The second half of Chapter 3 “Principles of Teaching Number” in Kamii’s Number in Preschool and Kindergarten describes “counting” as an emergent skill that can be observed in stages.  Citing research from two studies that looked at children’s counting conducted by Greco (1962) and Meljac (1979) Kamii found that even though children may be able to count in the correct sequence (1,2,3,4,5…) they may not use this skill when presented with a counting task.

If a child is presented with a configuration of 8 coins placed in a random pattern on the table and asked to put the same number of coins out as the model, the child will respond in one of the following 4 ways:

The four levels are:

Level 0: Inability to even understand the adult’s request.

Level I: Rough, visual estimation or copy of the spatial configuration.

Level II: Methodical one-to-one correspondence

Level III: Counting

“Counting does not become a perfectly dependable tool for young children until the age of 6,” reports Kamii.  Children need to be able to 1) say the words in the correct sequence, 2) count objects – make one-to-one correspondence between the words and the objects, and 3) final, choose counting as a dependable and desirable tool to come up with the correct answer.  Before these skills are secure, the child may feel more confident in choosing a different method for coming up with the answer.

If the 8 coins look like this;

The child may place coins so they take up the same amount of space as the ones above. The child is not concerned with “number” here, he is concerned that it looks the same.

This makes perfect sense for a child who is tricked by appearances.  It “looks” the same to the child, so it is the same.

The child may also line up her coins so they match the model exactly.  By doing this, she is using one-to-one correspondence rather than counting as a preferred method of answering the question.

Here, you can see that the child made sure that for each coin, he placed a corresponding coin.  This method is quite appropriate and will result in the correct answer.  It isn’t the most efficient nor does it use “counting” as a tool to arrive at the answer.

Kamii uses these examples to encourage teachers to allow children to choose their preferred method for arriving at the answers.  The goal is that all children eventually choose counting as the most efficient, practical and mathematically appropriate avenue to accomplish a counting task.