9. Concrete materials

Research findings

Many studies show that the use of concrete materials can produce meaningful use of notational systems and increase student concept development. In a comprehensive review of activity-based learning in mathematics in kindergarten through grade eight, Suydam and Higgins concluded that using manipulative materials produces greater achievement gains than not using them. In a more recent meta-analysis of sixty studies (kindergarten through post-secondary) that compared the effects of using concrete materials with the effects of more abstract instruction, Sowell concluded that the long-term use of concrete instructional materials by teachers knowledgeable in their use improved student achievement and attitudes.

In spite of generally positive results, there are some inconsistencies in the research findings. As Thompson points out, the research results concerning concrete materials vary, even among treatments that were closely controlled and monitored and that involved the same concrete materials. For example, in studies by Resnick and Omanson and by Labinowicz, the use of base-ten blocks showed little impact on children’s learning. In contrast, both Fuson and Briars and Hiebert and Wearne reported positive results from the use of base-ten blocks.

The differences in results among these studies might be due to the nature of the students’ engagement with the concrete materials and their orientation towards the materials in relation to notation and numerical values. They might also be due to different orientations in the studies, with regard to the role of computational algorithms and how they should be developed in the classroom. In general, however, the ambiguities in some of the research findings do not undermine the general consensus that concrete materials are valuable instructional tools.

In the classroom

Although successful teaching requires teachers to carefully choose their procedures on the basis of the context in which they will be used, available research suggests that teachers should use manipulative materials in mathematics instruction more regularly in order to give students hands-on experience that helps them construct useful meanings for the mathematical ideas they are learning. Use of the same material to teach multiple ideas over the course of schooling has the advantage of shortening the amount of time it takes to introduce the material and also helps students to see connections between ideas.

The use of concrete material should not be limited to demonstrations. It is essential that children use materials in meaningful ways rather than in a rigid and prescribed way that focuses on remembering rather than on thinking. Thus, as Thompson says, ‘before students can make productive use of concrete materials, they must first be committed to making sense of their activities and be committed to expressing their sense in meaningful ways. Further, it is important that students come to see the two-way relationship between concrete embodiments of a mathematical concept and the notational system used to represent it.’