Generalization Across Multiple Mathematical Areas: Studies of Classrooms and Teaching

在EHR核心研究计划的资助下，这个项目将调查如何在整个课堂环境中支持富有成效的数学概括。泛化是识别数字之间关系中的模式的能力。推广的一个例子是认识到2+3=5，推广到把任何偶数加到任何奇数上就会产生奇数。概括是数学推理的重要组成部分。因此，研究人员和政策制定者建议，它应该成为所有年级学生教育的核心。然而，考虑到关于学生在作出适当概括方面的困难以及教师在支持课堂概括方面面临的挑战的研究基础有限，按照这一建议采取行动会带来严重的挑战。该项目旨在通过关注从六年级到大学的数学课来帮助解决这一差距。为此，项目团队将调查学生的概括，以及影响他们在代数、高等代数、三角、微积分和组合学中发展概括技能的能力的策略。该项目将建立一个框架，以描述和理解以研究为基础的、以课堂为基础的战略，以便与本科数学课程的实习教师和讲师一起在整个课堂一级促进泛化。该项目有两个目标：(1)系统地调查课堂概括的现状，确定对学生概括的现有支持；(2)研究如何支持教师/指导者促进有效的数学概括。这项研究实施了基于课堂的设计实验模式，以两个阶段的方法利用课堂观察和专业发展视频会议，第二阶段建立在第一阶段的基础上。研究活动将产生一系列关于概括的教学支持与学生概括的性质之间的关系的发现，从而有助于理论和对跨多个领域的典型教学背景下的概括的更深入的理解。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.