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

博士论文研究：数学研究和教育学中从抽象到物质和虚拟模型的转变

• 批准号: 1057311
• 负责人: David Kaiser
• 金额: $ 0.99万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-03-01 至 2013-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1057311&HistoricalAwards=false
• 关键词: Doctoral; Dissertation; Research; Shift; Abstraction

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.

引言在21世纪初的世纪，数学家寻求有形的经验与数学理论和对象，反过来，他们主张一个经验，体现，和感官介导的方法来数学培训。通过档案研究，深入访谈，并在会议，教室和实验室的参与观察，该项目旨在定位其历史和制度背景下的上述特点的转变。具体来说，这个项目询问是什么促使并使数学家能够寻求对数学思想的更具体和更深入的理解，然后跟踪这种变化可能对现在的数学交流和教学方式产生的影响。为了回答上述问题，博士生将研究从20世纪60年代开始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，用于说明抽象和难以想象的数学问题;（2）1987年的绘图手册，呼吁拓扑学家绘制图片并说明他们的工作;（3）1991年在明尼苏达大学建立了一个由国家科学基金会资助的数学研究中心，致力于为数学家开发可视化程序;（4）纤维艺术的挪用（如针织和钩编）来研究数学曲面的性质;以及（5）在过去二十年中出现的计算折纸，这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，“数学表现”指的是具体的、可论证的、可示范的、可展示的，数学研究中最有代表性的方面通过关注数学工作的经验方面，该项目旨在为数学史带来基于实践的调查和科学研究学者开发的理论工具，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该项目将通过研究一种未被研究的数学教育方法，它通过培养数学工作的经验性、实践性和具体性来补充公理化的抽象方法。通过关注计算机图形学在数学中的应用，该项目将提供一个新的视角，以了解数学界关于计算机技术应用于数学工作的争论。