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EMBODIED MATHEMATICAL IMAGINATION AND COGNITION (EMIC) WORKING GROUP

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Embodied cognition is growing in theoretical importance and as a driving set of design principles for curriculum activities and technology innovations for mathematics education. The central aim of the EMIC (Embodied Mathematical Imagination and Cognition) Working Group 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. A thriving, informed, and interconnected community of scholars organized around embodied mathematical cognition will broaden the range of activities, practices, and emerging technologies that count as mathematical. EMIC builds upon our prior working groups with a specific focus on how we can leverage emerging technologies to study embodied cognition and mathematics learning. In particular, we aim to develop new theories and extend existing frameworks and perspectives from which EMIC collaboration and activities can emerge.
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Working Groups
Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1449
EMBODIED MATHEMATICAL IMAGINATION AND COGNITION (EMIC)
WORKING GROUP
Erin R. Ottmar Edward Melcer Dor Abrahamson
Worcester Polytechnic Institute Univ. of California, Santa Cruz Univ. of California, Berkeley
erottmar@wpi.edu eddie.melcer@ucsc.edu dor@berkeley.edu
Mitchell J. Nathan Emily Fyfe Carmen Smith
Univ. of Wisconsin-Madison Indiana University University of Vermont
mnathan@wisc.edu efyfe@indiana.edu carmen.smith@uvm.edu
Embodied cognition is growing in theoretical importance and as a driving set of design
principles for curriculum activities and technology innovations for mathematics education. The
central aim of the EMIC (Embodied Mathematical Imagination and Cognition) Working Group
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. A thriving, informed, and interconnected
community of scholars organized around embodied mathematical cognition will broaden the
range of activities, practices, and emerging technologies that count as mathematical. EMIC
builds upon our prior working groups with a specific focus on how we can leverage emerging
technologies to study embodied cognition and mathematics learning. In particular, we aim to
develop new theories and extend existing frameworks and perspectives from which EMIC
collaboration and activities can emerge.
Keywords: Technology; Cognition; Informal Education; Learning Theory
Motivations for This Working Group
Recent empirical, theoretical and methodological developments in embodied cognition and
gesture studies provide a solid and generative foundation for the establishment of a regularly
held Embodied Mathematical Imagination and Cognition (EMIC) Working Group for
PME-NA. 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, learning in and outside of formal educational settings, and across the lifespan.
The interplay of multiple perspectives and intellectual trajectories is vital for the study of
embodied mathematical cognition to flourish. While there is significant convergence of
theoretical, technological, and methodological developments in embodied cognition, there is also
a trove of technological, methodological, and theoretical questions that must be addressed before
we can formulate and implement effective design principles. As a group, we aim to address basic
theoretical questions such as, What is grounding? And practical ones such as, How can we
reliably engineer the grounding of specific mathematical ideas? We need to understand how
variations in actions and perceptions influence mathematical reasoning, including self-initiated
vs. prescribed actions, and actions that take place in intrapersonal versus interpersonal
interactions; how gestural point-of-view when enacting phenomena from a first- versus third-
person perspective, including how gestures move through space, influences reasoning and
communication; how actions enacted by oneself, observed in others, or imagined influence
Working Groups
Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1450
cognition; and how gestures connect with external visual representations (Alibali & Nathan,
2012; in press).
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 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 also is a need to articulate evidence-based findings and principles of
embodied cognition to the research and development communities that are looking to generate
and disseminate innovative programs for promoting mathematics learning through movement
(e.g., Ottmar & Landy, 2016; Smith, King, & Hoyte, 2014). Generating, evaluating, and curating
empirically validated and reliable methods for promoting mathematical development and
effective instruction through embodied activities that are engaging and curricularly relevant is an
urgent societal goal. As embodied cognition gains prominence in education, so, too, does new
ways of using technology to support teaching and learning (Lee, 2015). These new uses of
technology, in turn, offer novel opportunities for students and scientists to engage in math
visualization, symbolization, intuition, and reasoning. In order for these designs to successfully
scale up, they must be informed by research that demonstrates both ecological and internal
validity.
