We explore how productive mathematical generalization can be supported in whole-classroom settings. We investigate students' classroom generalizations and the instructional, task, and pedagogical supports for fostering generalizing in the mathematical domains of algebra, advanced algebra, trigonometry, calculus, and combinatorics in Grades 6 - 16.
Generalization is a critical component of mathematical reasoning, with researchers and policymakers recommending that it be central to the education of all students at all grade levels. This recommendation poses serious challenges, however, given the research base identifying students' difficulties in creating and expressing appropriate generalizations and the challenges teachers face in supporting generalization in the classroom.
Incorporating Generalization in the Classroom
GAMMA-CAT addresses the challenge of incorporating generalization in the teaching and learning of mathematics by identifying research-informed, classroom-based strategies for fostering generalization at the whole-classroom level with practicing teachers and undergraduate instructors.
The project's objectives are twofold: (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 three-phase methodology, with each phase building on the prior phase. The research activities will produce a set of findings about the relationships between instructional supports for generalizing and the nature of student generalizations, and will contribute to a deeper understanding of generalization as it occurs in the context of typical instruction.
GAMMA-CAT results include:
1. A research- based model characterizing the nature of generalization in classroom settings in diverse content areas as well as the relationships between task features, instructional moves, classroom interactions, and student generalizations in those settings.
2. An identification of specific task features and instructional moves for promoting generalization in particular content domains and grade bands.
3. A set of classroom-tested instructional principles of promoting productive generalizing.
4. An investigation into an under- researched area of generalization - namely, the complex and demanding activity of generalizing in secondary and undergraduate topics and in open-ended, authentic tasks.
5. An extension of the research base on generalization to the whole-classroom level.
6. An improvement of students' understanding of mathematics through engagement in research-based generalization activities.
GAMMA-CAT outcomes include:
1. Broadening participation of underrepresented groups by including them as participants in the research and educational activities. The PI team will leverage existing partnerships with local districts with racially and socio-economically diverse populations to increase participation for students of color, girls, and students from low SES backgrounds.
2. Improving STEM education and educator development by integrating research activities into mathematics teaching at multiple sites in 3 states, working with 45 teachers/instructors, 60 pre-service teachers / TAs, and approximately 1,380 secondary students and 750 undergraduates. The project also offers an integrated dissemination plan that will share findings with local and national communities through research-based units and teacher materials, workshops, and state-wide mini-conferences.
3. Developing a diverse, globally competitive STEM workforce by generating knowledge leading to improved instruction in critical areas of mathematics, including topics foundational for STEM careers such as advanced algebra, trigonometry, calculus, and combinatorics. Project outcomes will provide instructional principles useful to policy makers at the district, state, and national levels.