Are We Overemphasizing Manipulatives in the Primary Grades to the Detriment of Girls

During the last twenty-five years, arithmetic curricula have included more hands-on activities with an increasing emphasis on problem solving. According to the Representation Standard in Principles and Standards, “instructional programs from prekindergarten through grade 12 should enable all students to use representations to model and interpret physical, social, and mathematical phenomena” (NCTM 2000, p. 136). Representations, such as manipulatives, help children develop understanding by building a foundation for the later use of symbols. When given a choice of how to solve a problem, children gravitate toward manipulatives because they can act out the situation or relationships in the problem. When children are free to choose strategies for problem solving in single-digit arithmetic, they typically progress from using concrete strategies to more abstract strategies. For example, consider the following problem: “Lisa had 8 seashells in her collection. Hal gave her 5 more seashells. How many seashells did Lisa have then?” To solve this problem, children in the beginning stage of development will make a set of 8 objects and another set of 5 objects, join the two sets, and count all the objects. As they progress in their development, children do not need to build the first set but can count on from 8 to get the answer. Eventually, children can solve the problem abstractly without requiring a concrete model of either quantity in the problem (see Carpenter et al. [1999] for a complete discussion of these developmental stages). This progression of strategies is quite natural, and even five-year-olds move from counting-all strategies to counting-on strategies without instruction (Groen and Resnick 1977). Some children progress along the same kind of developmental path for multidigit addition and subtraction, initially using concrete strategies in which they manipulate the quantities in the prob-