Recent developments in theory of cognitive sciences, interactive technological media, and empirical research instruments are enabling the Learning Sciences to investigate whether manipulation-based mathematics learning might be appropriate beyond early elementary school. Drawing on the embodiment turn in epistemology, the chapter supports and extends Piaget’s implication of sensorimotor activity as grounding conceptual development. If, as per enactivism, perception consists in perceptually guided action, and cognitive structures emerge from the recurrent sensorimotor patterns that enable action to be perceptually guided, then mathematics pedagogy should stage conditions that foster this learning process. After overviewing the rationales of embodied learning, the chapter discusses cumulative findings from a design-based research project evaluating the action-based genre of the embodied-design framework. Implementing embodied design would require systemic change in how we conceptualize cognition, what activities we create, how we facilitate these activities, how we prepare teachers, what classroom epistemic norms we sanction, and how we assess learning.

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.

In a retrospective analysis of my own pedagogical design projects over the past twenty years, I articulate and compare what I discern therein as two distinct activity genres for grounding mathematical concepts. One genre, "perception-based design," builds on learners' early mental capacity to draw logical inferences from perceptual judgment of intensive quantities in source phenomena, such as displays of color densities. The other genre, "action-based design," builds on learners' perceptuomotor capacity to develop new kinesthetic routines for strategic embodied interaction, such as moving the hands at different speeds to keep a screen green. Both capacities are effective evolutionary means of engaging the world, and both bear pedagogical potential as epistemic resources by which to build meaning for mathematical models of, and solution processes for situated problems. Empirical studies that investigated designs built in these genres suggest a two-step activity format by which instructors can guide learners to reinvent conceptual cores. In a primary problem, learners apply or develop non-symbolic perceptuomotor schemas to engage the task effectively. In a secondary problem, learners devise means of appropriating newly interpolated mathematical forms as enactive, semiotic, or epistemic means of enhancing, explaining, and evaluating their primary response. Whereas my analysis distills activities into two separate genres for rhetorical clarity, ultimately embodied interaction may interleave and synthesize the genres' elements. Copyright 2013 ACM.

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.

Design research is a broad, practice-based approach to investigating problems of education. This approach can catalyze the development of learning theory by fostering opportunities for transformational change in scholars’ interpretation of instructional interactions. Surveying a succession of design-research projects, I explain how challenges in understanding students’ behaviors promoted my own recapitulation of a historical evolution in educators’ conceptualizations of learning—Romantic, Progressivist, and Synthetic (Schön, Intuitive thinking? A metaphor underlying some ideas of educational reform (working paper 8). Division for Study and Research in Education, MIT, Cambridge, 1981)—and beyond to a proposed Systemic view. In reflection, I consider methodological adaptations to design-research practice that may enhance its contributions in accord with its objectives.

Metaphor is a useful way of explaining how to do things. The literature on metaphor in the learning of physical skill has generally explicated its efficacy by examining its actionable directives for motor enactment. And yet from the perspectives of phenomenological philosophy, ecological psychology, and enactivism, action is immanently intertwined with perception, so that models of metaphor-based learning should foreground the role of sensory activity modulating motor behavior. As such, metaphor is retheorized as a sensorial constraint one imaginarily projects into one’s action–perception phenomenological landscape. I present two metaphors from an instructional video on cello technique. Whereas these metaphors are couched in action language (what one should do), their potential impact, I argue, lies in emergent goal sensations (what one should feel). These explorative sensorimotor accommodations may, in turn, bring forth yet new scopes of latent sensations coupled to unanticipated performance possibilities, which suggest further modifying and calibrating enactment in the target domain. Attending to, achieving, and maintaining emergent intermediary goal sensations regulates instrumented action by forging new affordances that bring forth new motor coordination. As teacher and student co-imagine images for action, they should attend to sensory perceptions. And the same goes for scholars of metaphor.

