The two children were absorbed in making pictures using colourful geometrical pieces. Anisha made a big elephant and Aman a big flowerpot. They finished and counted 28 pieces in both. They seemed happy with their work.
“What if you have to make the picture bigger and have 50 pieces in it? Then how many more pieces would you need?” Both children understood the question. Aman took out his fingers and started counting, saying 28, 29, 30, 31…, but had difficulty in keeping track and paused. Anisha also started with her fingers, but gave up and thought for a while and gave 32 as the answer instead of 22. She had subtracted 20 from 50 and eight from 10, and added the two results.
Their responses would not have been surprising if they were Class 2 students. But they were in Classes 6 and 7. The fact is that these children would have got the correct answer if they had been given two bare numbers written one below the other to find the answer.
It is natural to count or move ahead to find the gap or difference between two numbers. These children tried to do it using their fingers. When children get sufficient opportunity to do counting in meaningful contexts, they start seeing the landmarks of our number system. They might mentally add two to reach 30 and move ahead to 40 and 50 to find the difference as 22. Some might also make a jump of 10 to reach 38 and then 48 and then add two. We see very young children using different strategies when they have had the opportunity to start seeing number relationships through counting experiences without being directly taught. It is as if a a mental number line gets established in the mind which supports visualisation and problem solving.
But alas, most children are taught the standard digit-based algorithm in Class 2, if not earlier. Numbers are split into digits and children are taught to follow the rule of starting adding from right to left. Sometimes “activities” are done to explain the logic of carry-over and borrowing. But in fact, this method goes against the natural tendency for us to add the bigger numbers first, rather than the smaller numbers. Thus for example, when adding 358 and 275, we would add by saying 300 and 200 is 500, rather than saying eight and five is 13. But this is the rule that children are being taught. In fact, mathematics teaching has become an exercise to learn rules and procedures with little space for sense-making activities and developing own solution methods.
One might wonder why there should be a problem with this method since we have all learnt in this way. When a child is adding five and eight to write three and carry over one, the child is not thinking in terms of “five 10s” and “eight 10s”, but merely as five and eight, just two digits. If asked what this ‘one’ in carry-over means, there is a rare child who can say that it is “10 10s” and therefore a 100! This should not surprise us since “five 10s” means “five times 10” and involves an understanding of multiplication. This is too much to expect from a child who is just learning numbers and their additive relationships.
In fact, the current standard procedures for the four operations which children are being taught are something which makes sense to us adults. The problem is not only that these procedures are not based on what makes sense to children but also that they have deleterious effects on the further learning of mathematics. Since the 1980s, many researchers have been pointing out the “harmful effects” of teaching the standard algorithms to children before they are ready.
The standard digit-based algorithm continues to be taught in schools in our country with apparently little thought on the negative effects it can have on the thinking processes of children. This practice gets further aggravated through the work of various agencies which use more and more sophisticated statistical tools for evaluation, but where the heart of the matter continues to be alien to children and to mathematics. Perhaps, the single step of postponing the standard algorithm till the time children have learnt to think multiplicatively can help in dealing with maths phobia! It could then free teachers to attend to children and their thinking processes.
The author has been a member of various expert and advisory committees of the NCERT and the NCTE for early mathematics