Past Meetings and Achievements of the EMIC Working Group
“Mathematics Learning and Embodied Cognition,” the first PME-NA meeting of the EMIC
working group, took place in East Lansing, MI in 2015. Our group has been growing ever since.
In addition to the PME-NA meeting each year, there are a number of ongoing activities that our
members engage in. We have built an active website which provides updates on projects, and
hosts resources. On this website, we have a list of members with their emails and bios,
information about our PME-NA presence, and short personal introduction videos. We’ve also
created a space for members to share information about their research activities – particularly for
videos of the complex gesture and action-based interactions that are difficult to express in text
format. In addition, we have a common publications repository to share files or links (including
to ResearchGate or Academia.edu publication profiles, so members don’t have to upload their
files in multiple places). Our members collaborate on ongoing projects, and have presented at
other pre-conference workshop events such as the Computer Supported Collaborative Learning
conference (Williams-Pierce et al., 2017), and the APA Technology, Mind, and Society
conference (Harrison et. al, 2018). Several research programs have formed to investigate the
embodied nature of mathematics (e.g., Abrahamson 2014; Alibali & Nathan, 2012; Arzarello et
al., 2009; De Freitas & Sinclair, 2014; Edwards, Ferrara, & Moore-Russo, 2014; Lakoff &
Núñez, 2000; Melcer & Isbister, 2016; Ottmar & Landy, 2016; Radford 2009; Nathan,
Walkington, Boncoddo, Pier, Williams, & Alibali, 2014; Soto-Johnson & Troup, 2014; Soto-
Johnson, Hancock, & Oehrtman, 2016), demonstrating a “critical mass” of projects, findings,
senior and junior investigators, and conceptual frameworks to support an ongoing community of
like minded scholars within the mathematics education research community.
Some of our collaborative accomplishments since 2015 include:
Working Groups
Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1451
1. Developing a group website using the Google Sites platform to connect scholars, support
ongoing interactions throughout the year, and regularly adding additional
resources/activities https://sites.google.com/site/emicpmena/home
2. Joint submission of several NSF grants by members who first met during the 2015 EMIC
sessions
3. Some senior members joining junior members’ NSF grant proposals as Co-PIs and
advisors
4. Submission to the IES CASL program to study the role of action in pre-college proof
performance in geometry (Funded 2016-2020 for Nathan & Walkington), as well of the
use of perceptual learning on algebra learning (Ottmar & Landy, 2018)
5. Submission of 2 NSF Proposals to host Workshops on Embodied Cognition (one for
researchers and one for K-16 math educators)
6. A joint symposium on Embodied Cognition with 6 members at the 2018 APA
Technology, Mind & Society Conference
7. Examining the potential for an NSF Research Coordination Network (RCN)
8. Application for a grant from Association for Psychological Science (APS) to develop a
better website and offer stipends for contributors
9. Presenting a pre-conference workshop to CSCL 2017 on the embodied tools to promote
STEM education (Williams-Pierce et al., 2017)
10. A Conference at Berkeley’s Graduate School of Education to examine the relevance of
coordination dynamics -- the non-linear perspective on kinesiological development -- to
individuals’ sensorimotor construction of perceptual structures underlying mathematical
concepts. https://edrl.berkeley.edu/cdme2018
Current Working Group Organizers
As the Working Group has matured and expanded, we have a broadening set of organizers
that represent a range of institutions and theoretical perspectives (and is beyond the limit of six
authors in the submission system). This, we believe, enriches the Working Group experience and
the long-term viability of the scholarly community. The current organizers for 2018 are
(alphabetical by first name):
Candace Walkington, Southern Methodist University
Carmen J. Petrick Smith, University of Vermont
Caro Williams-Pierce, University at Albany, SUNY
David Landy, Indiana University
Dor Abrahamson, University of California, Berkeley
Edward Melcer, University of California, Santa Cruz
Emily Fyfe, Indiana University
Erin Ottmar, Worcester Polytechnic Institute
Hortensia Soto–Johnson, University of Northern Colorado
Ivon Arroyo, Worcester Polytechnic Institute
Martha W. Alibali, University of Wisconsin-Madison
Mitchell J. Nathan, University of Wisconsin-Madison
Focal Issues in the Psychology of Mathematics Education
Emerging, yet influential, views of thinking and learning as embodied experiences have
grown from several major intellectual developments in philosophy, psychology, anthropology,
Working Groups
Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1452
education, and the learning sciences that frame human communication as multimodal 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). As Stevens
(2012, p. 346) argues in his introduction to the JLS special issue on embodiment of mathematical
reasoning, “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”.
