Collaborative Research: Upgrading Learning for Teachers in Real Analysis

本科教育迫切需要为职前高中数学教师候选人提供与数学知识相关的深度数学理解，以支持学生发展数学知识、技能和积极倾向。为了获得高中数学教师资格证书，职前教师和在职教师（pist）需要完成一些高等数学课程。然而，研究表明，pist在高等数学课程上的经历对他们的教学方式或学生随后的数学成绩的影响微乎其微。这个改善本科STEM教育（IUSE）参与式学生学习- 1级项目将通过引入一种替代模式来向pist教授高等数学来解决这个问题。这个问题是在Real Analysis课程的背景下调查的，这门课程通常需要获得中学数学教学认证。与传统的真实分析课程相反，这个项目中的变革模型将真实的分析置于高中教学的背景下，并在真实的分析内容和教学行动之间建立了明确的联系，从而使教师可以将其带入课堂。它还评估pist对实际分析的理解以及利用实际分析知识对中学数学课堂情况作出适当反应的能力。如果pist能够(i)证明《真实分析》中的重要定理，（ii）使用《真实分析》中的内容更彻底地回答有关高中数学的问题，以及（iii）使用《真实分析》中的内容对实际教学情况提供富有成效的回应，那么这种教学上的变化将被视为成功。这种新的真实分析课程模式和相关研究将有助于美国国家科学基金会的IUSE:EHR项目的目标，即支持研究和开发，从而提高STEM毕业生（包括未来教师）的质量。在这个合作项目中，来自罗格斯大学、哥伦比亚大学和天普大学的研究人员将共同为pist设计、实施和评估创新的真实分析课程。在该项目的模型中，pist首先呈现了一个真实的高中数学课堂情境，在这个情境中，教师需要对数学有深刻的理解才能做出适当的反应。从随后的讨论中，课程将“从教学实践中积累”，以解决在实际分析环境中发挥作用的潜在数学问题。在实际分析工作解决了这些数学问题后，pist“退到实践中”，并被要求重新审视和回应原始的和类似的课堂情况。在项目的头两年，教学方法将采用基于设计的研究范式来评估和提高教学质量。该项目的第三年将是一项定量比较研究，该研究将在三个地点实施该教学模式，并将pist在这些课程中的表现与完成罗格斯大学高等数学与高中数学连接的传统教学顶点课程的pist进行比较。如果获得成功，这项资助的工作将说明向pist教授高等数学的变革模式的有效性，并为中学教师的专业（内容）知识库提供进一步的见解，以便准备和专业发展可以设计为一个有意的数学内容序列，而不仅仅是一组任意的数学课程。

There is a critical need in undergraduate education to provide pre-service high school mathematics teacher candidates with a substantial depth of mathematical understanding linked to the mathematical knowledge for teaching necessary to support their students developing knowledge, skills, and positive dispositions in mathematics. To obtain certification to teach high school mathematics, preservice and in-service teachers (PISTs) are required to complete a number of courses in advanced mathematics. Yet research has shown that PISTs' experience in advanced mathematics courses has a minimal effect on how they teach or on their students' subsequent mathematics achievement. This Improving Undergraduate STEM Education (IUSE) Engaged Student Learning--Level 1 project will address this issue by introducing an alternative model to teaching advanced mathematics to PISTs. This issue is investigated in the context of a Real Analysis course, a course often required for certification to teach secondary mathematics. As opposed to traditional Real Analysis courses, the transformative model in this project situates the real analysis in the context of high school teaching and makes explicit links between the real analysis content and pedagogical actions that a PIST can take into a classroom setting. It also assesses PISTs both on their understanding of real analysis and their ability to draw on their real analysis knowledge to respond appropriately to classroom situations in secondary mathematics. This change in instruction will be viewed as successful if the PISTs can (i) prove important theorems in Real Analysis, (ii) use the content in Real Analysis to answer questions about high school mathematics more thoroughly, and (iii) use the content in Real Analysis to provide productive responses to practical pedagogical situations. This new model of a Real Analysis course and the associated research will contribute towards the National Science Foundation's IUSE:EHR program's goal of supporting research and development leading to and propagating interventions that improve the quality of STEM graduates, including future teachers. In this collaborative project, researchers from Rutgers University, Columbia University, and Temple University will work together to design, implement, and assess the innovative Real Analysis course for PISTs. In the project's model, PISTs are first presented with an authentic classroom situation from high school mathematics in which a teacher needs a deep understanding of mathematics to respond appropriately. From the discussion that ensues, the class will "build up from teaching practice" to tackle the underlying mathematical issues at play in a real analysis context. After the work in real analysis resolves these mathematical issues, PISTs "step down to practice" and are asked to revisit and respond again to the original and analogous classroom situations. In the first two years of the project, the instructional methods will be implemented using a design-based research paradigm to assess and improve the quality of the instruction. The third year of the project will be a quantitative comparison study in which this instructional model is implemented at the three sites and PISTs performance in these classes will be compared to PISTs who have completed a traditionally taught capstone course that connects advanced mathematics to high school mathematics at Rutgers University. If successful, the work in this grant will illustrate the efficacy of a transformative model for teaching advanced mathematics to PISTs, as well as provide further insight into the professional (content) knowledge base for secondary teachers, so that preparation and professional development can be designed as an intentional sequence of mathematics content, and not just an arbitrary set of mathematics courses.