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Different approaches to embodied learning—conceptual learning of curricular content grounded in a new capacity for enacting forms of purposeful physical movement in interaction with the environment—have become increasingly central to mathematics-education research. This research forum provides participants with an up-to-date overview of diverse and complementary theoretical perspectives on embodied learning, principles derived from these perspectives governing the design of environments for learning various mathematical content, and demonstrations thereof. We speculate on promising directions for future embodied design research.

Research on technology and mathematics education has been a longstanding interest of the PME community. In this paper we revisit the interplay between technology and conjecturing within the process of problem-solving with an intention to capture different aspects of the processes in which students make and explore mathematical conjectures, and roles that both technology and teachers can play in this process. The focus is two-fold: first, to discuss different interpretations of conjectures and conjecturing within mathematics education, as reflected selected current works in mathematics education research; and second, to offer a discussion on progress in the implementation of these ideas with considerations of developments in technology, and our wider understandings of the role of the teachers.

Some forms of movement, such as rocking, are common means of sensory self-regulation yet are discouraged in classroom contexts. We analyze the ableist underpinnings of this approach, and ask instead: could sensory-attractive activities such as rocking instead be solicited to serve content learning? Our design-based research project, Balance Board Math, fosters sensory-cognitive opportunities for the regulatory movement of rocking to serve as an intrinsic interaction resource for exploring mathematical graphical representations, with a focus on properties of sinusoidal functions such as frequency and amplitude. We analyze the role of balance/vestibular activation in pilot participants' exploration of graphing before, during, and after Balance Board Math activities.

Designers of educational modules for conceptual learning often rely on procedural frameworks to chart out interaction mechanics through which users will develop target understandings. To date, however, there has been no systematic comparative evaluation of such frameworks in terms of their consequences for learning. This lack of empirical evaluation, we submit, is due to the intellectual challenge of pinning down in what fundamental sense these various frameworks differ and, therefore, along which parameters to conduct controlled comparative experimentation. Toward an empirical evaluation of educational-technology design frameworks, this conceptual paper considers the case of dynamic mathematics environments (DME), interactive modules for learning curricular content through manipulating virtual objects. We consider user activities in two paradigmatic DME genres that utilize similar HCI yet different mechanics. To compare these mechanics, we draw from complex dynamic systems theory a constraint-based model of embodied interaction. Task analyses suggest that whereas in one DME genre (GeoGebra) the interaction constraints are a priori inherent in the environment, in another DME genre (Mathematics Imagery Trainer) the constraints are ad hoc emergent in the task. We conjecture differential learning effects of these distinct constraint regimes, concluding that ad hoc emergent task constraints may better facilitate the naturalistic development of cognitive structures grounding targeted conceptual learning. We outline a future empirical research design to compare the pedagogical entailments of these two design frameworks.

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.

We report a subset of results from an exploratory study that modeled mathematics learning using a dynamical systems lens. This study applied Recurrence Quantification Analysis to model participants’ interactions with a touchscreen-based embodied-design learning environment for proportionality, conducting both qualitative (case study) and quantitative (linear regression) analyses. Findings indicate an abrupt change in the RQA meanline metric associated with increased fluency, suggesting a phase transition into a new mode of interaction. These findings suggest theoretical and methodological traction for modeling embodied math learning as phase transitions in a human–technology dynamical system.

In Peru, national assessments repeatedly rank Indigenous mathematics students as the lowest performing across the entirety of Latin America and South America. Whereas lack of financial resources often predicts low measures, the history of educational practice teaches us that students’ poverty need not predict their educational outcomes–creative instructional approaches may turn the tables. Here we report on an innovative, body-based arithmetic technique, the Botetano Method, that has been enabling poverty rural children from remote mountainous regions of Peru to match and even greatly surpass their urban peers on comparable test items. The article explains the method's guiding humanistic and cognitive principles and then reports on findings from explorative action research that implemented and evaluated the method. Using observational methodologies, we argue that the students developed in their conceptual understanding of the content as well as in their attraction to the discipline, their professional identity, their personal pride in their achievement, and their general epistemic capacity for concentration and self-regulation. Throughout, we emphasize the methodological limitations of this grassroots proof-of-concept action research, which threaten the validity of the assertions. We speculate on early extensions of the method to literacy studies.

