capacity is not changing with age. In both children and adults, richer knowledge bases result in larger memory capacities (Chi, 1978). There is more evidence, however, for an underlying maturational component for changes in processing speed (although some aspects are also clearly affected by experience) (Kail, 1991; Luna et al., 2004; Travis, 1998).
Children’s understanding of the simple mechanics of bounded objects undergoes considerable change during the elementary school years. One area of dramatic change concerns an appreciation of how to interrelate variables that are concerned with trajectories, most notably, distance, speed, and duration. Many years ago, the Swiss psychologist, Jean Piaget, demonstrated confusions among these variables in young children (Piaget, 1946a, 1946b; Piaget and Inhelder, 1948), but only recently have more systematic studies documented the ways in which children come to make sense of each of these variables and their interrelations. Those studies now suggest that even young preschool children distinguish distance, speed, and time in some contexts (in contrast to Piaget’s claim that these notions are initially completely undifferentiated) (Acredelo, Adams, and Schmid, 1984; Matsuda, 1994, 2001; Wilkening, 1981). Some of the differences across tasks depend on what criteria children use in judging each task. Many of the tasks require them to use qualitative criteria (e.g., comparing starting and stopping times); some give them direct information in some symbolic form and examine their ability to integrate it (a clock that says 10 versus 20 seconds; a distance strip; two animals that are known to be fast or slow, like turtles and horses). Development here, however, does not occur in a vacuum. Consider the normal developmental progressions for children in two different cultures that vary in their approach to science and math education. Chinese third graders seem to have no difficulty reasoning about inverse relations, whereas American third graders often do; American fifth graders achieve performance more like Chinese third graders (Zhou et al., 2000). Although further research is needed to confirm the reliability of this difference and to understand its sources, it may reflect differences in the quality of early mathematics and science education. In China, in contrast to the United States, the skills of argument and proof are taught as early as the first grade and mathematics and science topics are pursued more deeply and thoroughly. In addition, the elementary teachers are more highly trained in the teaching of mathematics.
Thus, although there is a clear age trend in learning to understand inverse relations, there can be dramatic differences in the age at which most children understand such relations as a function of educational and cultural