### Learning is Moving in New Ways [CDME2018]

UCB Committee on Research: Faculty Research Grants, 2008-2011 [$21k]

**Overview**

At EDRL we have become increasingly interested in the idea that “learning is moving in new ways.” This phrase has two meanings: It is both about students learning new concepts by physically moving in new ways and about the field’s theoretical advances in conceptualizing the phenomenon of learning. Through our design-research studies, we are investigating for relations between moving and learning, particularly as these relations may be of relevance to mathematics education.

Since Fall 2008 we have been building and evaluating a type of interactive technological system we call the Mathematics Imagery Trainer. The idea is for students to discover and practice a new way of moving, a purposeful mirco-choreography that is situated in the immediate context of achieving some task objective involving a particular dynamical form of manipulating objects. For example, students sitting at a desk learn to move their hands up from the desk in such a way that the hands’ respective heights over the desk maintain some goal proportion (e.g., 1:2). It is not easy to move this way. One has to figure something out, usually through trial and error. All the while, the students receive automatic feedback on whether or not they are moving correctly. The children learn to move in a new way, and then they describe this motion. The descriptions become mathematical discourse expressing new concepts.

We are curious about these potential relations between learning to move in this new way and learning the mathematics that these actions might be said to model, such as proportional equivalence. Moreover, we are curious how teachers guide this process of learning to move in new ways; how they frame this new way of moving as a conceptual performance. That is, we are engaged in research on the educational transition from action to concept.

Our premise is that everyday life does not occasion opportunities for children to enact what could then serve as the meaning of some mathematical notions. For example, we can all intuitively gesture what “addition” means and, at a stretch, what “multiplication” means. But what does “proportion” mean or look like? *Our design principle is to create interaction problems wherein a student’s coordinated dynamical solution enacts a new sensorimotor scheme underlying the targeted concept*. For example, if we program the solution choreography to be proportional motion of two hands, students may develop principled strategies for enacting this physical coordination; if we further interpolate into the learning environment a set of mathematical symbolic artifacts (a Cartesian grid, numerals, etc.), then students may be able to mathematize these emerging strategies in the form of quantitative propositions.

Indeed, this is what we are seeing in empirical data from hundreds of Grade 4-6 students, who participated either in individual, paired, or classroom activities. To date we have worked with Wii, Kinect, and touchscreen (both tablet and large). Particularly exciting findings have come from using eye-tracking technology to study how new mathematical objects emerge into the child’s consciousness (in collaboration with researchers from Utrecht University). This project has also resulted in an NSF-funded effort to create a virtual teacher (GEVAMT).

We have published our findings in numerous peer-reviewed journal articles and conference proceedings, have received grants from federal agencies to fund this line of research, and have been increasingly collaborating on this line of research with other scholars, both in the US and worldwide.

This research on how goal-oriented action on objects gives rise to conceptual notions is theoretically fecund. It invites perspectives from diverse traditions of educational scholarship going back to John Dewey or Lev Vygotsky as well as the many design luminaries who created materials and activities for children to play and learn, such as Maria Montessori or Zoltan Diénès. The action/concept relation is treated by phenomenology philosophers, cognitive linguists, neuroscientists, kinesiologists, and many others. Theoretical models arising from this scholarship are often contested, and certainly their implications to educational practice are heatedly debated. As scientists like to say, “more research is needed.”

Our own research has brought us to the shores of Geneva, so to speak. That is, we realized that our findings resonate strongly with the work of Swiss cognitive-developmental psychologist Jean Piaget — his theory of genetic epistemology. In particular, we have come to characterize learning to move in new ways as developing sensorimotor schemes (how you move) oriented on constructed realities in the field of interaction (what you move). That is, students working with the Mathematical Imagery Trainer apparently develop two interlinked things at the same time: a new bimanual coordination for engaging the field of action, and a new phenomenal category upon or through which this coordination operates. Perhaps most striking are the data visualizations of students’ eye-gaze foci and paths as they figure out and master the interaction problem. We, the researchers, literally see new objects and gaze patterns emerge in the child’s interaction space that were not there before (INTED 2015, HD 2016). As students look at new places and in new ways, they get better at manipulating the interactive device. Soon after, they report on the new objects and patterns they are seeing, and then we help them make mathematical sense of these new objects by way of introducing mathematical frames of reference.

EDRL’s research has become quite interdisciplinary. Some of our closest collaborators are dancers, martial artists, and somatic therapists. The biggest recent boost came from the quarters of sports science. Currently, we are corroborating our findings, refining our theoretical models, developing principles for pedagogical design, and looking to expand our work to other concepts and domains. These are exciting times, and we are constantly inspired. It seems that our learning, too, is moving in new ways.

