David A. Reid is a professor of mathematics education at the University of Agder. His research interests include teaching and learning mathematical proof and proving, and methodological implications of the work of Maturana and Varela. Examples of his work include the book Proof in Mathematics Education: Research, Learning and Teaching (2010), co-authored with Christine Knipping, and a special issue of the journal ZDM on “Enactivist methodology in mathematics education research” (2015), co-edited with Laurinda Brown, Alf Coles and Maria-Dolores Lozano.

Reid D. A. (1996) Enactivism as a methodology. In: Puig L. & Gutiérrez A. (eds.) Proceedings of the Twentieth Annual Conference of the International Group for the Psychology of Mathematics Education (PME-20), Volume 4. PME, Valencia: 203–210. https://cepa.info/2519

As research is learning, theories for learning and research methodologies in mathematics education overlap. For the Enactivist Research Group, enactivism is both the theoretical framework and the methodology for our research. Key ideas such as autopoesis, structure determinism, structural coupling, and coemergence are used to make sense of the learning of all participants in research, researchers included. This paper describes these key ideas and enactivist research methodology in mathematics education.

Reid D. A. (2007) Observations on an Observer’s Attachment to the Idea of Reality. Constructivist Foundations 3(1): 9–10. https://cepa.info/46

Open peer commentary on the target article “Arguments Opposing the Radicalism of Radical Constructivism” by Gernot Saalmann. First paragraph: As Maturana (e.g., 1987) has often reminded us, everything said is said by an observer. What I say here I say as an observer and reflects who I am, what I can perceive and what sense I am prepared to make of that. Similarly, what Gernot Saalmann says in his article is said by an observer and reflects who he is, what he can perceive and what sense he is prepared to make of that. I describe myself as an English speaking mathematics educator with a family background in neuroscience. Saalmann does not describe himself in his article but appears to be (to me as an observer) a German speaking sociologist. Given that we are different observers it is no surprise that we make different observations about radical constructivism.

Reid D. A. (2011) Enaction: An Incomplete Paradigm for Consciousness Science. Review of “Enaction: Toward a New Paradigm for Cognitive Science” edited by John Stewart, Olivier Gapenne and Ezequiel A. Di Paolo. Constructivist Foundations 7(1): 81–83. https://cepa.info/247

Upshot: According to its introduction, the aim of Enaction is to “present the paradigm of enaction as a framework for a far-reaching renewal of cognitive science as a whole.” While many of the chapters make progress towards this aim, the book as a whole does not present enactivism as a coherent framework, and it could be argued that enactivism’s embrace of phenomenology means it is no longer a theory of cognition.

Reid D. A. (2014) The coherence of enactivism and mathematics education research: A case study. Avant 5(2): 137–172. https://cepa.info/7105

This article addresses the question of the coherence of enactivism as a research perspective by making a case study of enactivism in mathematics education research. Main theoretical directions in mathematics education are reviewed and the history of adoption of concepts from enactivism is described. It is concluded that enactivism offers a ‘grand theory’ that can be brought to bear on most of the phenomena of interest in mathematics education research, and so it provides a sufficient theoretical framework. It has particular strength in describing interactions between cognitive systems, including human beings, human conversations and larger human social systems. Some apparent incoherencies of enactivism in mathematics education and in other fields come from the adoption of parts of enactivism that are then grafted onto incompatible theories. However, another significant source of incoherence is the inadequacy of Maturana’s definition of a social system and the lack of a generally agreed upon alternative.

Open peer commentary on the article “Problematizing: The Lived Journey of a Group of Students Doing Mathematics” by Robyn Gandell & Jean-François Maheux. Abstract: Describing group problematizing involves attending all-at-once to both individual and group thinking, without forgetting that what is being described is an ephemeral doing. I question whether Ingold’s metaphors of pathways and meshworks help or hinder our attending to these points and I offer an alternative way of describing group problematizing.

Reid D. A. & Mgombelo J. (2015) Key concepts in enactivist theory and methodology. ZDM 47(2): 171–183. https://cepa.info/2520

This article discusses key concepts within enactivist writing, focussing especially on concepts involved in the enactivist description of cognition as embodied action: perceptually guided action, embodiment, and structural coupling through recurrent sensorimotor patterns. Other concepts on which these concepts depend are also discussed, including structural determinism, operational closure, autonomy, autopoiesis, consensual domains, and cognition as effective action. Some related concepts that follow from an enactivist view of cognition are considered, in particular bringing forth a world and languaging. The use of enactivism as a methodology in mathematics education is also outlined. References to mathematics education research reported in this issue and elsewhere are used throughout to provide illustrations.

Reid D. A. & Mgombelo J. (2015) Survey of key concepts in enactivist theory and methodology. ZDM Mathematics Education 47(2): 171–183.

This article discusses key concepts within enactivist writing, focussing especially on concepts involved in the enactivist description of cognition as embodied action: perceptually guided action, embodiment, and structural coupling through recurrent sensorimotor patterns. Other concepts on which these concepts depend are also discussed, including structural determinism, operational closure, autonomy, autopoiesis, consensual domains, and cognition as effective action. Some related concepts that follow from an enactivist view of cognition are considered, in particular bringing forth a world and languaging. The use of enactivism as a methodology in mathematics education is also outlined. References to mathematics education research reported in this issue and elsewhere are used throughout to provide illustrations.