As technologies that put the body at the center of mathematics learning enter formal and informal learning spaces, we still know little about the teaching methods educators can use to support students’ learning with these specialized systems. Drawing on ethnomethodology and conversation analysis (EMCA) and the Co-Operative Action framework, we present three multimodal ways that educators can be responsive to learners’ embodied ideas and help them transform sensorimotor patterns into mathematically significant perceptions. These techniques include (1) encouraging learners to use gesture to express and reflect on their ideas, (2) presenting multimodal candidate understandings to check comprehension of learners’ embodied ideas, and (3) co-constructing multimodally expressed embodied ideas with learners. We demonstrate how these techniques create opportunities for learning and discuss implications for a multimodal, embodied practice of responsive teaching.

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.

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.

The core of human connection is embodied action, with synchrony, coordinated movement, and affective attunement through the body present from infancy. Yet whereas all students have the capacity for co-presence, a common focus of formal educational institutions on spoken language for interaction makes communication inaccessible to some students, thus impeding their participation in learning. As such, toward providing resources for nonverbal autistic students we must ask: How do we design for inclusive social participation of students with diverse interactional modalities? This paper outlines the development of an embodied-design solution that centers the dynamic body as the nexus of social interaction, thus reclaiming the natural versatility of multimodal interpersonal communication. The Magical Musical Mat is a domain-general platform that allows people to interact through the non-speaking modalities of touch and sound. It removes interactional asymmetry between diverse interlocutors and surfaces the basic human need and capacity to connect with one another, in school and beyond.

Vygotsky conceptualized the teaching/learning process as inherently collaborative. We extend prior evaluations of this claim by enlisting eye-tacking instruments to monitor the perceptual activity of four teacher–student dyads, as the student solves a challenging manipulation problem designed to ground the scientific notion of parabolas in their new sensorimotor routines. Analyzing each dyad's gaze paths led us to model the teaching/learning process as the emergence and dynamic transformation of intersubjective coupling between the student and tutor perception–action systems. While the student's sensory-motor coordination gradually gravitates toward an effective routine, the tutor's perception is iteratively launched from the student's current action, until the tutor detects an optimal moment for verbal intervention. In this micro-zone of proximal development, the student's motor action comes to align with the tutor's cultural-perspective strategy. Our elaboration of the cultural–historical approach to teaching/learning draws on research on joint attention and joint action from the cognitive sciences as well as the embodied-design approach from the educational sciences and demonstrates a compatibility of Vygotsky's heritage and complex dynamic systems theory. Finally, we discuss the educational value of the observed student–tutor intersubjective coupling phenomena, thus grounding the contribution of this multidisciplinary study within educational concerns.

Blind and visually impaired mathematics students must rely on accessible materials such as tactile diagrams to learn mathematics. However, these compensatory materials are frequently found to offer students inferior opportunities for engaging in mathematical practice and do not allow sensorily heterogenous students to collaborate. Such prevailing problems of access and interaction are central concerns of Universal Design for Learning (UDL), an engineering paradigm for inclusive participation in cultural praxis like mathematics. Rather than directly adapt existing artifacts for broader usage, UDL process begins by interrogating the praxis these artifacts serve and then radically re-imagining tools and ecologies to optimize usability for all learners. We argue for the utility of two additional frameworks to enhance UDL efforts: (a) enactivism, a cognitive-sciences view of learning, knowing, and reasoning as modal activity; and (b) ethnomethodological conversation analysis (EMCA), which investigates participants’ multimodal methods for coordinating action and meaning. Combined, these approaches help frame the design and evaluation of opportunities for heterogeneous students to learn mathematics collaboratively in inclusive classrooms by coordinating perceptuo-motor solutions to joint manipulation problems. We contextualize the thesis with a proposal for a pluralist design for proportions, in which a pair of students jointly operate an interactive technological device.

