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.

摘要

泛化是数学推理的一个重要组成部分，研究人员建议它是所有年级教育的核心。然而，对学生概括的研究揭示了在创建和表达概括性陈述方面普遍存在的困难，这强调了更好地理解可以支持更有成效的概括的过程的必要性。作为回应，我们报告了在解决复杂问题的同时，在代数、高级代数、三角学/微积分和组合学领域对 93 名初中到大学的参与者进行了 146 次采访的结果。我们的研究结果产生了关联-形成-扩展 (RFE) 框架，该框架区分了多种相关的泛化形式和类型。我们还提出了促进生成性概括的心理活动的两个方面：操作活动，