Four major ideas exemplify the plurality of ways that embodied cognition perspectives are
relevant for the study of mathematical understanding: (1) Grounding of abstraction in perceptuo-
motor activity as one alternative to representing concepts as purely amodal, abstract, arbitrary,
and self-referential symbol systems. 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. (2) Cognition emerges from perceptually guided action (Varela, Thompson, & Rosch,
1991). This tenet implies that things, including mathematical symbols and representations, are
understood by the actions and practices we can perform with them, and by mentally simulating
and imagining 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
detached from the circulation of bodily-based feelings and interpretations surrounding our
encounters with them. (4) Mathematical ideas are conveyed using rich, multimodal forms of
communication, including gestures and tangible objects in the world.
In addition to theoretical and empirical advances, new technical advances in multi-modal and
spatial analysis have allowed 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. 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), developments in brain imaging (e.g., Barsalou, 2008; Gallese
& Lakoff, 2005), multimodal learning analytics (Worsley & Blikstein, 2014), and data logs
generated from embodied math learning technologies that interacts with touch and mouse-based
interfaces (Manzo, Ottmar, & Landy, 2016).
Theme: Looking Back, Looking Ahead: Celebrating 40 Years of PME-NA
Inspired by the PME-NA 2018 theme, we will specifically focus on the ways in which the
field of embodied cognition has developed and how new emerging technologies and innovative
pedagogies can influence mathematics teaching and learning. This effort will be crafted to align
with recent developments in the embodiment literature, and new theoretical frameworks tying
various perspectives on embodiment to different forms of physicality in educational technology
(Melcer & Isbister, 2016; see Figure 1 below).
Working Groups
Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1453
Figure 1. Five distinct approaches to facilitating embodiment through bodily action, objects,
and the surrounding environment in educational technology.
Examples include: coding videos of pre-service teachers’ distributed gestures to explore a
mathematical conjecture (Walkington et al., 2018); exploring mathematical transformations
while using a dynamic technology tool (Ottmar & Landy, 2016), having students and teachers
play and create embodied technology games to teach mathematics and computational thinking
(Arroyo et al., 2017; Melcer & Isbister, 2018); usng dual eye tracking (Shvarts & Abrahamson,
2018), and a teacher guiding the movements of a learner exploring ratios (Abrahamson &
Sánchez-García, 2016). Through these examples, we will explore questions such as: what role
does technology play on supporting the connections between the mind, body, and action? During
the conference, participants in our EMIC workshop will engage in dedicated activities and
guided reflections as a basis for exploring the role of technology, action, and embodiment in the
emergence of mathematics learning.
Plan for Active Engagement of Participants
Our formula from prior PME-NA working groups proved to be effective: By inviting
participants into open ended math activities at the beginning of each session, we were rapidly
drawn into those very aspects of mathematics that we find most rewarding. We plan to facilitate
collaborative EMIC activities, followed by group discussions (and we now have many activities
and members who can trade off in these roles!) that will help us all to “pull back” to the
theoretical and methodological issues that are central to advancing math education research.
