Counting Many people take counting for granted. They think it is merely reciting of a list of words in order and once a child can recite the number names in order he can count. However there are many principles inherent in counting, such as : The number tags used for counting must be in a repeatable order so that the same list can be used for any collection The shape, size or colour of items does not affect the list of number tags used for counting The one-to-one principle: There should be as many number tags as the number of items being counted The cardinal principle: The last tag in the list represents the number of items in the collection being counted. Order of counting: Order of counting is irrelevant. You get the same answer no matter which order you count in. These are some of the principles children need to understand before they can fully master counting in its full meaning. We do not teach these as rules neither do we follow these principles one by one when we use the act of counting. These are implicit principles which are grasped and assimilated by children as they develop their counting skills. Observing little children counting shows how difficult these rules can prove to be to a beginner and how children at different stages fail to use some of these principles. Some samples of children counting are shown here. (You will need Quicktime 3 to view these movies).   Task 1 Jane can count from 1 to 12 confidently. CLICK HERE TO SEE HOW JANE COUNTS   Task 2 Now 10 items are placed in a circle and Jane is asked to count them. CLICK HERE TO SEE HOW JANE COUNTS It proves harder than we thought. Of course with experience they learn to overcome this difficulty. CLICK HERE TO SEE JANE'S NEXT ATTEMPT   Task 3 John too could count the items that were shown to him. CLICK HERE TO SEE HOW JOHN COUNTS John has no difficulty in counting the seven items shown to him. But has he grasped all the principles of counting? John is probed further to test his understanding of the task. CLICK HERE TO SEE JOHN'S RESPONSE John counts the items without realising that he can find out the number of items in the collection. He seems to attach number tags to items on a one-to-one basis. Here are some further tasks given to John.   Task 4 Now six items are placed in a row and John is asked to count starting at the YELLOW LADYBIRD. CLICK HERE TO SEE HOW JOHN COUNTS  Notice how John is not sure how to do this. He finally goes 1, 2, 3, 4, 5 starting at the yellow ladybird and leaving the item to the left of the yellow ladybird. John still attaches number tags one-to-one to the items from where he started because the cardinal number holds no meaning for him. So leaving out the first item is of no significance to John.   Task 5 John is now asked to count the same objects again making the green ladybird number 5. CLICK HERE TO SEE HOW JOHN COUNTS John is unable to do this at this stage. He is confused by this fairly complex task and actually assigns two different number tags to the green ladybird. Without grasp of the cardinal principle, it does not make a difference to John. Notice that the one-to-one principle also falls out once the order is changed. Tracey who has mastered the principles of counting is assigned the same task and has no difficulty in counting as instructed. CLICK HERE TO SEE HOW TRACEY COUNTS   Task 6 Lesley who is nearly three knows number names as well but see how he counts. CLICK HERE TO SEE HOW LESLEY COUNTS Lesley is at a stage when number names do not hold a meaning to him. He merely assigns names to items.   Task 7 Allen who is two recites number names without assigning them to any item or attempting to count items. CLICK HERE TO SEE HOW ALLEN COUNTS Here is very young Adrian who is learning number names. He tries to count by recalling the number names and assigning them to objects. He wants to know more numbers which he has forgotten and goes "what that name?" until he is told the next number. CLICK HERE TO SEE HOW ADRIAN COUNTS These examples show that children go through varying stages before assimilating the full meaning of counting. Some recite number names at first without attaching any meaning to them. Others attach names to items but do not realise that names should have a one-to-one correspondence with the items in a collection. Even when they understand the one-to-one principle, they may not realise that the last number in their list gives the number of items counted. Thus counting is a complex task and many of the components we take for granted are in fact acquired with practice, experience and over time.