can discuss general patterns they see in addition and subtraction, such as +1 is just the next counting number or −1 is the number just before. Children can discuss adding and subtracting 0 and the pattern it gives: adding or subtracting 0 does not change the original number, so the result (the answer) is the same as the original number. Many children can now informally use the commutative property (A + B = B + A) especially when one number is small (e.g., Baroody and Gannon, 1985; Carpenter et al., 1993; DeCorte and Verschaffel, 1985; for a review of the literature, see Baroody, Wilkins, and Tiilikainen, 2003). Experience with put together addition situations in which the addends do not have different roles provides better support for learning the commutative property than does experience with the change situation (see research described in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b) because these addends have such different roles in the action. To the child, it actually feels different to have 1 and then get 8 more than to have 8 and get 1 more. It feels better to gain 8 instead of gaining 1, even though you end up with the same amount. In contrast, the numerical work on put together/take apart partners facilitates understanding that the order in which one adds does not matter. Looking at composed/decomposed triads with the same addends also enables children to see and understand commutativity in these examples (for example, see that 9 = 1 + 8 and 9 = 8 + 1 and that the addends are just switched in order but still total the same).

All of the work on the relations/operation core in kindergarten serves a double purpose. It helps children solve larger problems and become more fluent in their Level 1 direct modeling solution methods. It also helps them reach fluency with the number word list in addition and subtraction situations, so that the number word list can become a representational tool for use in the counting on solution methods.

Different children learn and remember some sums and differences at each level, and it is very useful to know these for small numbers, for example for totals ≤ 8. But the more important step at the kindergarten level is that children are learning general numerical solution methods that they can extend to larger numbers. Simultaneously they are becoming fluent with these processes and with the number word list, so that they can advance to the Level 2 counting on methods that are needed to solve single-digit sums and differences with totals over ten. Children later in the year can begin to practice the number word list prerequisite for counting on by starting to count at a given number instead of always at one.

Kindergarten children are also working on all of the prerequisites for the Level 3 derived fact methods, such as make-a-ten (see Box 5-11). One prerequisite, seeing the tens in teen numbers, was discussed in the number core. The other two prerequisites involve knowing partners of numbers



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