about using counting to compare is that a number that is said later in the counting word list corresponds to a collection that has a greater number of objects than does a collection corresponding to a number earlier in the sequence. For example, one knows that there are more beads in a collection of eight black beads than there are in a collection of seven white beads because 8 occurs later in the counting list than 7 (see the bottom of Figure 2-4). Counting thus provides a more advanced way to compare sets of things than direct matching because it relies on knowledge about how numbers compare. Counting is also a more powerful way to compare sets of things than direct matching because it allows sets that are not in close proximity to be compared.
A key point about comparing collections of objects is that counting can be used to do so, and it relies on the link between the number list and cardinality: Numbers later in the list describe greater cardinalities than do numbers earlier in the list. Finding out which collection is more than another collection is easier than determining exactly how many more that collection has than the other, which can be formulated as an addition or subtraction problem. This more specific version of comparison is discussed in the next section.