What does it mean to say that a child understands numbers? There are many early milestones in number learning, and parents sometimes say that a toddler who can count to five or ten “knows” those numbers. Similarly, young children in literate environments learn to identify the written digits 0–9 along with letters of the alphabet and thus, in a sense, “know” the numbers. But what does it mean to *understand* numbers, in some important conceptual way? One operational definition comes from Piaget (1952). In the Piagetian tradition, children understand numbers when they pass the conservation-of-number task, around age 5 or 6 years. For Piaget, the key number concept is *equinumerosity* (sometimes called *exact equality*)—the idea that two sets have the same number of items, if and only if their members can be placed in perfect one-to-one correspondence (Frege, 1980 [1884]). The child's understanding of equinumerosity as an abstract principle is what the conservation task is supposed to measure.

A different operational definition of number knowledge arises in more recent work (e.g., Carey, 2009; Hurford, 1987; Klahr & Wallace, 1976). In this newer literature, children are said to understand numbers when they apply the cardinality principle of counting (Gelman & Gallistel, 1978) on the Give-N task (e.g., Condry & Spelke, 2008; Le Corre, Van de Walle, Brannon, & Carey, 2006; Sarnecka & Lee, 2009; Wynn, 1990, 1992). The cardinality principle states that the last word uttered in a (correct, rule-governed) count expresses the number of items in the whole set. It is the cardinality principle that gives number words their meanings, by making the cardinal meaning of any number word knowable from that word's ordinal position in the counting list. (For example, readers who do not speak Japanese—but do understand cardinality—can easily guess the meaning of the Japanese number word *nijuuichi* if they are told that it is the twenty-first word in the Japanese counting list.)

One current proposal about number development (Carey, 2009) is that children who understand cardinality (as measured by the Give-N task) also understand the key numerical concept of *succession* (often called the *successor principle* or *successor function*)*—*the idea that each number is generated by adding one to the previous number (Dedekind, 1901 [1872/1888]). One empirical study (Sarnecka & Carey, 2008) supports this claim for the numbers 5 and 6, although another study (Davidson, Eng, & Barner, 2012) finds that this early understanding is less robust for higher numbers, such as 25.

Integrating the older and newer notions of what it means to “understand” numbers, Izard and colleagues identified equinumerosity and succession as “two key concepts on the path toward understanding exact numbers” (Izard, Pica, Spelke, & Dehaene, 2008). But how do these concepts interact in development? Our proposal in this study is that children's understanding of cardinality (as measured by the Give-N task) predicts their understanding of equinumerosity (at least for the numbers *five* and *six*).

Note that this connection is not obvious. The traditional litmus test for understanding equinumerosity is Piaget's conservation-of-number task, which children pass at age 5 or 6. As Muldoon, Lewis, and Freeman (2009) noted,

The developmental puzzle is that up to the age of six, even some 2 years after they have mastered procedural counting, many children have yet to grasp that two sets with the same cardinal number must, by virtue of logical necessity, be equivalent, and that sets with different cardinals must by the same logic be numerically different. (pp. 203–204)

We will argue that Piaget's classic conservation-of-number task underestimated children's knowledge because it asked children about the abstract entity *number*, rather than about particular numbers such as *five* and *six*. (In other words, Piaget asked questions such as “Are there the same number of flowers and vases?” rather than “There are five flowers. Are there five vases, or six?”)

Piaget, of course, asked the question this way because he was interested in abstract and explicit knowledge that the child could articulate. But more recent theories of number-concept development (e.g., Carey, 2009) hold that children first learn about *particular* number words, and only later generalize their knowledge to the superordinate category of *numbers*. The counting list (*one, two, three*, etc.) is learned as a placeholder structure (something like the chant *eenie, meenie, minie, mo*), with little or no numerical content. The child acquires the deep numerical concepts (e.g., cardinality, equinumerosity, succession) during the process of assigning (or constructing, or discovering, depending on your theoretical bent) meanings for those number words. This is the process known as *conceptual-role bootstrapping* (Carey, 2009; Block, 1986; see also Quine, 1960). If children first learn about equinumerosity in the context of particular number words such as *five* and *six,* then measuring equinumerosity knowledge outside the context of particular number words (e.g., using only the word “number” as Piaget did) may obscure the early development of this knowledge.

