Generalization Across Multiple Mathematical Domains: Relating, Forming, and Extending

Abstract

Generalization is a critical component of mathematical reasoning, with researchers recommending that it be central to education at all grade levels. However, research on students’ generalizing reveals pervasive difficulties in creating and expressing general statements, which underscores the need to better understand the processes that can support more productive generalizations. In response, we report on results from 146 interviews with 93 participants in middle school through college in the domains of algebra, advanced algebra, trigonometry/pre-calculus, and combinatorics while solving complex problems. Our findings yielded the Relating-Forming-Extending (RFE) Framework, which distinguishes multiple related forms and types of generalizing. We also present two aspects of mental activity that promote generative generalizations: operative activity, and building and refining activity.

Acknowledgements

The views expressed do not necessarily reflect official positions of the Foundation. The authors wish to thank the anonymous reviewers for their helpful guidance and reactions to earlier drafts of this manuscript. We also thank Andrew Gatza, Biyao Liang, Zackery Reed, Brandon Singleton, Irma Stevens, and Halil Tasova for assistance with data collection, coding, and valuable conversations that contributed to the project.

Additional information

Funding

This research was supported by the National Science Foundation [grant no. DRL-1419973].