Simple Counting 2

Once teachers understand these aspects of counting, they can watch children practicing counting and ask appropriate questions to nudge & help as well as expand their thinking.

Common Counting Mistakes Done by Children

Children make three mistakes in the counting process. In each of the cases, the “quantity” as counted out would be different from the actual quantity.

1. Calling out the sequence of numbers wrongly. Eg. Saying One, Two, Four, Five…… This is a mistake younger children do if they cannot recall the number sequence correctly. Regular recitation of the number name sequence would avoid this mistake.

2. Calling out the sequence too fast before associating it with a member of the collection. Here some of the numbers are not associated with an object in the collection. E.g we touch an object when we say “six” and then say “seven” quickly and say “eight” while touching the next object. This will result in the “counted” quantity being more than the actual.

   1. After some experience, physical touching can be replaced with focused "mental touching" so that the reciting of the number sequence in in synch with the one-to-one-correspondence process

3. Leaving out some objects OR touching some object more than once. This usually happens when the child does not keep track of the touched items i.e those which have already been (touched) counted. This can happen even to adults if the objects to be counted as spread out in a haphazard way.

Avoiding Such Mistakes

Normally in schools, teachers spend a lot of time in training children in step 1, which can be called reciting the number sequence. At this stage, children recite the number sequence like a song that needs to be memorised. They do not necessarily understand the purpose of counting.

Children should also be given a lot of practice in steps 2 and 3 in counting fairly large collections. The collections should both be ordered (like in a string of beads) or spread out randomly as tokens on a table. These would help children to adopt strategies to avoid mistakes detailed in points 2 & 3 above.

Insisting on touching an object and then calling out a number name can rectify the 2nd type of mistake.

Separating any object which has been touched, away from the original collection, into a different pile, will avoid the 3rdtype of mistake.

Number Sense Precedes Counting

Because children are always asked to count to find the quantity of a set, teachers normally assume that Number Sense comes out of the counting process. But if we think a little deeply, we would realise that it is the other way around.

Where does the sequence one, two, three … itself come from? In what way is it different from the sequence of the English alphabet a,b,c……?

In the sequence a,b,c …. there is no logic. It is just a man-made convention, generally accepted by the English-speaking population.

But the sequence one, two, three … has an inherent logic.

It is the logic that each of these words represents a quantity and the next number always represents a quantity which is one more than the previous one. That is One, Two, Three … represent collections that are in ascending order with each being One more than the previous.

It is our Number Sense which helps us associate each number word with a quantity. Hence we can see that Number Sense leads to a process of counting. Counting reinforces number sense.

We can summarise this process as follows.

Small Collections -> Subitizing -> Number Sense -> Perceptual Numbers -> One More Concept -> Natural Number Sequence -> Counting -> One to One Correspondence- >Cardinality of Large Collections.

Pre-Number Skills & Counting

In order to master counting a firm grasp of some pre-number skills are necessary.

1. Classification & Grouping

   1. As they count, children need to physically or mentally classify things continuously as "already counted" and "yet to be counted. This is an abstract skill using an abstract concept.

2. Seriation

   1. Before starting counting, children need to have a physical or mental map of the sequence of counting. The sequence should be easy enough to follow and avoid short counting or double-counting.

Training Children in Counting

Children can be trained in simple counting by giving them strings of beads, where the teacher already knows the number of beads.

When the child comes back to the teacher with the answer, the teacher can immediately see if it is correct or not, without having to count herself. In case of an initial mistake, the child can be given 2 or more chances to count again. If the mistake persists, then a string with a smaller number of beads can be given. In this way, the exact nature of the mistakes committed by the child, in counting, can be identified and corrective remedial measures taken.

In the next chapter we would see some counting tasks which can be called "complex".

Counting Sequence and the Brain

Recent studies are also pointing to the fact that the human brain seems to have a normal tendency to associate an increasing sequence of numbers from left to right.

This could also be the reason why in most cultures we read text from left to right.

This also could be the reason for the universal adoption of a number line where positive numbers proceed from the left to the right.

Fingers & Counting

For a long time in counting with fingers, the focus has been on the math aspect. How to count a certain number?

But recently there have been cognitive scientific studies which seem to point out that different cultures use fingers in different sequences to count. Some start with the thumb and others start with the little fingers.

The reasons for these differences are yet to be studied. They may reveal a fundamental difference in which different cultures understood the idea of a number!