Interventions to improve equational reasoning: replication and extension of the Cuisenaire-Gattegno curriculum effect

Abstract

Introduction: The ability to reason about equations in a robust and fluent way requires both instrumental knowledge of symbolic forms, syntax, and operations, as well as relational knowledge of how such formalisms map to meaningful relationships captured within mental models. A recent systematic review of studies contrasting the Cuisenaire-Gattegno (Cui) curriculum approach vs. traditional rote schooling on equational reasoning has demonstrated the positive efficacy of pedagogies that focus on integrating these two forms of knowledge.

Methods: Here we seek to replicate and extend the most efficacious of these studies (Brownell) by implementing the curriculum to a high degree of fidelity, as well as capturing longitudinal changes within learners via a novel tablet-based assessment of accuracy and fluency with equational reasoning. We examined arithmetic fluency as a function of relational reasoning to equate initial performance across diverse groups and to track changes over four growth assessment points.

Results: Results showed that the intervention condition that stressed relational reasoning leads to advances in fluency for addition and subtraction with small numbers. We also showed that this intervention leads to changes in problem solving dispositions toward complex challenges, wherein students in the CUI intervention were more inclined to solve challenging problems relative to those in the control who gave up significantly earlier on multi-step problems. This shift in disposition was associated with higher accuracy on complex equational reasoning problems. A treatment by aptitude interaction emerged for both arithmetic equation reasoning and complex multi-step equational reasoning problems, both of which showed that the intervention had greatest impact for children with lower initial mathematical aptitude. Two years of intervention contrast revealed a large effect (d = 1) for improvements in equational reasoning for the experimental (CUI) group relative to control.

Discussion: The strong replication and extension findings substantiate the importance of embedding these teaching aides within the theory grounded curricula that gave rise to them.

Keywords: Cuisenaire rods; Cuisenaire-Gattegno; aptitude-treatment interactions; arithmetic fluency; pre-algebra.