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.
