Simple Counting

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We saw in chapter 3.4 "Perceptual Numbers" that when the quantity of a collection is small, then it can be perceived just by sight. But when the quantity of a large collection is to be determined, we need to "count" it. What exactly is counting?

What Exactly is Counting?

All of us count. But if we look closely at the process of counting, we can see that it is made up of three steps & processes.


 * 1) We orally recite the sequence of numbers one after another, in a particular order, beginning with One (One, Two, Three ... )
 * 2) As we "call out" each number, we mentally or physically associate it with any one member of the collection to be counted. Younger children can be encouraged to touch or point to the object. We then move to the next number and another member of the collection.
 * 3) When we thus reach the "last" member of the collection, we stop calling out. The number that we "call out" last will be the "quantity" or "cardinality" of the collection.

Children make several mistakes in this process, which can be classified into 3 types. In each of the cases, the "quantity" as counted out would be different from the actual quantity.

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.
 * 1) Calling out the sequence of numbers wrongly. Eg. Saying One, Two, Four, Five&hellip;&hellip; This is a mistake younger child 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 &ldquo;six&rdquo; and then say &ldquo;seven&rdquo; quickly and say &ldquo;eight&rdquo; while touching the next object. This will result in the &ldquo;counted&rdquo; quantity being more than the actual.
 * 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.

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 &amp; 3 above.

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

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

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 &hellip; itself come from? In what way is it different from the sequence of the English alphabet a,b,c&hellip;&hellip;? In the sequence a,b,c &hellip;. there is no logic. It is just a man-made convention, generally accepted by the English-speaking population.

But the sequence One, Two, Three &hellip; 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 &hellip; 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. We can summarise this process as follows.

Small Collections -&gt; Number Sense -&gt; Perceptual Numbers -&gt; One More Concept -&gt; Infinite Number Sequence -&gt; Counting -&gt; One to One Correspondence- &gt;Cardinality of Large Collections.

Understanding Counting


 * 1) At every stage the number “spoken out” indicates the cardinality (quantity) of all the “items” counted until that point.
 * 2) The last number “spoken out” indicates the cardinality of the entire set.
 * 3) The cardinality of the set does not depend on the order of counting.
 * 4) The cardinality of a collection does not change whichever way the objects are arranged – spread out, close by, orderly, random etc.
 * 5) The cardinality of a collection does not change, unless there is a change due to external circumstances. This principle, which was identified by Piaget, is known as "conservation of number"

Children can be trained in 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. With this process, the exact nature of the mistakes committed by the child, in counting, can be identified and corrective remedial measures taken.

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.

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