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• Index 371-386

of the raised fingers. (See Box 5-4 for a discussion of different conventions for counting on fingers.)

Some children learn at home or in a care center to put the addends on separate hands, while others continue on to the next fingers for the second addend. The former method makes it easier to see the addends, and the latter method makes it easier to see the total. Both methods can be modeled by the teacher. As children become more and more familiar with which group of fingers makes 4 or 5 or 7 fingers, they may not even have to count out the total because they can feel or see the total fingers. Similarly, children using the method of putting fingers on separate hands eventually can just raise the fingers for the addends without counting out the fingers. But they do need initially to count the total. Children who put addends on separate hands may have difficulty with problems with addends over 5 (e.g., 6 + 3) because one cannot put both such numbers on a separate hand. They can, however, continue raising fingers from 6 fingers. Because these problems involve adding 1 or 2, such continuations of 1 and 2 are relatively easy.

By now children who have had experience with adding and subtracting situations when they were younger can generalize to solve decontextualized problems that are posed numerically, as in Two and two make how many? (Clements and Sarama, 2007; Fuson, 1988). For some small numbers, children may have solved such a problem so many times that they know the answer as a verbal statement: Two and two make four. If such knowledge is fluent, children may be able to use it to solve a more complex unknown addend problem. For example, Two and how many make four? Two.

For larger numbers, children will need to use objects or fingers to carry out a counting all or taking away solution procedure (Box 5-11) (see research summarized in Fuson 1992a, 1992b). Children will learn new composed/decomposed numerical triads as they have such experiences. The doubles that involve the same addends (2 is 1 and 1, 4 is 2 and 2, 6 is 3 and 3, 8 is 4 and 4) are particularly easy for children to learn because the perceptual and verbal task is simplified by have the same addends (e.g., see research summarized in Fuson, 1992a, 1992b). The visual 5-groups (e.g., 8 is made from 5 and 3) discussed for the number core are also useful. Research about powerful patterns for conceptual subitizing for very small numbers would be helpful, including the extent to which flexibility is important beyond a single powerful visual core that will work for all numbers.

The put together/take apart situations, and especially the take apart situation, can be used to provide varied numerical experiences with given numbers that help children see all of the addends (partners) hiding inside a given number. For example, children can take apart five to see that it can be made from a three and two and also from four and one. Later on these decomposed/composed triads can be symbolized by equations, such as 5 =

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