This article concerns the purpose, function, and mechanisms of students’ rhythmic behaviors as they solve embodied-interaction problems, specifically problems that require assimilating quantitative information structures embedded into the environment. Analyzing multimodal data of one student tackling a bimanual interaction design for proportion, we observed the (1) evolution of coordinated movements with stable temporal–spatial qualities; (2) breakdown of this proto-rhythmic form when it failed to generalize across the problem space; (3) utilization of available resources to obtain greater specificity by way of measuring spatial spans of movements; (4) determination of an arithmetic pattern governing the sequence of spatial spans; and (5) creation of a meta-rhythmic form that reconciles continuous movement with the arithmetic pattern. The latter reconciliation selectively retired, modified, and recombined features of her previous form. Rhythmic enactment, even where it is not functionally imperative, appears to constitute a tacit adaptation goal. Its breakdown reveals latent phenomenal properties of the environment, creating opportunities for quantitative reasoning, ultimately supporting the learning of curricular content.

Sensor stream data, particularly those collected at the millisecond of granularity, have been notoriously difficult to leverage classifiable signal out of. Adding to the challenge is the limited domain knowledge that exists at these biological sensor levels of interaction that prohibits a comprehensive manual feature engineering approach to classification of those streams. In this paper, we attempt to enhance the assessment capability of a touchscreen based ratio tutoring system by using Recurrent Neural Networks (RNNs) to predict the strategy being demonstrated by students from their 60hz data streams. We hypothesize that the ability of neural networks to learn representations automatically, instead of relying on human feature engineering, may benefit this classification task. Our RNN and baseline models were trained and cross-validated at several levels on historical data which had been human coded with the task strategy believed to be exhibited by the learner. Our RNN approach to this historically difficult high frequency data classification task moderately advances performance above baselines and we discuss what implication this level of assessment performance has on enabling greater adaptive supports in the tutoring system.

© 2017, Springer Science+Business Media B.V., part of Springer Nature. Whereas some educational designers believe that students should learn new concepts through explorative problem solving within dedicated environments that constrain key parameters of their search and then support their progressive appropriation of empowering disciplinary forms, others are critical of the ultimate efficacy of this discovery-based pedagogical philosophy, citing an inherent structural challenge of students constructing historically achieved conceptual structures from their ingenuous notions. This special issue presents six educational research projects that, while adhering to principles of discovery-based learning, are motivated by complementary philosophical stances and theoretical constructs. The editorial introduction frames the set of projects as collectively exemplifying the viability and breadth of discovery-based learning, even as these projects: (a) put to work a span of design heuristics, such as productive failure, surfacing implicit know-how, playing epistemic games, problem posing, or participatory simulation activities; (b) vary in their target content and skills, including building electric circuits, solving algebra problems, driving safely in traffic jams, and performing martial-arts maneuvers; and (c) employ different media, such as interactive computer-based modules for constructing models of scientific phenomena or mathematical problem situations, networked classroom collective “video games,” and intercorporeal master–student training practices. The authors of these papers consider the potential generativity of their design heuristics across domains and contexts.

Forty 4th and 9th grade students participated individually in tutorial interviews centered on a problem-solving activity designed for learning basic algebra mechanics through diagrammatic modeling of an engaging narrative about a buccaneering giant burying and unearthing her treasure on a desert island. Participants were randomly assigned to experimental (Discovery) and control (No-Discovery) conditions. Mixed-method analyses revealed greater learning gains for Discovery participants. Elaborating on a heuristic activity architecture for technology-based guided-discovery learning (Chase and Abrahamson 2015), we reveal a network of interrelated inferential constraints that learners iteratively calibrate as they each refine and reflect on their evolving models. We track the emergence of these constraints by analyzing annotated transcriptions of two case-study student sessions and argue for their constituting role in conceptual development.

In this preliminary report, we propose a previously unidentified role that instructors’ gestures may play in helping students evaluate existing computer code. We find that instructors use gesture to animate processes encoded in the static inscriptions of computer programs in order to make invisible, dynamic phenomena perceptible to students. Our findings contribute to a better understanding of the embodied instructional work of teaching programming.