**Publications**

*Educational Designer*.

*Mathematics Imagery Trainer for Proportion*, students learn to perform pre-determined motor-control tasks by developing new perceptuomotor coordinations for enacting goal movements. In another program,

*Playground Physics*, students use real playground equipment, such as a swing, and virtual playground play performances in the app to determine their own task, such as swinging as high as possible. As such, task-initiation design considerations are tightly related to designers’ overall rationales that, in turn, emanate from assumptions concerning, for example, the epistemic constitution of STEM content, the affective allure of STEM practice, manifestations of agency in STEM problem solving, and other contextual details, such as logistical, architectural, institutional, and curricular constraints governing student and teacher experience.

Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020). The future of embodied design for mathematics teaching and learning [Original Research]. *Frontiers in Education*, 5(147). https://doi.org/10.3389/feduc.2020.00147

ABSTRACT: 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.

Abrahamson, D., Nathan, M. J., Williams–Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (in press). The future of embodied design for mathematics teaching and learning. In S. Ramanathan & I. A. C. Mok (Eds.), Futures of STEM education: Multiple perspectives from researchers (Special issue). *Frontiers of Education.*

ABSTRACT: 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.

Flood, V. J., Shvarts, A., & Abrahamson, D. (2020). Teaching with embodied-design technologies for learning mathematics. *ZDM Mathematics Education*.

Shvarts, A., & Abrahamson, D. (2019). Dual-eye-tracking Vygotsky: A microgenetic account of a teaching/learning collaboration in an embodied-interaction technological tutorial for mathematics. *Learning, Culture, and Social Interaction*, 22, 100316. https://doi.org/10.1016/j.

ABSTRACT: 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 simultaneously with their verbal and gestural utterance 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 towards an effective routine, the tutor’s perception is iteratively launched from the student’s current position, 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; the intervention is thus accomplished by smoothly soliciting from the student a new conceptualization and articulation of their emerged sensory-motor coordination. 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.

Abrahamson, D., Flood, V. J., Miele, J. A., & Siu, Y.-T. (2019). Enactivism and ethnomethodological conversation analysis as tools for expanding Universal Design for Learning: The case of visually impaired mathematics students. *ZDM Mathematics Education, 51*(2), 291-303. https://doi.org/10.1007/s11858-018-0998-1

ABSTRACT: 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.

Abrahamson, D., & Shulman, A. (2017). Constructing movement in mathematics and dance: An interdisciplinary pedagogical dialogue on subjectivity and awareness. *Feldenkrais Research Journal*.

ABSTRACT: A physical movement can be construed in many ways. For some researchers of mathematics education informed by embodiment theories this is important, as they perceive a mathematical concept as a polysemous structure grounded in multiple interrelated sensorimotor constructions. In this dance is no different. Similarly in both disciplines, the more ways one has of thinking about a movement and the more connections one builds across these different constructions, the deeper and richer one’s understanding and proficiency in enacting the movement and the greater one’s capacity to transpose the learning to new contexts. In both mathematics and dance, instructors thus seek to create conditions for students to develop diverse subjective constructions of the movements they are learning to enact and to explore relations across these different constructions. Any pedagogical discussion of movement, whether in dance or mathematics, must be a discussion of the individual’s subjective phenomenology and increasing awareness. In reflection, the very possibility of the authors’ interdisciplinary dialogue is testimony to the cohesive potential in systemic conceptualizations of human movement.

Palatnik, A., & Abrahamson, D. (2018). Rhythmic movement as a tacit enactment goal mobilizing the emergence of mathematical structures. *Educational Studies in Mathematics, 99*(3), 293–309.

ABSTRACT: 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: (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.

Rosen, D. M., Palatnik, A., & Abrahamson, D. (2018). A better story: An embodied-design argument for generic manipulatives. In N. Calder, N. Sinclair, & K. Larkin (Eds.), *Using mobile technologies in the learning of mathematics *(pp. 189-211). New York: Springer.