Within this structure of beginning with mathematical activities and facilitated discussions, on
Day 1 we plan to begin with four different group activities that highlight the interplay of
mathematics content, cognition, physicality, and action. These activities will serve as the
foundation for a broad group discussion about the varied roles of technology in EMIC. See
Figure 2 below for examples of collaborative activities from PME-NA 2017.
Figure 2. Collaborative activities. Participants explore geometric rotation and reflection(left);
two groups act out and prove mathematical conjectures (middle and right).
Working Groups
Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1454
The full first session will generally be taken up by introductions and a round of open-ended
activities followed by discussion. On Day 2, we will begin the session with technology-based
collaborative activities, with four stations that pairs of participants rotate through. Examples of
three of those stations are in Figure 3. Continuing with the routine established in Day 1, a full
group discussion will follow, with a particular focus on designing EMIC digital contexts to
support ongoing collaboration.
Figure 3. Technology activities. The Hidden Village (left); Graspable Math (middle); Bots &
(Main)Frames (right).
After the discussion, we will discuss different EMIC activity ideas, with the goal of developing
additional collaborative activities that can be used in various research and learning contexts. The
final activities will be shared on the EMIC website.
Day 3 is agenda-setting day, where we all discuss how we will keep the momentum going,
such as developing an NSF Research Coordination Network (RCN) to build the networked
community of international scholars from which many fruitful lines of inquiry can emerge. A
second group may draft a proposal for a special issue of the Journal of Research in Mathematics
Education that focuses on creating an integrated theoretical framework or sharing the different
theoretical perspectives, research activities, and operationalization of EMIC by the working
group members.
In order to find common ground for the RCN submission and the JRME special issue, we
will perform a live concept mapping activity that is displayed for all participants to explore the
range of EMIC topics and identify common conceptual structure. We will discuss different
general foci, such as teacher professional development with EMIC, designing EMIC games or
museum exhibits, etc. Building on the four major ideas that we developed earlier, possible topics
for organizing this activity will be explored, such as:
1. Grounding Abstractions
a. Conceptual blending (Tunner & Fauconnier, 1995) & metaphor (Lakoff & Núñez,
2000)
b. Perceptuo-motor grounding of abstractions (Barsalou, 2008; Glenberg, 1997;
Ottmar & Landy, 2016; Landy, Allen, & Zednik, 2014)
c. Progressive formalization (Nathan, 2012; Romberg, 2001) & concreteness fading
(Fyfe, McNeil, Son, & Goldstone, 2014)
d. Use of manipulatives (Martin & Schwartz, 2005)
2. Cognition emerges from perceptually guided action: Designing interactive learning
environments for EMIC
Working Groups
Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1455
a. Development of spatial reasoning (Uttal et al., 2009)
b. Math cognition through action (Abrahamson, 2014; Nathan et al., 2014)
c. Perceptual boundedness (Bieda & Nathan, 2009)
d. Perceptuomotor integration (Ottmar, Landy, Goldstone, & Weitnauer, 2015;
Nemirovsky, Kelton, & Rhodehamel, 2013)
e. Attentional anchors and the emergence of mathematical objects (Abrahamson &
Bakker, 2016; Abrahamson & Sánchez–García, 2016; Abrahamson et al., 2016;
Duijzer et al., 2017)
f. Mathematical imagination (Nemirovsky, Kelton, & Rhodehamel, 2012)
g. Students’ integer arithmetic learning depends on their actions (Nurnberger-Haag,
2015).
3. Affective Mathematics
a. Modal engagements (Hall & Nemirovsky, 2012; Nathan et al., 2013)
b. Sensuous cognition (Radford, 2009)
4. Gesture and Multimodality
a. Gesture & multimodal instruction (Alibali & Nathan 2012; Cook et al., 2008;
Edwards, 2009)
b. Bodily activity of professional mathematicians (Nemirovsky & Smith, 2013;
Soto-Johnson, Hancock, & Oehrtman, 2016)
c. Simulation of sensory-motor activity (Hostetter & Alibali, 2008; Nemirovsky &
Ferrara, 2009)
We will also discuss the implications of this work and the different areas of the concept map for
teaching, and discuss ideas for bridging the gap between research and practice.
