Measurement of length: The need for a better approach to teaching

School Science and Mathematics, Mar 1997 by Constance Kamii, Faye B Clark

Three hundred and eighty-three children in grades 1-5 were individually interviewed to find out at when they construct unit iteration out of transitive reasoning as described by Piaget, Inhelder, and Szeminska (1960). The results indicated that most children (72%) construct transitive reasoning by second grade and that most (76%) construct unit iteration out of transitive reasoning by fourth grade. The article explains why traditional instruction produces the poor results revealed by the National Assessment of Educational Progress. It also suggests a better approach to the teaching of measurement that presents problems and encourages children to modify their ways of thinking.

Measurement of length is introduced in kindergarten and taught repeatedly in subsequent years according to most state curriculum guides and nationally distributed mathematics textbooks. However, the outcome of this instruction is disappointing. For example, the results of the 1985-86 National Assessment of Educational Progress (NAEP) revealed that only 14% of the third graders and 49% of the seventh graders who responded to the question shown in Figure 1 gave the correct answer of 5 cm (Lindquist & Kouba,1989). In the 1990 NAEP, one of the items asked for the length of an object placed next to a portion of a ruler that had been broken. The percentages of correct responses were 24% at grade four and 61 % at grade eight (Mullis, Dossey, Owen, & Phillips,1991). Something is clearly wrong with the instruction given in measurement of length. The purpose of this article is to evaluate the way in which measurement of length is taught and to suggest a better approach. This evaluation is done in light of Piaget's research and theory (Piaget, Inhelder, & Szeminska, 1960), our research based on their findings, and kindergartners' approach to measurement described by Paley (1981).

The instruction typically found in textbooks begins by asking children to compare two or three objects. Students are then asked to align paper clips, cubes, or other items end to end along an object such as an unsharpened pencil and to count them. After thus using nonstandard units, children are introduced to rulers and standard units to find the length in centimeters (or inches) of an object.

To evaluate the preceding instruction, examine how kindergartners compare lengths. Paley (1981) provides an excellent example from her kindergarten class. The class was about to act out "Jack and the Beanstalk" when Wally and Eddie had a disagreement.

Wally: The big rug is the giant's castle. The small one is Jack's house.

Eddie: Both rugs are the same.

Wally: They can't be the same

Eddie: ...You have to measure it. You need a ruler

Wally: We have a ruler.

Eddie: Not that one. Not the short kind. You have to use the long kind that gets curled up in a box.

Wally: Use people. People's bodies. Lying down in a row.

Eddie: That's a great idea. I never even thought of that. (pp. 13-14)

Wally announced a try-out for "rug measurers." He added one child at a time until both rugs were covered-four children end to end on one rug and three on the other. Everyone in the class was now satisfied, and the play continued with Wally as the giant on the rug that came to be called "the four-person rug." The next day, Eddie measured the rugs again. He used himself, Wally, and two other children, but the line of four children was too short.

Wally: You're too short. I mean someone is too short. We need Warren. Where's Warren?

(Wally was told that Warren was absent.) ...

Eddie: Then we can't measure the rug.

Teacher: You can only measure the rug when Warren is here?

Jill: Because he's longer.

Deana: Turn everyone around. Then it will fit. (p. 14)

Eddie rearranged the children, and each was now lying in a different position in the line. The group was surprised to see that the total length remained unchanged.

Eddie: No, it won't work. We have to wait for Warren.

Deana: Let me have a turn. I can do it.

Jill: You're too big, Deana. Look at your feet sticking out....

Teacher: Is there a way to measure the rug so we don't have to worry about people's sizes?

Kenny: Use short people.

Teacher: And if the short people aren't in school?

Rose: Use big people

Deana: Use rulers. Get all the rulers in the room. I'll get the box of rulers.... This isn't enough rulers.

Wally: Use the dolls.

Teacher: So this rug is ten rulers and two dolls long? (Silence.) Here's something we can do. We can use one of the rulers over again, this way.

Eddie: Now you made another empty space.

Teacher: Eddie, you mentioned a tape measure before. I have one here.

(We stretch the tape along the edge of the rug, and I show the children that the rug is 156 inches long. The lesson is done. The next day Warren is back in school.)

Wally: Here's Warren. Now we can really measure the rug.

Teacher Didn't we really measure the rug with the ruler?

Wally: Well, rulers aren't really real, are they? (pp. 14-16)

Paley's account of kindergartners' thinking is difficult to interpret without Piaget's theory. To interpret Paley's example and to introduce our study, we now describe the research and theory of Piaget and his colleagues (1960).