ABSTRACT: Mathematics education practitioners and researchers have long debated best pedagogical practices for introducing to students new concepts. We worked with interactive technology of our own design to evaluate an instructional methodology whereby students: (1) first learn to move physically in a new way by solving a coordination problem of moving two virtual objects at the same time, one in each hand; and only then (2) ground these new movements in informal narratives about the objects; as well as (3) generalize these movements as formal mathematical rules. We compared students’ learning gains under two conditions of this methodology, where the virtual objects were either generic (non-representational, not signifying specific contexts, e.g., a circle) or situated (representational, signifying specific contexts, e.g., a hot-air balloon). We summarize findings from analyzing the behaviors of 25 Grade 4 – 6 students who participated individually in tutorial trials with one of the authors. The situated objects gave rise to richer stories than the generic objects, presumably because the students could bring to bear their everyday knowledge of these objects’ properties, scenarios, and typical behaviors. However, in so doing, the students treated the objects’ movements only as framed by those particular stories rather than considering other possible interpretations of those movements, even when those other interpretations would have advanced the learning process, such as by creating productive challenges. We argue that these findings can be explained in light of theoretical models that conceptualize knowledge as emerging from goal-oriented sensorimotor interaction with objects in the environment. We caution that designers and teachers should be aware of the double-edged sword of rich situativity: Familiar objects are perhaps more engaging but can also limit the scope of learning. We advocate for our instructional methodology of entering mathematical concepts through the action level.

Abrahamson, D., & Shulman, A. (2017). *Constructing movement in mathematics and dance: An interdisciplinary pedagogical dialogue on subjectivity and awareness*. Paper presented at the annual meeting of Movement: Brain, Body, Cognition, Oxford, UK.

ABSTRACT: A physical movement can be construed in many ways. For some researchers of mathematics education informed by embodiment theory this is important, as they perceive a mathematical concept as a polysemous structure grounded in multiple interrelated sensorimotor constructions. In this dance is no different. Similarly in both disciplines, the more ways one has of thinking about a movement and the more connections one builds across these different constructions, the deeper and richer one’s understanding and proficiency in enacting the movement and the greater one’s capacity to transpose the learning to new contexts. In both mathematics and dance, instructors thus seek to create conditions for students to develop diverse subjective constructions of the movements they are learning to enact and to explore relations across these different constructions. Any pedagogical discussion of movement, whether in dance or mathematics, must be a discussion of the individual’s subjective phenomenology and increasing awareness. In reflection, the very possibility of the authors’ interdisciplinary dialogue is testimony to the cohesive potential in systemic conceptualizations of human movement.

Palatnik, A., & Abrahamson, D. (2017). Taking measures to coordinate movements: Unitizing emerges as a method of building event structures for enacting proportion. In E. Galindo & J. Newton (Eds.), *“Synergy at the crossroads” — Proceedings of the 39th annual conference of the North-American chapter of the International Group for the Psychology of Mathematics Education* (Ch. 13 [Theory and research methods], pp. 1439-1442). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.

ABSTRACT: Rhythm is a means of production—a scheme for coordinating the enactment of real or imagined physical movements over time, space, material resources, and concerting participants. In activities requiring the coordination of two or more continuous motor actions, rhythmic re-assembly of the actions creates a goal event structure mediating the enactment. Yet building that structure requires first unitizing continuity. Unitizing could thus be conceptualized as a cultural–historical strategy for supporting mundane routines by parsing, distributing, and codifying activity as a sequence of iterated actions of equivalent magnitude. Ipso facto, unitizing shifts us from naive to disciplinary activity: articulated rhythm is an ontogenetic achievement driving cognitive growth. We present empirical data of a student spontaneously measuring continuous actions as her means of organizing the enactment of a bimanual task designed for proportions.

Duijzer, A. C. G., Shayan, S., Van der Schaaf, M. F., Bakker, A., & Abrahamson, D. (2017). Touchscreen tablets: Coordinating action and perception for mathematical cognition. *Frontiers in Psychology, 8*(144).

ABSTRACT: 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 encourage transitions to 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 attentional anchors?; 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 utterance). We conclude that touchscreen applications have the potential to create interaction conditions for coordinating action and perception for mathematical cognition.

Abrahamson, D., & Bakker, A. (2016). Making sense of movement in embodied design for mathematics learning. In N. Newcombe & S. Weisberg (Eds), Embodied cognition and STEM learning (Special issue). *The Psychonomic Society—Cognitive Research: Principles and Implications (CRPI), 1*(1), Article #33.

**Abstract**: 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: (a) distal movement is the technologically extended effect of physical movement on the environment; (b) proximal movement is the physical movements themselves; and (c) sensorimotor schemes are the routinized patterns of cognitive activity that becomes 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.

