Generalization Across Multiple Mathematical Areas: Studies of Classrooms and Teaching

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With funding from the EHR Core Research Program, this project will investigate how productive mathematical generalization can be supported in whole-classroom settings. Generalization is the ability to recognize patterns in relationships between numbers. An example of generalization is recognizing that 2 + 3 = 5, and generalizing that adding any even number to any odd number will produce an odd number. Generalization is a critical component of mathematical reasoning. As a result, researchers and policymakers recommend that it should be central to the education of all students at all grade levels. However, acting on this recommendation poses serious challenges, given the limited research base about students' difficulties in making appropriate generalizations and the challenges teachers face in supporting generalization in the classroom. This project aims to help address this gap by focusing on math classes from Grade 6 through college. To do so, the project team will investigate students' generalizations, as well as the strategies that affect their ability to develop generalization skills in algebra, advanced algebra, trigonometry, calculus, and combinatorics. The project will build a framework to describe and understand research-informed, classroom-based strategies for fostering generalization at the whole-classroom level with practicing teachers and instructors of undergraduate mathematics courses. The project has two objectives: (1) systematically investigate the current state of classroom generalization, identifying existing supports for students' generalizing; and (2) examine the ways in which teachers/instructors can be supported to foster productive mathematical generalization. The study implements a classroom-based design experiment model, leveraging classroom observations and videotaped professional development sessions in a two-phase methodology, with the second phase building on the first phase. The research activities will produce a set of findings about the relationships between instructional supports for generalizing and the nature of student generalizations, thus contributing to theory and a deeper understanding of generalization as it occurs in the context of typical instruction across multiple domains.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.