Cardinality

The process of identifying the number of items in a small set (cardinality) has been called subitizing. We will call it perceptual subitizing to differentiate it from the more advanced form we discuss later for larger numbers called conceptual subitizing (see Clements, 1999). For humans, the process of such verbal labeling can begin even before age 2 (see Chapter 3). It first involves objects that are physically present and then extends to nonpresent objects visualized mentally (for finer distinctions in this process, see Benson and Baroody, 2002). This is an extremely important conceptual step for attaching a number word to the perceived cardinality of the set. In fact, there is growing evidence that the number words are critical to toddlers’ construction of cardinal concepts of even small sets, like three and four and possibly one and two (Benson and Baroody, 2002; Spelke, 2003; also see Baroody, Lai, and Mix, 2006; and Mix, Sanhofer, and Baroody, 2005).

Children generally learn the first 10 number words by rote first and do not recognize their relation to quantity (Fuson, 1988; Ginsburg, 1977; Lipton and Spelke, 2006; Wynn, 1990). They do, however, begin to learn sets of fingers that show small amounts (cardinalities). This is an important process, because these finger numbers will become tools for adding and subtracting (see research literature summarized in Clements and Sarama, 2007; Fuson, 1992a, 1992b). Interestingly, the conventions for counting on fingers vary across cultures (see Box 5-4).

In order to fully understand cardinality, children need to be able to both generalize and extend the idea. That is, they need to generalize from a specific example of two things (two crackers), to grasp the “two-ness” in any set of two things. They also need to extend their knowledge to larger and larger groups—from one and two to three, four, and five, although these are more difficult to see and label (Baroody, Lai, and Mix, 2006; Ginsburg, 1989). Children’s early notions of cardinality and how and when they learn to label small sets with number words are an active area of research at present. The timing of these insights seems to be related to the grammatical structure of the child’s native language (e.g., see the research summarized in Sarnecka et al., 2007).

Later on, children can learn to quickly see the quantity in larger sets if these can be decomposed into smaller subitized numbers (e.g., I see two and three, and I know that makes five). Following Clements (1999), we call such a process conceptual subitizing because it is based on visually apprehending the pair of small numbers rather than on counting them. Conceptual subitizing requires relating the two smaller numbers as addends within the conceptually subitized total. With experience, the move from seeing the smaller sets to seeing and knowing their total becomes so rapid that one can experience this as seeing 5 (rather than as seeing 2 and 3). Children may also learn particular patterns, such as the 5 pattern on a die. Because these



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