What Makes Mathematics Manipulatives Effective? Lessons From Cognitive Science and Montessori Education

Abstract

Manipulatives are ubiquitous in early childhood classrooms; yet, findings regarding their efficacy for learning mathematics concepts are inconsistent. In this article, we present four general principles that have emerged from cognitive science about ways to ensure that manipulatives promote learning when used with young children. We also describe how Montessori instruction offers a concrete example of the application of these principles in practice, which may, in turn, explain the high levels of mathematics achievement among children who attend Montessori programs during early childhood. The general principles and concrete examples presented in this article should help early childhood programs maximize the benefits of using manipulatives for developmentally appropriate mathematics instruction.

Walk into any early childhood classroom and you are sure to see mathematics manipulatives. Manipulatives are concrete materials (e.g., blocks, tiles) used to demonstrate a mathematics concept or to support the execution of a mathematical procedure. They have become a mainstay of mathematical instruction in America as well as internationally (e.g., Correa, Perry, Sims, Miller, & Fang, 2008; Puchner, Taylor, O’Donnell, & Fick, 2008). In a study of two school districts, the average elementary teacher reported using manipulatives nearly every day (Uribe-Flórez & Wilkins, 2010).

Research examining the advantages of instruction using manipulatives, however, is inconsistent: Some studies find that manipulatives promote learning, whereas others find that they hinder it. A recent meta-analysis of 55 studies that compared instruction with or without manipulatives suggests that manipulatives can benefit learning, but only under certain conditions (Carbonneau, Marley, & Selig, 2013). For instance, differences in the benefits of manipulatives were associated with the content being taught; manipulatives were more advantageous for learning about fractions than for learning arithmetic. The results also indicated that instruction with manipulatives was least effective for children between the ages of 3 and 6 years, with very small and sometimes negative effects. These findings suggest that the efficacy of manipulatives for promoting learning may depend on the conditions under which they are used.

Given the lack of clear evidence supporting the use of manipulatives, should they be used to teach mathematics in early childhood? We believe the answer is yes—if careful consideration is given to what research has identified about the conditions under which when manipulatives are likely to promote, rather than hinder, learning. Cognitive science research, in particular, has generated a considerable amount of knowledge that could be useful for improving instruction so that all young children can acquire the mathematics knowledge necessary for success, as described in National Council of Teachers of Mathematics (NCTM) standards and Common Core standards (Laski, Reeves, Ganley, & Mitchell, 2013; NCTM, 2006; National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010; Newcombe et al., 2009; Siegler, 2003). In this article, we discuss the findings from cognitive science relevant to the use of manipulatives in early childhood math instruction, and synthesize them into four principles for maximizing the effective use of mathematics manipulatives.

To demonstrate how early childhood instruction can reflect these principles, we offer examples from Montessori instruction. Maria Montessori (Montessori & Simmonds, 1917) was among the first educators to develop materials specifically designed to instantiate mathematics concepts. She developed a wide array of materials designed to help children understand concepts, such as place value (Lillard, 2005). Children who attend Montessori programs in early childhood demonstrate high levels of mathematics achievement. Children who were randomly selected to attend a Montessori program scored higher on a standardized math test than children who had not been selected and attended a non-Montessori program (Lillard & Else-Quest, 2006). In fact, close adherence to the Montessori approach seems to promote better math learning: Children who attend high-fidelity Montessori programs are more likely to have higher standardized math scores than those who attend lower fidelity Montessori or traditional early childhood programs (Lillard, 2012). The benefits of the Montessori approach to mathematics learning in early childhood may, at least in part, be due to its effective use of manipulatives.