**Significance**: Engineering developments in computational technology have created unprecedented opportunities for industry to build and disseminate mathematics-education applications (“apps”). Thousands of these applications are now literally at the fingertips of any child who can access a tablet, smartphone, or personal computer with responsive touchscreen. Educational researchers could contribute to the quality of these ubiquitous consumer products by offering design frameworks informed by theories of learning. However, existing frameworks are derived from interaction theories drawing on epistemological assumptions that are no longer tenable, given the embodiment turn in the cognitive sciences. A proposed systemic reconceptualization of mathematical objects as grounded in sensorimotor schemes for material interaction offers educational designers heuristics for creating activities in which students learn by discovering motion patterns.

Shayan, S., Abrahamson, D., Bakker, A., Duijzer, A. C. G., & Van der Schaaf, M. F. (2017). Eye-tracking the emergence of attentional anchors in a mathematics learning tablet activity. In C. A. Was, F. J. Sansosti, & B. J. Morris (Eds.), *Eye-tracking technology applications in educational research* (pp. 166-194). Hershey, PA: IGI Global.

ABSTRACT: Little is known about micro-processes by which sensorimotor interaction gives rise to conceptual development. Per embodiment theory, these micro-processes are mediated by dynamical attentional structures. Accordingly this study investigated eye-gaze behaviors during engagement in solving tablet-based bimanual manipulation tasks designed to foster proportional reasoning. Seventy-six elementary- and vocational-school students (9-15 yo) participated in individual task-based clinical interviews. Data gathered included action-logging, eye-tracking, and videography. Analyses revealed the emergence of stable eye-path gaze patterns contemporaneous with first enactments of effective manipulation and prior to verbal articulations of manipulation strategies. Characteristic gaze patterns included consistent or recurring attention to screen locations that bore non-salient stimuli or no stimuli at all yet bore invariant geometric relations to dynamical salient features. Arguably, this research validates empirically hypothetical constructs from constructivism, particularly reflective abstraction.

Morgan, P., & Abrahamson, D. (2016). Cultivating the ineffable: The role of contemplative practice in enactivist learning.* For the Learning of Mathematics,**36*(3), 31-37.

ABSTRACT: We consider designs for conceptual learning where students first engage in pre-symbolic problem solving and then articulate their solutions formally. An enduring problem in these designs has been to support students in accessing their pre-symbolic situated process, so that they can reflect on it and couch it in mathematical form. Contemplative practices may offer practical solutions to this epistemic bottleneck by orienting students not on their explicit thoughts but on nuanced somatic sensations within pre-conceptual liminal space. We support this proposal for a contemplative mathematics program by drawing on Mason and Roth and by citing findings from pilot studies.

Rosen, D. M., Palatnik, A., & Abrahamson, D. (2016). Tradeoffs of situatedness: Iconicity constrains the development of content-oriented sensorimotor schemes. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), *Sin fronteras: Questioning borders with(in) mathematics education – Proceedings of the 38th annual meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA)* (Vol. 12, “Technology”, pp. 1509-1516). Tucson, AZ: University of Arizona.

ABSTRACT: Mathematics education practitioners and researchers have long debated best pedagogical practices for introducing new concepts. Our design-based research project evaluated a heuristic framework, whereby students first develop acontextual sensorimotor schemes and only then extend these schemes to incorporate both concrete narratives (grounding) and formal mathematical rules (generalizing). We compared student performance under conditions of working with stark (acontextual) vs. iconic (situated) manipulatives. We summarize findings from analyzing 20 individually administered task-based semi-structured clinical interviews with Grade 4 – 6 participant students. We found tradeoffs of situatedness: Whereas iconic objects elicit richer narratives than stark objects, these narratives may detrimentally constrain the scope of potential sensorimotor schemes students develop in attempt to solve manipulation problems.

Abrahamson, D., Sánchez-García, R., & Trninic, D. (2016). Praxes proxies: Revisiting educational manipulatives from an ecological dynamics perspective. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), *Sin fronteras: Questioning borders with(in) mathematics education – Proceedings of the 38th annual meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA)* (Vol. 13, “Theory and research methods”, pp. 1565-1572). Tucson, AZ: University of Arizona.

ABSTRACT: The recent proliferation of technological devices with natural user interfaces (e.g., touchscreen tablets) is regenerating scholarship on the role of sensorimotor interaction in conceptual learning. Some researchers of mathematical education have adopted views from constructivism, phenomenology, enactivism, and ecological dynamics to interpret implicit sensorimotor schemes as both forming and manifesting disciplinary competence. Drawing on these views, this theoretical paper discusses what it means to develop a new skill by way of task-oriented interaction with objects. Beginning with sports then moving to mathematics, we focus on a subcategory of pedagogical artifacts that serve students as effective proxies for developing normative application of target schemes. We argue for the contribution of these views to designing artifacts for learning mathematical content.

