The PascGalois Project: Visualization in Abstract Mathematics

索尔兹伯里大学与加州州立大学弗雷斯诺分校、洪堡州立大学、肯尼索州立大学、佛罗里达新学院和匹兹堡州立大学合作，正在开发在线材料，以加强几门数学课程，包括那些面向未来教师的课程。学生们经常发现抽象的数学很难形象化。他们很少看到课程作业和独立数学探究之间的联系。pasgalois项目提供了一类有趣的离散动力系统，这些系统是在代数结构上生成的，例如群，以及这些系统的彩色图形表示，使学生能够看到与动力学交织在一起的代数和数论性质。该项目的主要目标是：开发包括Java小程序在内的在线活动，为抽象代数、离散数学、数论、动力系统和计算机图形学等课程提供可视化组件；培养学生对困难数学概念的直观理解；为未来的数学教师提供有效使用技术的榜样，使他们更好地为实施NCTM标准做好准备；并通过提供有趣和易于获取的研究项目，在高级课程和本科研究之间架起一座桥梁。该项目的智力价值在于为高级数学课程设计的创新活动，揭示了课程之间的联系，否则可能看起来不相关，以及由此产生的本科研究项目。两个夏季夏令营突出了本科生的研究工作。通过MathDL等场所在线传播Java小程序和活动将产生更广泛的影响。这些活动将贯穿整个数学课程。

Salisbury University, in collaboration with California State University Fresno, Humbolt State University, Kennesaw State University, New College of Florida, and Pittsburg State University, is developing online materials to enhance several mathematics courses, including those for prospective teachers. Students frequently find abstract mathematics difficult to visualize. They rarely see links between coursework and independent mathematical inquiry. The PascGalois Project provides an interesting class of discrete dynamical systems generated over algebraic structures, such as groups, and color graphical representations of these systems that allow students to see algebraic and number theoretic properties intertwined with the dynamics.The primary objectives of this project are to: develop online activities including Java applets that provide a visual component for courses including abstract algebra, discrete mathematics, number theory, dynamical systems, and computer graphics; develop students' visual and intuitive understanding of difficult mathematical concepts; model effective use of technology for prospective mathematics teachers and better prepare them to implement the NCTM standards; and create a bridge between upper-level courses and undergraduate research by providing a source of interesting and accessible research projects.The intellectual merit of the project is in the innovative activities designed for upper-level mathematics courses, revealing connections between courses that otherwise may appear unrelated, and the undergraduate research projects that are generated. Two summer PascGalois retreats highlight the undergraduate research efforts.Broader impacts follow from online dissemination of the Java applets and activities through a venue such as MathDL. These activities will be integrated throughout the mathematics curriculum.