The PascGalois Project: Visualizing Abstract Algebra

PascGalois 项目：可视化抽象代数

• 批准号: 0087644
• 负责人: Michael Bardzell
• 金额: $ 7.5万
• 依托单位: Salisbury University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-01-01 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0087644&HistoricalAwards=false
• 关键词: PascGalois; Project; Visualizing; Abstract; Algebra

Mathematical Sciences (21) The PascGalois Project is developing materials that use computing technology to enhance the teaching of abstract algebra which has traditionally been one of the most difficult and least visual subjects in the undergraduate mathematics curriculum. These materials provide an interesting class of objects and computer generated representations that allow students to "see" numerous algebraic concepts. This project has its origin in a simple exercise with Pascal's triangle. The interest in this construction lies in the fact that Pascal's triangle mod n is the group multiplication for the cyclic group Z . This construction is being generalized using other finite groups. Like Pascal's triangle mod n, PascGalois triangles can have self-similar properties. Many of these properties can be described using subgroups, quotients, and automorphisms of the group G. The project is developing software to create these images and higher dimensional generalizations (e.g. automata, similar to Conway's Game of Life) on demand so that students can investigate these patterns and their relationships to group structure. The primary objectives of the project are: to develop laboratory exercises that provide a visual component for the junior/senior level Abstract Algebra courses; to develop students' visual and intuitive understanding of difficult concepts; to model effective use of technology for prospective mathematics teachers and thus better prepare them to implement the National Council of Teachers of Mathematics standards; to stimulate interest in abstract algebra and encourage graduate study in mathematics, and to increase participation in undergraduate research in mathematics by providing a source of interesting and accessible research projects.

数学科学（21）Pascgalois项目正在开发使用计算技术来增强抽象代数的教学的材料，该学位传统上是本科数学课程中最困难，最不视觉的主题之一。这些材料提供了一类有趣的对象和计算机生成的表示形式，使学生可以“查看”众多代数概念。该项目起源于帕斯卡尔（Pascal）的三角形的简单练习。对这种构建的兴趣在于，帕斯卡尔的三角形模式是循环组z的组乘法。使用其他有限群体将这种结构概括。像帕斯卡（Pascal）的三角模式一样，帕斯加罗瓦（Pascgalois）三角形可以具有自相似的特性。这些属性中的许多可以使用G组的子组，商和自动形态描述。该项目正在开发软件来创建这些图像和更高的维度概括（例如自动机，类似于Conway的生活游戏），以便学生可以研究这些模式及其与组结构的关系。 该项目的主要目标是：开发实验室练习，为初级/高级级别摘要代数课程提供视觉组成部分；发展学生对困难概念的视觉和直觉理解；将有效利用技术用于预期数学教师，从而更好地为他们做好准备，以实施全国数学标准教师理事会；为了激发对抽象代数的兴趣并鼓励数学研究生研究，并通过提供有趣且易于访问的研究项目的来源来增加对数学研究的参与。