There are various levels of mathematical sophistication in the ways children use counting and it is important for teachers to have some understanding of these. Teachers who have a good knowledge of these are better able to develop instructional approaches focused on developing more sophisticated arithmetical strategies in their children.
In Number Words and Numerals we did not use the word counting to describe the process. It was referred to as FNWS. This is because we find it useful to distinguish between, on the one hand, the activity of merely reciting the sequence of number words and, on the other hand, the activity of saying the words in a context where the child is purposefully keeping tract of items.
Refers to situations in which the items to be counted are available for the child to see. Counting that is not perceptual counting refers to situations in which the child knows how many counters are in a small collections that is screened.
Perceptual counting is where students can see the objects referred to. More advanced levels of counting occur when children count to solve problems involving items which are not seen. If items were screened in two different groups and briefly displayed and then the child asked to determine the total. The adult prospective this is trivial but from the child it is problematic. The arithmetical idea of addition is currently somewhat remote for the child. This is an additive task involving two screened collections of counters.
In counting a child coordinated each number word with an item. The perceptual, figurative and counting-on child coordinate number words with items. The difference is that the perceptual counter is directly perceiving(seeing) the items, whereas the figurative counter and the counting-on child imagine or visualize the items to be counted. For this reason the figurative and counting child is regarded as cognitively and arithmetically more advanced than perceptual counting.
There are three areas of focus in this section: Emergent, Perceptual and Figurative Counting. The term 'emergent' is used in cases where the child is not able to count perceptually. Thus the child is unable to count a collection of say 12 or 16 counters. They might not know the number word sequence or might not correctly coordinate each number word with tn item to be counted.
It is not unusual for children to use a level of counting less advanced than they are capable of. It is our task to elicit the most advanced kind of counting that the child can use spontaneously and with certitude. Only classify a child as perceptual if you are certain that they are not able of figurative counting or counting-on to solve a task. Typically, it is necessary to present the child with several tasks rather than just one, in order to elicit the most advanced kind of counting.