The PascGalois Project: Visualizing Abstract Algebra

数学科学(21)PascGalois项目正在开发利用计算机技术加强抽象代数教学的材料，抽象代数传统上是本科数学课程中最难和最不直观的科目之一。这些材料提供了一个有趣的对象类和计算机生成的表示法，使学生能够“看到”大量的代数概念。这个项目起源于一个简单的帕斯卡三角练习。这种构造的有趣之处在于，Pascal三角mod n是循环群Z的群乘。这一结构正在使用其他有限群进行推广。与Pascal的三角形mod n一样，PascGalois三角形也可以具有自相似性质。这些性质中的许多可以用群G的子群、商和自同构来描述。该项目正在开发软件，以根据需要创建这些图像和更高维的泛化(例如自动机，类似于康威的生命游戏)，以便学生能够研究这些模式及其与群结构的关系。该项目的主要目标是：开发实验室练习，为初级/高级抽象代数课程提供视觉内容；培养学生对困难概念的视觉和直觉理解；为未来的数学教师示范有效利用技术，从而更好地为他们实施全国数学教师委员会标准做准备；激发对抽象代数的兴趣，鼓励研究生学习数学，并通过提供有趣和可获得的研究项目来源，增加对数学本科生研究的参与。

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.