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.

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.

What evolutionary account explains our capacity to reason mathematically? Identifying the biological provenance of mathematical thinking would bear on education, because we could then design learning environments that simulate ecologically authentic conditions for leveraging this universal phylogenetic inclination. The ancient mechanism coopted for mathematical activity, I propose, is our fundamental organismic capacity to improve our sensorimotor engagement with the environment by detecting, generating, and maintaining goal-oriented perceptual structures regulating action, whether actual or imaginary. As such, the phenomenology of grasping a mathematical notion is literally that – gripping the environment in a new way that promotes interaction. To argue for the plausibility of my thesis, I first survey embodiment literature to implicate cognition as constituted in perceptuomotor engagement. Then, I summarize findings from a design-based research project investigating relations between learning to move in new ways and learning to reason mathematically about these conceptual choreographies. As such, the project proposes educational implications of enactivist evolutionary biology.

Key words: constructivism, design, education, enactivism, mathematics

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

Enactivism theorizes thinking as situated doing. Mathematical thinking, specifically, is handling imaginary objects, and learning is coming to perceive objects and reflecting on this activity. Putting theory to practice, Abrahamson’s embodied-design collaborative interdisciplinary research program has been designing and evaluating interactive tablet applications centered on motor-control tasks whose perceptual solutions then form the basis for understanding mathematical ideas (e.g., proportion). Analysis of multimodal data of students’ hand- and eye- movement as well as their linguistic and gestural expressions has pointed to the key role of emergent perceptual structures that form the developmental interface between motor coordination and conceptual articulation. Through timely tutorial intervention or peer interaction, these perceptual structures rise to the students’ discursive consciousness as “things” they can describe, measure, analyze, model, and symbolize with culturally accepted words, diagrams, and signs – they become mathematical entities with enactive meanings. We explain the theoretical background of enactivist mathematics pedagogy, demonstrate its technological implementation, list its principles, and then present a case study of a mathematics teacher who applied her graduate-school experiences in enactivist inquiry to create spontaneous classroom activities promoting student insight into challenging concepts. Students’ enactment of coordinated movement forms gave rise to new perceptual structures modeled as mathematical content.

A rising epistemological paradigm in the cognitive sciences – embodied cognition – has been stimulating innovative approaches, among educational researchers, to the design and analysis of STEM teaching and learning. The paradigm promotes theorizations of cognitive activity as grounded, or even constituted, in goal-oriented multimodal sensorimotor phenomenology. Conceptual learning, per these theories, could emanate from, or be triggered by, experiences of enacting or witnessing particular movement forms, even before these movements are explicitly signified as illustrating target content. Putting these theories to practice, new types of learning environments are being explored that utilize interactive technologies to initially foster student enactment of conceptually oriented movement forms and only then formalize these gestures and actions in disciplinary formats and language. In turn, new research instruments, such as multimodal learning analytics, now enable researchers to aggregate, integrate, model, and represent students’ physical movements, eye-gaze paths, and verbal–gestural utterance so as to track and evaluate emerging conceptual capacity. We – a cohort of cognitive scientists and design-based researchers of embodied mathematics – survey a set of empirically validated frameworks and principles for enhancing mathematics teaching and learning as dialogic multimodal activity, and we synthetize a set of principles for educational practice.

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

This paper is a report of a study conducted with preservice elementary teachers at the University of Wyoming during the summer of 1993. The study had two purposes: (1) to observe the effectiveness of using a constructivist approach in teaching mathematics to preservice elementary teachers, and (2) to focus on teaching probability using a constructivist approach. The study was conducted by one instructor in one class, The Theory of Arithmetic II, a required mathematics class for preservice elementary teachers.

Key words: educational methods, teacher education, concept formation, constructivist teaching, preservice teacher education, problem solving, classroom techniques, mathematical concepts, teacher education programs

The major purpose of this study was to investigate factors and challenges that hindered effective teaching of mathematics for understanding in senior secondary schools in the Omusati Education Region in Namibia. The study investigated how the participants dealt with identified challenges in the mathematics classrooms in selected senior secondary schools. Further, the study attempted to establish necessary support and / or training opportunities that mathematics teachers might need to ensure effective application of teaching mathematics for understanding in their regular classrooms. The sample was made up of eight senior secondary schools out of the population of 12 senior secondary schools in the Omusati Education Region. The schools were selected from the school circuits using maximum variation and random sampling techniques. Twenty out of 32 mathematics teachers from eight selected senior secondary schools in the Omusati Education Region responded to the interviews and two lessons per participant were observed. Interviews and observations were used to collect data from the 20 senior secondary school mathematics teachers with respect to teaching mathematics for understanding. Frequency tables, pie charts and bar graphs were used to analyze the data collected. The results indicated that teaching for understanding was little observed in mathematics classrooms. Part of the challenges identified were, overcrowded classrooms, lack of teaching and learning resources, lack of support from advisory teachers, and automatic promotions, among others. Mathematics teachers needed induction programmes, in-service training opportunities, and advisory services amongst others in order to be able to teach mathematics effectively. The study recommended that teaching for understanding should be researched in all subjects in Namibian classrooms and should be made clearly understood by all teachers in order to be able to use and apply it during their teaching. New teachers should be provided with induction programmes to give them support and tools at the beginning of their teaching careers. Further research on teaching for understanding should be conducted in other school subjects in Namibia in order to ensure teaching for understanding across the curriculum.

Key words: teaching, understanding.