Manipulatives as symbols: a new perspective on the use of concrete objects to teach mathematics

This article offers a new perspective on the use of concrete objects to teach mathematics. It is commonly assumed that concrete manipulatives are effective because they allow children to perform mathematics without understanding arbitrary, written mathematical symbols. We argue that the sharp distinction between concrete and abstract forms of mathematical expression may not be justified. We believe instead that manipulatives are also symbols; teachers intend for them to stand for or represent a concept or written symbol. Consequently, research on how young children comprehend symbolic relations is relevant to studying their comprehension of manipulatives. We review evidence that many of the problems that children encounter when using manipulatives are very similar to problems that they have using other symbol systems such as scale models. Successful use of manipulatives depends on treating them as symbols rather than as substitutes for symbols. A persistent dilemma for teachers of mathematics concerns how to help children understand abstract concepts, such as addition and multiplication, and the symbols that are used to represent these concepts (Hiebert & Carpenter, 1992; Resnick & Ford, 1984). Teachers face a double challenge. Symbols may be difficult to teach to children who have not yet grasped the concepts that they represent. At the same time, the concepts may be difficult to teach to children who have not yet mastered the symbols. Not surprisingly, both teachers and mathematics researchers have called for better techniques to help children learn mathematical concepts and symbols