Embodied Mathematical Imagination and Cognition (EMIC)

Recent empirical, theoretical and methodological developments in embodied cognition and gesture studies provide a solid and generative foundation for the establishment of

an Embodied Mathematical Imagination and Cognition (EMIC) Working Group

for PME-NA 2015 and future meetings. The central aim of EMIC is to attract engaged and inspired colleagues into a growing community of discourse around theoretical, technological, and methodological developments for advancing the study of embodied cognition for mathematics education, including, but not limited to studies of mathematical reasoning, instruction, the design and use of technological innovations, and learning in and outside of formal educational settings and across the lifespan. The interplay of multiple perspectives and intellectual trajectories is vital for the flourishing of embodied mathematical cognition. Partial confluences and differences have to be kept throughout the conversations; this is because instead of being oriented towards a single and unified theory of mathematical cognition, we are aiming towards a philosophical/educational "playground" in which entrenched dualisms, such as mind/body, language/materiality, or signifier/signified are subject to an ongoing and stirring criticism. A thriving, informed, and interconnected community of scholars organized around embodied mathematical cognition will broaden the range of activities and emerging technologies that count as mathematical and to envision alternative forms of engagement with mathematical ideas and practices (e.g., De Freitas & Sinclair, 2014). This broadening is particularly important at a time when schools and communities in North America face persistent achievement gaps between groups of students from many ethnic backgrounds, geographic regions, and socioeconomic circumstances (Ladson-Billings, 1995; Moses & Cobb, 2001; Rosebery, Warren, Ballenger & Ogonowski, 2005). There is also need to articulate evidence based findings and principles of embodied cognition to the research and development communities that are starting to generate and disseminate programs for promoting mathematics learning through movement (e.g., Petrick Smith, King, & Hoyte, 2014). Generating, evaluating, and curating empirically validated

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methods for promoting mathematical development through embodied activities that have engaging social bearing and curricular relevance is an urgent societal objective.

The submitters are proposing a new working group to PME-NA. They draw, however, on earlier interest in this topic from the International PME community (ESM, 2009), and several years of collaboration and intellectual exchange exploring the nature of EMIC from multiple methodological perspectives. As early as 2003, several of the submitters proposed the idea of a Science of Learning Center to NSF organized around the nascent research on embodied cognition for mathematics education, and worked on a funded NSF "catalyst" grant to begin to curate and publicly share the scholarship distributed across multiple disciplines, such as linguistics, psychology, anthropology, philosophy, and neuroscience. This early work led to, among other things, a 2007 AERA symposium on mathematics learning and embodied cognition entitled, "Mathematics Learning and Embodied Cognition." This early experience led to a 6-year NSF-REESE grant, "Tangibility for the Teaching, Learning, and Communicating of Mathematics," starting in 2008, in which collaboration across three campuses (San Diego State University, University of Wisconsin-Madison, and Vanderbilt) instituted a multidisciplinary approach to the empirical investigation of embodied mathematical thinking, drawing equally on ethnomethodology, phenomenology, and experimental psychology. The three campus-based research teams ran a coordinated series of empirical and design studies that focused on learning the mathematics of space, scale, modality, and motion. The investigations took place in professional workplaces such as medical response teams, formal K-12 math and engineering classrooms, mathematics professors' offices, museums, football fields, bike paths and city streets -- places and contexts where people were learning, communicating, and doing mathematics. Some of the research findings have been reported in a special issue of The Journal of the Learning Sciences (2012), "Modalities of Body Engagement in Mathematical Activity and Learning," and an NCTM 2013 research pre-session keynote panel, "Embodied cognition: What it means to know and do mathematics," as well as a series of academic presentations, book chapters, and journal articles, as well as several masters' theses and doctoral dissertations. Along with this group, numerous other research efforts formed to investigate the embodied nature of mathematics (e.g., Abrahamson 2014; Arzarello et al., 2009; De Freitas & Sinclair, 2014; Edwards, Ferrara, & Moore-Russo, 2014; Lakoff & Núñez, 2000; Radford 2009). There is good reason at this point in time, to proclaim that there is a "critical mass" of projects, senior and junior investigators, research findings, and conceptual frameworks to support an on-going community of likeminded scholars within the mathematics education research community.

Emerging, yet still influential, views of thinking and learning as embodied experiences have grown from several major intellectual developments in philosophy, psychology, anthropology, education, and the learning sciences that frame human communication as multi-modal interaction, and human thinking as multi-modal simulation of sensory-motor activity (Clark, 2008; Hostetter & Alibali, 2008; Lave, 1988; Nathan, 2014; Varela et al., 1992; Wilson, 2002). These views acknowledge the centrality of both unconscious and conscious motor and perceptual processes for influencing conscious awareness, and of embodied experience as following/producing pathways through social and cultural space. As Stevens (2012, p. 346) argues in his introduction to the JLS special issue on embodiment of mathematical reasoning,

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it will be hard to consign the body to the sidelines of mathematical cognition ever again if our goal is to make sense of how people make sense and take action with mathematical ideas, tools, and forms.

