earlier age and in more complex detail than originally thought. By preschool, they benefit from learning about a variety of shapes, both typical and atypical, and this knowledge is impacted by their acquisition of spatial language. Language input and spatial activities appear to be highly influential in the development of spatial categories and spatial skills during the preschool years.

DEVELOPMENT OF MEASUREMENT

Measurement is a fundamental aspect of mathematics, which “bridges two main areas of school mathematics—geometry and number” through the attachment of number to spatial dimensions (National Council of Teachers of Mathematics, 2000). The development of measurement skills usually starts with directly comparing objects along one dimension. Thus, children generally succeed in measuring length prior to area and volume (Hart, 1984; but see Curry and Outhred, 2005, for early success in measuring volume when the task involves successive filling of a container).

Certain skills, such as sensitivity to variations in amount, can be thought of as precursors to mature measurement skills and have been observed in infants. The ability to directly compare the lengths of objects is an early emerging skill and initially appears to be perceptually based (Boulton-Lewis, 1987). Infants demonstrate awareness of variations in amount in one dimension (e.g., noticing height) as early as 4 months (Baillargeon, 1991) and can discriminate between two objects based on height at 6 months (Gao, Levine, and Huttenlocher, 2000). For example, 6-month-old infants and 2-year-old toddlers are able to discriminate the length of dowels when they appear in the presence of a constant, aligned standard but not when there is no standard available with which to compare them (Huttenlocher, Duffy, and Levine, 2002).

Subsequent studies show that infants and toddlers are responding to the relative size of the standard and the test objects (Duffy, Huttenlocher, and Levine, 2005a, 2005b). This result is in line with the theory (Bryant, 1974) that relative coding precedes absolute coding. The ability to discriminate lengths in a more precise manner (distinguishing two heights that are fairly close without a present, aligned standard) develops some time between ages 2 and 4. However, even by age 4, children’s sensitivity to variations in size is often influenced by the relation between two objects.

This early reliance on a standard to assess size may seem to contrast with findings by Piaget and his colleagues showing that young children do not spontaneously use a standard to measure objects (Piaget, Inhelder, and Szeminska, 1960). Piaget and colleagues argue that before school age, children’s ability to encode metric information is limited because they lack the ability to make transitive inferences that are involved in measurement—that



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