Dor Abrahamson

New and radically reformative thinking about the enactive and embodied basis of cognition holds out the promise of moving forward age-old debates about whether we learn and how we learn. The radical enactive, embodied view of cognition (REC) poses a direct, and unmitigated, challenge to the trademark assumptions of traditional cognitivist theories of mind—those that characterize cognition as always and everywhere grounded in the manipulation of contentful representations of some kind. REC has had some success in understanding how sports skills and expertise are acquired. But, REC approaches appear to encounter a natural obstacle when it comes to understanding skill acquisition in knowledge-rich, conceptually based domains like the hard sciences and mathematics. This paper offers a proof of concept that REC’s reach can be usefully extended into the domain of science, technology, engineering, and mathematics (STEM) learning, especially when it comes to understanding the deep roots of such learning. In making this case, this paper has five main parts. The section “Ancient Intellectualism and the REC Challenge” briefly introduces REC and situates it with respect to rival views about the cognitive basis of learning. The “Learning REConceived: from Sports to STEM?” section outlines the substantive contribution REC makes to understanding skill acquisition in the domain of sports and identifies reasons for doubting that it will be possible to apply the same approach to knowledge-rich STEM domains. The “Mathematics as Embodied Practice” section gives the general layout for how to understand mathematics as an embodied practice. The section “The Importance of Attentional Anchors” introduces the concept “attentional anchor” and establishes why attentional anchors are important to educational design in STEM domains like mathematics. Finally, drawing on some exciting new empirical studies, the section “Seeing Attentional Anchors” demonstrates how REC can contribute to understanding the roots of STEM learning and inform its learning design, focusing on the case of mathematics.

n our response, we provide theoretical clarifications on how the functional dynamic systems approach allows a monist theorization of offline cognition and imagination; highlight the importance, for ontogenetic cultural development, of the system’s sensorimotor dynamics as well as this dynamics’ reification; elaborate on the notions of “field of promoted action,” “zone of proximal development,” “scaffolding,” and “repetition without repetition” as well as purported relations among these notions. We stress the transformative potential both of culture’s proleptic anticipation and, conversely, individuals’ non-linear rebellion against it. We also address questions regarding educational applicability, design, and classroom practices. Finally, we unpack the ethical and transformative role of our theoretical work in 21st-century educational science.

Context: Radical embodied approaches to cognition propose a drastic alternative to representation-based models of the mind by way of theorizing and empirically demonstrating the constitutive roles of perception-action loops in human behavior. However, applying those approaches to higher-order processes - such as mathematical thinking and learning - remains one of the hottest debates within contemporary cognitive science. Problem: How might a radical embodied perspective theoretically explain semiotic mediation? For example, how can we theorize the understanding of trigonometric relations expressed through the symbolic form of “sin 2a = sin a? Method: Revisiting the nature of semiotic mediation, we pursue a historically grounded theoretical analysis that integrates perspectives from Lev Vygotsky and Nikolai Bernstein; to make the theoretical proposal accessible, we illustrate it by empirical data from a dual-eye-tracking study on teaching/learning trigonometry. Results: We show how semiotic mediation of mathematical ideas is constituted as direct intercorporeal sensorimotor coordination between interlocutors. We treat semiotic actions as directly transforming an environment for the interlocutor, inviting new sensorimotor processes. New sensorimotor routines potentially lead to the emergence of pedagogically desired perception and action orientations, that is, new enactive capacity as the cognitive vehicle of mathematical reasoning. As such, Vygotskian cultural-historical ideas of semiotic mediation become a natural continuation of a radical embodied project developed by Bernstein, Kelso, Turvey, and others. Semiotic actions do not represent mind-independent reality as standalone tokens - rather, they present the environment itself, for the interlocutor, in a new way. Implications: The proposed perspective avoids ontologically problematic views of pedagogical discourse as the negotiation of minds, instead focusing educators’ attention on transformations of the students’ environment that foster desired perceptions and actions. Constructivist content: We develop an alternative to a social-constructivist reading of cultural-historical ideas, thus contributing to the understanding of higher-order cognition as direct extension of perception and action.

Whereas I empathize with Penny’s grave concern over current modalist instructional technology - “modalist” in the sense of privileging one modality, predominantly vision, at the expense of all others - I do not quite share his bleak assessment of future offerings. Following some hopefully inspiring words from historical philosophers of education, I showcase the Quad, a haptic-tactile mechatronic device built by three US-based laboratories collaborating to create modally expansive learning tools for classrooms that are inclusive of sensorially diverse students. While the Quad is “digital” in the familiar computational sense, it is at once “digital” in the corporeal sense of evoking the fingers - it reintroduces mutimodal engagement into mathematics learning.

This chapter is an overview of central research-based perspectives that support teaching-learning for understanding and for fluency. We summarize the Class Learning Path Model that integrates two theoretical foci – a Piagetian focus on learning and a Vygotskiian focus on teaching – and specifies phases in learning that reflect Vygotsky’s assertion about the move from spontaneous to scientific concepts. Major aspects of the model were drawn from national research-based reports. This model connects understanding and fluency with a focus on mathematically important but also accessible methods in the middle and on maths drawings and other supports for understanding these methods. Such methods can be generated by students and can bridge from less-advanced student methods to formal methods that are unnecessarily complex. For three maths domains in Grades Kindergarten through Grade 6, we illustrate and discuss methods in the middle and drawings that support these methods: problem solving and especially the full range of word problem situations with each quantity the unknown; multidigit addition, subtraction, multiplication, and division; and ratio and proportion. Central features of the Common Core State Standards Mathematical Practices in these domains are identified, and how these can support understanding and fluency are briefly discussed. Further aspects of how the pedagogical supports help students move through the Class Learning Path in their own individual ways, and implications for research and for designing maths programmes are then discussed.