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Background This article is focused on the role of the facilitator in the professional development of mathematics teachers. It is a feature of several frameworks for using video that teachers are invited to comment on the detail of what they saw and heard, and base comments on evidence from the video. It is also a common finding that it is hard to establish ‘speaking evidentially’ as a way of working and at the same time, such an expectation is seen to be critical to making discussion productive. This article draws on a paradigmatic case of one video club, during which seven teachers met over a 3-month period, and shared video recordings of their own classrooms. The group of teachers learn how to focus on the detail of video, avoiding judgement, from the first session. The way this is achieved and what it occasions is analysed, within an enactivist methodology. Results The analysis of the first meeting of the video club shows how the facilitator has a focus on the kinds of comments made by teachers as well as their content. If a teacher makes a contribution that is not of the ‘kind’ required within the way of working, the facilitator is observed to highlight that the participant is not focused on the detail of events, and re-direct the group back to the intended task. This sequence of moves is not captured in current frameworks of how to facilitate video discussion. There is a suggestion that it is the very distinction between observation and interpretation or judgement that is significant for teachers and allows them to re-think their own teaching practices. Conclusions This article aims to share awarenesses of the facilitator, concerning how it is possible to focus a group on the detail of events when using video. What the facilitator is doing, in highlighting when a norm has been breached, is making a judgement about the kind of comment made by participants, not judging the comment’s content. The meta-focus of the facilitator seems to allow enabling contraints to be placed on discussion while letting conversation follow the interests of participants.

The goal of this study is to characterise how pre-service teachers recognise students' reasoning, based in the unitising process, when students solve ratio comparison problems and, how they consider the unitising process as a Key Development Understanding to propose teaching decisions. Ninety-one preservice primary teachers interpreted primary school students' answers to a ratio comparison problem and proposed teaching decisions to improve students' reasoning. Findings indicate that pre-service teachers recognised features of the unitising process as a Key Development Understanding but had difficulties to propose teaching decisions addressed to improve the students' mathematical reasoning. These findings display the role played by specific mathematical content domain when pre-service teachers make teaching decisions considering the students' mathematical reasoning.

In this article, we present a re-framing of teacher development that derives from our convictions regarding the enactive approach to cognition and the biological 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 effectiveness 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.

A problem identified in the literature around task design is the persistence of a gap between teacher intention and student activity. We show how principles designed around the making of distinctions and having an explicit language of mathematical thinking can eliminate the ''gap'' by guiding teacher planning, teacher actions in the classroom and student activity in the classroom. We show how our task design principles have developed during the time of our collaboration, over a period of 20 years, across several research projects. We argue for the importance, in task design, of an explicit theory of change and an explicit image of mathematical thinking, where the theory of change is applicable to researchers, teachers and students.

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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.

This article addresses two main questions, how do mathematics teachers learn from using video? and, what is the role of the didactician? A common problem is reported in the difficulty of keeping teacher discussion of video away from judgment and evaluation. A review of mathematics education literature revealed four existing models for the use of video with teachers. Drawing on enactivist ideas (Rosch 1999), there are reasons why these models are likely to be productive and therefore suggestions for how teachers can learn from video. However, little is known about the role of the didactician in supporting learning. From empirical data, there is evidence that didacticians need to engage in a particular form of attention that I label a 'heightened listening' since there are simultaneous foci (on the content of teachers' contributions and what kind of a comment is being made), in order to establish discussion norms and to support the development of new ways of seeing and acting in the classroom.