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Peripheral sensorimotor stimming activity, such as rocking and fidgeting, is widely considered irrelevant to and even distracting from learning. In this critical-pedagogy conceptual paper, we argue that stimming is an intrinsic part of adaptive functioning, interaction, and cognitive dynamics. We submit that when cultural resources build from students’ own sensorimotor dynamics, rather than subjugating them to hegemonic corporeal norms, learners’ intrinsic sensorimotor behaviors may be embraced and empowered as mental activity. This call for transformative inclusive pedagogy is of particular importance for neurodivergent children whose sensorimotor engagements have historically been ostracized as disruptive. Following a conceptual analysis of stimming that builds on a range of neuro-cognitive empirical studies drawing on post-cognitivist embodied cognition theory, we imagine inclusive educational futures that disrupt sedentary instructional design to elevate minoritized learners’ sensorimotor activity. As proof of concept, we present an example inclusive embodied activity, balance board math, a pedagogical tool designed to elicit stimming as thinking. We propose a set of design heuristics for realizing stimming’s pedagogical potential.

As AI increasingly enters classrooms, educational designers have begun investigating students' learning processes vis-à-vis simultaneous feedback from active sources—AI and the teacher. Nevertheless, there is a need to delve into a more comprehensive understanding of the orchestration of interactions between teachers and AI systems in educational settings. The research objective of this paper is to identify the challenges and opportunities when AI intertwines with instruction and examine how this hybrid teaching intelligence is being perceived by the students. The insights of this paper are extracted by analysing a case study that utilizes an AI-driven system (MOVES-NL) in the context of learning integer arithmetic. MOVES-NL is an advanced interactive tool that deploys whole-body movement and immediate formative feedback in a room-scale environment designed to enhance students' learning of integer arithmetic. In this paper, we present an in-situ study where 29 students in grades 6–8 interacted individually with MOVES-NL for approximately 1 hour each with the support of a facilitator/instructor. Mixed-methods analyses of multimodal data sources enabled a systematic multifaceted account of students' cognitive–affective experiences as they engaged with MOVES-NL while receiving human support (eg, by asking students to elaborate on their digital actions/decisions). Finally, we propose design insights for instructional and technology design in support of student hybrid learning. The findings of this research contribute to the ongoing discourse on the role of hybrid intelligence in supporting education by offering practical insights and recommendations for educators and designers seeking to optimize the integration of technology in classrooms. Practitioner notes What is already known about this topic Students and teachers develop different relations with and through AI, beyond just interacting with it. AI can support and augment the teachers' capabilities. Hybrid intelligence (HI) has already demonstrated promising potential to advance current educational theories and practices. What this paper adds This research identifies the important learning opportunities and adversities emerging when AI intertwines with instruction and examines how learners perceive those moments. The results show that the system and the facilitator's feedback were complementary to the success of the learning experience. AI-enabled students to reflect upon and test their previous knowledge and guided teachers to work with students to consolidate challenging topics. Findings provide insights into how the teacher–AI collaboration could engage and motivate students to reflect conceptually upon mathematical rules. Implications for practice and/or policy This study encourages practitioners and scholars to consider hybrid teaching intelligence when designing student-centred AI learning tools, focusing on supporting the development of effective teacher–AI collaborative technologies.

When teachers, researchers, and students describe productively responding to moments of failure in the learning process, what might this mean? Blending prior theoretical and empirical research on the relationship between failure and learning, and empirical results from four data sets that are part of a larger design-based research project, we investigate the heterogeneous processes teachers and students value and pursue following moments in which computer bugs thwart their immediate progress on an activity. These include: (1) resolving moments of failure; (2) avoiding recurring failures; (3) preparing for novel failures; (4) engaging with authority; and (5) calibrating confidence/efficacy. We investigate these processes taking into account the personal, social, and material context in which students and teachers collaborate when encountering broken computer programs, in addition to teachers’ planning efforts and the community’s reflections on past debugging experiences. We argue that moments of failure are not simply occasions for seeking resolutions. They are points of departure for decisions about how and what to foreground and interleave among a range of valued processes. Overall, this study aims to support research on the heterogeneous processes that shape how students new to a discipline such as computer programming respond to getting stuck.