Flood, V. J., Harrer, B. W., & Abrahamson, D. (2016). The interactional work of configuring a mathematical object in a technology-enabled embodied learning environment. In C.-K. Looi, J. L. Polman, U. Cress, & P. Reimann (Eds.), *“Transforming learning, empowering learners,” Proceedings of the International Conference of the Learning Sciences (ICLS 2016)* (Vol. 1, “Full Papers”, pp. 466-473). Singapore: International Society of the Learning Sciences.

ABSTRACT: We present a detailed account of interactional mechanisms that support participation in STEM disciplinary practices as an adult and a child explore a technologyenabled embodied learning environment for mathematics. Drawing on ethnomethodological studies of technology-rich 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.

Abrahamson, D., & Sánchez-García, R. (2016). Learning is moving in new ways: The ecological dynamics of mathematics education. *Journal of the Learning Sciences, **25*(2), 203-239.

ABSTRACT: 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 you act with to things you think with. 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. [Shorter version appears in PME-NA 2015; See also ICLS 2016 on metaphors as constraints on action.]

Abrahamson, D., Shayan, S., Bakker, A., & Van der Schaaf, M. F. (2016). Eye-tracking Piaget: Capturing the emergence of attentional anchors in the coordination of proportional motor action. *Human Development, 58*(4-5), 218-244.

ABSTRACT: The combination of two methodological resources—natural-user interfaces (NUI) and multimodal learning analytics (MMLA)—is creating opportunities for educational researchers to empirically evaluate theoretical models accounting for the emergence of concepts from situated sensorimotor activity. 76 participants (9-12 yo) 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 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. (Shorter version appears in ICLS 2016; even shorter version in JPS 2016; read Allen & Bickhard (2016) Commentary.)

Hutto, D. D., Kirchhoff, M. D., & Abrahamson, D. (2015). The enactive roots of STEM: Rethinking educational design in mathematics. In P. Chandler & A. Tricot (Eds.), Human movement, physical and mental health, and learning (Special issue). *Educational Psychology Review, 27*(3), 371-389.

ABSTRACT: 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.

Shayan, S., Abrahamson, D., Bakker, A., Duijzer, A. C. G., & Van der Schaaf, M. F. (2015). The emergence of proportional reasoning from embodied interaction with a tablet application: An eye-tracking study. In L. Gómez Chova, A. López Martínez, & I. Candel Torres (Eds.), *Proceedings of the 9th International Technology, Education, and Development Conference (INTED 2015)* (pp. 5732-5741). Madrid: International Academy of Technology, Education, and Development.

ABSTRACT: Embodied cognition is emerging as a promising approach in educational technology. This study is based on the conjecture that a potentially powerful methodology for researching embodied mathematics design would be to gather empirical data on children’s shifting visual attention as they learn to operate the technological devices so as to solve the situated problems. The aim of the study is to gain insight into the role of visual attention in the emergence of new sensorimotor schemes underlying mathematical concepts, especially proportion tasks. The research question is: How does visual attention change in the emergence of sensorimotor schemes during proportional reasoning tasks? An exploratory study was conducted amongst eight 5th- and 6th-grade students (age 10-12 years). Based on the original Mathematical Imagery Trainer for Proportion (MIT-P) designs for Wii and iPad (Abrahamson), a new tablet app has been developed. While students completed the hands-on proportion tasks, a Tobii x2-30 and an external camera were used for real-time processing and storage of both video and gaze data, resulting in integrated videos of hand- and eye movements. Data analysis revealed cross-participant variation in exploration path, progress rate, and inferences. Yet across all participants insight coincided with a shift from: (a) random finger movements accompanied by gazing at salient figural contour; to (b) new bimanual coordinations accompanied by gazing at new non-salient figural features. The study thus supports classical constructivist claims that mathematical concepts are grounded in operatory schemes. In particular, the data literally show the dynamical emergence of attentional anchors for situated problem solving as mediating the development of mathematical concepts. Thus new research methodologies stand to validate claims from embodiment.

Flood, V. J., & Abrahamson, D. (2015). *Refining mathematical meanings through multimodal revoicing interactions: The case of “faster.”* Paper presented at the annual meeting of the American Educational Research Association, Chicago, April 16-20.

