measurement of time: Children's construction of transitivity, unit iteration, and conservation of speed, The

School Science and Mathematics, Mar 2001 by Long, Kathy, Kamii, Constance

One hundred twenty children in kindergarten and grades 2, 4, and 6 were individually interviewed with five Piagetian tasks to determine the grade level at which most have constructed transitive reasoning, unit iteration, and the conservation of speed. The responses were categorized as "successful, " "unsuccessful, "or "transitional. "By combining the "successful" and "transitional" categories, it was found that the children reasoned transitively by second grade (70. 0%) and demonstrated unit iteration and conservation of speed by sixth grade (70.0% and 83.3%, respectively). It was concluded that the construction of the logic necessary to make sense of the measurement of time is generally not complete before sixth grade.

Primary school mathematics textbooks in the U.S. usually include a chapter on measurement (Boswell et al., 1998; Clements, Jones, Moseley, & Schulman, 1998). The subject of time is likely to appear in this chapter, beginning with how to read a clock. Educators familiar with Piagetian theory immediately notice that this instruction focuses mainly on the conventional aspects of time, such as the ability to read a clock. Reading a clock is different from measuring time.

To study the measurement of time from a Piagetian perspective, educators must understand some of the basic notions underlying the construction of knowledge. Piaget (1967/1971, 1945/1962) made a fundamental distinction between three kinds of knowledge according to their ultimate sources: physical, logico-- mathematical, and social knowledge.

Physical knowledge is knowledge of objects observable in external reality. The observation that water runs out of a container having a hole in the bottom is an example of physical knowledge. A source of physical knowledge is in objects (Piaget, 1967/1971).

Social knowledge is knowledge of conventions created by people. It, too, has a source outside the individual, because social knowledge requires input from people. Examples of social knowledge are the words used to identify various objects, such as "clock" and "watch," and the fact that there are 60 minutes in an hour or that, in reading the minutes, the numbers 1, 2, and 3 on a clock must be read as 5, 10, and 15.

Logico-mathematical knowledge refers to the mental relationships children construct in their heads. If two vials seem to be the same because both are made of transparent plastic and both are the same in shape, color, and size, "the same" is an example of logico-- mathematical knowledge. The color and "plasticness" of the vials are physical knowledge, but the sameness of these objects is a mental relationship created by the individual who puts the two vials into a relationship. The source of logico-mathematical knowledge is thus in the child (Kamii & DeVries, 1993).

The purpose of measurement is to make indirect comparisons (relationships). Piaget (1946/1969; Piaget, Inhelder, & Szeminkska, 1948/1960) demonstrated that children must use two kinds of reasoning to measure objects: transitive reasoning and unit iteration. Transitive reasoning involves comparing a whole quantity to another whole quantity. Unit iteration requires the ability to think about each whole as consisting of equal parts.

Kamii and Clark (1997) studied children's construction of transitive reasoning and unit iteration with respect to measurement of length. They showed children an inverted T made of two 20-cm lines photocopied on a sheet of paper. The inverted T created a perceptual illusion, making the vertical line appear longer than the horizontal line. The researchers asked children if the two lines had the same length or if one was longer than the other. Whatever the child's response, they offered a 40 cm by 2 cm strip of tagboard and asked if the strip could be used to prove this judgment.

The children who had not constructed transitive reasoning could not think of a way to use the strip, but the others knew exactly what to do. They laid the strip against one of the lines and made a pencil mark on the strip to show the exact length of that line. They then laid the strip against the second line to compare its length to the length marked on the strip. Most children (72%) were found to have constructed transitive reasoning by second grade, as Piaget had found (Piaget et al., 1948/1960).

Kamii and Clark (1997) then offered a small block (4.5 cm x 2.24 cm x 0.64 cm) and asked the children if it could be used to prove what they believed about the two lengths. The children who had not constructed unit iteration could not think of a way to use the block. For those who had constructed the part-whole relationship between the length of the block and the length of each line, this question was easy. These children used the block as a unit and stepped it along each line, without any gaps or overlaps, and counted how many "parts" made up the "whole." Most children (76%) were found to have constructed unit iteration by fourth grade. Only when children can make the part-whole relationship of unit iteration can they be said to understand a unit of measurement.