There are hints of this in the findings reported by Sarnecka and Gelman (2004). That study investigated children's understanding of the *specificity* of number words. This is the idea that every number word picks out a specific, unique numerosity (Wynn, 1990, 1992). Some proposals had claimed that children did not understand this property of number words until they mastered the cardinality principle of counting (as measured by the Give-N task). Sarnecka and Gelman developed three new tasks to measure children's knowledge of specificity, two of which children passed before understanding the cardinality principle. Thus, Sarnecka and Gelman concluded that children understood specificity before cardinality.

This article revisits the third task—the one children failed until they understood cardinality. This was the “Compare-Sets” task. In it, children were shown two pictures, representing the snacks given to a pair of animals. The pictures were either identical or differed by one item. The children were told how many items one set had, and then were asked about the other set (e.g., “Frog has five peaches. Does Lion have five, or six?”).

The authors intended the Compare-Sets task to measure the child's knowledge that number words are specific. When non-CP-knowers (children who do not yet understand cardinality as measured by the Give-N task) passed two other “specificity” tasks but failed Compare-Sets, the authors concluded that the task was simply too difficult, and predicted that if the procedural demands could be reduced, the performance gap between CP-knowers and non-CP-knowers would disappear.

The present work tests that prediction and concludes that it was wrong. A new, simplified version of the Compare-Sets task actually makes the performance gap between CP-knowers and non-CP-knowers even more obvious. In light of this finding, we revisit the question of what the Compare-Sets task actually measures and argue for an answer that was not considered in the 2004 study: that the task does not primarily measure the child's knowledge of *specificity*, but of *equinumerosity*. So while children may indeed see number words as specific (or simply as being about quantity—another possibility consistent with the 2004 results), they do not understand *equinumerosity* (i.e., that any and only two sets with the same number word can be placed in one-to-one correspondence with each other) until they become CP-knowers.

Earlier studies have reported findings that are consistent with this possibility, although none have tested it directly. For example, Sophian (1988) presented 3- and 4-year olds with two sets of objects (e.g., a group of jars arranged in a circle, with a pile of spoons in the middle). In half the trials, children were told the number of each set, and then asked about their correspondence. For example, “There are *n* jars. There are *m* spoons. Can every jar have its own spoon?” In the other trials, children were told about the correspondence and given the number of one set, and then were asked about the number of the other set. For example, “Every jar has its own spoon. There are *n* jars. Are there *n* spoons?” Sophian reported that about 30–40% of 3-year-olds, and 70–75% of 4-year-olds, succeeded on both types of trial. These are approximately the proportions of Sophian's (relatively high-SES) sample that we would expect to be CP-knowers if they were tested on the Give-N task.

Frydman and Bryant (1988) reported a similar finding. In that study, 4-year-olds were asked to divide (“share out”) a set fairly, and to count one of the resulting portions. Having done that, many of the 4-year-olds were able to infer the number of another, uncounted portion. (See Izard et al., 2008 for another related finding.)

This study revisits the Compare-Sets task and tests Sarnecka and Gelman's (2004) explanation for the performance gap between CP-knowers and other children (i.e., that the task was too procedurally difficult). A simplified version of the task greatly reduces the burden on attention and memory by leaving the sets visible the whole time.1 But contrary to Sarnecka and Gelman's prediction, simplifying the task does not eliminate the performance gap between CP-knowers and other children.

In light of these findings, we reconsider how this task should be interpreted. We suggest that the gap in performance between CP-knowers and non-CP-knowers may reflect CP-knowers’ understanding of equinumerosity—and that equinumerosity itself may be (along with understanding of the cardinality principle and the successor function) a manifestation of a broad conceptual achievement: the “exact numbers” idea—or at least the idea of the exact numbers *five* and *six*.