more. These children focused either on length or on density, but they could not notice and coordinate both. However, when asked to count in such situations, many 4-year-olds can count both rows accurately, remember both count words, and change them to cardinal numbers and find the order relation on the cardinal numbers (Fuson, 1988). Thus, many 4-year-olds need encouragement to count in more than/less than/equal to situations, especially when the perceptual information is misleading.

To use matching successfully to find more than/less than, children may need to learn how to match by drawing lines visually to connect pairs or draw such matching lines if the compared sets are drawn on paper. Then they need to know that the number with any extra objects is more than the other set. It is also helpful to match using actual objects.

To use counting successfully, children need to be able to count both sets accurately and remember the first count result while counting the second set. Here is another example of the need for fluency in counting (see Box 5-1). Without such fluency, some children forget their first count result by the time they have counted the second set. They need more counting practice in such situations. Children also need to know order relations on cardinal numbers. They need to learn the general pattern that most children do derive from the order of the counting words: The number that tells more is farther along (said later) in the number word list than the smaller number (e.g., Fuson, Richards, and Briars, 1982). Activities in which children make sets for both numbers, match them in rows and count them, and discuss the results can help them establish this general pattern.

There was an early period in which the counting and matching research had not been done and many researchers and educators suggested that teachers had to wait until children conserved number (said that rows in the classic Piagetian task were equal even in the face of misleading perceptual transformations) to do any real number activities, such as adding and subtracting. However, newer research shows that there is a crucial stage for 4- and 5-year-olds in which using counting and matching are important to learn and can lead to correct relational judgments (see the research summarized in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b). It is true that children typically do not understand that the rows are equal out of a logical necessity until age 6 or 7 (sometimes not until age 8). These older children (ages 6-7) judge the rows to be equal based on mental transformations that they apply to the situation. They do not see the need to count or match after one row is made shorter or longer by moving objects in it together or apart to see that they are equal. They are certain that simply moving the objects in the set does not change the numerosity. This is what Piaget meant by conservation of number. But children can work effectively with situations involving more and less long before they demonstrate this meaning of conservation of number.

For progress in relations, it is important that children hear, and try



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