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

等价性是数学中最重要的概念之一，即两个物体在某种程度上可以被认为是“相同的”。从小学到研究生院的学习者在许多数学主题上都在许多方面遇到了等价性。不幸的是，在更高级的数学课程中，学生在理解等价性概念方面可能会有困难。一个可能的挑战是，每次在课程内或课程之间引入等值时，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，这个项目旨在开发一种关于学生如何在两个数学学科：组合学和抽象代数上进行等价性推理的理论。为了收集数据作为理论的基础，该项目将检查当前关于等价性的文献，分析教科书，并对数学家和学生进行采访。这项研究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跨数学领域的等价性。这项工作也可能有长期的好处：这样的理论可以指导课程材料的设计，帮助各级学生以更一致、更有联系的方式看到等价性的实例。等价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。它的重要性在中学后阶段尤为明显，在从微积分到抽象代数的几乎每一个领域中，等价性都表现出来并发挥着关键作用。尽管等价性很普遍，也很重要，但本科生可能会被挑战去理解等价性的例子，特别是如果类似的概念是以一种脱节的方式引入的。此外，研究中对等价性的描述通常是隐含的或特定于领域的，这表明需要认知模型，这些模型可能在数学学科内部或跨数学学科被证明是有用的。该项目将致力于建立一种可在多种情况下应用的等价性的横切理论。该项目的主要研究问题集中在组合学和抽象代数领域，主要研究问题是：(1)本科生在抽象代数和组合数学领域内的等价性思维方式是什么？(2)本科生在这些领域的等价性思维方式是什么？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，第一年关注抽象代数，第二年关注组合学，第三年关注这两个领域。该项目由EHR核心研究(ECR)计划资助，该计划支持推进STEM学习和学习环境的基础研究，扩大对STEM的参与，这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.