From a teacher's point of view, the goal of using a manipulative is to provide support for learning more general mathematical concepts. However, this is no guarantee that children will see the manipulative in this way. Previous work on the use of manipulatives has documented numerous examples of mismatches between teachers' expectations and students' understandings. Even when young children do learn to perform mathematical operations using manipulatives, their knowledge of the two ways of solving the problems may remain encapsulated; that is, children often fail to see the relation between solving mathematics problems via manipulatives and solving the same or similar problems via abstract symbols (Uttal, Scudder, & DeLoache, 1997). For example, children may not see that the solutions to two-digit subtraction problems that they derive from manipulatives are also relevant to similar but written versions of the same problems. In the child's mind, the task of doing manipulatives-based arithmetic may be completely separate from doing written arithmetic.

An analogy to our scale model illustrates the differences between how students and teachers may view the relation between manipulatives and written representations of numbers. Our model is extremely concrete, and parents are amazed when the task proves difficult for intelligent, interested children. We believe that similar issues may arise when older children are asked to use manipulatives; to the teacher, the relation between the manipulative and a more abstract concept may be direct and obvious, but this relation may be, and may remain, obscure to young children, particularly if the relation is not pointed out explicitly.

Evidence that children often fail to draw connections between manipulatives and more traditional forms of mathematical symbols comes from Resnick and Omanson's (1987) intensive studies of children's use of manipulatives and their understanding of mathematical concepts. Resnick and Oman-son systematically evaluated third-grade children's ability to solve problems both with and without manipulatives. Much of the work involved Dienes Blocks, which are a systematic set of manipulatives that are designed to help children acquire understanding of base 10 concepts. Most of the children understood what was asked of them and appeared to enjoy working with the blocks. Unfortunately, however, the children's ease with and knowledge of the blocks was not related to their understanding of similar kinds of problems expressed in more formal mathematical terms. The children did not relate approaches they had used to solve problems with manipulatives to the solution of similar problems involving written symbols. For example, children who were successful in using Dienes Blocks to solve subtraction problems involving two or three digits had trouble solving simpler written problems. Indeed, the child who performed best with the Dienes Blocks performed worst on the standard problems. Clearly, success with a manipulative did not guarantee success with written symbols; in fact, success with one form of mathematical expression was unrelated to success with the other.

Other researchers have provided additional evidence of the nonequivalence of concrete and more abstract forms of mathematical expressions. For example, Hughes (1986) investigated young elementary school children's ability to use simple blocks or bricks to solve addition and subtraction problems. What is most interesting about this study for the current discussion is that the children were explicitly asked to draw connections between solutions involving concrete objects and those involving more abstract, written problems. The children were asked to use the bricks to represent the underlying concepts that were expressed in the written problems. For example, the children were asked to use bricks to solve written problems, such as 1 + 7 = ?. The experimenter and the teachers expected that the children would use the bricks to show how the two numbers could be combined. For example, children might be expected to show 1 brick and a pile of 7 bricks. The process of addition could be represented by combining the single brick and the pile of 7 bricks to form one pile with 8 bricks. But this is not what happened. Overall, the children performed poorly. Regardless of whether they could solve the written problems, they had difficulty representing the problems with the bricks. Moreover, the children's errors demonstrated that they failed to appreciate that the bricks and written symbols were two alternate forms of mathematical expression. Many children took the instructions literally, using the bricks to physically spell out the written problems (Figure 8.3). For example, they

Typical soiution

Figure 8.3 An example of how children use small bricks to represent the problem 1 + 7 = 8. The children often copied written problems with the bricks rather than using the bricks as an alternate representational system. From Children and Numbers: Difficulties in Learning Mathematics, by M. Hughes, 1986, pp. 99-103. Copyright 1986 by Basil Blackwell. Adapted with permission.

made a line of bricks to represent the "1" and two intersecting lines to represent the "+" and so on. These results again demonstrate that children may treat solutions involving manipulatives and those involving written mathematical symbols as cognitively distinct entities.

The research on children's understanding of manipulatives also highlights the conditions under which manipulatives are likely to be effective. Specifically, the results of several studies suggest that manipulatives are most effective when they are used to augment, rather than to substitute for, instructions involving written symbols. In successful cases of manipulative use, teachers have drawn specific connections between children's use of a manipulative and the related expression of the underlying concept in written form. For example, consider Wearne and Hiebert's (1988) program. It focuses on fractions, but the results are relevant to other mathematical concepts. At all stages of the program, the teacher draws specific links between manipulatives and written symbolic expressions. The manipulative is used as a bridge to the written expressions rather than as a substitute or precursor for written symbols. As a result, a scaffold is provided to assist children in learning written representations. The program gradually leads them away from a focus on concrete manipulatives and toward a focus on written representations. Thus, the focus of this and similar successful programs is on the relation between manipulatives and other forms of mathematical expression. Similarly, the Building Blocks curriculum (Sarama & Clements, 2002, 2004) uses manipulatives to help children gain insight into mathematical concepts, but it also includes activities to link manipulatives to other forms of representation. There is extensive use of concrete manipulatives, but the activities with the manipulatives are designed with the end goal of facilitating children's understanding of written representations. What makes this curriculum special, if not unique, is that there is, by design, a systematic formulation by which children grow out of using manipulatives. The materials are progressively layered, meaning that activities at earlier levels are designed to lay the foundation for later activities. In this way, the curriculum establishes linkages, both implicitly and explicitly, between manipulative-based solutions and written solutions.

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