This paper combines recent developments in theories of knowledge (complex dynamic systems), technologies (embodied interactions), and research tools (multimodal data collection and analysis) to offer new insights into how conceptual mathematical understanding can emerge. A complex dynamic system view models mathematics learning in terms of a multimodal agent who encounters a set of task constraints. The learning process in this context includes destabilizing a systemic configuration (for example, coordination of eye and hand movements) and forming new dynamic stability adapted to the task constraints. To test this model empirically, we applied a method developed to study complex systems, recurrence quantification analysis (RQA), to investigate students’ eye–hand dynamics during a touchscreen mathematics activity for the concept of proportionality. We found that across participants (n = 32), fluently coordinated hand-movement solutions coincided with more stable and predictable gaze patterns. We present a case study of a prototypical participant’s hand–eye RQA and audio–video data to show how the student’s cognitive system transitioned out of prior coordination reflective of additive thinking into a new coordination that can ground multiplicative thinking. These findings constitute empirical substantiation in mathematics education research for cognition as a complex system transitioning among dynamic equilibria.

> 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.

What could possibly be a meaningful conversation between educational researchers and movement scientists? Curiously, they have much in common. Both groups of researchers increasingly (1) appreciate the human capacity to enact perceptually guided movement as an overarching psychological model of thinking, problem-solving, and learning; (2) theorize the development of perceptual structures, including actual and imaginary percepts, as a key epistemic vehicle for solving motor-control problems; and (3) promote a view of abstract thinking as movement-grounded and movement-oriented perceptual dynamics. Probing toward theoretical synergy between these traditionally disparate fields of research, the present article is built as an interdisciplinary conversation between two researchers—of mathematics education and movement science, respectively—who become aware of their intellectual alignment, garner new insights and inspirations from each other’s work, and speculate on implications of this concordance for their fields. Future exploration into the unity of movement and cognition could enrich dialogue between manifold disciplines, with the overall goal of clarifying, developing, and integrating an interdisciplinary common foundation and framework for the benefit of education.

Two nascent lines of inquiry in the Learning Sciences are contributing to research and development of interactive digital resources for STEM education. One is embodied design, a research program to create theoretically driven and empirically validated technological learning environments where students ground STEM concepts in new perceptual capacity they develop through solving motor-control problems. The other is multimodal learning analytics, a methodological approach to investigating learning processes through gathering, analyzing, triangulating, and presenting data from multiple measures of students’ actions and sensations. This special issue looks at a set of articles reporting on pioneering efforts to coordinate these parallel lines of inquiry into a theoretically coherent research program informing an integrated design framework. The following editorial frames and motivates these research efforts, surveys the set of papers, and speculates on possible futures for the learning analytics of embodied design.

As educational approaches increasingly adopt digital formats, data logs create a newfound wealth of information about student behaviors in those learning environments for educators and researchers. Yet making sense of that information, particularly to inform pedagogy, remains a significant challenge. Data from digital sensors that sample at the millisecond level of granularity, such as computer mouses or touchscreens, is notoriously difficult to computationally process and mine for patterns. Adding to the challenge is the limited domain knowledge of this biological sensor level of interaction which prohibits a comprehensive manual feature engineering approach to utilize those data streams. In this paper, we attempt to enhance the assessment capability of a touchscreen-based tutoring system by using Recurrent Neural Networks (RNNs) to predict students’ strategies 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 classification models (including majority class) were trained and cross-validated at several levels on historical data which had been human coded with learners’ strategies. Our RNN approach to this difficult classification task moderately advances performance above logistic regression. We discuss the implications of this performance for enabling greater tutoring system autonomy. We also present visualizations that illustrate how this neural network approach to modeling sensor data can reveal patterns detected by the RNN. The surfaced patterns, regularized from a larger superset of mostly uncoded data, underscore the mix of normative and seemingly idiosyncratic behavior that characterizes the state space of learning at this high frequency level of observation.

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.

