Discussant: Jim Slotta

Justin Dimmel - The Geometry of Movement: Designing a Gesture-Based Virtual Environment for Making and Transforming Spatial Inscriptions

Spatial inscriptions are representations that are inscribed into space itself, rather than onto some two-dimensional surface. My research lab has been designing and developing mathematics-specific virtual environments that allow learners to make and transform spatial inscriptions. One such environment is HandWaver, a virtual mathematical making environment in development since 2017. I describe the current minimum viable prototype of HandWaver in terms of its gesture-based user-interface. I report on an in-progress study in which we are attempting to capture the natural movements people use to create mathematical figures in space. We prompted undergraduates to act out primitive geometric figures (e.g., lines, planes, circles, spheres) and recorded their resulting movements. The goal of this work is to develop a geometry of movement, whereby people could use their bodies to create spatial inscriptions that represent geometric figures.

Caro Williams-Pierce - Designing for - and Seeing - Mathematical Play

Rolly's Adventure is a mathematics learning game designed to support mathematical play and learning, which players manifest through spoken language, physical gestures, and digital actions. Mathematical play is defined as ""voluntary engagement in cycles of mathematical hypotheses with occurrences of failure,"" and occured due to five principles for designing provocative objects (Williams-Pierce & Thevenow-Harrison, in revision). One of the primary components of provocative objects is that mathematical notation is introduced late, which causes players to frequently use physical gesture and digital action in order to communicate about the underlying mathematical behavior of the game. In this presentation, mathematical play, learning (as generalizing), provocative objects, and embodied cognition will be discussed.

Erin Ottmar - Graspable Math: Integrating Perceptual Learning, Gesture, and Action within Algebra Problem Solving

Mathematics is driven by the study of patterns or relations among entities. Algebraic notation is a pervasive and powerful means of capturing these patterns, but notation presents a major hurdle for students learning algebra. Substantial empirical work has demonstrated that notation reading and manipulation involves not just acquiring abstract rules, but also learning appropriate perceptual processes .Yet, typical instruction often focuses on interpretation and manipulation of symbols, neglecting the importance of perceptual learning to mathematical skill. Based on perceptual and embodied theory, my colleagues and I have developed Graspable Math (GM), an interactive notation tool, which makes the implicit structure of mathematical objects overtly visual symbols by turning symbols into tactile objects whose structure can be appreciated through exploration and manipulation, by grounding algebraic expressions and transformations in space and action. This presentation will introduce GM, its theoretical underpinnings, and data suggesting that interventions that incorporate embodied, object-based understandings of formal symbolic systems (a dynamic algebra) may help increase students’ mathematical proficiency and engagement and help facilitate high quality mathematical discourse and instruction in the classroom.

Alik Palatnik - 3D Sketching Approach to Solid Geometry Learning

A rise of new mediums provides an alternative to what was called by Seymour Papert “a paper math”. We present preliminary findings about the affordances of 3D-sketching (drawing in three dimensions with a handheld device extruding melted thermoplastic) approach to learning and teaching spatial geometry. The empirical context for presented study is a design-based educational research project evaluating possible added value of 3D-sketching while solving solid geometry tasks requiring introduction of auxiliary elements. The working hypothesis is that the 3D sketching mobilizes latent sensorimotor and haptic resources of the learner and, thus, facilitate conceptual learning of spatial geometry. We present micro-genetic analysis of multimodal data gathered during clinical interviews focusing on 3D-sketching artifacts produced by students, students' physical actions and multimodal utterance around the available media. 3D sketching provided students with valuable insights on spatial properties of figures and solids in the tasks they face.

Tyler Marghetis - Doing Math as Design: How Math-Doers Create Their Own Ecosystems for Thinking

Math experts actively design their own physical environments. A typical practioner might stand at a blackboard, chalk in hand… and then sketch, scribble, and inscribe, thus crafting the ecosystem in which they’re thinking. I refer to this as ‘notational niche construction,’ on analogy with the way organisms engage in ‘niche construction’ within evolutionary biology. In this talk, I describe two projects that look at this process, at a massive scale and with precise granularity. In one, we investigate how the public creates and transforms their own ‘notational niches,’ using a huge corpus of hundreds of thousands of algebraic expressions created using free, online dynamic algebra software. In the other, we use tools from complexity and network science to describe how PhD students create and interact with inscriptions while generating proofs. Together, these projects shed light on how even the most mundane mathematical practice requires designing one’s own cognitive ecosystem.

Ivon Arroyo - Learning Technologies for Embodied Math Learning

We present a technology-based paradigm to support embodied mathematics educational games, using wearable devices in the form of SmartPhones for physically active mathematics learning, for full classes of students in formal in- school education settings. The Wearable Learning Cloud Platform is web based infrastructure that enables students to carry (or wear) one mobile device per child, as they embark on individual or team-based activities that require physical engagement with the environment. These Wearable Tutors serve as guides and assistants while students manipulate, measure, estimate, discern, discard, find and/or place  mathematical objects that satisfy specified constraints. Multi-player math games that use this infrastructure have yielded both cognitive and affective benefits. Beyond math game play, the Wearable Games Engine Authoring Tool enables students and teachers to create games themselves for other students to play; in this process, students engage in computational thinking and learn about finite-state machines. We present the idea, the infrastructure, games, and results for a series of experiments on both game play and game creation.