Doctoral Dissertation Research: The Shift from Abstraction to Material and Virtual Models in Mathematical Research and Pedagogy

在21世纪初，数学家寻求数学理论和数学对象的有形经验；反过来，他们提倡一种经验的、具体化的、感官介导的方法来进行数学训练。通过档案研究、深度访谈以及在会议、教室和实验室中的参与者观察，本项目旨在将上述转变定位于其历史和制度背景中。具体来说，这个项目询问是什么促使并使数学家们寻求对数学思想更具体和更深入的理解，然后跟踪这种变化对现在的数学交流和教学方式可能带来的影响。为了回答上述问题，博士候选人将研究从20世纪60年代到现在的五个案例研究：(1)20世纪60年代后期制作的数学视频，以说明抽象和难以想象的数学问题；(2) 1987年的绘图手册，要求拓扑学家绘图并说明他们的工作；(3)由美国国家科学基金会资助的明尼苏达大学数学研究中心，成立于1991年，致力于为数学家开发可视化程序；(4)利用纤维艺术（如编织和钩针编织）来研究数学表面的性质；(5)过去二十年来计算折纸的出现，将古老的折纸艺术纳入了专业数学的范畴。所有五个案例都代表了“数学表现”的例子，它表示数学研究的具体的、可展示的、示范的、可展示的和可呈现的方面。通过关注数学工作的经验方面，该项目旨在将科学研究学者开发的基于实践的数学调查和理论工具的历史带入数学实践，并为数学实践理论提供独特的理论干预。潜在的更广泛的影响本项目将通过研究一种尚待研究的数学教育方法来促进科学教育学，该方法通过培养数学工作的经验、动手和具体化的性质来补充公理抽象方法。通过关注计算机图形学在数学中的应用，该项目将为理解数学界关于计算机技术在数学工作中的应用的争论提供一个新的视角。

IntroductionAt the beginning of the twenty-first century, mathematicians sought tangible experiences with mathematical theories and objects; in turn, they advocated an experiential, embodied, and sensorial mediated approach to mathematical training. Intellectual MeritThrough archival research, in-depth interviews, and participant observation in conferences, classrooms, and labs, this project aims to locate the shift characterized above in its historical and institutional context. Specifically, this project asks what prompted and enabled mathematicians to seek a more embodied and procession understanding of mathematical ideas, and then tracks what this change might entail for the way mathematics is now communicated and taught.To answer the above questions, PhD candidate will examine five case studies starting in the 1960s and leading up to the present: (1) mathematical videos produced in the late 1960s to illustrate otherwise abstract and difficult to imagine mathematical problems; (2) a 1987 drawing manual that called on topologists to draw pictures and illustrate their work; (3) An NSF-funded mathematical research center at the University of Minnesota, established in 1991, that was dedicated to developing visualization programs for mathematicians; (4) the appropriation of fiber arts (such as knitting and crocheting) to investigate the properties of mathematical surfaces; and (5) the emergence over the past two decades of computational origami, which brings the ancient art of origami under the purview of professional mathematics.All five cases represent examples of "mathematical manifestation," which denotes the concrete, demonstrable, exemplary, displayable, and presentable aspects of mathematical research. By focusing on the experiential aspects of mathematical work, this project seeks to bring to the history of mathematics practice-based investigations and theoretical tools developed by science studies scholars, as well as offer a unique theoretical intervention into theories of mathematical practice.Potential Broader ImpactsThis project will contribute to scientific pedagogy by examining a under examined approach to mathematical education, which supplements the axiomatic abstract method by cultivating the experiential, hands-on, and embodied nature of mathematical work. By focusing on the application of computer graphics in mathematics, the project will provide a new perspective from which to understand debates within theMathematical community regarding the application of computer technology to mathematical work.