Collaborative Research: Upgrading Learning for Teachers in Real Analysis

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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.