Open peer commentary on the article “From Problem Solving to Problem Posing, and from Strategies to Laying Down a Path in Solving: Taking Varela’s Ideas to Mathematics Education Research” by Jérôme Proulx & Jean-François Maheux. Upshot: The aim of this commentary is to extend the work of Proulx and Maheux to include consideration of the teacher-observer whose role (in part) in the mathematics classroom is to ensure that curriculum goals are being met.

In this paper, we explore the implications of adopting (and developing the capacities necessary to adopt) an empathic second-person research perspective. Such a perspective aims to mediate participants’ access to their own experience, thereby providing a rich source of first-person data as well as a powerful pedagogical tool. Working within the enactivist tradition (Maturana and Varela 1987; Varela et al. 1991), we acknowledge and welcome the co-evolution and intertwining of awareness, description, and experience that such an approach necessarily entails, and we further note a blurring of the distinction between teacher and researcher that occurs as the research method prompts changes in the very aspects of experience we are observing. We begin by weaving together insights based on Varela’s “empathic coach” (Varela and Shear 1999) and Gendlin’s (1962, 1978, 1991) Philosophy of the Implicit and practice of Focusing. We describe how we developed and refined our own use of these methods to prompt and describe learners’ evolving experiences of mathematical doubt and certainty. We then further elaborate the nature of the empathic coach and close with a discussion of implications for teaching mathematics.

Key words: enactivism, complexity, second-person observer, pre-reflective awareness, mathematical experience of doubt and certainty

The paper explores three topics: interpretations and the world of significance; problem solving and problem posing; and knowledge as acquisition and knowledge as being. Trough these topics we illustrate the distance of enactivism from constructivism and the various extensions of it. After having discussed each theme and the issues they raise, we will highlight our appreciation of learning, knowledge, teaching and curriculum, proposing conceptualizations that offer paths of understandings and research.

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 the authors reflect on the contents of this current issue of ZDM and ask why focus an entire issue on enactivism as a research methodology in mathematics education. In their synthesis of the papers they distinguish and explicate what they observe as three moves in the enactivist research discussed. The first move (and the one that receives much of the attention in the papers) is that of the observer. Enactivism proposes the observer is one who arises in the act of observing and whose knowing is explained through the mechanism she describes. The second move is an understanding that all knowing is perceptually guided action that brings forth a world of significance. The third is a consequence of the first two: All knowing has implications. Hence that third move is towards ethics. The observer is not neutral; her observations bring forth worlds of significance that intersect with the worlds of others. They conclude, that the strength of enactivism as a methodological frame for mathematics education research is that it is a form of research that is incomplete. Incomplete in that from this framework there is necessarily always more to be said and different grounds for the saying about the phenomena under investigation in mathematics education.

Key words: enactivism, interaction, observer, methodology