Ordering and Ordinal Numbers
There is frequent confusion in the research literature in the use of the terms ordered or ordering, ordinal number, and order relation. Some of this confusion stems from the fact that adults can flexibly and fluently use the counting, cardinal, and ordinal meaning of number words without needing to consciously think about the different meanings. As a result, they may not be able to differentiate the meanings very clearly. But young children learn the meanings separately and need to connect them.
When counting to find the total number in a set, the order for connecting each number word to objects is arbitrary and could be done in any order. As noted previously, the last number takes on a cardinal meaning and refers to the total numbers of items counted. Thus, the cardinal meaning of a number refers to a set with that many objects. Cardinal numbers can be used to create an order relation. That is the idea that one set has more members than another set. An order relation (one number or set is less than or more than another number or set) tells how two quantities are related. This order relation produces a linear ordering on these numbers or sets. An ordinal number tells where in the ordering a particular number or set falls. A child can subitize for the small ordinal numbers (see whether an object in an ordered set is first, second, or third), but needs to count for larger ordinal numbers and shift from a count meaning to an ordinal meaning (e.g., count one, two, three, four, five, six, seven [count meaning]. That person is seventh [ordinal meaning and ordinal work] in the line to buy tickets.).
We have not emphasized ordinal words in this chapter because they are so much more difficult than are cardinal words, and children learn them much later (e.g., Fuson, 1988). Although 4- and 5-year-olds could learn to use the ordinal words first, second, and last, it is not crucial that they do so. The ordinal words first through tenth could wait until Grade 1.
Many researchers have noted how the number word list turns into a mental representational tool for adding and subtracting. A few researchers have called this a mental number line. However, for young children this is a misnomer, because children in kindergarten and Grade 1 are using the number word list (sequence) as a count model: Each number word is taken as a unit to be counted, matched, added, or subtracted. In contrast, a number line is a length model, like a ruler or a bar graph, in which numbers are represented by the length from zero along a line segmented into equal lengths. Young children have difficulties with the number line representation because they have difficulty seeing the units—they need to see things, so they focus on the numbers instead of on the lengths. So they may count the starting point 0 and then be off by one, or they focus on the spaces and are confused by the location of the numbers at the end of the spaces. The report Adding It Up: Helping Children Learn Mathematics (National