting?” within such a frame. We set out our enactivist theoretical stance, in particular pointing to implications for how we can learn from experience and showing the role of “deliberate analysis.” We then describe, drawing on education literature, what is generally seen as the purpose of reflection and review some existing conceptualizations in mathematics education, pointing out where we draw distinctions. To illustrate how we do reflecting, we offer excerpts from two lessons of an expert teacher and the writing of a prospective teacher. We exemplify how reflecting as deliberate analysis leads to a way of working with teachers supporting them in handling multiple views and ambiguity, their actions being contingent upon their students’ actions in learning mathematics.

Key words: reflection, deliberate analysis, enactivism, mathematics education.

This article is an early step in the development of a methodological approach to the study of language deriving from an enactivist theoretical stance. Language is seen as a co-ordination of co-ordinations of action. Meaning and intention cannot easily be interpreted from the actions and words of others; instead, careful attention can be placed in not going beyond what is observable within the text itself, for example by focusing on patterns in word use. Conversations are highly ritualised affairs and from an enactivist perspective these rituals can be read in terms of pattern. The notion of the ‘structural coupling’ of systems, which will inevitably have taken place in a classroom, means that the history and context of communication needs to be taken into account. The methodological perspective put forward in this article is exemplified with an analysis of two classroom incidents (involving different teachers) in which almost identical words are used by the teachers, but markedly different things happen next. The analysis reveals a complexity within the classroom that, although available to direct observation, only became apparent using an approach to studying language that took account of the context and history of communication in a recursive process of data collection and analysis.

Key words: language, enactivism, mathematical learning, methodology