Development of Concrete Transitivity of Length in Children1

The purpose of the present paper is to analyze the criteria for a diagnosis of concrete transitivity of length and to present some developmental data obtained with a new technique. subject is said to have concrete transitivity of length when he is able to infer from the observations A is longer than and is longer than that must be longer than C or when he is able to infer from the observations A is equally long as and is equally long as that must be equally long as C. In what follows, only transitivity of inequalities will be considered. Concrete transitivity, involving inferences from actual observations, should not be confused with formal transitivity which permits the subject to make inferences from verbally stated, hypothetical premises. The first study directly relevant to transitivity of length was made by Piaget et al. (2, ch. 2). It concerned the development of measuring procedures, which by definition presuppose transitivity. Children were asked to build with blocks a tower equal in height to a tower already built by the experimenter. The given tower stood on a table which was higher than the table on which the subject was to build his tower. Sticks of various lengths were available to the subjects. Observation of children's procedures in evaluating the relative heights of the towers showed a very gradual change towards adequate measurement. The majority of the children appeared to understand and master the measuring technique only after the age of 7 or 8 years. Piaget's procedure was highly flexible and unsystematic, and no complete information is available on the number and age of the subjects and on their exact performance. In a recent study, severely critical of Piaget, Braine (i) employed a nonverbal technique. His subjects were instructed to find candy, which was always hidden under the longer (or shorter) of any two upright pieces of wood. Having learned to depend on relative length as a cue to the location of the candy, the subjects were given a series of transitivity items. They were shown that an upright was longer than a measuring stick B and that the