Inspired by the current embodiment turn in the cognitive sciences, researchers of STEM teaching and learning have been evaluating implications of this turn for educational theory and practice. But whereas design researchers have been developing domain-specific theories that implicate the role of physical movement in conceptual learning, the field has yet to agree on a conceptually coherent and empirically validated framework for leveraging and shaping students’ capacity for physical movement as a socio–cognitive educational resource. This symposium thus convenes to ask, “What is movement in relation to concepts such that we can design for learning?” To stimulate discussion, we highlight an emerging tension across a set of innovative technological designs with respect to the framing question of whether students should discover an activity’s targeted movement forms themselves or that these forms should be cued directly. Our content domains span mathematics (proportions, geometry), physics, chemistry, and ecological system dynamics (predator–prey, bees).

We present a detailed account of the interactional work between a programming instructor and a middle school student that leads to the resolution of an elusive error in the student’s code. By tracing the fine details of how this resolution came to be, we demonstrate how learning to debug in face-to-face interactions resembles a process of enskilment.

The use of video for in-service and pre-service teacher development has been gaining acceptance, and yet video remains a challenging and understudied tool. Many projects have used video to help pre-service and in-service teachers reflect on their own teaching processes, examine teacher–student interactions, and develop their professional vision. But rarely has video been used in ways more akin to qualitative education research that is focused on student learning. Even more rarely has this focus occurred at the earliest stages of pre-service teaching when students have not yet decided to pursue teaching careers. Yet here we argue that there are benefits to our approach. We examine a course for prospective pre-service math and science teachers at the University of California, Berkeley, that engages participants in qualitative video analysis to foster their reflective practice. This course is unique in that the prospective pre-service teachers engage in qualitative video analysis at a level characteristic of professional educational research, in that their analysis focuses on student learning of math and science content. We describe classroom activities that provide opportunities for the preservice teacher participants to better observe, notice, and interpret their students’ sociocognitive activity. The course culmination project involves participants developing and teaching lessons in a high school classroom. The participants then videotape the lessons and conduct qualitative video analysis. Results include detailed examples of two selected prospective pre-service teachers demonstrating coherent and effective approaches to conceptualizing the learning and teaching of mathematical and science content along with some potential design principles for building reflective practices through qualitative video projects. © 2018 Association for Science Teacher Education.

Theories of embodiment offer challenges to educational research and practice in ways that could potentially both reveal and support processes of teaching and learning in populations otherwise underserved. In particular, we focus on the 2017 conference theme Making a Difference: Prioritizing Equity and Access in CSCL by sharing with the CSCL community our varied approaches for designing learning contexts that provide diverse students movement-based experiential entry points to STEM content. In our pursuit, we recognize that core content notions may initially emerge for students through participating in problem-solving activities that complement traditional verbal and symbol sign systems with corporeal–dynamical modalities. Drawing on our workshop participants’ research goals, we will facilitate activities oriented on grasping key ideas for theory, methods, and design.

This paper presents a pedagogical agent designed to support students in an embodied, discovery-based learning environment. Discovery-based learning guides students through a set of activities designed to foster particular insights. In this case, the animated agent explains how to use the Mathematical Imagery Trainer for Proportionality, provides performance feedback, leads students to have different experiences and provides remedial instruction when required. It is a challenging task for agent technology as the amount of concrete feedback from the learner is very limited, here restricted to the location of two markers on the screen. A Dynamic Decision Network is used to automatically determine agent behavior, based on a deep understanding of the tutorial protocol. A pilot evaluation showed that all participants developed movement schemes supporting proto-proportional reasoning. They were able to provide verbal proto-proportional expressions for one of the taught strategies, but not the other.

