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

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.

Automated summary

Generalization is crucial in mathematical reasoning, but students often struggle to create and express general statements, highlighting the need to better understand processes that can support more productive generalizations. Through interviews with middle school to college students in algebra and related domains, the Relating-Forming-Extending (RFE) Framework was developed to distinguish different forms of generalizing. Two mental activities, operative activity and building/refining activity, were identified as promoting generative generalizations.