Levels in Children’s Numerical Solution Methods
Level 1: Direct modeling of all quantities in a situation; used at the first three number/operation levels:
Counting all: Count out things or fingers for one addend, count out things or fingers for the other addend, and then count all of the things or fingers.
Take away: Count out things or fingers for the total, take away the known addend number of things or fingers, and then count the things or fingers that are left.
Level 2: Count on can be done in first grade (some children can do so earlier): They use embedded number understanding to see the first addend within the total and so see that they do not need to count all of the total, but instead could make a cardinal-to-count shift and count on from the first addend.
Count on to find the total: On fingers or with objects or with conceptual subitizing, children keep track of how many words to count on so that they stop when they have counted on the second addend number of words and the last word they say is the total:
6 + 3 = ? would be “six, seven, eight, nine, so the total is nine. I counted on 3 more from 6 to make 9.”
After learning counting on from the first addend, children learn to count on from the larger addend.
Count on to find the unknown addend: Children stop counting when they say the total, and the fingers (or other keeping track method) tell the answer (the unknown addend number of words they counted on past the first addend).
6 + ? = 9 would be “six, seven, eight, nine, so I added on 3 to 6 to make 9.
I counted on 3 more from 6 to make 9. Three is my unknown addend.”
(situations expressed in words, perhaps with an accompanying picture), oral numerical problems such as three plus two, and written numerical problems such as 3 + 2. Chapter 4 summarized research reporting that more children from low-income families had trouble with the last three kinds of problems than with the first kind and than did their middle-income peers. Therefore, such children especially need help and practice in generating models using objects or fingers for such situations.
At Grade 1, children who have not yet moved to the Level 2 general counting on methods (see Box 5-8 and Box 5-11 for more details) can do so with help. In these methods, children shift from the cardinal meaning of the first addend to the counting meaning as they count on from it: For 5 + 2, they think five, shift to the counting word five in the number word list, and count on two more words—five, six, seven. This ability to count on can be