Teaching Abstract Algebra for Understanding

数学科学 (21) 该项目正在开发教师材料，以支持通过一系列研发 (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，平衡重塑阶段（学生根据直觉、非正式策略和先验知识发展概念）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，这两个阶段的一系列连续使用有助于学生将对概念的直观理解与形式理论联系起来。该课程已在 PI 机构的标准抽象代数入门课程中成功试点，目前正在对部分材料进行改编，用于当地社区学院的创新过渡课程，该课程旨在为职前教师教育学生准备基于证明的数学课程。除了开发一套教师指南和相关课程实施材料外，该项目还计划进行研究，以深入了解不同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增强学生学习的新知识。该项目的智力价值不仅在于其以当前数学教育研究文献为基础，还在于正在开发的教师材料作为一种工具，将项目自身关于学生抽象代数概念学习的研究成果传达给广大数学教师。该项目正在产生广泛的影响，因为抽象代数和“过渡到证明”课程是数学课程的主要内容，特别是对于准备成为教师的大量学生而言。此外，数学家、数学教育者和两年制大学水平的同行之间的合作为其他机构与其“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.