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Though counting starts as a simple idea, in real-life situations it can get complicated requiring sophisticated methods of counting. We saw that even simple counting is actually a sophisticated skill consisting of 3 separate steps.
We will first see how the steps in simple counting themselves can be made more complex and sophisticated.
The sophistication of the second step can be varied to develop the ability to think visually & logically.
Firstly physical touching the items to be counted can be discouraged. This means the one-to-one correspondence has to be done mentally and the sequence remembered. The process can be started with a small number and then increased as the student masters the "mental" counting process.
A set of tokens can be spread randomly on a table and the student asked to find the total without touching the tokens.
Next step can be to count a distinct subset of a given set. For example counting the "red" tokens from a set of "red & green" tokens spread randomly on the table.
Need for a Logical Counting Strategy
Next step can be counting embedded shapes. For example, counting the number of squares in the shape given below.
The figure contains squares of 3 different sizes. To count correctly, a logical procedure would be helpful to avoid double counting or missing a square. The different sized squares can be counted one after the other, for example 3 X 3 then 2 X 2 & then 1 X 1.
While counting the 2 X 2 squares, a high level of visual discrimination would be required to count all possible squares at the same time avoiding double counting.
The result can be captured as a formula connected to the size of the square (here 3) and extended to square of any dimension.
This is an example to abstracting a pattern from a given result, demonstrating the power of mathematical thinking.
Permutations & Combinations
From these basic ideas, the idea of counting has developed into the topics of permutations & combinations and combinatorics.
Permutations is finding the number of ways in which a given set of "things" can be arranged, subject to specified restrictions.
The idea of permutations can be captured by this problem. Using numbers 1, 2 & 3, how many different 3-digit numbers can be formed?
How many numbers can be formed if a restriction that "no digits should be repeated in a number" is imposed?
Pingala & Fibonacci
One of the earliest contributions to this science was the study of combinations of various "verse meters" in the hymns of the Vedas by Pingala in his Chandas-Sutra written in 2nd century BC.
In Sanskrit poetry, there are long and short syllables. Long is twice as long as short.
If you want to work out how many there are that take a length of time of three, you can have short, short, short, or long, short, or short, long. There are three ways to make three.
There are five ways to make a length-four phrase. And there are eight ways to make a length-five phrase.
This sequence is one where every term is the sum of the previous two. You exactly reproduce what we nowadays call the Fibonacci sequence. But this was centuries before Fibonacci.
Combinatorics
The idea of complex counting also led to a new branch of math called Combinatorics, which is an important branch of modern mathematics. In simple words, it is the math of counting things.
Counting also leads to the topic of probability.
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