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

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.

In this article we analyze the methodological commitments of a radical embodied cognition (REC) approach to social interaction and social cognition, specifically with respect to the explanatory framework it adopts. According to many representatives of REC, such as enactivists and the proponents of dynamical and ecological psychology, sociality is to be explained by (1) focusing on the social unit rather than the individuals that comprise it and (2) establishing the regularities that hold on this level rather than modeling the sub-personal mechanisms that could be said to underlie social phenomena. We point out that, despite explicit commitment, such a view implies an implicit rejection of the mechanistic explanation framework widely adopted in traditional cognitive science (TCS), which, in our view, hinders comparability between REC and these approaches. We further argue that such a position is unnecessary and that enactive mechanistic explanation of sociality is both possible and desirable. We examine three distinct objections from REC against mechanistic explanation, which we dub the decomposability, causality and extended cognition worries. In each case we show that these complaints can be alleviated by either appreciation of the full scope of the mechanistic account or adjustments on both mechanistic and REC sides of the debate.

Key words: enactivism, mechanistic explanation, social cognition

In this paper we examine an enactive approach to social cog- nition, a species of radical embodied cognition typically pro- posed as an alternative to traditional cognitive science. Ac- cording to enactivists, social cognition is best explained by reference to the social unit rather than the individuals that par- ticipate in it. We identify a methodological problem in this approach, namely a lack of clarity with respect to the model of explanation it adopts. We review two complaints about a mechanistic explanatory framework, popular in traditional cognitive science, that prevent enactivists from embracing it. We argue that these complaints are unfounded and propose a conceptual model of enactive mechanistic explanation of so- cial cognition.

Key words: enactivism, social cognition, mechanistic expla- nation

Open peer commentary on the article “Decentering the Brain: Embodied Cognition and the Critique of Neurocentrism and Narrow-Minded Philosophy of Mind” by Shaun Gallagher. Abstract: Simulation theory and theory theory are interaction accounts of theory of mind that have been neurocentrically characterized. A hybrid of these theories approximates the interaction theory of social cognition, and can be described in an indexical-symbolic processing framework.

This paper examines the shared roots and crossed paths between Jean Piaget’s constructivism, what Seymour Paper refers to as “constructionism,” and socio-cultural theories as epitomized by Lev Vygotsky. We do so in the light of more situated, pragmatic, and ecological approaches to human cognition. All these views are developmental (stressing the genesis children’s interests and abilities over time), experiential (in the sense that knowledge is rooted in sensori-motor activity) and interactionist (people are seen as constructing their knowledge by transforming the world). Yet, the views also differ, each highlighting some aspects of how children grow and learn, while leaving other questions unanswered. Piaget’s main contribution was to flesh out what is common in children’s ways of thinking at different stages of their cognitive development and, more important, how consistent, robust, and generally “adapted” their views are. The theory stresses the progressive de-contextualization of knowledge (from here-and-now to then-and-there) and identifies some of the hidden mechanisms (internal reorganizations) that drive human cognitive development. Papert, in contrast, stresses how individuals learn in context and how they use their own – and other people’s – externalizations as objects to think with, especially as their convictions break down. His approach is more situated. Papert is particularly interested the role of new media in human learning. Both Papert and Vygotsky shed light on the articulations between direct and mediated experience (from action and tool-use to enactments, language, and symbol-use). Yet Vygotsky and the Russian school have paid much closer attention to the role of caring adults and peers in a child’s initiation to her culture. They remind us that it takes a whole village to raise a child. Integrating the views helps rethink how children come to make sense of their experiences, and how they find their own places – and voices – in the world. At once world-makers, world-readers, and dwellers in the world, human infants are granted from birth with the abilities to optimize exchanges with people and things by moving in and out of contexts, by shifting perspectives, and by switching roles or standpoint. They are extraordinary learners, and much can be learned from them. Lastly, while mostly inner-driven and curious, children need caring adults, secure grounds, and engaging peers and props to thrive and grow. Tools, media, and cultural artifacts are the tangible forms through which they explore their surrounds, express their thoughts, and share the fun with others – and the traces left by those who came before (cultural heritage) become a terrain for newcomers to create their paths.

Key words: Constructivism, Piaget, Papert, Vygotsky, situated learning, embodied cognition, ecology of mind

Recent work in cognitive science has suggested that there are actual cases in which cognitive processes extend in the physical world beyond the bounds of the brain and the body. We argue that, while transcranial cognition may be both a logical and a nomological possibility, no case has been made for its current existence. In other words, we defend a form of contingent intracranialism about the cognitive.

Philosophical interest in situated cognition has been focused most intensely on the claim that human cognitive processes extend from the brain into the tools humans use. As we see it, this radical hypothesis is sustained by two kinds of mistakes, confusing coupling relations with constitutive relations and an inattention to the mark of the cognitive. Here we wish to draw attention to these mistakes and show just how pervasive they are. That is, for all that the radical philosophers have said, the mind is still in the head.