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Murray Klamkin, an American mathematician known worldwide as a prolific proposer of professionally challenging mathematical problems, once noted that formulating a problem well is often more difficult than solving it. Yet, despite the fact that generating problems to solve, conjecturing theorems to prove, and identifying situations to model are central to mathematical thinking and learning, historically, there has been little emphasis on these kinds of mathematical activities in school mathematics. In fact, research has revealed that widely used curriculum materials fail to incorporate problem posing in a substantial and consistent way, leaving teachers with sparse resources to enact this process.
To solve the problem of problem posing in school mathematics, mathematics education faculty in the Department of Mathematical Sciences proposed a 4-year longitudinal research project to the National Science Foundation. In March, 2021, Professors Jinfa Cai and Michelle Cirillo, along with Dr. Faith Muirhead from UD's Professional Development Center for Educators, were awarded $2.55M for the project: Supporting Teachers to Teach Mathematics through Problem Posing: An Early Stage Longitudinal Study. Dr. Stephen Hwang will serve as the Project Coordinator.
Problem posing refers to both the reformulation of given problems and the generation of novel problems. Theoretically, there are many benefits of engaging students in problem posing. For example, problem posing can foster students' positive mathematics identities and dispositions, as engaging in problem posing sparks curiosity, increases interest in mathematics, and develops agency by empowering and positioning students as explorers of mathematics. Second, engaging students in mathematical problem posing is a powerful approach to teaching and learning mathematics that fosters reasoning and sense-making. Third, through its flexibility, problem posing offers students of all abilities access to sense-making opportunities.
Empirically, there is a need to generate longitudinal data confirming the promise of problem posing. There are two major goals of the project. The first goal is to support teachers to teach mathematics through engaging their students in mathematical problem posing (problem-posing-based learning, or P-PBL). The second goal of the project is to longitudinally investigate the promise of supporting teachers to teach with P-PBL with respect to teachers' instructional practice and students' learning. The team will collaborate with local middle school teachers to engage students in mathematical problem posing. Through the ongoing partnership, a networked improvement community of teachers and researchers will integrate problem posing into daily mathematics instruction and continuously improve the quality of P-PBL through iterative task and lesson design. A quasi-experimental design coupled with design-based research and improvement science will be used to understand how, when, and why P-PBL works.
The intellectual merit of this project is in its contribution to research on supporting teachers to teach mathematics through problem posing. The broader impact of the project is in the generation of 20-30 research-based P-PBL cases of high-quality, problem-posing based, middle school algebra lessons that include critical details about implementation of P-PBL and the "lessons learned" from the lesson development process. In particular, this project will generate valuable scientific research findings about teaching through problem posing for district administrators, mathematics teachers, teacher educators, professional development providers, and researchers, as well as curriculum developers and policy makers.
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