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

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 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 article, I offer a synthesis and a clarification of important aspects of the constructivist theory of learning – under its “radical constructivist” version as developed and promoted by Ernst von Glasersfeld. Drawing from a few of the key principles of constructivism – specifically, concepts of fitting, of reality, of learning, and of the subjectivity-objectivity dialectic – I address issues that have emerged in recent efforts to adapt, assimilate, or apply the theory within varied educational and theoretical contexts. The point is then made that constructivism is not theory of teaching, but a theory of learning. As such, it is argued that constructivism brings a proscriptive discourse on teaching, one that sets boundaries in which to work, but does not prescribe teaching actions. In that sense, constructivism offers implications on pedagogy and on our ways in which we approach the teaching act, and not prescriptions or already made solutions.

The theory of cognition of Varela and Maturana differs in specific aspects from constructivist theories and so should not be seen or interpreted as another form of constructivism. To encourage the emergence of a discussion on important differences between both theories, this paper aims at highlighting three of these specific aspects, namely the biological roots of cognition, its phylogenic and ontogenic basis, and the nature of reality and knowledge. In many regards, it is possible that the first two points were seen as extensions of constructivism, and had not been theorized previously as distinctions, as is done in the paper. The third point concerning the ideas of “bringing forth a world” represents a clear conceptual shift from the visions inherent in constructivism, and should not be neglected in discussions on epistemology and the nature of knowledge and reality. This third fundamental point brings us to see Varela and Maturana as being different than constructivists, rather seeing them as “bring forthists.”

In this paper, I use Maturana and Varela’s (e.g., 1992) theoretical construct of structural determinism as a lens to better understand and discuss specific events of teachers’ learning or non-learning in various situations. Through excerpts from the literature and data from one of my projects, I illustrate teachers’ personal orientations that guide their potential learning. These interpretations have implications for teacher educators, who need to become more than facilitators or guides in order to trigger learning opportunities for/in teachers.

In this article, I present and build on the ideas of John Threlfall [(Educational Studies in Mathematics 50:29–47, 2002)] about strategy development in mental mathe- matics contexts. Focusing on the emergence of strategies rather than on issues of choice or flexibility of choice, I ground these ideas in the enactivist theory of cognition, particularly in issues of problem posing, for discussing the nature of the solving pro- cesses at play when solving mental mathematics problems. I complement this analysis and conceptualization by offering two examples about issues of emergence of strategies and of problem posing, in order to offer illustrations thereof, as well as to highlight the fruitfulness of this orientation for better understanding the processes at play in mental mathematics contexts.

Key words: mental mathematics, strategy development, emergence of strategies, enactivist theory of cognition.

In this article, I present and build on the ideas of John Threlfall [(Educational Studies in Mathematics 50:29–47, 2002)] about strategy development in mental mathematics contexts. Focusing on the emergence of strategies rather than on issues of choice or flexibility of choice, I ground these ideas in the enactivist theory of cognition, particularly in issues of problem posing, for discussing the nature of the solving processes at play when solving mental mathematics problems. I complement this analysis and conceptualization by offering two examples about issues of emergence of strategies and of problem posing, in order to offer illustrations thereof, as well as to highlight the fruitfulness of this orientation for better understanding the processes at play in mental mathematics contexts.

Key words: Mental mathematics, strategy development, emergence of strategies, enactivist theory of cognition

Open peer commentary on the article “Constructivist Model Building: Empirical Examples From Mathematics Education” by Catherine Ulrich, Erik S. Tillema, Amy J. Hackenberg & Anderson Norton. Upshot: The target article by Ulrich et al. is a good example of constructivist research in mathematics education, and illustrates how constructivism can ground a research endeavour toward modelling students’ mathematical understandings. I propose to delve into these issues of model building and reflect on Maturana’s notion of the observer. I do this through discussing issues about the language used in the article and of data description, and of analysis of students’ mathematical activity.