Undergraduate Students' Reasoning about Equivalence in Multiple Mathematical Domains: Exploration and Theory-Building

本科生对多个数学领域等价性的推理：探索与理论构建

• 批准号: 2055590
• 负责人: John Paul Cook
• 金额: $ 50万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055590&HistoricalAwards=false
• 关键词: Undergraduate; Students; Reasoning; about; Equivalence

Equivalence, the idea that two objects can be considered “the same” in some way, is one of the most important concepts in mathematics. Learners from elementary through graduate school encounter equivalence in many ways across many mathematical topics. Unfortunately, students can have difficulties in understanding ideas of equivalence in more advanced mathematics courses. One possible challenge is that equivalence is often treated as a new concept each time it is introduced within or across courses. In addition, students’ ways of thinking about this fundamental idea are not yet well understood. To begin to fill this knowledge gap, this project aims to develop a theory about how students reason with equivalence across two mathematical disciplines: combinatorics and abstract algebra. To gather data on which to base the theory, the project will examine the current body of literature on equivalence, analyze textbooks, and conduct interviews with mathematicians and students. The theory that emerges from this research will help researchers and educators better understand different ways to reason about equivalence across mathematical domains. This work also may have long-term benefits: such a theory could inform the design of curricular materials to help students at all levels see instances of equivalence in a more consistent, linked fashion.Equivalence is one of the most fundamental, far-reaching concepts in all of mathematics and an essential component of the K-16 mathematics curriculum. Its importance is particularly evident at the postsecondary level, where equivalence manifests and plays a key role in virtually every domain from calculus to abstract algebra. Despite its prevalence and importance, undergraduate students can be challenged to understand instances of equivalence, especially if similar concepts are introduced in a disconnected way. Moreover, characterizations of equivalence in research are often implicit or domain-specific, speaking to the need for cognitive models that might prove useful within and across mathematical disciplines. This project will work toward a crosscutting theory of equivalence that could be applied in multiple contexts. Focusing on the domains of combinatorics and abstract algebra, the project’s primary research questions are: (1) What is entailed in undergraduate students’ ways of thinking about equivalence within the domains of abstract algebra and combinatorics? (2) What is entailed in undergraduate students’ ways of thinking about equivalence across these domains? To answer these questions, the project will leverage existing literature, textbook analysis, and interviews with mathematicians to develop an initial theory and then rigorously refine that theory via sequences of exploratory and targeted task-based clinical interviews with students, focusing on abstract algebra in Year 1, combinatorics in Year 2, and both domains in Year 3. This project is funded by the EHR Core Research (ECR) program, which supports work that advances fundamental research on STEM learning and learning environments, broadening participation in STEM, and STEM workforce development.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.

等价性，即两个对象在某种程度上可以被认为是“相同的”，是数学中最重要的概念之一。从小学到研究生院的学习者在许多数学主题中以许多方式遇到等价。不幸的是，学生在理解高等数学课程中的等价概念时会有困难。 一个可能的挑战是，每次在课程内或跨课程引入等值概念时，等值往往被视为一个新概念。 此外，学生对这一基本思想的思考方式还没有得到很好的理解。为了开始填补这一知识空白，本项目旨在开发一个关于学生如何在两个数学学科：组合数学和抽象代数中进行等价推理的理论。为了收集数据作为理论的基础，该项目将审查当前关于等效性的文献，分析教科书，并与数学家和学生进行访谈。从这项研究中出现的理论将帮助研究人员和教育工作者更好地理解不同的方法来推理数学领域的等价性。这项工作也可能有长期的好处：这样的理论可以为课程材料的设计提供信息，帮助各个层次的学生以更一致、更联系的方式看待等价的实例。等价是所有数学中最基本、影响最深远的概念之一，也是K-16数学课程的重要组成部分。它的重要性在中学后阶段尤为明显，在那里，等价性在从微积分到抽象代数的几乎每个领域都表现出来并发挥着关键作用。尽管它的普遍性和重要性，本科生可能会受到挑战，以了解等效的实例，特别是如果类似的概念是在一个断开的方式介绍。此外，研究中的等价性特征通常是隐含的或特定于领域的，这说明需要在数学学科内部和跨学科的认知模型。这个项目将致力于建立一个可以应用于多种背景的横向对等理论。 以组合数学和抽象代数领域为研究对象，主要研究的问题是：（1）大学生对抽象代数和组合数学领域中的等价性的思考方式包含了什么？(2)本科生在思考这些领域的等价性时，会有什么样的想法？为了回答这些问题，该项目将利用现有的文献，教科书分析和数学家访谈来开发初始理论，然后通过对学生进行探索性和有针对性的基于任务的临床访谈序列来严格完善该理论，重点是抽象代数在第一年，组合数学在第二年，这两个领域在第三年。 该项目由EHR核心研究（ECR）计划资助，该计划支持推进STEM学习和学习环境的基础研究，扩大STEM参与，以及STEM劳动力发展的工作。该奖项反映了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。

项目成果

How do we disentangle equality from equivalence? Well, it depends

我们如何区分平等和对等？

• 发表时间: 2023
• 期刊: Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education
• 作者: Reed, Zackery;Cook, John Paul;Lockwood, Elise;Richardson, April
• 通讯作者: Richardson, April

Explicating Interpretations of Equivalence in Measurement Contexts

解释测量环境中的等价性解释

• 发表时间: 2021
• 期刊: For the Learning of Mathematics
• 作者: Cook, J.P.;Dawkins, P.C.;Reed, Z.
• 通讯作者: Reed, Z.

Using conceptual analyses to resolve the tension between advanced and secondary mathematics: the cases of equivalence and inverse

使用概念分析解决高等数学和中等数学之间的紧张关系：等价和逆的情况

• DOI: 10.1007/s11858-023-01495-2
• 发表时间: 2023
• 期刊: ZDM – Mathematics Education
• 作者: Cook, John Paul;Richardson, April;Reed, Zackery;Lockwood, Elise
• 通讯作者: Lockwood, Elise

A framework for analyzing students' reasoning about equivalence across undergraduate mathematics

分析学生对本科数学等价性推理的框架

• 发表时间: 2022
• 期刊: Proceedings of the 24th Conference on Research in Undergraduate Mathematics Education
• 作者: Reed, Z.;Cook, J.P.;Lockwood, E.;Richardson, A.
• 通讯作者: Richardson, A.

An initial framework for analyzing students’ reasoning with equivalence across mathematical domains

用于分析学生推理与跨数学领域的等价性的初始框架

• DOI: 10.1016/j.jmathb.2022.100935
• 发表时间: 2022
• 期刊: The Journal of Mathematical Behavior
• 作者: Cook, John Paul;Reed, Zackery;Lockwood, Elise
• 通讯作者: Lockwood, Elise