Three-dimensional figures can now be represented as diagrams that appear to extend into space in ways that are free of material or physical constraints. They can be rendered at any size, in any orientation, and at any position in space, and can thereby realize a far more varied set of mathematical concepts than what is possible with physical models. The goal of this project is to investigate the transformative educational potential of these representations and to generate a knowledge base that teachers, teacher educators, and researchers can use to reimagine the learning and teaching of geometry.

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CAREER: A Transformative Approach for Teaching and Learning Geometry by Representing and Interacting with Three-Dimensional Figures

People think about space, objects in space, and the relationships of those objects all of the time. Reasoning about space is fundamental to tasks ranging from engine repair and moving furniture to multivariable calculus, linear algebra, and other advanced areas of mathematics. In the physical world, spatial reasoning is linked to physical movement, such as turning one’s head to get a different view of an assembly or rotating an object in space to fit it around a corner. In the realm of mathematics, spatial reasoning is associated with mental manipulations of two-dimensional diagrams. The disconnect between the strategies used to explore the world and the spatial abilities used in mathematics classrooms is fundamentally a representation problem: Because of the constraints of drawing on flat surfaces, three-dimensional figures are compressed into flat diagrams. But the emergence of extended reality (XR) technologies, such as virtual and augmented reality, offers a profound shift in our capacities for representing and interacting with information. Three-dimensional figures can now be represented as diagrams that appear to extend into space in ways that are free of material or physical constraints. They can be rendered at any size, in any orientation, and at any position in space, and can thereby realize a far more varied set of mathematical concepts than what is possible with physical models. The goal of this project is to investigate the transformative educational potential of these representations and to generate a knowledge base that teachers, teacher educators, and researchers can use to reimagine the learning and teaching of geometry.

The educational potential of spatial inscriptions will be explored through two parallel strands of research. A teaching strand will characterize the existing representations for three-dimensional figures and analyze how those representations feature in the instructional practices of high school geometry teachers. The teaching strand has two components. (1) A semiotic and conceptual analysis of contemporary high school geometry textbooks will investigate how three-dimensional figures are typically represented. (2) A semiotic analysis of the words, symbols, diagrams, and models that high school geometry teachers use to communicate about three-dimensional geometric figures will investigate how three-dimensional figures are typically represented in classrooms. A complementary learning strand of research will investigate how the unique affordances of diagrams inscribed in immersive spaces can be leveraged to create new opportunities for high school students to explore three-dimensional figures. The methods for these investigations will be textbook analysis, classroom observations, survey experiments, and task-based interviews of students in immersive environments. Through the learning strand of research, multimodal analyses of student explorations of diagrams will generate a first approximation of a theory that explains how movement-based interactions with spatially-inscribed diagrams can contribute to the learning and teaching of high school geometry. Through the teaching strand of research, analyses of the extant representations of three-dimensional figures in textbooks and high school geometry classrooms will provide a foundation for embedding the emerging representations of spatial inscriptions in instructional practices.

## Project Materials

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