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

人们一直在思考空间，空间中的物体，以及这些物体之间的关系。从发动机修理、搬家具到多变量微积分、线性代数和其他高级数学领域，对空间的推理是各种任务的基础。在物理世界中，空间推理与物理运动有关，比如转动一个人的头以获得一个组件的不同视图，或者在空间中旋转一个物体以适应它在拐角处的位置。在数学领域，空间推理与二维图形的心理操作有关。用于探索世界的策略与数学课堂中使用的空间能力之间的脱节，从根本上说是一个表征问题：由于在平面上绘图的限制，三维图形被压缩成平面图表。但扩展现实（XR）技术的出现，如虚拟现实和增强现实，为我们表示和交互信息的能力提供了深刻的转变。三维图形现在可以用图表来表示，这些图表似乎可以以不受材料或物理限制的方式延伸到空间中。它们可以以任何尺寸、任何方向和空间中的任何位置呈现，因此可以实现比物理模型更多样化的数学概念集。这个项目的目标是调查这些表示的变革教育潜力，并产生一个知识库，教师、教师教育工作者和研究人员可以使用它来重新想象几何的学习和教学。空间铭文的教育潜力将通过两个平行的研究方向来探索。一个教学链将描述三维图形的现有表征，并分析这些表征在高中几何教师的教学实践中的特点。教学链有两个组成部分。(1)对当代高中几何教科书的符号学和概念分析将探讨三维图形的典型表现方式。(2)对高中几何教师用来交流三维几何图形的文字、符号、图表和模型进行符号学分析，将探讨三维图形在课堂上的典型表现。研究的补充学习链将调查如何利用沉浸式空间中雕刻的图表的独特可视性，为高中生探索三维图形创造新的机会。这些调查的方法将包括教科书分析、课堂观察、调查实验和学生在沉浸式环境中的任务访谈。通过研究的学习链，对学生对图表探索的多模态分析将产生一个理论的初步近似，该理论解释了基于运动的相互作用与空间嵌入的图表如何有助于高中几何的学习和教学。通过教学研究，分析教科书和高中几何课堂中现有的三维图形表征，将为在教学实践中嵌入新兴的空间铭文表征提供基础。该奖项部分由2021年美国救援计划法案（公法117-2）资助。该奖项还部分由探索研究preK-12计划（DRK-12）资助，该计划旨在通过研究和开发创新资源、模型和工具，显著提高preK-12学生和教师对科学、技术、工程和数学（STEM）的学习和教学。DRK-12计划中的项目建立在STEM教育的基础研究和先前的研究和开发努力的基础上，为拟议的项目提供了理论和实证依据。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.This award is funded by in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The award is also funded in part by the Discovery Research preK-12 program (DRK-12) which seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers, through research and development of innovative resources, models and tools. Projects in the DRK-12 program build on fundamental research in STEM education and prior research and development efforts that provide theoretical and empirical justification for proposed projectsThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.