Design researchers should inform the commercial production of educational technology by explicating their tacit design practice in workable structures and language. Two activity genres for grounding mathematical concepts are explained: "perception-based design" builds on learners' early mental capacity to draw logical inferences from perceptual judgment of intensive quantities in source phenomena, such as displays of color densities; "action-based design" builds on learners' perceptuomotor capacity to develop new kinesthetic routines for strategic embodied interaction, such as moving the hands at different speeds to keep a screen green. In a primary problem, learners apply or develop non-symbolic perceptuomotor schemas to engage the task effectively; In a secondary problem, learners devise means of appropriating newly interpolated mathematical forms as enactive, semiotic, or epistemic means of enhancing, explaining, and evaluating their primary response. In so doing, learners heuristically determine either inferential parity (perception-based design) or functional parity (action-based design) as epistemic grounds for reconciling naïve and scientific perspectives. Ultimately embodied-learning activities may interleave and synthesize the genres' elements. This taxonomy opens design practice into richer dialog with the learning sciences. An appendix lays out the embodied-design framework in a "how to" form amenable for replication both within the domain of mathematics and beyond.

Embodiment perspectives from the cognitive sciences offer a rethinking of the role of sensorimotor activity in human learning, knowing, and reasoning. Educational researchers have been evaluating whether and how these perspectives might inform the theory and practice of STEM instruction. Some of these researchers have created technological systems, where students solve sensorimotor interaction problems as cognitive entry into curricular content. However, the field has yet to agree on a conceptually coherent and empirically validated design framework, inspired by embodiment perspectives, for developing these instructional resources. A stumbling block toward such consensus, we propose, is an implicit disagreement among educational researchers on the relation between physical movement and conceptual learning. This hypothesized disagreement could explain the contrasting choices we witness among current designs for learning with respect to instructional methodology for cultivating new physical actions - whereas some researchers use an approach of direct instruction, such as explicit teaching of gestures, others use an indirect approach, where students must discover effective movements to solve a task. Prior to comparing these approaches, it may help first to clarify key constructs. In this theoretical essay we draw on embodiment and systems literature as well as findings from our design research so as to offer the following taxonomy that may facilitate discourse about movement in STEM learning: (1) distal movement is the technologically extended effect of physical movement on the environment; (2) proximal movement is the physical movements themselves; and (3) sensorimotor schemes are the routinized patterns of cognitive activity that become enacted through proximal movement by orienting on so-called attentional anchors. Attentional anchors are goal-oriented phenomenological objects or enactive perceptions ("sensori-") that organize proximal movement to effect distal movement ("-motor"). All three facets of movement must be considered in analyzing embodied learning processes. We demonstrate that indirect movement instruction enables students to develop new sensorimotor schemes including attentional anchors as idiosyncratic solutions to physical interaction problems. These schemes are, by necessity, grounded in students' own agentive relation to the world while also grounding target content such as mathematical notions.

Embodiment perspectives from the cognitive sciences offer a rethinking of the role of sensorimotor activity in human learning, knowing, and reasoning. Educational researchers have been evaluating whether and how these perspectives might inform the theory and practice of STEM instruction. Some of these researchers have created technological systems, where students solve sensorimotor interaction problems as cognitive entry into curricular content. However, the field has yet to agree on a conceptually coherent and empirically validated design framework, inspired by embodiment perspectives, for developing these instructional resources. A stumbling block toward such consensus, we propose, is an implicit disagreement among educational researchers on the relation between physical movement and conceptual learning. This hypothesized disagreement could explain the contrasting choices we witness among current designs for learning with respect to instructional methodology for cultivating new physical actions - whereas some researchers use an approach of direct instruction, such as explicit teaching of gestures, others use an indirect approach, where students must discover effective movements to solve a task. Prior to comparing these approaches, it may help first to clarify key constructs. In this theoretical essay we draw on embodiment and systems literature as well as findings from our design research so as to offer the following taxonomy that may facilitate discourse about movement in STEM learning: (1) distal movement is the technologically extended effect of physical movement on the environment; (2) proximal movement is the physical movements themselves; and (3) sensorimotor schemes are the routinized patterns of cognitive activity that become enacted through proximal movement by orienting on so-called attentional anchors. Attentional anchors are goal-oriented phenomenological objects or enactive perceptions ("sensori-") that organize proximal movement to effect distal movement ("-motor"). All three facets of movement must be considered in analyzing embodied learning processes. We demonstrate that indirect movement instruction enables students to develop new sensorimotor schemes including attentional anchors as idiosyncratic solutions to physical interaction problems. These schemes are, by necessity, grounded in students' own agentive relation to the world while also grounding target content such as mathematical notions.