Pre-Number Concepts 3

The activities in the previous chapter enable children to see objects as part of various collections and leads to the idea of sets. Let us now see the development of concepts related to sets.

Concepts Related to Sets

1. Cardinality - looking at the number (quantity) of things and not the sizes of individual things

2. Perceptual Numbers - Ability to Perceive & show (by fingers) numbers up to 5 by sight (without counting)

3. Visual Comparison of small sets or sets widely different in their cardinality (More, Less, Same as)

4. Comparisons using vocabulary - More Than, Less Than or Equal To

5. Comparing using One-to-One Correspondence

6. Seriation - Arranging sets in order - increasing or decreasing cardinality

Cardinality - By seeing collections of different sizes, the idea emerges that "quantity" can also be seen as a property of sets. A collection of "four" triangles, "four" pencils, "four" tokens can be seen to have a common property which can be called "fourness".

An important experience for children is to realise the following when comparing a handful of seeds with a jackfruit. The jackfruit is "more in size" than the seeds but the seeds are "more in numerosity" compared to the jackfruit!

Once cardinality can be seen as a property of a set, then sets can be compared by their cardinality.

Numbers with Fingers - A child develops the ability (without counting, just by sight) to recognize numbers by the number of fingers shown or show a number with appropriate number of fingers. This is a very important skill which can develop into complex representations of numbers with fingers.

Visual Comparison - Collections which are significantly different in their cardinality can be compared just by sight. Even a small child can pick out a larger collection of sweets without counting. It leads to the idea of greater/ smaller.

One-To-One Correspondence- Small sets can be compared by sight. But larger sets need the idea of One-To-One correspondence for comparison. For example, assume that we have a collection of cups & spoons. We can put one spoon each in a cup. If spoons are over and there are still empty cups, then it means that the collection of cups is more. On the other hand if all the cups are filled and there are more spoons left then the collection of spoons is more than that of cups.

This is a very powerful idea of comparing sets without finding out their individual quantities. In higher math, different kinds of infinities are compared using this idea. Georg Cantor proved that the infinity of counting numbers (1, 2, 3 & so on and rational numbers) is smaller than the infinity of irrational numbers by using the one-to-one correspondence idea.

Counting is a process of matching a collection of things with the names of the number series (one, two, three etc) using the principle of "one-to-one-correspondence".

More Than/ Less Than/ Equal To OR Bigger than/ Smaller than- Children need to understand and use such vocabulary to express results of comparing 2 sets.

Seriation- is arranging a collection of things in increasing or decreasing order of their quantities. It is the extension of comparison of 2 objects using their physical characteristics. Seriation is ordering a given set of collections either in the increasing (ascending) or decreasing (descending) order of cardinality.

Seriation gives an insight as to the increasing cardinality of numbers. It provides the reason why numbers are arranged in the sequence 1,2,3,4.... etc. Every subsequent number is "1 more" than the previous number!

It also gives an insight that there is no "biggest number". Given any number we can always generate a number which is more than it by 1.

This further leads to the insight that there are an infinity of numbers.

Seriation enables the understanding of logic behind "ordinal numbers" - the meaning of the words first, second, third etc.

It also shows that there is a clear logic in the way numbers (1,2,3...) are arranged. This shows the deep difference between numbers and letters of the alphabet in any language.

There is no logic in the way the letters of the alphabet (a, b, c etc) are arranged. It has been derived by convention.

Learning the above Concepts & Skills

Teachers need to plan activities (refer Chapter 4.5) for checking the understanding of all the above concepts & abilities. They are not listed chronologically in the manner in which they develop. They are all inter-connected and reinforce one another. Hence, they have to be taught, repeated and assessed at frequent intervals.

If a child has difficulty in understanding a concept, teachers need to first check if the difficulty is due to language. They need to give the instructions for an activity in a language the child can understand. It is also better to use activity materials with which the child is familiar because of his home experiences.

The vocabulary used for learning these concepts; One, Many, More, Less, Share etc may also be unfamiliar to the children. However, they need to be remembered. Hence the teacher should deliberately use these words, in the appropriate context, many times and ask children to use them many times, so they would be "remembered by repeated use".

The normal sequence for learning anything is Observing, Listening, Introspecting, Speaking, Reading and Writing.

Applying it to the process of learning numbers, we have the following steps.

1. The child should observe an activity/process,

2. Listen to the teacher explaining that process and the related number names,

3. Understand what is happening, if necessary by thinking and introspecting

4. Ask for clarifications or express opinions by speaking

5. Identify (read) numbers associated with that process in any representation and only after that

6. Write a number in numeral form.