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This study investigates how learners collaboratively construct embodied geometry knowledge in shared VR environments. Three groups of in-service teachers collaboratively explored six geometric conjectures with various virtual objects (geometric shapes) under the guidance of a facilitator. Although all the teachers were in different physical locations, they logged into a single virtual classroom with their respective groups and were able to see and manipulate the same geometric shapes as well as see their collaborators' avatars and actions on the shapes in real time in the shared virtual space. This paper introduces a novel multimodal data analysis method for analyzing participants' interactive patterns in collaborative forms of actions, gestures, movements, and speech. Results show that collaborative speech has a strong simultaneous relationship with actions on virtual objects and virtual hand gestures. They also showed that body movements and positions, which often focus on virtual objects and shifts in these movements away from or around the object, often signal key interactional collaborative events. In addition, this paper presents five emergent multimodality interaction themes showing participants' collaborative patterns in different problem-solving stages and their different strategies in collaborative problem-solving. The results show that virtual objects can be effective media for collaborative knowledge building in shared VR environments, and that structured activity design and moderate realism may benefit shared VR learning environments in terms of equity, adaptability, and cost-effectiveness. We show how multimodal data analysis can be multi-dimensional, visualized, and conducted at both micro and macro levels.

Materialist design is presented as an embodied perspective on educational design that can be applied to redesign of classroom-based learning environments. Materialist design is informed by a framework of materialist epistemology, which positions material innovation on equal placement with symbol-based formal theory. Historical examples of Einstein's conceptual reliance on trains for his Theory of Relativity and the Wright brothers' use of wind tunnels in aeronautics illustrate how materialist design drives progress on complex design problems. A key aspect is the application of scale-down methodology, where complex systems are reconceptualized as interactions among nearly decomposable subsystems that can be redesigned and integrated back into the entire system. The application of materialist design is illustrated with the redesign of an embodied video game that uses real-time motion capture technology to promote high school geometry reasoning and proof, following its use in an ethnically and linguistically diverse classroom. Our embodied perspective offers particular insights for understanding and implementing designs of complex learning environments, and assessing their influences on educational practices and student outcomes. [This article was published in "Educational Technology Research and Development" (EJ1310136).]

Theories of grounded and embodied cognition offer a range of accounts of how reasoning and body-based processes are related to each other. To advance theories of grounded and embodied cognition, we explore the "cognitive relevance" of particular body states to associated math concepts. We test competing models of action-cognition transduction to investigate the cognitive relevance of directed actions to students' mathematical reasoning in the area of geometry. The hypotheses we test include (1) that cognitively relevant directed actions have a direct effect on performance (direct cognitive relevance hypothesis), (2) that cognitively relevant directed actions lead to more frequent production of gestures during explanations, which leads to improved performance (mediated cognitive relevance hypothesis), and (3) that performance effects of directed actions are influenced by the presence or absence of gesture production during mathematical explanations (moderated cognitive relevance hypothesis). We explore these hypotheses in an experiment where high school students (N = 85) evaluated the truth of geometry conjectures after performing cognitively relevant or cognitively irrelevant directed actions while playing a movement-based video game. Contrary to the direct and mediated cognitive relevance hypotheses, we found no overall differences in performance or gesture production between relevant and irrelevant conditions. Consistent with the moderated cognitive relevance hypothesis, cognitive relevance influenced mathematical performance, as measured by the accuracy of students' intuitions, insights, and the validity of their proofs, provided that students produced certain kinds of gestures during mathematical explanations (i.e., with explanatory gestures as the moderator). Implications for theories of grounded and embodied cognition and the design of embodied forms of educational interventions are discussed.

We develop a theory of grounded and embodied mathematical cognition (GEMC) that draws on action-cognition transduction for advancing understanding of how the body can support mathematical reasoning. GEMC proposes that participants' actions serve as inputs capable of driving the cognition-action system toward associated cognitive states. This occurs through a process of transduction that promotes valuable mathematical insights by eliciting dynamic depictive gestures that enact spatio-temporal properties of mathematical entities. Our focus here is on pre-college geometry proof production. GEMC suggests that action alone can foster insight but is insufficient for valid proof production if action is not coordinated with language systems for propositionalizing general properties of objects and space. GEMC guides the design of a video game-based learning environment intended to promote students' mathematical insights and informal proofs by eliciting dynamic gestures through in-game directed actions. GEMC generates several hypotheses that contribute to theories of embodied cognition and to the design of science, technology, engineering, and mathematics (STEM) education interventions. Pilot study results with a prototype video game tentatively support theory-based predictions regarding the role of dynamic gestures for fostering insight and proof-with-insight, and for the role of action coupled with language to promote proof-with-insight. But the pilot yields mixed results for deriving in-game interventions intended to elicit dynamic gesture production. Although our central purpose is an explication of GEMC theory and the role of action-cognition transduction, the theory-based video game design reveals the potential of GEMC to improve STEM education, and highlights the complex challenges of connecting embodiment research to education practices and learning environment design.

