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

人们无时无刻不在思考空间、空间中的物体，以及这些物体之间的关系。关于空间的推理是从发动机维修和搬运家具到多变量微积分、线性代数和其他高级数学领域的基本任务。在物理世界中，空间推理与物理运动联系在一起，比如转动头部以获得组件的不同视图，或者在空间中旋转对象以适应角落。在数学领域，空间推理与对二维图的心理操作有关。用来探索世界的策略和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现问题：由于在平面上绘画的限制，三维图形被压缩成平面图。但虚拟和增强现实等扩展现实(XR)技术的出现，使我们表达信息和与信息互动的能力发生了深刻的变化。三维图形现在可以表示为图表，这些图表似乎以不受材料或物理限制的方式延伸到空间中。它们可以在任何大小、任何方向和空间中的任何位置进行渲染，因此可以实现比物理模型更多样化的一组数学概念。这个项目的目标是调查这些表征的变革教育潜力，并生成一个知识库，教师、教师教育工作者和研究人员可以利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线索来探索。一条教学线索将描述现有的三维图形的表征，并分析这些表征如何在高中几何教师的教学实践中表现出来。教学链由两个组件组成。(1)当代高中几何教科书的符号学和概念分析将调查三维图形是如何典型地表现出来的。(2)对高中几何教师用来交流三维几何图形的文字、符号、图表和模型的符号学分析将调查三维图形在课堂上通常是如何表现的。一个补充学习链的研究将调查如何利用在沉浸式空间中刻画的图表的独特启示，为高中生创造探索三维图形的新机会。这些调查的方法将是教科书分析、课堂观察、调查实验和对身临其境环境中的学生进行基于任务的访谈。通过研究的学习线索，对学生探索图表的多模式分析将产生一种理论的第一近似值，该理论解释了基于运动的互动与空间刻画图表如何有助于高中几何的学习和教学。通过教学研究，对教科书和高中几何教室中现有的三维图形表示的分析将为将新兴的空间铭文表示法嵌入教学实践提供基础。该奖项的部分资金来自《2021年美国救援计划法案》(公共法律117-2)。该奖项的部分资金来自发现研究预科-12项目(DRK-12)，该项目旨在通过研究和开发创新资源、模型和工具，显著提高预科-12学生和教师的科学、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工作的基础上，为拟议的项目提供了理论和经验上的证明。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.