ABSTRACT: How do learners come to connect embodied experience with cultural-historical definitions? We compare two cases of a learner-instructor dyad negotiating the meaning of “faster” to investigate the role embodied, multimodal discourse plays in processes of collaborative semiosis within a technology enhanced discovery-based mathematics learning context (the Mathematical Imagery Trainer for Proportion). We implicate and characterize two new forms of multimodal revoicing interactions: (a) Selective gestural repetition with co-timed elaborated verbal content and (b) Elaborated gestural content with co-timed repeated verbal content in this process. Closer investigation of these forms may lead to deeper understanding of how responsive teaching supports embodied learning.

Abrahamson, D., & Trninic, D. (2015). Bringing forth mathematical concepts: Signifying sensorimotor enactment in fields of promoted action. In D. Reid, L. Brown, A. Coles, & M.-D. Lozano (Eds.), Enactivist methodology in mathematics education research (Special issue). *ZDM Mathematics Education, 47*(2), 295–306.

ABSTRACT: 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 motoraction 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-a`-vis seminal contributions to mathematics education research. (See also Abrahamson & Trninic, 2015, in diSessa, Levin, & Brown.)

Abrahamson, D., Lee, R. G., Negrete, A. G., & Gutiérrez, J. F. (2014). Coordinating visualizations of polysemous action: Values added for grounding proportion. In F. Rivera, H. Steinbring, & A. Arcavi (Eds.), Visualization as an epistemological learning tool (Special issue). *ZDM Mathematics Education, 46*(1), 79-93.

ABSTRACT: 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.

**Students coordinating dynamical visualizations of the Mathematical Imagery Trainer.**

Negrete, A. G., Lee, R. G., & Abrahamson, D. (2013). Facilitating discovery learning in the tablet era: rethinking activity sequences vis-à-vis digital practices. In M. Martinez & A. Castro Superfine (Eds.), *“Broadening Perspectives on Mathematics Thinking and Learning”—Proceedings of the 35th Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 35) *(Vol. 10: “Technology,” p. 1205). Chicago, IL: University of Illinois at Chicago.

ABSTRACT: Reflecting on empirical results from a design-based research pilot intervention, we frame our study as paradigmatic of what may occur when digital devices are introduced into a classroom without anticipating students’ preexisting cultural practices surrounding a technological medium. In a previous study, we evaluated a Wii-mote-based design for proportions and we used an interview protocol to guide students through a sequence of activities that implemented our design rationale. We wished to replicate this design in tablet form. However, tablets afford users free access to all interaction features. The Mathematical Imagery Trainer for Proportion (MIT-P) is an embodied-learning device. Users solve motor-action problems and articulate their solutions prior to the introduction of formal notation. They manipulate two bars (Fig. 1a) that turn green if the bars compare in height by a “secret” ratio (e.g., 1:2). They bootstrap principles of proportional equivalence by developing strategies for making green. Various tools—grid, numerals, and table—scaffold progressive mathematization of these strategies. Users can modify the activity’s ratios, feedback, and appearance. Three 9th grade students participated in a 25 min. interview. The students, all fluent tablet consumers, engaged the medium in ways that inadvertently derailed our intended activity sequence and therefore undermined our design rationale. Scaling up to classrooms, we modified our activity rationale so as to accommodate students’ digital practices.

Lee, R. G., Hung, M., Negrete, A. G., & Abrahamson, D. (2013, April). *Rationale for a ratio-based conceptualization of slope: Results from a design-oriented embodied-cognition domain analysis.* Paper presented at the annual meeting of the American Educational Research Association (Special Interest Group on Research in Mathematics Education), San Francisco, April 27 – May 1.

ABSTRACT: We report encouraging results from the Planning Phase of a co-participatory designbased research project that brings together researchers and teachers interested in incorporating educational technology into high school mathematics classrooms. The following two ideas, which were pivotal to aligning our perspectives, co-emerged in our discourse following curricular reviews and cognitive domain analyses that specifically investigated the potential role of proportional reasoning in learning algebra content: (a) product-based conceptualization of proportional equivalence is pedagogically advantageous over process-based conceptualization; and (b) ratio-based conceptualization of slope is pedagogically advantageous over rate-based conceptualization. We detail findings from our collaborative reflection process and outline principles for the Design Phase, during which our embodied-interaction systems will be further developed to incorporate a product-based conceptualization of slope.

Abrahamson, D., Gutiérrez, J. F., Charoenying, T., Negrete, A. G., & Bumbacher, E. (2012). Fostering hooks and shifts: Tutorial tactics for guided mathematical discovery. *Technology, Knowledge, and Learning, 17*(1-2), 61-86.