This classroom-based investigation sought to document how, in real time, STEM teachers and students attempt to locate the invariant mathematical relations that are threaded through the range of activities and representations in these classes, and how highlighting this common thread influences student participation and learning. The authors conducted multimodal discourse analyses of teacher-student interactions during multiday observations in 3 urban high school STEM classes. The focal lessons were in electrical engineering and mechanical engineering (within Project Lead the Way), and precollege geometry. Across 3 cases, teachers and students actively built and maintained cohesion of invariant mathematical relations across activities and representations. Pre- and post-lesson interviews revealed that teachers intentionally managed cohesion to provide the continuity across the curricular activities that teachers believed would promote student understanding. The findings contribute to ways of fostering STEM integration and ways of grounding abstractions to promote meaning making and transfer. [This paper was published in "Journal of Education Research" (EJ1134489).]

Dynamic geometry software (DGS) has long been studied in mathematics education as a way for students to explore and interact with geometric objects and figures. Recent advances in Augmented Reality (AR) technologies that allow dynamic three-dimensional mathematical objects to appear in students' environment as holograms have changed the nature of what is possible for a DGS, particularly with respect to embodiment. New forms of embodied interactions may arise in AR-based DGS, as students gesture and move their bodies through their environment, taking different perspectives to interact with these immersive shapes projected in three dimensions. In the present study, we examine videos of 28 high school students interacting with an AR-based version of the DGS GeoGebra, while wearing the Microsoft HoloLens 2 headsets. We document the novel kinds of embodied interactions that the AR environment affords, relating to (1) perspective and orientation, (2) scale, (3) three dimensions. Based on our analysis, we give important directions for future research on DGS and implications for the design of the next generation of holographic DGS. • Augmented Reality-based Dynamic Geometry Software has novel affordances. • We examine 28 high students interacting with an AR-based version of GeoGebra. • We found new interactions related to perspective, scale, and 3 dimensions. • Students used gesture, actions, and body movements in novel ways. [ABSTRACT FROM AUTHOR]

Gestures are associated with powerful forms of understanding; however, their causative role in mathematics reasoning is less clear. We inhibit college students' gestures by restraining their hands, and examine the impact on language, recall, intuition, and mathematical justifications of geometric conjectures. We test four mutually exclusive hypotheses: (1) gestures are facilitative, through cognitive off-loading, verbal support, or transduction, (2) gestures are not facilitative, but being inhibited from gesturing increases cognitive demands, (3) gestures are a byproduct of reasoning processes that would take place with or without the gestures' overt presence, and (4) gestures can cause learners to focus on concrete, salient representations, inhibiting abstraction. We find support for the third hypothesis, concluding that learners making or being inhibited from making gestures does not seem to impact their mathematical problem-solving, cognitive, or language processes. This suggests that being unable to overtly perform personally-generated gestures is not a hindrance to learners in this context; however this would not necessarily hold for directed or structured gestures. • Inhibiting hand gestures does not impair undergraduate performance on geometry tasks. • Inhibiting hand gestures does not impact language use in math justifications. • Gesture may be correlated with valid reasoning, but is not causatively related. • Making dynamic gestures was associated with improved insight and proof. [ABSTRACT FROM AUTHOR]

• Learners use collaborative gestures when proving mathematical conjectures. • Types of collaborative gestures include mirror, echo, alternate, and joint gestures. • Collaborative gestures show how mathematical cognition becomes extended. • Gestures should be considered alongside collaborative talk moves. Gestures have been shown to play a key role in mathematical reasoning and to be an indicator that mathematical understanding is embodied – inherently linked to action, perception, and the physical body. As learners collaborate and engage in mathematical discussions, they use discourse practices like explaining, refuting, and building on each other's reasoning. Here we examine how gestural embodied actions become distributed over multiple learners confronting mathematical tasks. We define collaborative gestures as gestural exchanges that take place as learners discuss and explore mathematical ideas, using their bodies in concert to accomplish a shared goal. We identify several ways in which learners' gestures can be used collaboratively and explore patterns in how collaborative gestures arise while proving geometric conjectures. Learners use collaborative gestures to extend mathematical ideas over multiple bodies as they explore, refine, and extend each other's mathematical reasoning. With this work, we seek to add to notions of important talk moves in mathematical discussions to also include a consideration of important gesture moves. [ABSTRACT FROM AUTHOR]