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

等效性是数学中最重要的概念之一，它指的是两个物体在某种程度上可以被认为是“相同的”。从小学到研究生的学习者在许多数学主题中都会遇到等值。不幸的是，在更高级的数学课程中，学生在理解等值的概念时可能会遇到困难。一个可能的挑战是，每次在课程内或跨课程引入等效性时，它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，该项目旨在发展一种关于学生如何在两个数学学科（组合学和抽象代数）中进行等效推理的理论。为了收集建立理论基础的数据，该项目将检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。从这项研究中产生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效性。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、更有联系的方式看到等效的实例。等效性是所有数学中最基本、最深远的概念之一，也是K-16数学课程的重要组成部分。它的重要性在高等教育水平上尤为明显，在从微积分到抽象代数的几乎每个领域中，等效性都表现出来并发挥着关键作用。尽管等效性很普遍，也很重要，但本科生在理解等效性的实例方面可能会遇到挑战，尤其是当以一种不连贯的方式引入类似概念时。此外，研究中等效性的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中证明有用的认知模型。这个项目将致力于一个可以应用于多种情况的横切等价理论。聚焦于组合学和抽象代数领域，该项目的主要研究问题是：(1)在抽象代数和组合学领域中，本科生对等价的思考方式涉及什么？(2)本科生对这些领域的等效性的思考方式包含了什么？为了回答这些问题，该项目将利用现有文献、教科书分析和对数学家的采访来开发一个初始理论，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来严格完善该理论，重点关注一年级的抽象代数、二年级的组合学和三年级的这两个领域。该项目由EHR核心研究（ECR）计划资助，该计划支持推进STEM学习和学习环境基础研究、扩大STEM参与和STEM劳动力发展的工作。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.