conclusion on counting in math

Conclusion on counting in math: Kindergarten 1
(7 children, ages: 4 to 5 years old)

Abstract.
The children in Kindergarten 1 could not tell how many cubes they have after counting them.

Introduction.
Counting objects is different than understanding the concept of quantity. Counting is a repetition of a sequence of digits stored in the long-term memory. Quantity is the ability to conclude how much we have in all. It means that it is a process of addition. For example: the understanding that number three is not only the third position in a reciting set of sequence of digits (Figure 1, Method A) but it represents the addition of 1+1+1 (Figure 1, Method B). In Creative Foundation School we chose method B, which is adequate to most children. However, the children could not understand the meaning of quantity.

Figure 1: Different ways to teach counting.

Task: Counting items and expressing their quantity by mentioning the last number only.

The children learnt math since the last term of Nursery 2. They started by learning how to count 1 to 5 items such as discs, cubes, matchsticks, and beans, which was not an easy task for them. In Kindergarten 1, they learnt to say the total number of items after counting them. The problem was that when they were asked “How many discs do you have?” they could only answer: “One, two, three”. They could not answer “Three” no matter if the teacher put the discs in a line or in a group (Figure 2), how many times they repeated the operation, and even though they learnt the task in their mother tongue.

Figure 2: Counting discs.

By the end of Kindergarten 1 all the pupils could answer the question: How many items they counted by summarizing it in one number, the last number that they have counted. But in all, we had the strong feeling that even though we invested a lot of time in explanations and tried to make the teaching as practical as possible, the children themselves could grasp some knowledge only by reciting.
When a child learns counting, it can be done by a mechanic operation which he or she can memorize. However, this mechanical operation has no understanding because the numbers by themselves do not have a meaning. Number 5 for example is not a total of 5 items but a sequence of digits learnt to be recited in a certain order. In this way, the names of the numbers are stored in the long-term memory without the working memory processing them farther. The set of 5 digits: 1, 2, 3, 4, 5, would be ready to be used only when the child will be ordered to count again and not when the child will be asked how much is 1+2 for example. The children of Kindergarten 1 in Creative Foundation School could not summarized how many items they have counted because this cognitive process involves understanding the concept of addition which is the foundation of quantity, and it could not be grasped by them at that age.
It is also possible that the rote learning of reciting without conceptual understanding caused the surprising forgetfulness we witnessed in the school with the children when they went to Easter vacation for three weeks at the end of the second term of Kindergarten 1. In the first lesson of the third term after the vacation, they even forgot how to count up to five. The teacher was so amazed. She revised with them how to count items again, and from the second lesson all their memories regarding the work they did in math in the second term returned to normal. It is possible that the mechanic operation of counting without understanding the meaning of quantities was not stored properly in the long-term memory but was placed there without any association or connection to something meaningful and therefore it could not be used even to count the items again. The fact that the memories of the children regarding what they leant in math in the previous term returned to normal in the second lesson may point that the knowledge they learnt in math was indeed placed in the long-term memory but not in an organized way where related concepts are connected, and the information is organized in the hierarchical way. Maybe the refreshing knowledge of the first lesson of the term was helping them to reconnect scattered information in the long-term memory and to modify it so it will be useful. In all, it shows that the working memory of these children was weak.