Rethinking "Concrete" Manipulatives

Article excerpt

Close your eyes and picture students doing mathematics. Like many educators, the mental pictures may include manipulative objects, such as cubes, geoboards, or colored rods. Does the use of such concrete objects really help students learn mathematics? What is meant by "concrete"? Are computer displays concrete and can they play an important role in learning? By addressing these questions, the authors hope to change the mental picture of what manipulatives are and how they might be used effectively.

Are Manipulatives Helpful?

Helpful, yes . . . Students who use manipulatives in their mathematics classes usually outperform those who do not (Driscoll 1983; Sowell 1989, Suydam 1986). This benefit holds across grade level, ability level, and topic, given that using a manipulative makes sense for the topic. Manipulative use also increases scores on retention and problem-solving tests. Finally, attitudes toward mathematics are improved when students are instructed with concrete materials by teachers knowledgeable about their use (Sowell 1989).

. . . but no guarantee. Manipulatives, however, do not guarantee success (Baroody 1989). One study showed that classes not using manipulatives outperformed classes using manipulatives on a test of transfer (Fennema 1972). In this study, all teachers emphasized learning with understanding.

In contrast, students sometimes learn to use manipulatives only in a rote manner. They perform the correct steps but have learned little more. For example, a student working on place value with beans and bean sticks used the bean as 10 and the bean stick as 1 (Hiebert and Wearne 1992).

Similarly, students often fail to link their actions on base-ten blocks with the notation system used to describe the actions (Thompson and Thompson 1990). For example, when asked to select a block to stand for 1 then put blocks out to represent 3.41, one fourth grader put out three flats, four longs, and one single after reading the decimal as "three hundred forty-one."

Although research suggests that instruction begin concretely, it also warns that concrete manipulatives are not sufficient to guarantee meaningful learning. This conclusion leads to the next question.

What Is Concrete?

Manipulatives are supposed to be good for students because they are concrete. The first question to consider might be, What does concrete mean? Does it mean something that students can grasp with their hands? Does this sensory character itself make manipulatives helpful? This view presents several problems.

First, it cannot be assumed that when children mentally close their eyes and picture manipulative-based concepts, they "see" the same picture that the teacher sees. Holt (1982, 138-39) said that he and his fellow teacher "were excited about the rods because we could see strong connections between the world of rods and the world of numbers. We therefore assumed that children, looking at rods and doing things with them, could see how the world of numbers and numerical operations worked. The trouble with this theory is that [my colleague] and I already knew how the numbers worked. We could say, 'Oh, the rods behaved just the way numbers do.' But if we hadn't known how numbers behaved, would looking at the rods enable us to find out? Maybe so, maybe not."

Second, physical actions with certain manipulatives may suggest mental actions different from those that teachers wish students to learn. For example, researchers found a mismatch when students used the number line to perform addition. When adding 5 and 4, the students located 5, counted "one, two, three, four," and read the answer. This procedure did not help them solve the problem mentally, for to do so they must count "six, seven, eight, nine" and at the same time count the counts - 6 is 1, 7 is 2, and so on. These actions are quite different (Gravemeijer 1991, 59). These researchers also found that students' external actions on an abacus did not always match the mental activity intended by the teacher. …