Overview

Table of Contents

• Overview
• Description and History
• Support from Research
• Assessment for Learning
• Summary
• References
• Downloadable Version

Support from Research

Researchers over the past forty years have generally found that manipulatives are a powerful addition to mathematics instruction. Meta-analyses by Suydam & Higgins (1977), Parham (1983), and Sowell (1989) found that achievement in mathematics could be increased by the long-term use of manipulatives. Well over 100 studies are included among these three analyses.

Three more recent publications confirm similar findings. In 2001, the National Research Council released the book Adding It Up: Helping Children Learn Mathematics. This book reflects a comprehensive review of mathematics education research and includes this statement:

“The evidence indicates, in short, that manipulatives can provide valuable support for student learning when teachers interact over time with the students to help them build links between the object, the symbol, and the mathematical idea both represent.” (p 354)

The 2009 What Works Clearinghouse report Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools includes using visual models and manipulatives as one of its research-based recommendations. In addition, Appendix D of the report summarizes research supporting the use of the Concrete–Representational–Abstract (CRA) approach for helping students make connections between concrete manipulatives and abstract mathematical ideas. In this approach, built on Bruner’s (1966) work, students first use concrete materials to solve problems and look for patterns and generalizations. As students need to record their work, they do so first by sketching pictures (representations) of the manipulative models and then finally move to using abstract (and more formal) mathematical notations for their work. Moore (2013) provides an example of this structure in action.

Wenglinsky’s (2000) analysis of NAEP data suggests the value of interaction over time, another key idea from Adding It Up. By examining data about classroom activity, this analysis suggests that, “when students are exposed to hands-on learning on a weekly rather than a monthly basis, they prove to be 72% of a grade level ahead in mathematics” (p 27). Not only is it essential to help students make the connections between concrete manipulatives and abstract mathematics, but it is also essential that this instructional strategy be a frequent element of classroom practice.

Assessment for Learning

Manipulatives are a powerful tool for supporting classroom assessment. The literature review by Black & Wiliam (1998) found that formative assessment practices yielded effect sizes of one-half to one full standard deviation. More importantly, “improved formative assessment helps low achievers more than other students and so reduces the range of achievement while raising achievement overall.” (p 141) In other words, formative assessment increases student learning substantially while reducing achievement gaps. There are many definitions of formative assessment. Stiggins’ (2005) description suggests the critical element – teachers use classroom assessment to make changes in instruction in real time to result in real increases in learning.

In order to implement formative assessment well, Wiliam (2011) shares five key strategies. Three are particularly well-supported by manipulative-based instruction: classroom activities and learning tasks which elicit evidence of learning, activating learners as instructional resources for one another, and activating learners as owners of their own learning. In all three cases, by actively engaging students in the doing of mathematics, manipulatives provide a foundation which encourages discussion and student ownership of their work. This provides teachers with a vivid current picture of student understanding and guides teachers in determining appropriate next steps.

Summary

Building on the learning theory work of Piaget and Bruner, a solid history of research supports the regular use of manipulatives in classroom mathematics instruction. While children can remember, for short periods of time, information taught through books and lectures, deep understanding and the ability to apply learning to new situations requires conceptual understanding that is grounded in direct experience with concrete objects. It is also important to note the critical role of the teacher in helping students connect their manipulative experiences, through a variety of representations, to essential abstract mathematics. Together, excellent teachers and regular experiences with hands-on learning can provide students with powerful learning in mathematics.

References