Concreteness Dual Representation And Educational Symbols

In the preschool and early elementary school years, children are asked to master a variety of symbol systems, such as letters, numbers, maps, and musical notation. Symbolic reasoning is thus fundamentally important for educational achievement, and children who fail to become skilled in even one of the major symbol systems are at serious risk of being left behind.

The difficulty that children sometimes have in acquiring an understanding of these important symbol systems has led to a variety of materials that are designed to help children learn the relevant information. For example, teachers often use concrete objects as substitutes for abstract symbolic representations. These objects are often referred to as manipulatives. Examples of concrete, three-dimensional objects include Dienes Blocks, Base 10 blocks, Digi-Blocks, and Cuisenaire Rods. In addition, teachers use many household objects as informal manipulatives, including cereal, money, and paper clips (Stevenson & Stigler, 1992). Outside the classroom, parents can purchase a vast array of attractive objects of a symbolic nature in the hope that such objects will help their children acquire early literacy and number skills. Magnetic letters and numbers cover a large proportion of the refrigerators in the homes of young American children, stuck there to encourage early learning.

Manipulatives have been touted as solutions for children of a wide range of ages and ability levels; they have been offered as appropriate for all ability levels, ranging from the disabled to the gifted (Clements & McMillen, 1996; Sowell, 1989; Wearne & Hiebert, 1988). Indeed, faith in the value of manipulatives is almost a defining characteristic of modern approaches to early childhood education. Unfortunately, however, research on the effectiveness of manipulatives has not confirmed the anticipated benefits. Several studies have shown, at best, inconsistent or weak advantages for manipulatives in comparison to more traditional techniques for teaching mathematics to children (Ball, 1992; Clements, 1997; Clements & McMillen, 1996; Hughes, 1986). Longitudinal and intensive studies of the use of manipulatives in classrooms have shown that children often fail to establish connections between manipulatives and the information that the manipulatives are intended to communicate (Sarama & Clements, 2002, 2004; Sowell, 1989). Put simply, although manipulatives can facilitate thinking, they are not a panacea.

We suggest that part of the reason that manipulatives have not been shown to support symbol-based solutions involves challenges that are very similar to those that younger children encounter when using a scale model. There are at least two general similarities between what is required to succeed in our model task and what is required to effectively use a manipulative. The first is that the relation between a manipulative and what it is intended to represent may not be transparent to young children. In other words, the concreteness of a manipulative (or of our model) does not guarantee that children will understand that it is intended to represent something other than itself. To a teacher or parent, the relation between a manipulative-based solution and a more traditional written solution may seem obvious or even transparent. But the same may not be true in the minds of young children. As we discuss, the relation between manipulatives and other types of representations may be opaque to young children.

The second similarity between children's difficulties with our model and with manipulatives is that dual representation is relevant to both. As was true of our model, manipulatives have a dual nature; they are intended to be used as representations of something else, but they also are objects in their own right. In the next section, we review some difficulties that children encounter when using manipulatives, difficulties that parallel younger children's problems with our scale model and that are consistent with the dual representation perspective.

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