A Class 4 teacher was going to teach her class fractions. At the beginning of the term she asked the children, "If you had three chocolates, and wanted to divide them equally among five people, how would you do it?" Most of the children could think of one or more ways of doing this. By the end of the term, when the children had been taught how to deal with fractions, the teacher again asked them the same question. And this time, most of the children couldn't do it! Instead of reality and their own common sense, they now had "rules", which they could never understand or remember how to apply.

 Reversibility: the principle that action taken on objects, if reversed in sequence, will return the object to its original state.

 Conservation: the principle that quantity (number, mass, liquid) remains the same regardless of the spatial shape it may assume.

This example reminds us that only supplying readymade rules to children, without explaining why the rules work, usually blocks their thinking. Often, if children are encouraged to see patterns themselves, they find it easy to accept the formal rules of arithmetic that you may be trying to teach them.

Coming to symbols, various experimental studies show that even children as old as 9 have difficulty in representing the operations of addition and subtraction (+ and - signs). Most primary schoolchildren are uncomfortable with the conventional operator signs of arithmetic. This is because symbols (and algorithms, etc.) are taught in a way that makes no sense to the children, as they are not related to the children's reality. Therefore, the mechanics of dealing with the symbols, etc., doesn't interest them.

What can we do to help our learners acquire abstract concepts? To begin with, we mast remember that no amount of explanation will enable any of us to relate an unfamiliar symbol system with reality. We must go the other way, that is, from concrete examples to the symbol system. Relating abstract concepts and symbols to the everyday experiences of our learners seems to be the easiest way to learn/teach them. Indeed, we all learn this way. Or don't we? Try the following activity and judge for yourself.

E1) Add 4 and 5 in base 5. What processes did you follow in making sense of this task? What difficulties did you face? Do you think the task of a learner beginning to learn mathematics is more or less difficult than this?

While doing this exercise, how much of the difficulty that you faced was because you felt that you didn't have enough previous knowledge to do the task? The point brought out by this question is important, namely, the readiness of the learner to comprehend a particular concept or to do a particular task. For example, Class 2 children cannot completely grasp the idea that the digit 2 in 26 means 20, even though they can write and recognise 26 and can also identify it as a number smaller than 62. But the teachers often assume that the children have understood the concept of place value, and force them to start solving problems with "large" numbers by using standard algorithms. This is of no pedagogic value. In fact, teaching children strategies and methods of solving problems that they are not ready for stops them from thinking, simply because they get preoccupied with the mechanical task of arriving at an answer.

What we have discussed so far also adds weight to the following observation of child psychologists.