Collaborative Research: Upgrading Learning for Teachers in Real Analysis

在本科教育中，迫切需要为职前高中数学教师候选人提供与必要的数学知识教学相关的数学理解的实质性深度，以支持学生发展数学知识，技能和积极的性格。 为了获得高中数学教师资格证，在职教师和在职教师（PIST）必须完成一些高等数学课程。 然而研究表明，PIST在高等数学课程中的经验对他们的教学方式或学生随后的数学成绩影响很小。 这个改进本科STEM教育（IUSE）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。 这个问题是调查的背景下，真实的分析过程中，当然往往需要认证教中学数学。 相对于传统的真实的分析课程，在这个项目中的转型模式定位在高中教学的背景下的真实的分析，并使明确的联系之间的真实的分析内容和教学行动，一个PIST可以进入课堂设置。 它还评估PISTs对真实的分析的理解和他们利用真实的分析知识对中学数学课堂情况做出适当反应的能力。 如果PIST能够（i）证明真实的分析中的重要定理，（ii）使用真实的分析中的内容更彻底地回答有关高中数学的问题，以及（iii）使用真实的分析中的内容为实际教学情况提供富有成效的反应，则这种教学变化将被视为成功。 这种真实的分析课程的新模式和相关的研究将有助于实现美国国家科学基金会的IUSE：EHR计划的目标，即支持研究和开发，以提高STEM毕业生（包括未来的教师）的质量。 在这个合作项目中，来自罗格斯大学，哥伦比亚大学和坦普尔大学的研究人员将共同设计，实施和评估PIST的创新真实的分析课程。 在该项目的模型中，PIST首先呈现高中数学的真实课堂情境，其中教师需要对数学有深入的了解才能做出适当的反应。 从讨论中，england，类将“建立从教学实践”，以解决潜在的数学问题，在发挥在一个真实的分析背景。 在真实的分析工作解决了这些数学问题后，PIST们“退下来练习”，并被要求重新审视和回应原始的和类似的课堂情境。 在该项目的头两年，教学方法将使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个定量的比较研究，在这三个地点实施这种教学模式，并在这些类PIST的性能将进行比较PIST谁已经完成了传统的教学顶点课程，连接高等数学到高中数学在罗格斯大学。 如果成功的话，这项资助的工作将说明一个变革性的模式对PIST教授高等数学的有效性，并为中学教师的专业（内容）知识基础提供进一步的见解，以便准备和专业发展可以被设计为数学内容的有意序列，而不仅仅是一套任意的数学课程。

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.