Concrete Operational Stage :  Piaget describes a five-year-old boy playing with a collection of pebbles. First, he laid them in a line and counted along the line from left to right. There were ten. Then he counted them from right to left. To his great astonishment, the total was again ten! He put them in a circle, counting them clockwise, and then anticlockwise.

Full of enthusiasm he found that there were always ten. Whichever way he counted, the number of objects was always the same. He was discovering that the number of objects remains the same even if the way they are placed changes. Slowly the child was ridding himself of his own earlier idea that the number of objects in a set depends on the way they are laid out. He was now ready to conserve. He had also achieved the understanding of the mathematical idea that one conserves quantity even when one partitions a set of things into subsets. Conservation is achieved by a child at the concrete operational stage, which a child passes through approximately between the ages of 6 and 10. It is this understanding that creates a qualitative difference in the concrete operational child's thinking.

Around the age of six or seven, the child can count and compare two sets of objects, and perform the more complex operations of adding and subtracting objects. The numerical operations gradually become internalised, but not at the level of abstraction. 7 to 10-year-olds remain essentially related to physical objects. They can conserve and intuitively grasp many basic ideas of mathematics. But this grasp is only in terms of concrete operations. This is why Piaget refers to this stage as the stage of 'concrete operations'.

Thus, if you give a concrete operational child the choice of solving a problem of addition by adding objects rather than numerals, she would prefer to count objects because she trusts her intuition and concrete experience rather than symbolic operations. For example, take six-year-old Kavita. She worked out the following problem of subtraction:

16 - 31= 25

When asked how she did it, she replied, "1 take away 6 is 5, 3 take away one is 2." When asked if it was right, she said, she did not really know. However, given the problem of '31 take away 16' to do the way she wanted to, she soon came up with the answer, 15. She even exclaimed that this was done by different methods.

Similarly, Amit (8 years) could not do the division problem ' 45 ÷3 ', but found it easy to share 45 sweets between three people.

Children have problems with conventional methods because the formal code is much too abstract to master at this stage. Relating the problem to a concrete real-life experience helps children to rely on their own intuitive understanding, and thus invent a strategy to arrive at a solution.

There is also another reason why children in early primary school have a problem with handling formal arithmetic. In the formal code of arithmetic the operations proceed from right to left, whereas reading in English proceeds from left to right.

Many primary school children continue to make the error in going from left to right while doing arithmetical operations. This indicates the need to develop their spatial thinking ability before and along with other arithmetical abilities.