Whereas emerging technologies, such as touchscreen tablets, are bringing sensorimotor interaction back into mathematics learning activities, existing educational theory is not geared to inform or analyze passages from action to concept. We present case studies of tutor–student behaviors in an embodied-interaction learning environment, the Mathematical Imagery Trainer. Drawing on ecological dynamics—a blend of dynamical-systems theory and ecological psychology—we explain and demonstrate that: (a) students develop sensorimotor schemes as solutions to interaction problems; (b) each scheme is oriented on an attentional anchor—a real or imagined object, area, or other aspect or behavior of the perceptual manifold that emerges to facilitate motor-action coordination; and (c) when symbolic artifacts are introduced into the arena, they may both mediate new affordances for students’ motor-action control and shift their discourse into explicit mathematical re-visualization of the environment. Symbolic artifacts are ontological hybrids evolving from things with which you act to things with which you think. Students engaged in embodied-interaction learning activities are first attracted to symbolic artifacts as prehensible environmental features optimizing their grip on the world, yet in the course of enacting the improved control routines, the artifacts become frames of reference for establishing and articulating quantitative systems known as mathematical reasoning.

The combination of two methodological resources-natural user interface and multimodal learning analytics-is creating opportunities for educational researchers to empirically evaluate theoretical models accounting for the emergence of concepts from situated sensorimotor activity. Seventy-six participants (9-14 years old) solved tablet-based presymbolic manipulation tasks designed to foster grounded meanings for the mathematical concept of proportional equivalence. Data gathered in task-based semi-structured clinical interviews included action logging, eye-gaze tracking, and videography. Analysis of these data indicates that successful task performance coincided with spontaneous emergence of stable dynamical gaze path patterns soon followed by multimodal articulation of strategy. Significantly, gaze patterns included unmanipulated, non-salient screen locations. We present cumulative evidence that these gaze patterns served as "attentional anchors" mediating participants' problem solving. By way of further contextualizing our claim, we also present case studies from the various experimental conditions. We interpret the findings as enabling us to revisit, support, refine, and perhaps elaborate on seminal claims from Piaget's theory of genetic epistemology and in particular his insistence on the role of situated motor-action coordination in the process of reflective abstraction.

Learning scientists have been considering the validity and relevance of arguments coming from philosophy and cognitive science for the embodied, enactive, embedded, and extended nature of individual learning, reasoning, and practice in sociocultural ecologies. Specifically, some design-based researchers of STEM cognition and instruction have been evaluating activities for grounding subject content knowledge in interactive sensorimotor problem solving. In so doing, we submit, the field stands greatly to avail of theoretical models and pedagogical methodologies from disciplines oriented explicitly on understanding, fostering, and remediating motor action. This conceptual paper considers potential values of ecological dynamics, a perspective originating in kinesiology, as an explanatory resource for tackling enduring Learning Sciences research problems. We support our position via an ecologicaldynamics reexamination of the function of metaphor in the instruction of sports skills, somatic awareness, and mathematics. We propose a view of metaphors as productive constraints reconfiguring the dynamic system of learner, teacher, and environment.

The digital age has introduced a host of new challenges and opportunities for the learning sciences community. These challenges and opportunities are particularly abundant in multimodal learning analytics (MMLA), a research methodology that aims to extend work from Educational Data Mining (EDM) and Learning Analytics (LA) to multimodal learning environments by treating multimodal data. Recognizing the short-term opportunities and longterm challenges will help develop proof cases and identify grand challenges that will help propel the field forward. To support the field's growth, we use this paper to describe several ways that MMLA can potentially advance learning sciences research and touch upon key challenges that researchers who utilize MMLA have encountered over the past few years.

There is increasing recognition amongst learning sciences researchers of the critical role that the body plays in thinking and reasoning across contexts and across disciplines. This workshop brings ideas of embodied learning and embodied cognition to the design of instructional environments that engage learners in new ways of moving within, and acting upon, the physical world. Using data and artifacts from participants' research and designs as a starting point, this workshop focuses on strategies for how to effectively leverage embodiment in learning activities in both technology and non-technology environments. Methodologies for studying/assessing the body's role in learning are also addressed.

We present a detailed account of interactional mechanisms that support participation in STEM disciplinary practices as an adult and a child explore a technology-enabled embodied learning environment for mathematics. Drawing on ethnomethodological studies of technologyrich workplaces, we trace the process of transforming a vague reference into a mutually available mathematical object: a covariant variable. Our analysis reveals that this mathematical object is an interactional achievement, configured via a reciprocal process of instructing one another's attention. In particular, we demonstrate how participants' explicit responsiveness to indexical and multimodal resources achieves this object.

