cussed, this is often the only meaning of = that students in the United States know, and this interferes with their use of algebra. So it is really important that they also see and use forms like 5 = 3 + 2 to show the numbers hiding inside a number, the partners (addends) that make that number.

Step 4 (Grade 1)

At this step, children build on their earlier number and relations/operation knowledge and skills to advance to Level 2 counting on solution methods. They also come to understand that addition is related to subtraction and can think of subtraction as finding an unknown addend (see Box 5-10).

Grade 1 addition and subtraction is the culmination of all of the number core and relations/operation core experiences and expertise that have been building since birth, for those who have been given sufficient opportunities to build such competence. Foundational and achievable relations and operations content for Grade 1 children is summarized in Box 5-9.

For all of the earlier experiences to come together into the Level 2 counting on solution methods, some children may still need some targeted practice in beginning counting at any number instead of always starting at one (one of the prerequisites for counting on). It is also helpful to begin counting on in some kind of structured visual setting, so that children can conceptualize the relationships between the counting and cardinal meanings of number words.

Counting on is not a rote method. It requires a shift in word meaning for the first addend from its cardinal meaning of the number in that first addend to a counting meaning, as children count on from that first addend to the total. Children then must shift from that last counted word to its cardinal meaning of how many objects there are in total. For example, seeing circles for both addends in a row with the problem printed above enables children to count both addends and then count all to find the total (their usual Level 1 direct modeling solution method). But after several times of counting all, they can be asked what number they say when they count the last circle in the group of 6 and whether they need to count all of the objects or could they just start at 6. Going back and forth between this counting on and the usual counting all enables children to see that counting on is just an abbreviation of counting all, in which the initial counts are omitted (e.g., Fuson, 1982; Fuson and Secada, 1986; Secada, Fuson, and Hall, 1983).

 

6 + 3

Six is a cardinal number.

o o o o o o

o o o

 

1 2 3 4 5 6

7 8 9

Six here is a count number when counting all.



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