Follow-up Activities
We envision an emergent process for the specific follow-up activities based on participant
input and our multi-day discussions. At a minimum, we will continue to develop a list of
interested participants and grant them all access to our common discussion forum and literature
compilation. Those that are interested in the NSF RCN plan will work to form the international
set of collaborations and articulate the intellectual topics that will knit the network together; and
those that are interested in the JRME special issue proposal will outline a specific timeline for
progressing. One additional set of activities we hope to explore is to create a series of
instructional activities that can be used to introduce educational practitioners at all levels of
administration and across the lifespan to the power and utility of the EMIC perspective.
In the past several years, we have seen a great deal of progress. This is perhaps best
exemplified by coming together of the EMIC website, the ongoing collaborations between
members, and the annual workshops, which each draws across multiple institutions. We thus will
strive to explore ways to reach farther outside of our young group to continually make our work
relevant, while also seeking to bolster and refine the theoretical underpinnings of an embodied
view of mathematical thinking and teaching.
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Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.). (2018). Proceedings of the 40th annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics
Education. Greenville, SC: University of South Carolina & Clemson University.
1456
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The book establishes a common language for, and understanding of, embodiment as it applies to mathematical thinking, and to link mathematics education research to recent work in gesture studies, cognitive linguistics and the theory of embodied cognition. The purpose is to acknowledge the multimodal nature of mathematical knowing, and to contribute to the creation of a model of the interactions and mutual influences of bodily motion, spatial thinking, gesture, speech and external inscriptions on mathematical thinking, communication and learning.
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This classic book, first published in 1991, was one of the first to propose the “embodied cognition” approach in cognitive science. It pioneered the connections between phenomenology and science and between Buddhist practices and science-claims that have since become highly influential. Through this cross-fertilization of disparate fields of study, The Embodied Mind introduced a new form of cognitive science called “enaction," in which both the environment and first person experience are aspects of embodiment. However, enactive embodiment is not the grasping of an independent, outside world by a brain, a mind, or a self; rather it is the bringing forth of an interdependent world in and through embodied action. Although enacted cognition lacks an absolute foundation, the book shows how that does not lead to either experiential or philosophical nihilism. Above all, the book’s arguments were powered by the conviction that the sciences of mind must encompass lived human experience and the possibilities for transformation inherent in human experience. This revised edition includes substantive introductions by Evan Thompson and Eleanor Rosch that clarify central arguments of the work and discuss and evaluate subsequent research that has expanded on the themes of the book, including the renewed theoretical and practical interest in Buddhism and mindfulness. A preface by Jon Kabat-Zinn, the originator of the mindfulness-based stress reduction program, contextualizes the book and describes its influence on his life and work. © 1991, 2016 Massachusetts Institute of Technology. All rights reserved.
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Learning algebra is difficult for many students, in part due to an emphasis on the memorization of abstract rules. Algebraic reasoners across expertise levels often rely on perceptual-motor strategies to make these rules meaningful and memorable. However, in many cases, rules are provided as patterns to be memorized verbally, with little overt perceptual support. Although most work on concreteness focuses on conceptual support through examples or analogies, we here consider notational concreteness—perceptual/motor supports that provide access into the dynamic structure of a representation itself. We hypothesize that perceptual support may be maximally beneficial as an initial scaffold to learning, so that later static symbol use may be interpreted using a dynamic perspective. This hypothesis meshes with other findings using concrete analogies or examples, which often find that fading these supports over time leads to stronger learning outcomes. In an experiment exploring this hypothesis, we compare gains from the fading out of dynamic concrete physical motion of symbols during instruction with the introduction of motion over the course of instruction. In line with our theoretical perspective, concreteness fading led to significantly higher achievement than concreteness introduction after Day 2 of the intervention.