Three major ideas mesh with embodied cognition and serve as useful examples of the ways that embodied cognition perspectives are relevant for the study of mathematical understanding: (1) Grounding of abstraction in perceptuo-motor activity. This conception shifts the locus of "thinking" from a central processor to a distributed web of perceptuo-motor activity situated within a physical and social setting. It suggests that to make meaning people ground seemingly abstract concepts in modality-specific, sensory-motor systems, as an alternative to representing concepts as purely amodal, abstract, arbitrary, and self-referential symbol systems. (2) Cognition is for action. This tenet proposes that things, including mathematical symbols and representations, are understood by the actions and practices we can perform with them, and by mentally simulating the actions and practices that underlie or constitute them. (3) Mathematics learning is always affective: there are no purely procedural or "neutral" forms of reasoning to be learned in a manner detached from the circulation of bodily-based feelings and interpretations surrounding our encounters with them. Alongside these theoretical developments have been technical advances in multi-modal and spatial analysis, which allow scholars to collect new sources of evidence and subject them to powerful analytic procedures, from which they may propose new theories of embodied mathematical cognition and learning. Just as the "linguistic turn" in the social sciences was largely made possible by the innovation that enabled scholars to collect audio recordings of human speech and conversation in situ , growth of interest in multi-modal aspects of communication have been enabled by high quality video recording of human activity (e.g., Alibali et al., 2014; Levine & Scollon, 2004), motion capture technology (Hall, Ma, & Nemirovsky, 2014; Sinclair, 2014), and developments in brain imaging (e.g., Barsalou, in press; Gallese & Lakoff, 2005). New and still largely under-explored advances in location-aware technologies and spatial analysis, such as wearable GPS devices that capture personal space-time paths, expand our opportunities to study human activity and learning in natural settings and at an increasing broader set of spatial and temporal scales (Christensen, 2003; Miller, 2007). A growing suite of geographic information system ( GIS ) software tools is now available and can be employed to analyze path structures in relation to other spatially anchored data (e.g., the accessibility of cultural resources that provide children with learning opportunities; Ma, Hall, & Leander, 2010).

The formation of this new scholarly group requires some initial structure and the cultivation of distributed leadership. EMIC scholars will meet for 90 minutes each of three days during the PME- NA 2015 conference. We propose commencing the first gathering on Day 1 of the EMIC working group by engaging participants one or more embodied activities for promoting mathematical reasoning (a sample of the activities we propose is presented in Figure 1). Each of the activities has been used in formal and informal learning settings. The implementation and points of discussion and reflection for each activity will be informed by our prior experiences. One activity (Figure 1a) fosters deeper understanding of the Cartesian coordinate system through physically pushing and pulling handles that control coordinate displays by perceptuomotor integration of the physical and the semiotic. This has been used in interactive museum exhibits. A second activity (Figure 1b) engages participants in a projection activity using Alberti's Window. Participants look through the eyehole to draw projected images on the vertical drawing pane. A third activity uses a walking scale geometry activity (Figure 1c) to facilitate geometric reasoning in new ways by placing participants within the inscriptions (e.g., an isosceles triangle) rather than looking over them on paper from a bird's-eye-

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view, using everyday objects (string), and participating in large scale imagery that is necessarily collaborative. Finally, an activity engages one's body to experience spatial relations of geometric forms through body mechanics in order to influence their proof practices (e.g., for the Triangle Inequality Theorem). Day 2 of the working group would invite participants to provide brief (5 to 10 mins) synopses of their research interests and goals for the group. The balance of Day 2 and a portion of Day 3 would be devoted to generating and discussing topics of interest. A partial list of seed topics is presented below. The final time on Day 3 would focus on practicalities, including assigning action items and roles for specific participants.

Sample Seed Topics

• Conceptual blending (Tunner & Fauconnier, 1995) & metaphor (Lakoff & Núñez, 2000)
• Development of spatial reasoning (Uttal et al., 2009)
• Gesture & multimodal instruction (Alibali & Nathan 2012; Alibali et al., 2014Cook et al.; Edwards, 2009)
• Gesture in mathematical imagination (Nemirovsky, Kelton, & Rhodehamel, 2012)
• Perceptuo-motor grounding of abstractions (Barsalou, 2008; Glenberg, 1997)
• Mathematical cognition through action. (Abrahamson, 2014; Nathan et al., 2014)
• Modal engagements (Hall & Nemirovsky, 2012; Nathan et al., 2013)
• Perceptual boundedness (Bieda & Nathan, 2009)
• Perceptuomotor integration (Nemirovsky, Kelton, & Rhodehamel, 2013).
• Progressive formalization (Nathan, 2012;Romberg, 2001) & concreteness fading (Fyfe, McNeil, Son, & Goldstone, 2014)
• Sensuous cognition (Radford, 2009)
• Simulation of sensory-motor activity (Hostetter & Alibali, 2008; Nemirovsky & Ferrara, 2009).
• Use of manipulatives (Martin & Schwartz, 2005)
• Bodily activity among professional mathematicians (Nemirovsky & Smith, 2013)
• Ways spaces are be mathematized by architects and urban designers
• Ways bodies are deployed (or suppressed) in the diversity of conceptual practices we call mathematics

We envision an emergent process for the specific follow-up activities based on participant input. At a minimum, we will develop a list of interested participants and grant them all access to a common, closed discussion forum. We also offer these proposed activities as outcomes for extending the intellectual work of the group beyond the PME-NA meeting. The proposed activities include:

• Develop a multicampus course or MOOC
• Compile/curate literature, activities demonstrating principles of embodied cognition for promoting mathematical thinking, learning, and communicating
• Propose a sharable corpus of video data focused on gesture and action during math learning and instruction (e.g., hosted on Databrary.org)
• Exchange resources/interventions for embodied mathematical cognition for students or teachers, potentially conducting cross-institutional research studies on their use.

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• Propose and edit a journal special issue.
• Sponsor the participation of colleagues from outside the community of research in mathematics education (e.g. philosophers, technologists)