Naïve and sophisticated counting strategies
When children add or subtract they initially make use of objects or replacements for the objects, such as fingers, combined with knowledge of counting sequences. To add or subtract it is possible count from one, three times, to form each quantity. For example, to find 8 + 3 you can count from one to eight and then from one to three, combine the two quantities and then count from one to eleven to find the total. It is also possible to use a similar process to take away one quantity from another. Methods involving counting from one three times can be described as naïve counting strategies.
The number words, representing completed counts, can also be used themselves to make this process shorter. For example, if a student is told that some counters have been removed from a group of eleven counters and eight counters remain, the student can use the number word eleven as the end product of counting the objects. The student can then count down to eight to determine the number of counting words that have been removed from the sequence.
Counting down to eight
Of course there is more than one way to think about subtraction. If I take three away from eleven I can think about taking away the 11th item, then the 10th item and the 9th item so that eight items remain. Alternatively, I can say that there are eleven and when I count back one I am at ten, count back another one and I am at nine, and count back a third one and I am at eight. In this way I am working with the number words.
The progression of increasing sophistication in the use of counting strategies to solve a range of addition and subtraction problems is summarised in the following table.