Combinatorics Problems: a Constructive Resource for Finding Volumes of Fractional Dimension?

Abstract

Fractions and volume are two challenging domains, which initially come together in the Common Core State Standards in Mathematics (CCSS-M) in the 7th grade where students learn about volumes of rectangular prisms with fractional dimensions. However, relatively little research has been conducted on how students’ reason about these volumes. To address this dearth of research, we designed a teaching experiment based on a central conjecture that combinatorics problems could be a constructive resource in the development of volumes with fractional dimension. In this paper, we demonstrate the central conjecture by providing two cases of how pre-service secondary teachers (PSSTs) reasoned with volumes of fractional dimension. A contribution of this study is that it offers an expansion and novel combination of a number of empirically grounded theoretical constructs.