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As technologies that put the body at the center of mathematics learning enter formal and informal learning spaces, we still know little about the teaching methods educators can use to support students’ learning with these specialized systems. Drawing on ethnomethodology and conversation analysis (EMCA) and the Co-Operative Action framework, we present three multimodal ways that educators can be responsive to learners’ embodied ideas and help them transform sensorimotor patterns into mathematically significant perceptions. These techniques include (1) encouraging learners to use gesture to express and reflect on their ideas, (2) presenting multimodal candidate understandings to check comprehension of learners’ embodied ideas, and (3) co-constructing multimodally expressed embodied ideas with learners. We demonstrate how these techniques create opportunities for learning and discuss implications for a multimodal, embodied practice of responsive teaching.

Design and communication materials for connecting tangible devices to PhET Interactive Simulations. Collaboration between Saint Louis University, University of California Berkeley, and University of Colorado at Boulder.

Vygotsky conceptualized the teaching/learning process as inherently collaborative. We extend prior evaluations of this claim by enlisting eye-tacking instruments to monitor the perceptual activity of four teacher–student dyads, as the student solves a challenging manipulation problem designed to ground the scientific notion of parabolas in their new sensorimotor routines. Analyzing each dyad’s gaze paths led us to model the teaching/learning process as the emergence and dynamic transformation of intersubjective coupling between the student and tutor perception–action systems. While the student’s sensory-motor coordination gradually gravitates toward an effective routine, the tutor’s perception is iteratively launched from the student’s current action, until the tutor detects an optimal moment for verbal intervention. In this micro-zone of proximal development, the student’s motor action comes to align with the tutor’s cultural-perspective strategy. Our elaboration of the cultural–historical approach to teaching/learning draws on research on joint attention and joint action from the cognitive sciences as well as the embodied-design approach from the educational sciences and demonstrates a compatibility of Vygotsky’s heritage and complex dynamic systems theory. Finally, we discuss the educational value of the observed student–tutor intersubjective coupling phenomena, thus grounding the contribution of this multidisciplinary study within educational concerns.

This paper presents a pedagogical agent designed to support students in an embodied, discovery-based learning environment. Discovery-based learning guides students through a set of activities designed to foster particular insights. In this case, the animated agent explains how to use the Mathematical Imagery Trainer for Proportionality, provides performance feedback, leads students to have different experiences and provides remedial instruction when required. It is a challenging task for agent technology as the amount of concrete feedback from the learner is very limited, here restricted to the location of two markers on the screen. A Dynamic Decision Network is used to automatically determine agent behavior, based on a deep understanding of the tutorial protocol. A pilot evaluation showed that all participants developed movement schemes supporting proto-proportional reasoning. They were able to provide verbal proto-proportional expressions for one of the taught strategies, but not the other.

The combination of two methodological resources - natural user interface and multimodal learning analytics - is creating opportunities for educational researchers to empirically evaluate theoretical models accounting for the emergence of concepts from situated sensorimotor activity. Seventy-six participants (9-14 years old) solved tablet-based presymbolic manipulation tasks designed to foster grounded meanings for the mathematical concept of proportional equivalence. Data gathered in task-based semi-structured clinical interviews included action logging, eye-gaze tracking, and videography. Analysis of these data indicates that successful task performance coincided with spontaneous emergence of stable dynamical gaze path patterns soon followed by multimodal articulation of strategy. Significantly, gaze patterns included unmanipulated, non-salient screen locations. We present cumulative evidence that these gaze patterns served as ‘‘attentional anchors'' mediating participants' problem solving. By way of further contextualizing our claim, we also present case studies from the various experimental conditions. We interpret the findings as enabling us to revisit, support, refine, and perhaps elaborate on seminal claims from Piaget's theory of genetic epistemology and in particular his insistence on the role of situated motor-action coordination in the process of reflective abstraction.

In the course of developing an experimental algebra unit, the researchers noted variability in their design's instructional potential across a set of implementation media. In an effort to explain this variability, we revisited the classical theoretical construct of transparency. Transparency is the perceptual and conceptual accessibility of the mechanism, logic, and application of a tool. Corroborating earlier literature, it appears that participants saw only what they had built - transparency is a subjective achievement of a learner rather than an inherent feature of a device. Our first design prompted students with an algebraic proposition, for example "3x+2=4x-1". The two equivalent expressions were to be interpreted by students as alternative quantifications of a single linear spatial interval; namely, the path that a giant took on two separate occasions to bury and recover treasure. Problem solving required manually adjusting the modeling media to coordinate two types of equivalence: (a) the total length represented by each expression; and (b) the length represented by the variable and known units. The researchers found that successful coordination was predicated on the subjective transparency of the models' perceptual. Therefore, in redesigning the activity as a computer-based application we will have learners first construct tools and only then automatize them. Copyright 2013 ACM.

In a retrospective analysis of my own pedagogical design projects over the past twenty years, I articulate and compare what I discern therein as two distinct activity genres for grounding mathematical concepts. One genre, "perception-based design," builds on learners' early mental capacity to draw logical inferences from perceptual judgment of intensive quantities in source phenomena, such as displays of color densities. The other genre, "action-based design," builds on learners' perceptuomotor capacity to develop new kinesthetic routines for strategic embodied interaction, such as moving the hands at different speeds to keep a screen green. Both capacities are effective evolutionary means of engaging the world, and both bear pedagogical potential as epistemic resources by which to build meaning for mathematical models of, and solution processes for situated problems. Empirical studies that investigated designs built in these genres suggest a two-step activity format by which instructors can guide learners to reinvent conceptual cores. In a primary problem, learners apply or develop non-symbolic perceptuomotor schemas to engage the task effectively. In a secondary problem, learners devise means of appropriating newly interpolated mathematical forms as enactive, semiotic, or epistemic means of enhancing, explaining, and evaluating their primary response. Whereas my analysis distills activities into two separate genres for rhetorical clarity, ultimately embodied interaction may interleave and synthesize the genres' elements. Copyright 2013 ACM.

We suggest that a commitment to political Participatory Design defines a new role for children in participatory practices – the role of protagonist. In this role, the objective extents beyond giving children a voice in design, to more broadly address how they can be empowered to shape technological development and critically reflect on the role of technology in their practices. This re-accentuation of political Participatory Design and the formulation of the protagonist role is important in terms of deepening our understanding of how children may be empowered through design, and is timely in relation to current challenges pertaining to training children within the context of 21st century skills. Through a case study we demonstrate how the role of protagonist, based on political Participatory Design, changes the objective, process, and outcome measures of the design process.