Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthropology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an “instrumented field of promoted action”. Children operating in this field first develop environmentally coupled motor-action coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disciplinary semiotic systems. We exemplify our thesis with two selected excerpts from our videography of Grade 4–6 volunteers participating in task-based clinical interviews centered on the Mathematical Imagery Trainer for Proportion. In particular, we present and analyze cases of either smooth or abrupt transformation in learners’ operatory schemes. We situate our design framework vis-à-vis seminal contributions to mathematics education research.

Key words: mathematics, education, motor learning, conceptual learning, motor problem, conceptual metaphor

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.

Key words: mathematics, teacher prospective, teacher collaborative group, structural coupling, mathematics lesson

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.

Key words: mathematics teacher development, awareness, purposes, collaborative group, enactivism, expertise.

This article is an early step in the development of a methodological approach to the study of language deriving from an enactivist theoretical stance. Language is seen as a co-ordination of co-ordinations of action. Meaning and intention cannot easily be interpreted from the actions and words of others; instead, careful attention can be placed in not going beyond what is observable within the text itself, for example by focusing on patterns in word use. Conversations are highly ritualised affairs and from an enactivist perspective these rituals can be read in terms of pattern. The notion of the ‘structural coupling’ of systems, which will inevitably have taken place in a classroom, means that the history and context of communication needs to be taken into account. The methodological perspective put forward in this article is exemplified with an analysis of two classroom incidents (involving different teachers) in which almost identical words are used by the teachers, but markedly different things happen next. The analysis reveals a complexity within the classroom that, although available to direct observation, only became apparent using an approach to studying language that took account of the context and history of communication in a recursive process of data collection and analysis.

Key words: language, enactivism, mathematical learning, methodology

As we witness a push toward studying spatial reasoning as a principal component of mathematical competency and instruction in the twenty first century, we argue that enactivism, with its strong and explicit foci on the coupling of organism and environment, action as cognition, and sensory motor coordination provides an inclusive, expansive, apt, and fit framework. We illustrate the fit of enactivism as a theory of learning with data from an ongoing research project involving teachers and elementary-aged children’s engagement in the design and assembly of motorized robots. We offer that spatial reasoning with its considerations of physical context, the dynamics of a body moving through space, sensorimotor coordination, and cognition, appears different from other conceptual competencies in mathematics. Specifically, we argue that learner engagements with diverse types of informationally ‘dense’ visuo-spatial interfaces (e.g., blueprints, programming icons, blocks, maps), as in the research study, afford some of the necessary experiences with/in a vast number of cases described by Varela et al. (1991) that enable the development of other mathematical competencies.

Key words: enactivism, sensorimotor coordination, spatial reasoning, robotics

My purpose in this paper is to illustrate the way in which an enactivist methodological approach guided me as I conducted a two-case longitudinal study where the learning of algebra was explored in different contexts throughout time. Three groups of students in two different schools in the city of Puebla, Mexico, were followed from the last year of primary school (Year 6) to the second year of their secondary education (Year 8). Learning was characterised as the ongoing structural change that allows individuals or groups to act effectively in a changing environment [Maturana (GAIA, a way of knowing: political implications of the new biology. Lindisfarne, New York, pp 65–82, 1987)]. An enactivist methodology, which revolves around the idea of research being a form of learning [Reid (Proceedings of the 20th conference of the international group for the psychology of mathematics education. PME, Valencia, pp 203–209, 1996)], implied that, as I carried out the study, what I was doing was learning about how people learn algebra. My initial questions arose from my experiences with the teaching and learning of mathematics, which gave me a sense of the complexity involved in the learning processes. Later, as I became immersed in the process of investigation of the teaching and learning of algebra, my conceptions continually evolved. The meaning of the phrase ‘algebraic learning’, which I used as a way of maintaining a wide perspective that allowed me to explore the events in the classroom in a complex way, arose as I engaged with enactivist ideas about learning, with the research literature on the learning of algebra, in conversations with people and in interactions with the participants of my project. I went through a process of continual development and change which I describe in this paper.

Key words: enactivism, algebraic learning

In this paper, we emphasize the methodological challenges of analyzing data with/in an enactivist framework in which (1) all knowing is doing is being, while focusing on, (2) how an entry by the observer transforms usual ways of analyzing data. In this sense, the association of knowing with doing suggests a shift in attention away from what students might “know” toward attending to the active, dynamic, enacted mathematical activity as knowing, bringing to bear the local and emergent character of the mathematical activity. In addition to illustrating data analysis along those lines, we discuss the methodological significances and challenges of, and paradigm shifts required for, replacing questions of knowledge and acquisition with ones that concern mathematical doing alone.

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

This paper explores the learning of both individuals and organizations within the context of a 3-year professional development program for mathematics and science teachers in a middle school. We propose to extend the notion of awareness from individuals to autonomous systems as a means to study the learning of teachers, mentors, the school, and the organization that provided the program. We describe how the notions of structural determinism and co-evolution through structural coupling informed the enactment of the program, as well as how this perspective informed the design of research on teachers’ experiences of their deepening understanding of mathematics for teaching during this time. Then we elaborate on the levels of awareness developed by teachers, mentors, the school, and the organization as a result of the constant interactions and mutual influence along and beyond the program. Data consisted of post-interviews with eleven mathematics teachers, our own reflections, and the documents generated during the program.

Key words: structural determinism, co-evolution, noticing, teacher educators, autonomous systems

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.

Key words: mathematics education, operational closure, mindfulness, meditation, mathematics education, research, structural coupling.