This article examines the intellectual and institutional factors that contributed to the col- laboration of neuropsychiatrist Warren McCulloch and mathematician Walter Pitts on the logic of neural networks, which culminated in their 1943 publication, “A Logical Calculus of the Ideas Immanent in Nervous Activity.” Historians and scientists alike often refer to the McCulloch–Pitts paper as a landmark event in the history of cybernetics, and fundamental to the development of cognitive science and artificial intelligence. This article seeks to bring some historical context to the McCulloch–Pitts collaboration itself, namely, their intellectual and scientific orientations and backgrounds, the key concepts that contributed to their paper, and the institutional context in which their collaboration was made. Al- though they were almost a generation apart and had dissimilar scientific backgrounds, McCulloch and Pitts had similar intellectual concerns, simultaneously motivated by issues in philosophy, neurology, and mathematics. This article demonstrates how these issues converged and found resonance in their model of neural networks. By examining the intellectual backgrounds of McCulloch and Pitts as individuals, it will be shown that besides being an important event in the history of cybernetics proper, the McCulloch– Pitts collaboration was an important result of early twentieth-century efforts to apply mathematics to neurological phenomena.

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

Upshot: In view of Kenny’s clinical insights, Hug’s notes on the intricacies of rational vs. a-rational “knowing” in the design sciences, and Chronaki & Kynigos’s notice of mathematics teachers’ meta-communication on experiences of change, this response reframes the heuristic power of bisociation and suspension of disbelief in the light of Kelly’s notion of “as-if-ism” (constructive alternativism. Doing as-if and playing what-if, I reiterate, are critical to mitigating intra-and inter-personal relations, or meta-communicating. Their epistemic status within the radical constructivist framework is cast in the context of mutually enriching conversational techniques, or language-games, inspired by Maturana’s concepts of “objectivity in parenthesis” and the multiverse.

Excerpt: In 1995, the Leo Apostel Centre in Brussels, Belgium, organised an international conference called “Einstein meets Magritte”. Nobel prize winner Ilya Prigogine held the opening lecture at the conference, and Heinz von Foerster’s lecture was scheduled last… Heinz von Foerster was enchanted by the conference theme and – in the spirit of surrealist Belgian painter René Magritte – had chosen an appropriate title for his talk: “Ceci n’est pas Albert Einstein”. … [H]e was delighted to grant the organisers the following interview, in which he tells us about an even longer journey – that of his remarkable life and scientific career.

Key words: cybernetics, mathematics, interdisciplinarity, Heinz von Foerster

The paper deals with Aristotelian logic as the special case of more general epistemology and sociology of both science and common sense. The Aristotelian principles of identity, of noncontradiction, and of excluded middle are to be supplemented by the secondorder cybernetic, or cybernEthic principles of paradox, of ambivalence, and of control. In this paper we collect some ideas on how to evaluate the scope of Aristotelian logic with respect to the laws of thought they tried to determine and to do so within the historical moment of the impact of the invention of writing possibly triggering this determination. We look at some modern doubts concerning these laws and discovering an understanding of complexity that is not to be resumed under any principle of identity. The invention of sociology, epistemology, and the mathematics of communication follow suit in focusing not only on the observer but more importantly on the distinction between observers to further contextualize any talk of identities and operationalize both talk and fact of contradiction, paradox, and ambivalence.

Key words: Aristotle, Spencer-Brown, distinction, form, logic, mathematics, pistemology, sociology

Open peer commentary on the article “Problematizing: The Lived Journey of a Group of Students Doing Mathematics” by Robyn Gandell & Jean-François Maheux. Abstract: The aim of this commentary is to add to the conceptualization of problematizing by prompting a consideration of the teacher-observer as a wholly complicit, ethical participant in the mathematical journeys of their students.

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

Excerpt: The main thesis of this chapter is that the culture of teacher education must share the core characteristics of the desired mathematics classroom culture. In particular, this relationship must hold if teacher education is to exercise a reforming influence on in-service teachers and create a break in the circle of reproduction by which the weaknesses of the existing school system are perpetuated. It follows that more attention should be devoted to the relation between alternative classroom experiences and theoretical instruction, to the fundamental role of everyday language for the understanding of mathematics, the balance of self-directed work and the negotiation of meanings in seminars and tutorials, and to the development of reflection and self-monitoring.

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