This article reports on results from a study that quantitatively tested hypotheses arising from Les Steffe and John Olive’s Fractions Project. It affirms their work and scheme theory in general. For example, the study showed that additional mental operations are necessary for middle school students to generalize their partitive conceptions from unit fractions to other proper fractions.

Piagetian theory describes mathematical development as the construction and organization of mental operation within psychological structures. Research on student learning has identified the vital roles two particular operations – splitting and units coordination – play in students’ development of advanced fractions knowledge. Whereas Steffe and colleagues describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students’ psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students’ development with a structure that is isomorphic to the positive rational numbers under multiplication – “the splitting group.” We use a quantitative analysis of written assessments from 59 eighth grade students in order to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students’ constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students’ constructions of three levels of units coordination. Because three levels of units coordination also plays a vital role in other mathematical domains, such as algebraic reasoning, implications from the study extend beyond fractions teaching and research. Relevance: The paper uses constructivist theories of learning, including scheme theory and Piaget’s structuralism, to study how students construct mature conceptions of fractions.

The article describes a quantitative analysis that utilizes Piaget’s structuralist approach to mathematical development. Results affirm Steffe’s models of students’ constructions of fraction schemes and operations, particularly regarding the construction of the splitting operation.