Building bridges in mathematical reasoning: how students connect isomorphic problems in combinatorics
Creators: name: Foley, Ronald G.
Publisher: name: No Publisher Supplied
Year: 2024

description: This dissertation examines the development of combinatorial reasoning in high school students through an in-depth analysis of problem-solving sessions involving the Pizza Problem. The study, part of the long-term Rutgers-Kenilworth longitudinal research project, focuses on four 11th-grade students as they explore and connect concepts related to combinatorics, Pascal's Triangle, and isomorphic relationships between different problem contexts. The research is grounded in constructivist theories of learning and draws on the work of scholars such as Maher, Uptegrove, and Francisco in the field of mathematical reasoning and problem-solving. It also builds upon Goldin's framework of problem-solving and affective pathways, and Dienes' theory of multiple embodiments in mathematics education. Using a qualitative approach that combines Powell et al.'s (2003) seven nonlinear phases and a grounded theory approach, the research analyzes video data and student work to trace the evolution of students' mathematical reasoning. The study employs VMCAnalytics to create detailed video narratives that capture key moments in the students' problem-solving process. Four specific VMCAnalytics were developed, peer-reviewed, and published, each focusing on different aspects of the students' reasoning processes. Key Findings:1.The non-linear nature of students' learning, characterized by cycles of exploration, conjecture, and revision.2.The importance of multiple representations in developing structural understanding, particularly the role of Pascal's Triangle as a mediating tool.3.The critical role of collaborative discourse in mathematical sense-making and the development of shared understanding.4.The progressive development of students' ability to recognize and leverage isomorphic relationships between the Pizza Problem, the Towers Problem, and Pascal's Triangle.5.The emergence of sophisticated combinatorial thinking, including the recognition of the 2^n pattern in various problem contexts. The research highlights how students progressively recognize and leverage isomorphic relationships between a series of counting problems, demonstrating a sophisticated level of mathematical thinking. It provides insights into the processes by which students develop abstract mathematical concepts and transfer knowledge across different problem contexts. This study contributes to our understanding of how students develop advanced mathematical concepts, particularly in combinatorics. It offers insights into effective teaching strategies that foster deep, flexible, and transferable mathematical understanding, with implications for curriculum design, instructional practices, and assessment in mathematics education. The dissertation suggests several avenues for future research, including longitudinal studies tracking the development of isomorphic thinking across different mathematical domains, investigations into the role of technology in supporting structural understanding, and explorations of how teachers can effectively scaffold the development of combinatorial reasoning.