The combination of two methodological resources-natural-user interfaces (NUI) and multimodal learning analytics (MMLA)-is creating opportunities for educational researchers to empirically evaluate seminal models for the hypothetical emergence of concepts from situated sensorimotor activity. 76 participants (9-14 yo) solved tablet-based non-symbolic manipulation tasks designed to foster grounded meanings for the mathematical concept of proportional equivalence. Data gathered in task-based semi-structured clinical interviews included action logging, eye-gaze tracking, and videography. Successful task performance coincided with spontaneous appearance of stable dynamical gaze-path patterns soon followed by multimodal articulation of strategy. Significantly, gaze patterns included uncued non-salient screen locations. We present cumulative results to argue that these 'attentional anchors' mediated participants' problem solving. We interpret the findings as enabling us to revisit, support, refine, and elaborate on central claims of Piaget's theory of genetic epistemology and in particular his insistence on the role of situated motor-action coordination in the process of reflective abstraction.

Scaffolding is the asymmetrical social co-enactment of natural or cultural practice, wherein a more able agent implements or performs for a novice elements of a challenging activity. What the novice may not learn, however, is how the expert’s co-enactments support the activity. Granted, in many cultural practices novices need not understand underlying process. But where process is content, such as mathematics, scaffolding is liable to undermine tenets of reform-oriented pedagogy. We point to tensions between traditional conceptualizations of scaffolding and discovery-based pedagogical methodology for mathematics education. Focusing on co-enactment as a critical feature of scaffolding activities, we introduce “reverse scaffolding”, wherein experts enact for novices only what they know to do rather than what they do not know to do. We demonstrate our approach by discussing a novel technological learning activity, Giant Steps for Algebra, wherein students construct models of realistic narratives. We argue for the method’s potential via reporting on findings from mixed-methods analyses of a quasi-experimental implementation with 40 students.

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.

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.

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.

Little is known about the collaborative learning processes of interdisciplinary teams designing technology-enabled immersive learning systems. In this conceptual paper, we reflect on the role of digitally captured embodied performances as boundary objects within our heterogeneous two-team collective of learning scientists and computer scientists as we design an embodied, animated virtual tutor embedded in a physically immersive mathematics learning system. Beyond just a communicative resource, we demonstrate how these digitized, embodied performances constitute a powerful mode for both inter- and intra-team learning and innovation. Our work illustrates the utility of mobilizing the material conditions of learning.

Can reflective practice help an interdisciplinary team collaborate? When a new team begins negotiating their working process toward achieving project deliverables, members implicitly bring diverse professional practices to the table. Once minimal common ground has been established, members specify what they each need from the other in order to implement their respective expertise. This process of imposing mutual constraints results in adjusted workflow protocols that may modify participants’ regular course of action yet are vital for facilitating the collaboration. Yet, we argue, this discursive process of negotiating collaboration protocols in interdisciplinary projects may result in more than just surface reconfiguration of local practices. The negotiation may yield an articulated reification of implicit know-how in the form of new theoretical constructs bearing potential impact beyond the local context of the project. We support the argument by presenting and analyzing archived records gathered from an interdisciplinary project, in which educational researchers and technology engineers collaborated in creating new instructional media for young mathematics students. In the course of struggling to formulate a mutually coherent workflow, the team ‘stepped back’ to formulate new goals that would address their coordination challenges. In turn, these goals implicated a new theoretical architecture that we present.

