siiiixxx

7 8 9

To count on like this, a child must shift from the cardinal meaning above to the count meaning of six and then keep counting 7, 8, 9.

Trying this with different problems enables many children to see this general pattern and begin counting on. Transition strategies, such as counting 1, 2, 3, 4, 5, 6 very quickly or very softly or holding the 6 (siiiiiiixxxxxx), have been observed in students who are learning counting on by themselves; these can be very useful in facilitating this transition to counting on (e.g., Fuson, 1982; Fuson and Secada, 1986; Secada, Fuson, and Hall, 1983). Some weaker students may need explicit encouragement to trust the six and to let go of the initial counting of the first addend, and they may need to use these transitional methods for a while.

Counting on has two parts, one for each addend. The truncation of the final counting all by starting with the cardinal number of the first addend was discussed above. Counting on also requires keeping track of the second addend—of how many you count on so that you count on from the first addend exactly the number of the second addend. When the number is small, such as for 6 + 3, most children use perceptual subitizing to keep track of the 3 counted on. This keeping track might be visual and involve actual objects, fingers, or drawn circles. But it can also use a mental visual image (some children say they see 3 things in their head and count them). Some children use auditory subitizing (they say they hear 7, 8, 9 as three words). For larger second addends, children use objects, fingers, or conceptual subitizing to keep track as they count on. For 8 + 6, they might think of 6 as 3 and 3 and count with groups of three: 8 9 10 11 12 13 14 with a pause after the 9 10 11 to mark the first three words counted on. Other children might use a visual (I saw 3 circles and another 3 circles) or an auditory rhythm to keep track of how many words they counted on. So here we see how the perceptual subitizing and the conceptual subitizing, which begin very early, come to be used in a more complex and advanced mathematical process. This is how numerical ideas build, integrating the levels of thinking visually/holistically and thinking about parts into a complex new conceptual structure that relates the parts and the whole. Children can discuss the various methods of keeping track, and they can be helped to use one that will work for them. Almost all children can learn to use fingers successfully to keep track of the second addend.

Many experiences with composing/decomposing (finding partners hiding inside a number) can give children the understanding that a total is any number that has partners (addends) that compose it. When subtracting, they have been seeing that they take away one of those addends, leaving the other one. These combine into the understanding that subtracting means finding the unknown addend. Therefore, children can always solve



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