Simple Counting 1

Counting is a Critical Math Skill

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. This skill of “subitizing” is the first mathematical skill to be acquired by children.

But when the quantity of a large collection is to be determined, we need to “count” it. Counting is the second mathematical skill which needs to be mastered by children.

What exactly is counting?

If we look closely at the process of counting, we will see that it is made up of several physical processes & skills and mental concepts.

1. One-to-one correspondence

   1. We start counting by calling out number names starting with one.

   2. Each number word, starting from “one”, is matched with one (& only one) of the objects to be counted.

   3. This is called the principle of “one-to-one correspondence”. Each number word should be matched with an object and each object should be matched with a number word.

1. Counting in a particular Sequence

   1. If there are several objects to be counted, decisions of where we start from, which path we take and where we end counting is completely in our control.

   2. We count in a certain (mental) sequence in a manner that that “one-to-one correspondence” is maintained.

   3. In simple terms it means that all objects should be counted only once and no object should be missed out.

   4. This mental sequence can be changed, every time we count, depending on our convenience.

2. Use of Ordinal Numbers

   1. While counting we mentally arrange the objects in a particular sequence or order.

   2. We make this order explicit by using numbers in the ordinal sense.

   3. Though we say one-two-three… what we really mean is “first-second-third…”.

3. Cardinality/ Numerosity – If we start with 1, then the last number used to count a set gives the cardinality of the set being counted.

   1. Hence, even though the counting numbers are ordinal, the final number is ordinal as well as cardinal.

   2. If we end counting by saying “twenty-one”, what we intend to say is “twenty first”.

   3. It also means that the set we have counted has twenty-one objects

   4. It means the cardinality/ numerosity of the set is twenty-one.

4. Invariance

   1. The order of counting does not change the cardinality/ numerosity of the set

   2. Even if we start from a different object, take a different path and end at a different object, the last number to be counted will remain the same.

Abstract Concepts Absorbed Through Counting

Though counting looks like a physical activity, there are many abstract concepts about counting and number which children should absorb through counting.

1. Abstract Concepts about Counting

   1. Any collection of objects/ events which are clearly distinct from one another can be counted. We cannot count clouds or a stream of water as the particles they are made of have no distinct identities.

   2. Counting depends on the application of the idea of one-to-one correspondence

   3. The order of counting does not change the cardinality of a set.

   4. The arrangement of objects does not change the cardinality of a set. They could be arranged in a line or in a random fashion. The arrangement of objects could be "spread out" or "densely packed".

   5. This idea, credited to Piaget, is called “Principle of Conservation of Number”.

   6. This leads to the idea that every distinct set has a definite cardinality. Cardinality is denoted by a "countable number". It is a property of a distinct set.

   7. Cardinality of a set is independent of the person who is counting or the way in which s/he is counting.

   8. Two completely different sets, with different kinds of objects, can have the same cardinality. One set could be that of toys and the other could be of vegetables.

   9. Counting uses numbers in an ordinal sense. Though we use the words one, two, three etc. what we really mean in first, second, third etc.

2. Abstract nature of number

   1. A number has an identity of its own, not dependent on what is counted or how counting is done.

   2. A number can be thought of as a property common to all sets with a certain cardinality. We call the common property of a set of eyes, ears and bird’s legs as “two”

   3. A number is the numerosity of a set.

Children who practice counting "suddenly" wake up to the above aspects of counting sometime during their practice.

The power of counting and the idea of a number, fills them with wonder. It is like a revelation to them.

They start enjoying counting anything & everything that they come across. This is a common experience for parents.