Defining, Measuring, and Promoting Mathematical Flexibility in Undergraduate Education

研究数学的灵活性，即数学使用者认识到一个问题可以用多种方法解决，并选择最适合给定任务的方法，是本提案的重点。近年来，数学科学对全球经济的重要性急剧增加，部分原因是大数据革命和互联网及相关技术的持续快速扩张。数学过程的灵活知识支持概念理解，因此是数学学习的一个重要方面。关于数学程序灵活性和促进其发展的课堂策略的大部分已知内容都集中在中学生身上，通常涉及他们解决线性方程的过程。该项目将在现有文献的基础上进行扩展和创新，并有助于加深对程序灵活性如何发展以及如何在整个学生教育过程中教授这种灵活性的理解。主要的项目活动包括发展本科数学程序灵活性理论，创造一种工具来衡量学生的灵活性，以及一项研究，以了解教育者在课堂上的选择如何有助于增加程序灵活性。该项目还将对程序灵活性如何影响学生的数学信心进行初步研究。最终目标是开发必要的理论和测量工具，以支持简单但深刻的方法来培养本科生的灵活性，作为支持他们更广泛的数学学习和理解的手段。本项目将分几个阶段支持本科生对灵活的程序性数学知识的更广泛理解。首先，“灵活性”的可操作性定义将与本科数学特别相关。这一理论将通过分析现有的关于灵活性和真实的学生程序表演的研究来发展。基于已发展的理论，该项目下一步将开发观察和识别学生作业灵活性的方法，包括创建一个工具来衡量本科设置中的程序灵活性。这些理论和测量工具将允许对促进程序灵活性增长的教学实践进行更深入的研究。通过这些活动，本项目旨在为数学灵活性的理论基础和教学做出重大贡献。预期的项目成果包括扩展和推广以往的研究成果，具体来说，通过课堂实践中有目的、直接的改变，学生的程序灵活性以及数学内容知识可以得到良好的提高。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。

Investigating mathematical flexibility, where a user of mathematics recognizes that a problem can be solved in multiple ways and chooses an approach that is most appropriate to the given task, is the focus of this proposal. The importance of the mathematical sciences to the global economy has increased dramatically in recent years, due in part to the big data revolution and the continued rapid expansion of the Internet and related technologies. Flexible knowledge of mathematical procedures is known to support conceptual understanding and is therefore an essential aspect of mathematical learning. Much of what is known about procedural flexibility in mathematics and classroom strategies to promote its development has centered on middle school students, often involving their processes for solving linear equations. This project will expand and innovate on the existing literature and contribute to increased understanding of how procedural flexibility develops and how to teach for such flexibility throughout a student’s education. Major project activities include the development of a theory of procedural flexibility in undergraduate mathematics, the creation of an instrument to measure that flexibility in students, and a study to understand how educators’ choices in the classroom might contribute to increased procedural flexibility. The project will also undertake a preliminary study of how procedural flexibility impacts students’ mathematical confidence. The ultimate goal is to develop the theory and measurement tools necessary to support simple but profound ways of developing undergraduate students’ flexibility as a means to support their larger mathematical learning and understanding. This project will support a broader understanding of flexible procedural knowledge of mathematics at the undergraduate level in several stages. First, an operational definition of “flexibility” that is particularly relevant to undergraduate mathematics will be developed. This theory will be developed through analysis of both existing research on flexibility and authentic student procedural performances. Building from the developed theory, the project will next develop ways of observing and identifying flexibility in student work, including the creation of an instrument to measure procedural flexibility in undergraduate settings. These theoretical and measurement tools will allow for a deeper study of instructional practices that promote growth in procedural flexibility. Through these activities, this project aims to make significant contributions to both the theoretical underpinnings and the teaching of flexibility in mathematics. Expected project outcomes include expansion and generalization previous research findings, specifically that favorable gains in students’ procedural flexibility, and thereby their mathematical content knowledge, can be realized through purposeful, straightforward changes in classroom practice. 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.