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This study examines the consequences of whole-body, multi-party activityfor mathematics learning, both in and out of the classroom. We develop atheoretical framework that brings together contemporary theories related tosocial space, embodied cognition, and mathematical activity. Then, drawing onmicro-ethnographic and case-comparative techniques, we examine and juxtaposetwo cases of implementing whole-body, collaborative movement to engagelearners in the mathematics of number sense and ratio and proportion.Analytically foregrounding the interdependence among setting, embodiedactivity, and mathematical tools and practices, we illustrate how whole-bodycollaboration can transform how learners experience learning environments andmake sense of important mathematical ideas. The analysis enriches ourunderstanding of the changing spatial landscapes for learning and doingmathematics as well as how re-instating bodies in mathematics education canopen up new forms of collective mathematical sense-making and activity.

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Without questioning the distinctiveness and importance of hand-based tactile,gestural, and inscriptional practices for processes of mathematical thinkingand learning, we suggest that fully leveraging the potential of the embodiedturn in mathematics education requires an anatomical expansion toward moreholistic and encompassing attentiveness to whole bodies-always diverselyconstituted and abled-and their myriad, heterogeneously organized components,including but not limited to the hands. We consider forays into thisstill-too-rare consideration of bodies in a more holistic sense to include,inter alia, Gerofsky's (2010) attention to whole-body enactments offunctional graphs, Ma's (2017) analysis of whole-body interactions inwalking-scale geometry task settings, and studies of student engagement withmotion-detection graphing technologies (e.g., Nemirovsky, Tierney, &Wright, 1998; Radford, 2009a), although, interestingly, the latter analysesstill tend to privilege talk and gesture over graphical traces of walkingactivity rather than, say, the walking activity itself. Of particularinterest in the present study are (a) participants' bodily capacitiesthat extend beyond the hands, especially walking and other movements thatemanate from the waist and legs, and (b) the anatomical contextualization andenablement of hand movements and configurations in relation to, for example,torso inflections, crouching, and standing on tiptoes. Our intentions towardanalytic expansion focus primarily on a more holistic consideration of humanbodies-albeit dynamically constituted, diversely abled, and reconfigurablethrough tools and prosthetics-in contrast to new materialist and post-humanconceptualization of the body as more radically inclusive of human andmore-than-human elements, offered by de Freitas and Sinclair (2013) andfurther expanded by Ferrara and Ferrari (2017). At the same time, our framingof the present work resonates with new materialist approaches to embodimentin mathematics education in that we also conceptualize bodies as dynamic,heterogeneous multiplicities; by contrast, our attention to more-than-humanmateriality here is more directly found in our treatment of social space andbuilt environments than it is in a reconsideration of the ontology of bodies.

We begin by describing how the school gym was edited in the WSNL activity,both through instructors' material intervention (e.g., the placement oftape on the floor) and students' engagements in WSNL tasks. The WSNLsetting brought the gym-both its material arrangements and students'routine experiences there-into interaction with "familiar"mathematical tools (students were all familiar with number lines on paper).Features of the gym arena took on new meanings edited in the context of WSNLactivity. The open space typically used for play, competition, andperformance was transformed by tape and the designed activities. The floor ofthe gym, painted with lines for basketball and foursquare, was temporarilyaugmented with number lines of brightly colored tape running parallel to thelong wall and one long diagonal blue line, the "paper" forstudents' problem solving. The tape itself became a locus ofmathematical activity-on the lines, students' bodies enacted quantities,their movements representing arithmetic operations, while off-the-linestudents became observers with extrinsic views on the quantitative relations.The familiar mathematical tool of number lines was newly materialized aslarge-scale walkable physical phenomena embedded in the gym floor, tacitlyagreed-upon attributes (of number lines and of the gym) no longer so readilyavailable. For example, left and right, negative and positive wereexperienced variably, depending on individuals' embodied orientations inrelation to each other; even "to Thad's left" becameproblematic as soon as Thad turned around to face the other way. Staticaspects of the arena (the wall, the stage) became stand-ins for direction(left, right), taking on mathematical meanings in the service of performingand describing quantities and operations.

