Teaching Abstract Algebra for Understanding

数学科学（21）本项目正在开发教师材料，以支持通过一系列研究和开发（R D）工作开发的抽象代数创新课程材料的实施。课程材料让学生积极理解群论的基本概念，平衡再造阶段（学生根据直觉，非正式策略和先验知识开发概念）和演绎阶段（学生根据正式定义和先前建立的结果证明重要结果）。早期的研发工作证实，连续使用这两个阶段的顺序，帮助学生连接概念的直观理解的形式理论。该课程已在PI的机构中成功地试行了标准的抽象代数入门课程，目前正在对材料的一个子集进行调整，以便在当地社区学院的创新过渡课程中使用，该课程旨在为职前师范教育学生提供基于证明的数学课程。除了开发一套教师指南和相关的课程实施材料，该项目计划进行研究，以深入了解不同的教师实施课程时出现的挑战和机遇，并获得有关学生如何学习抽象代数以及课程材料如何提高学生学习的新知识。这个项目的智力价值在于它的接地在当前的数学教育研究文献和方式，其中正在开发的教师材料作为一种工具，沟通项目自己的研究结果对学生的抽象代数概念的学习回到更大的机构的数学教师。该项目正在发挥广泛的影响，因为抽象代数和“过渡到证明”课程是数学课程的主要内容，特别是对准备成为教师的大部分学生。此外，数学家，数学教育工作者和两年制大学的同行之间的合作为其他机构与他们的“支线”两年制学校密切合作提供了一个模式。

Mathematical sciences (21) This project is developing instructor materials to support the implementation of innovative course materials for abstract algebra developed through a series of research and development (R&D) efforts. The course materials engage students actively in understanding fundamental concepts of group theory, balancing reinvention phases (in which students develop concepts based on their intuition, informal strategies, and prior knowledge) and deductive phases (in which students prove important results based on formal definitions and previously established results). The earlier R&D efforts confirm that a sequence of successive uses of these two phases helps students connect an intuitive understanding of concepts to the formal theory. The curriculum has been piloted successfully at the PI's institution in a standard introductory abstract algebra course, and a subset of the materials is currently being adapted for use in an innovative transition course at a local community college designed to prepare pre-service teacher education students for proof-based mathematics courses. In addition to developing a set of instructor guides and associated curricular implementation materials, the project plans to conduct research to gain insight into the challenges and opportunities that emerge as different faculty implement the curriculum, and to obtain new knowledge about how students learn abstract algebra and how the course materials enhance student learning. The intellectual merit of this project lies both in its grounding in the current mathematics education research literature and the way in which the instructor materials under development serve as a vehicle to communicate the project's own research findings on student learning of abstract algebra concepts back to the larger body of mathematics faculty. The project is exercising broad impact since abstract algebra and "transition-to-proof" courses are a staple of the mathematics curriculum, particularly for the large subset of students preparing to be teachers. Furthermore, the collaboration among mathematicians, mathematics educators, and counterparts at the two-year college level offers a model for other institutions working closely with their "feeder" two-year schools.