A learning trajectory for enumerating permutations: Applying and elaborating a theory of levels of abstraction

Abstract

Permutations are fundamental to combinatorics and other areas of mathematics, and it is important that students develop efficient and conceptually supported ways of mentally constructing, listing, and enumerating them. To date, there is still much to learn about how students reason about enumerating permutations, and how instruction can support students’ conceptual development. We address this gap in the research literature by carefully tracing the evolution of two preservice middle school teachers’ permutation enumeration strategies and conceptualizations, which led to the formulation of levels of sophistication for combinatorial reasoning. These levels are explained by applying and extending a constructivist theory of levels of abstraction. Additionally, we outline an instructional approach that was instrumental in facilitating student learning. Together, the proposed levels and linked instructional approach constitute an initial learning trajectory for permutations that we believe could be useful for understanding and supporting post-secondary non-STEM students’ meaningful conceptualizations and enumerations of permutations.