This paper focusses on how researching is done through reflections about, or at a meta-level to, the practice over time of an enactivist mathematics education researcher. How are the key concepts of enactivist theory (ZDM Mathematics Education, doi: 10. 1007/s11858–014–0634–7, 2015) applied? This paper begins by giving an autobiographical account of the author’s engagement with enactivist ideas and the development of enactivist research projects. The rest of the paper then discusses principles of the design of enactivist followed by four themes of learning, observing, interviewing and find-ing(s). The spelling, find-ing(s), draws attention to the findings of enactivist research being processes not objects. In the case of the collaborative research group used as an exemplar throughout the paper, for instance, the find-ing(s) shed light onto the journeys of professional development travelled by the members of the group as they develop their teaching.
Brown L. & Coles A. (2011) Developing expertise: How enactivism re-frames mathematics teacher development. ZDM – Mathematics Education 43: 861–873. https://cepa.info/6861
Abstract In this article, we present a re-framing of tea- cher development that derives from our convictions regarding the enactive approach to cognition and the bio- logical 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 effec- tiveness 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.
Brown L. & Coles A. (2012) Developing “deliberate analysis” for learning mathematics and for mathematics teacher education: How the enactive approach to cognition frames reflection. Educational Studies in Mathematics 80: 217–231. https://cepa.info/6846
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.
Brown L. C. (2017) Francisco Varela’s Four Key Points of Enaction Applied to Working on Mathematical Problems. Constructivist Foundations 13(1): 179–181. https://cepa.info/4432
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: After a description of Varela’s four key points to a science of inter-being: embodiment, emergence, intersubjectivity and circulation, three questions are asked and briefly explored: Are these key points illustrated in the target article? What is a problem? And what could classrooms look like where knowing is doing?