Excerpt: A number of intentions triggered this research forum on enactivism and mathematics education research, and those are significant to highlight as they have in return structured the content and form that this forum takes. First, there has been and continues to be a substantial amount of research and writing on issues of enactivism undertaken by mathematics education researchers; thus we wanted to highlight and synthesize this body of research. At the same time, although much research has been conducted within the enactivist perspective, many of those contributions, and their authors, are not always well known and have often been seen merely as “interesting” orientations or “alternative” perspectives – but clearly not mainstream. Because we believe enactivism offers an insightful orientation which shows promise for enhancing our understanding of mathematics teaching and learning, we wanted to bring forth the nature and wide spectrum of enactivist contributions in order to share and create dialogue with the PME community about significant issues raised through this orientation. A third intention is in reaction to what might be thought of as a hegemony of constructivism in the mathematics education literature. We believe that enactivism, as a theory of cognition, offers a more encompassing and enlightening perspective on learning, teaching, and epistemology. Therefore, the following concerns will orient and be continuously present in the research forum unfoldings: retrospectives (as well as perspectives and prospectives) on research studies and writing done on enactivism in mathematics education will be shared; contributors will focus on insightful features that enactivism offers us; particularities of enactivism as a theory of cognition will permeate all discussions and presentations; and finally, but not least, interactions and discussions will take place about the ideas put forward.

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.