Similar to WSNL, W + H recruited participants' bodies and body partsinto mathematical objects, relations, and processes, making them part of adynamic materialization of relevant mathematics in which (a) W'sbimanual hand positioning embodied an interval-like whole, (b) H'ssingle hand embodied a half, and (c) the spatial relationship betweenW's and H's hands created a multi-party, body-based materializationof a part-whole quantitative comparison. Expressing and holding constant apart-whole relationship became a matter of intricate social and embodiedcoordination. Interactional breakdowns made particularly visible howparticipants were incorporating multiple bodies and the dynamic spatialrelations among them into mathematical objects and events. For example, justafter being told to begin the activity, Katie lingered at her desk, makingnotes while her partner, Claire, skirted around her desk to the front of theroom and, taking on the role of W, positioned her hands to materialize adiagonal whole in front of Katie's desk (Fig. 5a). But Katie, stillwriting in her notebook, left Claire hanging for about a quarter of a minute.Holding her hands still to keep the diagonal whole interval in place, Clairewaited for Katie, growing increasingly impatient, re-iterating theactivity's directive in physical terms ("stick your hand in betweenit"), and urging Katie to hurry up ("come on Katie"). As aresult of their bodily incorporation into the mathematics, this briefinteractional breakdown was simultaneously a breakdown in the mathematicsitself; without Katie's cooperation, Claire's whole lacked itscomparative half and the desired mathematical objects and relations couldonly be incompletely materialized.

We further wonder how multi-party, whole-body mathematical practices mighthinge on the relative mobilities of materialized mathematical objects. Inboth cases, mathematical tools and practices were reconfigured to incorporatewhole bodies; body parts; and new regions, materials, and features of thearena. Yet, the resulting reconfigured mathematical spaces were composed ofremarkably different material ecologies. In WSNL, much of the space wasdetermined by the tape on the gym floor and, as a result, had a relativelystatic quality. Bodies as points along the line became the mobile part of thematerial space, and the possibilities and constraints for making sense ofnumeric operations depended on the negotiated interplay between the staticframe of the tape-augmented gymnasium floor and collective physical movement.In contrast, in W + H, interval boundaries were constituted by moving handssuch that the emergent mathematical practices were less fixed to anyparticular aspect of the classroom arena. Where bodies went determined wherethe mathematics was. As intervals became unfixed and re-tethered to movingbodies, the students quickly and flexibly re-oriented, re-scaled, andrelocated interval boundaries in order to vary the mathematical task.Leveraging a newly mobilized mathematical space, the North Lake studentsspontaneously and opportunistically incorporated unanticipated materialelements of the arena into dynamically changing intervals, producing asetting in which unexpected bodily movements and regions of the classroommight suddenly become salient and saturated with mathematical significance.Bringing this comparison into broader dialog with the field, we suggest thatresearchers and practitioners attend more closely to the ways in whichdifferent patterns of mobilities in mathematical activity might affect thenegotiation and development of reconfigured mathematical practices in anyinstructional design.

In addition to these distinctions related to mobility of mathematicalcomponents, comparative analyses also revealed a second important dimensionof these activities. In particular, while both activities involvedre-purposing the arena in which they took place, upon closer examination,they appeared to set up different kinds of relationships between mathematicalactivity and setting. Thus, while WSNL deliberately capitalized on thehistories of participation associated with the gymnasium (whole-bodymovement), W + H was taken up in salient contrast to the embodied and spatialroutines of the classroom, as well as the material arrangements of theclassroom that both indexed and enabled those routines (e.g., compare Figs. 1and 4b). Thus, the design of WSNL predominantly made visible the ways inwhich instructional designs might reconfigure spaces by intentionallyhighlighting and leveraging-or augmenting-a setting for doing and learningmathematics. W + H, on the other hand, made salient how instructional designcan also lead to meaningful contrasts-or induce friction-with the built andpracticed environment in which it unfolds. Although dynamics of augmentationand friction were differentially salient across the cases, our approach tocomparative analysis also led us to consider how both qualities might bepresent in each case. Thus, for example, while W + H participants were oftenin friction with the built classroom, quite literally skirting around andbumping up against desks and chairs, they also dynamically augmented theenvironment by flexibly re-purposing mundane features of the classroom intocomponents of mathematical objects. Taking this comparative approach to thebroader work of analytic generalization (Yin, 2009), we offer these conceptsof friction and augmentation as tools for interrogating instructional designsmore broadly. We posit that any instructional design, especially those thatleverage new or alternative bodily repertoires, can strike a complex set ofresonances and dissonances with the built arenas-and attendant histories ofspatial practice-in which that design is implemented, and researchers andeducators should take this into account when considering instructionaldesign. ff782bc1db