We contribute to research on visualization as an epistemic learning tool by inquiring into the didactical potential of having students visualize one phenomenon in accord with two different partial meanings of the same concept. 22 Grade 4-6 students participated in a design study that investigated the emergence of proportional-equivalence notions from mediated perceptuomotor schemas. Working as individuals or pairs in tutorial clinical interviews, students solved non-symbolic interaction problems that utilized remote-sensing technology. Next, they used symbolic artifacts interpolated into the problem space as semiotic means to objectify in mathematical register a variety of both additive and multiplicative solution strategies. Finally, they reflected on tensions between these competing visualizations of the space. Micro-ethnographic analyses of episodes from three paradigmatic case studies suggest that students reconciled semiotic conflicts by generating heuristic logico-mathematical inferences that integrated competing meanings into cohesive conceptual networks. These inferences hinged on revisualizing additive elements multiplicatively. Implications are drawn for rethinking didactical design for proportions. © 2013 FIZ Karlsruhe.

We examine expert human mathematics-tutor gestures in the context of an interactive design for proportionality in order to design a virtual pedagogical agent. Early results implicate distinct gesture morphologies serving consistent contextual functionalities in guiding learners towards quantitative descriptions of proportional relations.

When designers set out to create a mathematics learning activity, they have a fair sense of its objectives: students will understand a concept and master relevant procedural skills. In reform-oriented activities, students first engage in concrete situations, wherein they achieve situated, intermediary learning objectives (SILOs), and only then they rearticulate their solutions formally. We define SILOs as heuristics learners devise to accommodate contingencies in an evolving problem space, e.g., monitoring and repairing manipulable structures so that they model with fidelity a source situation. Students achieve SILOs through problem-solving with media, instructors orient toward SILOs via discursive solicitation, and designers articulate SILOs via analyzing implementation data. We describe the emergence of three SILOs in developing the activity Giant Steps for Algebra. Whereas the notion of SILOs emerged spontaneously as a framework to organize a system of practice, i.e. our collaborative design, it aligns with phenomenological theory of knowledge as instrumented action.

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.

In the course of developing an experimental algebra unit, the researchers noted variability in their design's instructional potential across a set of implementation media. In an effort to explain this variability, we revisited the classical theoretical construct of transparency. Transparency is the perceptual and conceptual accessibility of the mechanism, logic, and application of a tool. Corroborating earlier literature, it appears that participants saw only what they had built - transparency is a subjective achievement of a learner rather than an inherent feature of a device. Our first design prompted students with an algebraic proposition, for example "3x+2=4x-1". The two equivalent expressions were to be interpreted by students as alternative quantifications of a single linear spatial interval; namely, the path that a giant took on two separate occasions to bury and recover treasure. Problem solving required manually adjusting the modeling media to coordinate two types of equivalence: (a) the total length represented by each expression; and (b) the length represented by the variable and known units. The researchers found that successful coordination was predicated on the subjective transparency of the models' perceptual. Therefore, in redesigning the activity as a computer-based application we will have learners first construct tools and only then automatize them. Copyright 2013 ACM.

Coronary calcium as detected by electron beam computed tomography always signifies at least some atherosclerosis, appears to be correlated with coronary risk factors, cardiac history, and overall angiographic severity of disease, but is inconsistently related to degree of atherosclerotic lesion stenosis in a given artery. Increasing evidence, however, suggests an association between coronary artery calcium, atherosclerosis, and coronary risk. But atherosclerosis is a very common condition, its prevalence increasing with age. No fully validated method for determining the quantity of coronary calcium is available, and we do not know whether the amount of calcium is a consistently accurate reflection of the amount of atherosclerosis or whether the amount of atherosclerosis reflects the degree of risk. Furthermore, the prognostic significance of coronary calcium in any given atherosclerotic lesion is not yet established. What is clear from cohort studies, however, is that at least three quarters of asymptomatic individuals, at least half of whom would have "positive" coronary calcium electron beam computed tomographic scans, will live for at least 10 years without cardiac problems of any kind. Investigation is needed to determine whether medical intervention may impact the clinical outcome of the rest of those identified with a positive scan but destined to suffer future clinical events. Despite lack of validation, this test has widespread appeal, both to the public as a means of being able to find out the condition of their coronary arteries "without injections or dye" and to hospitals and private medical groups who view this both as an innovation in cardiovascular diagnosis and as a potentially profitable diagnostic procedure.(ABSTRACT TRUNCATED AT 250 WORDS)