In this paper, we trace the development of our theorizing about students’ mathematical understanding, showing how the adoption of an enactivist perspective has transformed our gaze in terms of the objects of our studies and occasioned for us new methods of data analysis. Drawing on elements of Pirie–Kieren (P–K) Theory for the Dynamical Growth of Mathematical Understanding, together with aspects of improvisational theory and the associated notion of coactions, we describe the ways in which we have moved from a focus on the individual learner to that of the collective. In particular, we identify how our research methods and methodology have evolved to enable us to transform our data in ways that allow us to identify, consider, and discuss collective mathematical action. Using a brief transcription of an extract of video-recorded data in which three Grade 6 students work together to find the area of a parallelogram, we share and discuss successive iterations of our data analysis process. We identify the ways in which we manipulate and rework transcriptions of group discourse to reveal the relationship between enactivist thought and processes of engagement with data involving groups of mathematics learners.

Key words: enactivism, methodology, collective mathematical understanding

We reject a trajectory approach to teaching that classifies “good” and “bad” teaching actions and seeks to move teachers’ practices from one of these poles to the other. In this article we offer instead a conceptualisation of mathematics teaching actions as a “landscape of possibilities”. We draw together terms commonly used in the literature to describe teaching strategies, and add our own, to offer an expanded view of teaching actions. We illustrate each with data extracts drawn from our various studies of mathematics teachers and classrooms, and explain how a range of teaching actions can be woven into a coherent teaching practice. Note that we are not talking about growth in teaching in this paper, nor about change in teachers’ practice over time. We aim to simply talk about and conceptualise teaching in ways that can broaden our understanding of it.