ABSTRACT: How do instructors guide students to discover mathematical content? Are current explanatory models of pedagogical practice suitable to capture pragmatic essentials of discovery-based instruction? We examined videographed data from the implementation of a natural user interface design for proportions, so as to determine one constructivist tutor’s methodology for fostering expert visualization of learning materials. Our analysis applied professional-perception cognitive–anthropological frameworks. However, several types of tutorial tactics we observed appeared to ‘‘fall between the cracks’’ of these frameworks, due to the discovery-based, physical, and semantically complex nature of our design. We tabulate and exemplify an expanded framework that accommodates the observed tactics. The study complements our earlier focus on students’ agency in discovery (in Abrahamson et al., Technol Knowl Learn 16(1):55–85, 2011) by offering an empirically validated resource for researchers, instructors, and professional developers interested in preparing future teaching for future technology.

Abrahamson, D., Trninic, D., Gutiérrez, J. F., Huth, J., & Lee, R. G. (2011). Hooks and shifts: A dialectical study of mediated discovery. *Technology, Knowledge, and Learning, 16*(1), 55-85.

ABSTRACT: Radical constructivists advocate discovery-based pedagogical regimes that enable students to incrementally and continuously adapt their cognitive structures to the instrumented cultural environment. Some sociocultural theorists, however, maintain that learning implies discontinuity in conceptual development, because novices must appropriate expert analyses that are schematically incommensurate with their naive views. Adopting a conciliatory, dialectical perspective, we concur that naive and analytic schemes are operationally distinct and that cultural–historical artifacts are instrumental in schematic reconfiguration yet argue that students can be steered to bootstrap this reconfiguration in situ; moreover, students can do so without any direct modeling from persons fluent in the situated use of the artifacts. To support the plausibility of this mediated-discovery hypothesis, we present and analyze vignettes selected from empirical data gathered in a conjecture-driven design-based research study investigating the microgenesis of proportional reasoning through guided engagement in technology-based embodied interaction. 22 Grade 4–6 students participated in individual or paired semi-structured tutorial clinical interviews, in which they were tasked to remote-control the location of virtual objects on a computer display monitor so as to elicit a target feedback of making the screen green. The screen would be green only when the objects were manipulated on the screen in accord with a ‘‘mystery’’ rule. Once the participants had developed and articulated a successful manipulation strategy, we interpolated various symbolic artifacts onto the problem space, such as a Cartesian grid. Participants appropriated the artifacts as strategic or discursive means of accomplishing their goals. Yet, so doing, they found themselves attending to and engaging certain other embedded affordances in these artifacts that they had not initially noticed yet were supporting performance subgoals. Consequently, their operation schemas were surreptitiously modulated or reconfigured—they saw the situation anew and, moreover, acknowledged their emergent strategies as enabling advantageous interaction. We propose to characterize this two-step guided re-invention process as: (a) hooking— engaging an artifact as an enabling, enactive, enhancing, evaluative, or explanatory means of effecting and elaborating a current strategy; and (b) shifting—tacitly reconfiguring current strategy in response to the hooked artifact’s emergent affordances that are disclosed only through actively engaging the artifact. Looking closely at two cases and surveying others, we delineate mediated interaction factors enabling or impeding hook-and-shift learning. The apparent cognitive–pedagogical utility of these behaviors suggests that this ontological innovation could inform the development of a heuristic design principle for deliberately fostering similar learning experiences.

Howison, M., Trninic, D., Reinholz, D., & Abrahamson, D. (2011). The Mathematical Imagery Trainer: From embodied interaction to conceptual learning. In G. Fitzpatrick, C. Gutwin, B. Begole, W. A. Kellogg, & D. Tan (Eds.), *Proceedings of the annual meeting of The Association for Computer Machinery Special Interest Group on Computer Human Interaction: “Human Factors in Computing Systems” (CHI 2011), Vancouver, May 7-12, 2011* (Vol. “Full Papers”, pp. 1989-1998). New York: ACM Press.

ABSTRACT: We introduce an embodied-interaction instructional design, the Mathematical Imagery Trainer (MIT), for helping young students develop grounded understanding of proportional equivalence (e.g., 2/3 = 4/6). Taking advantage of the low-cost availability of hand-motion tracking provided by the Nintendo Wii remote, the MIT applies cognitive-science findings that mathematical concepts are grounded in mental simulation of dynamic imagery, which is acquired through perceiving, planning, and performing actions with the body. We describe our rationale for and implementation of the MIT through a design-based research approach and report on clinical interviews with twenty-two 4th–6th grade students who engaged in problem-solving tasks with the MIT.

