is much easier and can make subtraction as easy as addition (e.g., Fuson, 1986b; Fuson and Willis, 1988). It also emphasizes addition and subtraction as inverse operations.

The derived fact methods (Level 3) are mastered by some children at Grade 1, depending on how many of the prerequisites shown in Box 5-8 have been made accessible for 4- and 5-year-olds and then have been practiced so that they become fluent. These methods require recomposing the given numbers into a new, easier problem (e.g., 9 + 4 becomes 10 + 3). The make-a-ten methods are taught in East Asian countries and are very useful in multidigit computation (see the discussion in Chapter 2). The prerequisites are discussed later in the summaries of the 4- and 5-year-olds because children can begin building these prerequisites then. Enabling 4-and 5-year-olds to learn the prerequisites for the counting on and derived facts methods can help low-income children to learn more advanced strategies, which fewer of them do now. This can also help children with learning difficulties in mathematics because they often continue to use the Level 1 modeling methods for too many years unless they are helped to learn more advanced strategies. The general counting on methods for addition and subtraction can be learned meaningfully and done accurately and rapidly by most children in Grade 1 (Fuson, 2004).

Throughout the process of learning and using more advanced approaches to solving addition and subtraction problems, children also become fluent with individual sums and differences. Small numbers, such as plus 1 and minus one, and doubles (2 + 2, 3 + 3) become fluent early. Others become fluent over time.

Step 1 (Ages 2 and 3)

Children at this step use subitized and counted cardinality to solve situation and oral number word problems. They also use perceptual, length, and density strategies to find which is more with totals ≤ 5 (see Box 5-10).

Relations: More Than, Equal To, Less Than

Children ages 2 and 3 begin to learn the language involved in relations (Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; Ginsburg, 1977). More is a word learned by many children before they are 2. Initially it is an action directive that means: Give me more of this. But gradually children become able to use perceptual subitizing and length or density strategies to judge which of two sets has more things: She has more than I have. Such comparisons may not be correct at this age level if the sets are larger than three because children focus on length or on density and cannot yet coordinate these dimensions or use the strategies of matching or count-



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