Equivalent Fraction Learning Trajectories for Students with Mathematical Learning Difficulties When Using Manipulatives

Abstract

This study identified variations in the equivalent fraction learning of students with mathematical learning difficulties when using physical and virtual manipulatives. The study compared three interventions: physical manipulatives, virtual manipulatives, and a combination of physical and virtual manipulatives. The research used a mixed-method approach to collect and analyze data. Two types of learning trajectories were used to compare and synthesize the result. For this study, 43 fifth-grade students with mathematical learning difficulties participated in 10 sessions of equivalent fraction intervention. Pre- to postdata analysis indicated significant gains for all three interventions. Effect size scores were used to compare the effects of the three types of manipulative intervention at the total, cluster, and questions levels of the assessments. Daily assessment data were used to develop trajectories comparing mastery and achievement changes over the duration of the intervention. Data were also synthesized into an iceberg learning trajectory containing five clusters and three subcluster concepts of equivalent fraction understanding. The syntheses favored the use of physical manipulatives for instruction in two clusters, the use of virtual manipulatives for one cluster and the use of combined manipulatives for two clusters. The qualitative analysis indentified variations in students’ resolution of misconceptions and students’ use of strategies and representations. Variations favored virtual manipulatives for the development of students’ understanding of representations using only symbols. Physical manipulatives were favored for students’ understanding of set model representations. Results also suggested that the ability of students using virtual manipulatives to see the link between their manipulation of the objects and simultaneous changes in the symbolic representations of the building of equivalent fraction groups. Students using virtual manipulatives tended to have higher gains on questions that asked students to develop groups of three or more equivalent fractions. The results of this study demonstrated that the instructional benefits of physical and virtual manipulative instruction are specific to the different equivalent fraction subconcepts and that an understanding of the variations is needed to determine when and how each manipulative should be used in the sequence of instruction