Open peer commentary on the article “Enactive Metaphorizing in the Mathematical Experience” by Daniela Díaz-Rojas, Jorge Soto-Andrade & Ronnie Videla-Reyes. Abstract: Welcoming their scholarly focus on metaphorizing, I critique Díaz-Rojas, Soto-Andrade and Videla-Reyes’s selection of the hypothetical constructs “conceptual metaphor” and “enactive metaphor” as guiding the epistemological positioning, educational design, and analytic interpretation of interactive mathematics education purporting to operationalize enactivist theory of cognition - both these constructs, I argue, are incompatible with enactivism. Instead, I draw on ecological dynamics to promote a view of metaphors as projected constraints on action, and I explain how mathematical concepts can be grounded in perceptual reorganization of motor coordination. I end with a note on how metaphors may take us astray and why that, too, is worthwhile.

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

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.

Research in mathematics education is a discursive process: It entails the analysis and production of texts, whether in the analysis of what learners say, the use of transcripts, or the publication of research reports. Much research in mathematics education is concerned with various aspects of mathematical thinking, including mathematical knowing, understanding and learning. In this paper, using ideas from discursive psychology, I examine the discursive construction of mathematical thinking in the research process. I focus, in particular, on the role of researchers’ descriptions. Specifically, I examine discursive features of two well-known research papers on mathematical thinking. These features include the use of contrast structures, categorisation and the construction of facts. Based on this analysis, I argue that researchers’ descriptions of learners’ or researchers’ behaviour and interaction make possible subsequent accounts of mathematical thinking.

Key words: Mathematical thinking, research methodology, discursive psychology, contrast structures, descriptions, epistemology

Context: During the 1980s, Ernst von Glasersfeld’s reflections nourished various studies conducted by a community of mathematics education researchers at CIRADE, Quebec, Canada. Problem: What are his influence on and contributions to the center’s rich climate of development? We discuss the fecundity of von Glasersfeld’s ideas for the CIRADE researchers’ community, specifically in didactique des mathématiques. Furthermore, we take a prospective view and address some challenges that new, post-CIRADE mathematics education researchers are confronted with that are related to interpretations of and reactions to constructivism by the surrounding community. Results: Von Glasersfeld’s contribution still continues today, with a new generation of researchers in mathematics education that have inherited views and ideas related to constructivism. For the post-CIRADE research community, the concepts and epistemology that von Glasersfeld put forward still need to be developed further, in particular concepts such as subjectivity, viability, the circular interpretative effect, representations, the nature of knowing, errors, and reality. Implications: Radical constructivism’s offspring resides within the concepts and epistemology put forth, and that continue to be put forth, through a large number of new and different generations of theories, thereby perpetuating von Glasersfeld’s legacy.

Key words: radical constructivism, mathematics education, viability, reification, representation

Constructivism is a theory of learning, which claims that students construct knowledge rather than merely receive and store knowledge transmitted by the teacher. Constructivism has been extremely influential in science and mathematics education, but much less so in computer science education (CSE). This paper surveys constructivism in the context of CSE, and shows how the theory can supply a theoretical basis for debating issues and evaluating proposals. An analysis of constructivism in computer science education leads to two claims: (a) students do not have an effective model of a computer, and (b) computers form an accessible ontological reality. The conclusions from these claims are that: (a) models must be explicitly taught, (b) models must be taught before abstractions, and (c) the seductive reality of the computer must not be allowed to supplant construction of models.

Key words: computer science education, dominant theory, idiosyncratic version, accessible ontological reality, ective model, theoretical basis, seductive reality, store knowledge, mathematics education

Within the context of mathematics education, the current view of evaluation is open to criticism first, in terms of the rather behavioristic classification of the learning outcomes it identifies, and second, with regards to the prevalent mode of obtaining information, the written test. A constructivist approach affects our perspective of both the learner’s and the teacher’s role in a didactical situation, and also that of the subject matter’. In such s perspective, the need for formative evaluation becomes crucial since in order to follow the student’s thinking, the teacher requires feedback from him. To this effect, we have developed a new tool, the mini-interview. This paper describes an experiment investigating the problems involved in training teachers in the use of this tool for formative purposes.

This article discusses the central question of how to deal with the diversity and the richness of existing theories in mathematics education research. To do this, we propose ways to structure building and discussing theories and we contrast the demand for integrating theories with the idea of networking theories.

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.