After learning the concept "length," however, the child engages in a contrasting type of thought process. He or she learns to differentiate among longer objects and shorter objects, thus learning to measure length. Such comparisons are possible only because the length of a pencil is similar in kind to the length of a sofa, although they differ in degree. In adult language, we say that the two are "commensurable."

There are several varieties of measurement, among which the most important are cardinal and ordinal measurement. Like the cardinal numbers (1, 2, 3, . . .), cardinal measurements can be subjected to such arithmetic operations as addition, subtraction, and multiplication. Attributes such as length, weight, and duration are measured in this manner. Somewhat less familiar are ordinal measurements, which (like the ordinal numbers "first," "second," "third," . . .) identify only comparative relationships among entities. For example, concepts of "value," which will be explored at great length in later sections of this course, are subject only to ordinal measurement. We shall find that while values can be compared, they cannot be meaningfully added, subtracted, or multiplied. The child's measurements will probably start as simple ordinal comparisons; later he or she will learn to make cardinal measurements with the aid of a ruler.