Trninic, D., Reinholz, D., Howison, M., & Abrahamson, D. (2010). Design as an object-to-think-with: semiotic potential emerges through collaborative reflective conversation with material. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), *Proceedings of the Thirty-Second Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 32)* (Vol. VI, Ch. 18: Technology, pp. 1523 – 1530). Columbus, OH: PME-NA.

ABSTRACT: We chart a historical analysis of a collaborative design-based research project investigating the emergence of mathematical meaning from embodied interaction with a technological tracking system supporting the learning of proportionality. Recounting iterative cycles of a conceptually critical perceptual feedback element, we articulate three interconnected images of research based designers: (a) Janus the two-headed keeper of passageways who sees artifacts alternately as a student would or as an expert would; (b) an investigator searching to explicate design decisions coherently in light of learning-sciences theory; and (c) a reflective practitioner who embraces tradeoffs and is open to constructive criticism and to implementing radical changes to design and theory. Ultimately, we posit, we as researchers are continuously developing professional vision for our own design even as the design changes. (See also AERA2011-Embodied-Learning-Symp)

Reinholz, D., Trninic, D., Howison, M., & Abrahamson, D. (2010). It’s not easy being green: Embodied artifacts and the guided emergence of mathematical meaning. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), *Proceedings of the Thirty-Second Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 32)* (Vol. VI, Ch. 18: Technology, pp. 1488 – 1496). Columbus, OH: PME-NA.

ABSTRACT: This on-going design-based research study focuses on Grade 4-6 students’ guided task-based interaction with a novel computer-based hand-tracking system built to suggest the limitations of naïve additive schemes and create opportunities to develop core notions of proportionality as elaborations on these schemes, even before engaging numerical semiotic forms. Study participants struggled with canonical issues inherent to rational numbers. They formulated a string of insights leading up to a new type of equivalence class. Reported as a case study of Itamar, a 5th-grade middle-achieving student, our analyses reveal emergence of conceptually critical mathematical meanings in an activity that initially bears little mathematical significance.

Abrahamson, D., & Howison, M. (2010, May). *Embodied artifacts: Coordinated action as an object-to-think-with.* In D. L. Holton (Organizer & Chair) & J. P. Gee (Discussant), Embodied and enactive approaches to instruction: implications and innovations. Paper presented at the annual meeting of the American Educational Research Association, April 30 – May 4.

ABSTRACT: Building on growing evidence that human reasoning simulates multi-modal dynamical imagery drawn from lived experience, we conjectured that some mathematical concepts may be difficult to learn precisely because our everyday experience fails to provide adequate opportunities to develop the requisite body-based imagery underlying those specific concepts. To evaluate this conjecture, we are conducting a study in which we first provide students with a “ready-made” visual–kinesthetic basis for developing imagery pertaining to a difficult mathematical concept and then measure the effects of such provision on their emergent understanding of the targeted mathematical concept. This initially “meaningless” yet embodied experience (aka the “Karate Kid effect”) is to play a pedagogical role analogous to concrete artifacts, such as an abacus, pendulum, or dice, in terms of creating opportunities for guided reflection, mathematization, and reinvention of culturally knowledge. Our study is thus geared to comment on theoretical models that argue for an embodied basis of mathematical learning and reasoning. Prior arguments build either on theoretical analyses of human reasoning, empirical studies of human activity in general, interpretations of mathematics students’ behaviors, or specifically evidence of gestures accompanying speech utterances produced during the solution of mathematical problems. Whereas these studies furnish strong support for the potential viability of the embodied conjecture, and whereas they have demonstrated the plausibility of an embodied substrate for working memory, they have not established conclusively a sine qua nonrole of multi-modal dynamical imagery in the ontological development of mathematical concepts. Namely, it has yet to be shown compellingly that imagery plays more than a supportive or epiphenomenal role in the instruction of essentially abstract concepts. We chose the mathematical content domain of proportionality because rational numbers are fraught with conceptual challenges. We conjecture that students’ difficulties with proportionality, and in particular their “additive” or “same difference” errors, stem from their lack of a suitable dynamic image in which to ground their understanding of proportionality. We have developed a device that trains the user to perform arm motions describing proportional growth. Clearly, this is pioneering work in progress, and assessment tools are still under development. Yet, we feel confident that there is already sufficient theoretical as well as empirical material for sparking useful scholarly discourse.

Abrahamson, D., & Howison, M. (2008). *Kinemathics: Kinetically induced mathematical learning*. Paper presented at the UC Berkeley Gesture Study Group (E. Sweetser, Organizer), December 5, 2008. (“White paper” stating the tenets, rationale, and research and pragmatic objectives of the Mathematical Imagery Trainer)

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