Although cognitive activity has been modeled through the lens of dynamical systems theory, the field lacks robust demonstrations in the learning of mathematical concepts. One empirical context demonstrating potential for closing this gap is embodied design, wherein students learn to enact new movement patterns that instantiate mathematical schemes. Changes in students’ perceptuomotor behavior in such contexts have been described as bearing markers of systemic phase transitions, but no research to date has characterized these dynamics quantitatively. This study applied a nonlinear analysis method, continuous cross-Recurrence Quantification Analysis (RQA), to touchscreen data excerpts from 39 study participants working with the Mathematics Imagery Trainer on the Parallel Bars problem. We then conducted linear regression analysis of a panel of five RQA metrics on learning phase (Exploration, Discovery, and Fluency) to identify how nonlinear dynamics changed as fluency developed. Results showed an increase in determinism from the Exploration to the Discovery phase, and an increase in recurrence rate, trapping time, mean line length, and normalized entropy from Discovery to Fluency phases. To put these dynamics in context, we qualitatively contrasted the RQA metric trajectories of two case study participants who developed different degrees of fluency. Our results support the hypothesized existence of phase transitions in the human–technology dynamical system during a math learning task. More broadly, this study illustrates the purchase of nonlinear methods on multimodal mathematics learning data and reveals perceptuomotor learning dynamics informative for the design and use of embodied-interaction technologies.

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.

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.

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.

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.

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.

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.

Constructing auxiliary lines is an important component skill in solving geometry problems, and yet it is difficult to teach, precisely because these lines are ‘invisible’ until they are actually drafted. Is there any intuitive resource that geometry students could possibly draw on to develop this skill? Is there any domain of human activity where we all naturally entertain imaginary lines, even if we are not aware of doing this? And yet, if so, how would these tacit imaginary lines come forth to be geometrical auxiliary lines? It turns out that dancers spontaneously imagine linear structures, known as attentional anchors, to help them enact their movements. These attentional anchors are drawn out in the dancer's subjective perception and are, therefore, invisible to others. Notwithstanding, we have used an embodied design-based research framework to create a gridded floor mat where students can render their covert dance-oriented attentional anchors as overt geometry-oriented auxiliary constructions. Situated in the cultural context of traditional Balinese dance, this practitioner-oriented paper demonstrates several activities for global classroom use by way of sharing some empirical results from implementing this pedagogical approach with young learners. An appendix lists a set of additional activities for dance-based geometry exploration. [ABSTRACT FROM AUTHOR]

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.

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.

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.

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.

Proportional reasoning is important and yet difficult for many students, who often use additive strategies, where multiplicative strategies are better suited. In our research we explore the potential of an interactive touchscreen tablet application to promote proportional reasoning by creating conditions that steer students toward multiplicative strategies. The design of this application (Mathematical Imagery Trainer) was inspired by arguments from embodied-cognition theory that mathematical understanding is grounded in sensorimotor schemes. This study draws on a corpus of previously treated data of 9-11 year-old students, who participated individually in semi-structured clinical interviews, in which they solved a manipulation task that required moving two vertical bars at a constant ratio of heights (1:2). Qualitative analyses revealed the frequent emergence of visual attention to the screen location halfway along the bar that was twice as high as the short bar. The hypothesis arose that students used so-called "attentional anchors" (AAs)-psychological constructions of new perceptual structures in the environment that people invent spontaneously as their heuristic means of guiding effective manual actions for managing an otherwise overwhelming task, in this case keeping vertical bars at the same proportion while moving them. We assumed that students' AAs on the mathematically relevant points were crucial in progressing from additive to multiplicative strategies. Here we seek farther to promote this line of research by reanalyzing data from 38 students (aged 9-11). We ask: (1) What quantitative evidence is there for the emergence of AAs?; and (2) How does the transition from additive to multiplicative reasoning take place when solving embodied proportions tasks in interaction with the touchscreen tablet app? We found that: (a) AAs appeared for all students; (b) the AA-types were few across the students; (c) the AAs were mathematically relevant (top of the bars and halfway along the tall bar); (d) interacting with the tablet was crucial for the AAs' emergence; and (e) the vast majority of students progressed from additive to multiplicative strategies (as corroborated with oral utterances). We conclude that touchscreen applications have the potential to create interaction conditions for coordinating action and perception into mathematical cognition.

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.

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.

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.

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.