In the Common Core State Standards for Math, counting-on is considered “a strategy for finding the number of objects in a group without having to count every member of the group.” Counting-on is an efficient way to add and we want children to count-on. Yet, many young children begin by counting-all.
Teacher [placing cards with the values 8 & 4 on the table]: What would the total be?
Brittany [extending her fingers while counting out 1-2-3-4-5-6-7-8, then 9-10, then reuses two fingers to count 11-12]: 12!
The important questions are: why do children count from one and what can teachers do to help them count-on? Counting-on would seem a simple concept for kids to grasp. I’ve heard several times that all we need is for kids to put one number, usually the “big” number, in their head and count-on from there. But there is a problem. Children who have not constructed an understanding of number need to count-all in order to construct their meaning for a given number. As shown above, when Brittany saw the numeral 8, she counted from one to eight. This was necessary for her to give eight meaning. Once there, she could continue her count to determine the total. Dr. Les Steffe, in his article PSSM From a Constructivist Perspective (2001), states, “The major advance the child makes in constructing counting-on is that the child can use its counting scheme to generate an experience of counting without actually counting” (p. 229). The child moves from counting-all to counting-on. From her experiences of counting, Brittany would eventually shed her need to count from one, but she was not there yet.
During a different teaching session, we presented Brittany with problems in which we systematically increased the first addend while keeping the second addend constant, such as 7 & 6, 8 & 6, 13 & 6, etc. An interesting change occurred in Brittany’s thinking. Initially, she started counting-all, but then she adjusted her behavior. She began simultaneously extending fingers, such as nine fingers with 9 & 6, then she continued counting extending six more fingers. Later she changed again and began just saying the first addend and then counting-on. We found for the remainder of the teaching session, counting-on was the strategy she used. The following week she began by counting-all once again, but after only a few problems she changed her strategy to counting-on. She could now reflect on her previous counting experiences and not have to count-all. She was not taught to count-on but developed her understanding of number through counting experiences and counting-on was her construct.
When telling children to put the “big” number in their head and count-on from there we rob the child of their counting experiences. It gives them a prescribed method for arriving at a solution but undermines their construction of number. In fact, Steffe (2001) says that counting-on is a non-teachable scheme (p. 233). This does not mean the child will develop it in isolation, however. Rather, it is the responsibility of the teacher to provide situations and opportunities to develop counting-on.