Enhancing students’ fraction magnitude knowledge: A study with students in early elementary education

Abstract

The idea of magnitude is central to understanding fractional numbers. To investigate this relationship, we implemented a design research project in an urban school in the northeast of the US to examine the potential of a measuring perspective and the mathematical notion of fraction-of-quantity to enhance second-grade students’ conceptual understanding of fraction magnitude. We used ideas from the history of mathematics and mathematics education within a cultural-historical framework to define fractions and construct tasks. The research team consisted of a university professor, two doctoral students, one of whom was an administrator of the municipal board of education, eight elementary school teachers, and a parent. The research sessions involved 35 students divided into two classes, meeting one hour per session twice a week for 12 weeks or 24 hours. The students manipulated non-symbolic or non-numeric manipulatives (Cuisenaire rods) and learned to talk about specific relations they perceived among them. Through physical manipulations and discourse, students developed the idea that a fraction reports a multiplicative comparison between two commensurable quantities of the same kind. Our results indicate that second-grade students appropriated the concept of the magnitude of fractions-of-quantity and, based on mental manipulations of evoked non-numeric images, constructed symbolic expressions involving fractional comparisons.