**A. Concepts Emerge through Modal Engagements**

One of the most important challenges for a theory of embodied mathematical cognition is to articulate a viable perspective on the nature of concepts. Traditionally, concepts belong to the "mental" side of cognition, while perception remains attached to the "body" side providing no more than the content of concepts. On the basis of case studies, we argue that our engagements in activities with others, tools, and symbols constitute the fabric within which concepts emerge, not in terms of necessary and sufficient conditions, as in classic definitions, but as experiential family resemblances. Furthermore, we propose an analytic framework to describe engagements with others, tools, and symbols based on the discrimination among four aspects of embodied engagement:

- Embodied engagement—Place, spatiality, and body-based scale
- Encultured engagement—Horizon of availability/tool- and symbol-use
- Expressive engagement—Interactive stances
- Imaginary engagement—Temporality

If organized in terms of these four aspects, we refer to the descriptions of utterances, gestures, tool uses, and other dimensions of participants’ interaction in activities as accounts of "modal engagements." This framework is emerging as productive and insightful.

**B. Scale and Modality are Dynamically Related and Influence the Learning and Practice of Professionals**

Understanding modal engagements also involves understanding how such engagements are “scaled” for participants. Ethnographic studies have highlighted several key findings concerning the relationships between modality and scale in professional work practices. First, the relationship between modality and scale in conventional work practices influences what is available for the learning of newcomers. Second, “learning to see” in professional practice is also related to scaling practices with the body, where the body is re-scaled in order to perceive and constitute objects as salient and relevant. Third, scale is not merely a static feature of representations; rather, scaling and modal-izing are practices or processes. In this sense, scaling is itself a practice that manipulates time and space, a dynamic activity that should be studied as a form of mathematical thinking in the workplace. Fourth, changes in scale and modality are related such that the body can be extended through tool use from peripersonal space (what is reachable with the body) to extrapersonal space (what is visible and can be attended to) and further to navigable, mapped space (what is reachable through movement and way-finding). These relations also have a converse: just as the body can be extended into the world, the world can be collapsed into the scale of the body, using a suite of representational technologies that bring vast or minute scales into the gestural/visual stage.

**C. The Meaning of Symbols Is Embedded in the Physicality of Symbol Use**

One result emerging from our developing theoretical framework is what we term the “physicality of symbol use.” Rather than interpreting written mathematical symbols as being references to some kind of intangible idea, we suggest instead that the meaning of symbols can be seen as embedded in the way the symbol-user creates, modifies, and physically interacts with them. Physical interactions and manipulations with material objects give birth to routine as well as innovative symbolic meanings for learners, including high-level mathematicians. For instance, one of our subjects, a mathematician, animates diagrams of rotating spheres by positioning his hand immediately in front of the diagram in a shape as though holding the sphere and then rotating his hand as though spinning it. By viewing this gesture not just as indicative of how he thinks about the rotating sphere but as part of his understanding, we gain insight into how he may be thinking about that sphere and what he perceives it as being capable of. This technique of observing mathematicians’ embodied interactions with their symbols, as a means of understanding the meanings these symbols have for the mathematicians, has given us many promising leads on understanding how mathematicians experience their work.

**D. Modal Transitions Are Fundamental to STEM Education**

Through our analysis of high school engineering and geometry classrooms, we have identified three important ways in which modal transitions unfold in classroom activity. First, participants make *ecological shifts*, which involve the reorientation of the activity context to include different forms of modal engagement. Second, participants use *projection over time *to connect current modal engagements to modalities in past and future ecological contexts. Finally, participants engage in *coordination*, which is the juxtaposition and linking of co-occurring modal engagements and their associated representations, tools, and materials. We posit that many aspects of curriculum and instruction that we observe across our cases of secondary STEM education exist in order to engage relation-producing mechanisms to advance students’ perceptions of locally invariant properties so they serve as a common thread of coherence through STEM activities. We have investigated complex instructional activities by tracking how coherence is maintained by participants as mathematical ideas are “threaded through” different modal engagements. We found that an analysis of how teachers and students co-produce modality transitions and engage in modality-specific behavior is a productive approach to locating the “where” of the mathematics in STEM classrooms.

**E. Gesture Is Important in Producing Coherence for Learning and Communicating Mathematics **

Coherence-building actions like coordination and projection are accomplished through language and gesture. Body movements such as catchment, highlighting, and matching gestures, key participants in modal engagements, create a sense of continuity across modalities, linking ideas and visual elements. Gesture catchments are repeated hand shapes or movements that signify and reinvoke previously referenced ideas and events. Highlighting gestures make specific phenomena in a complex field salient by marking them in some manner. Matching gestures like pointing are used by participants to concurrently link different modal engagements. Teachers can support integration by invoking actions of the body that become signified, threading mathematical relations through different ecological contexts using gesture.

**F. Spatial Learning Trajectories Involve Embodied and Representational Practices toward the Production of Spatial Enclosures **

Learning in professional domains involving spatial analysis and modeling occurs along trajectories that are shaped by distinct temporal and spatial aspects of practice. Professional learning is distributed over place and time in ways that both produce increasingly refined objects of practice and newly capable practitioners. Newcomers to practice must learn to “see” as a competent member of the technical community, and to “show” their emerging competence in response to problems that arise in routine practice. Newcomers must also be able to show important objects of practice in conventional, recognizable ways. Work practices and infrastructure partition work into units we call “enclosed space.” yet these units must be coordinated together (both for modeled space and persons doing the modeling) into what we call a “spatial enclosure.” For example, in our study of archaeology enhanced by location aware technologies, one type of enclosed space is the particular structures, while a “spatial enclosure” is the entire settlement. Yet, enclosed spaces and spatial enclosures are comprised not only of the built environment, but also situated and embodied interactions, digital connections, representations of space and place, activity types and individuals’ unique senses of place. Modeling practices develop spatial enclosures toward particular ends, producing meanings and senses of activity types with relevant “inside” relations, borders, and “outside” realities. How spatial enclosures with relevance for learning are produced, “seen,” and “shown” vary across the cases in ways that shape how professional visions are learned and learning trajectories formed.

**G. Modality-Specific Epistemological Commitments Are a Challenge to STEM Integration**

In professional practices, the separations and later complex laminations of scales and modalities suggest that newcomers are progressively re-scaling and re-modalizing their ways of being and acting as they learn, while also increasingly articulating multiple scales and associated modalities as they move toward more full participation. In classrooms involving STEM integration, however, participants have a tendency to focus on salient, local, present forms that characterize their immediate experience. Students may struggle to “see” a mathematical idea in each of its settings and material, symbolic, and discursive instantiations. Participants’ modality-specific *epistemological commitments *pose a serious challenge to obtaining coherence of mathematical relations across modal forms, and thus to the goals of STEM education. Our study of modality transitions identifies cross-modality behaviors enacted by participants actively working to build and maintain coherence, and provides a basis for understanding how STEM integration can be successfully implemented in classrooms.

Other products generated will be theories of embodied mathematical cognition that will serve as a basis for the design and analysis of learning environments on the mathematics of motion and space. Instructional activities for middle and high school students will be produced and made available to other investigators on request.