Counting-on versus counting-all

As children progress in their counting development they reach a point where they are able to count-on from a given number.
For example, when five year old Jackson’s father says to him:

“Jack, can you see 8 crackers on the table?

How many more would you need to have 13 crackers?”

Jackson solves this problem by counting-on from 8: “nine, ten, eleven, twelve, thirteen”, and then says: “five!”

How is Jackson able to do this? What has happened in his counting development that allows him the flexibility to begin counting-on from eight, and recognize how many more times he counted to get to thirteen?

Jackson’s father notices that as Jackson counts he moves his jaw from side to side. He thinks Jackson counted how many times he moved his jaw as he said “nine, ten, eleven, twelve, thirteen”, and so concluded that 13 is 5 more than 8.

Jackson is at the very beginning stages of learning how to count-on. He uses body movements, coordinated with saying a number that does not begin at “one”. He seems to count his body movements as he does this.

Jackson has to focus his attention very carefully and coordinate two sets of movements: his speech and his jaw movements. He seems to count the movements of his jaw. But he is doing much more than that. He begins saying part of a number sequence, beginning at “nine” – the point he would have been at if he had counted from “one” through “eight”.

This indicates he can extract a number like “eight” from the sequence of counting numbers and begin his counting from there.

Children, like Jackson, who count-on have made a huge advance in counting, They are flexible with regard to where their counting begins.

Children who can count-on will also use counting-on to solve addition problems. For example, a child who can count-on will typically answer a question such as “What is 7 and 4 more?” by counting-on from “seven”: “eight”, “nine”, “ten”, “eleven”.

At first, children count-on to solve addition problems by starting from the first number presented to them: for 3 + 7 they will typically begin by counting-on from 3. Only later, as they have mastered this aspect of counting-on, will they begin by counting-on from the larger number.

It sometimes seems that children who are only just beginning to count-on are still focusing on images of objects to be counted, such as marbles, counters, or jelly babies. However we see with the example of Jackson, and other children who tap their fingers when they count-on, that sometimes children who count-on are focusing mainly on body actions. That is, the children make body actions such as tapping, moving their jaws or eyebrows, or biting their lips, as they count-on.

It may be that they do not have visual or other concrete images in their minds. Rather, they may be focusing on keeping in memory the number of taps or bites they are making.

When a child can count-on, a number such as “eight” has two meanings. First, it is the number in sequence immediately following “seven”.

Second, it is now a trigger for a memory of having counted “one, two, three, four, five, six, seven, eight”.

For a child who can count-on, a number name such as “eight” brings to mind these two sorts of memories. The first is a memory of “eight” as following “seven”. The second is a memory of “eight” as being the last name spoken in a sequence beginning with “one”.

Counting-on is a sure sign of developing flexibility in a child’s counting development. It is not something that can be explicitly taught, as such: a parent or teacher can help a child develop this level of flexible counting through appropriate activities, carried out over a period of time. We will write about what these activities might be in a follow-up post.

Counting-on is such a strong indicator of flexible development in counting that Eddie Gray, then at the University of Warwick in England, described it as “the parting of the ways”. By this, he meant that the divergence in counting ability between children who can count-on, and those who cannot, is so significant that a child’s later number sense can be impacted. Counting-on is a highly significant counting ability, yet it cannot be taught through explicit instruction, so a deeper understanding of how a child develops this flexible ability to count is critical for parents and teachers to understand.

• Read Eddie Gray’s original research.