Abrahamson D. & Trninic D. (2015) Bringing forth mathematical concepts: Signifying sensorimotor enactment in fields of promoted action. Mathematics Education%22\ title=\List all publications from ZDM Mathematics Education\>ZDM Mathematics Education 47(2): 295–306. Fulltext at https://cepa.info/6129

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.

Banting N. & Simmt E. (2017) From (Observing) Problem Solving to (Observing) Problem Posing: Fronting the Teacher as Observer. Constructivist Foundations 13(1): 177–179. Fulltext at https://cepa.info/4431

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.

Barwell R. (2009) Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics 72(2): 255–269. Fulltext at https://cepa.info/3731

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.

Bednarz N. & Proulx J. (2011) Ernst von Glasersfeld’s Contribution and Legacy to a Didactique des Mathématiques Research Community. Constructivist Foundations 6(2): 239–247. Fulltext at https://cepa.info/206

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.

Ben-Ari M. (2001) Constructivism in computer science education. Journal of Computers in Mathematics and Science Teaching 20(1): 45–73. Fulltext at https://cepa.info/3080

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.

Bikner-Ahsbahs A. & Prediger S. (2006) Diversity of theories in mathematics education: How can we deal with it? Zentralblatt für Didaktik der Mathematik 38(1): 52–57. Fulltext at https://cepa.info/3937

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.

Brown L. (2015) Researching as an enactivist mathematics education researcher. Mathematics Education%22\ title=\List all publications from ZDM Mathematics Education\>ZDM Mathematics Education 47(2): 185–196.

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. C. (2017) Francisco Varela’s Four Key Points of Enaction Applied to Working on Mathematical Problems. Constructivist Foundations 13(1): 179–181. Fulltext at 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?

Cifarelli V. V. (2017) Dynamic Connections between Problem Posing and Problem‑Solving: On the Usefulness of Multiple Perspectives. Constructivist Foundations 13(1): 172–173. Fulltext at https://cepa.info/4428

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: Drawing from the ideas of Varela, Proulx and Maheux, I propose a theoretical framework to examine problem-posing and problem-solving and provide evidence for their ideas with examples of student work from their research studies. I will draw comparisons between the approach taken by the researchers to the constructivist approach I have taken in my studies of problem-solving and those conducted with collaborators. My intent with these comments is not to argue the merits of one perspective over the other; rather, I look to point out and elaborate on these differences and make some specific suggestions to the researchers.

Cifarelli V. V. & Sevim V. (2014) Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis. Constructivist Foundations 9(3): 360–369. Fulltext at https://cepa.info/1093

Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated in the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed.