How much ash in eleven pits?

This is the third consecutive day of our visit to the brick kiln. By now, we are quite familiar with the work that goes on here. It is back-breaking work! The workers get up around 2 AM and start molding the bricks. By the time we reach there around 7.30 AM, they are busy wrapping up the clay work.

When Kishor and I reached the kiln today, we asked Rahul to call the other children. But no one turned up, except Amit. So we started teaching just the two of them. I asked them, “How many clay pits are here in all?” They had never counted the pits. Rahul went running around the kiln and came back with a number – eleven pits on this side of the road. Rahul had told me yesterday, that they put four ghamelas (small round metallic tubs) worth of ash in each pit. So I asked him, “If we have to put four ghamelas of ash in all the eleven pits, how much ash would be required?” Rahul said, “Mopaay lagel…(मोपाय लागंल) I will have to count it.” He went around counting and came back with the answer, “38 ghamelas”. Obviously, he made a mistake. I asked him to explain how he had counted it. This is how our dialogue went:

Rahul: Four and four, eight. Then eight and eight, sixteen. Sixteen and four, twenty.

I: Twenty ghamelas for how many pits?

Rahul: Five.

I: Now imagine, there are five pits on this side of the road and five on that side, and if you have to put four ghamelas of ash in all of them, how many ghamelas would you need? ( I asked this so that Rahul could see all the ten pits in the line of sight from where we stood.)

Rahul: Twenty for this side, and twenty for that side… forty?

I: Correct! Forty ghamelas in how many pits?

Rahul: Umm… ten.

I: But how many pits did you count on this side of the road?

Rahul: Eleven.

I: Forty ghamelas for ten pits, so how many for eleven pits?

Rahul: Umm… forty four.

Kishor and I were happy to see how Rahul had solved this problem. We realized that Rahul is able to count in fours. The problem was closely linked to his reality, hence Rahul was able to visualize and calculate, without a paper and pencil. Later, we asked, “If you put 15 ghamelas of Raabit (powder of unused bricks) in one pit, how much would you require for 11 pits?”. This number much larger than 4. Rahul and Amit drew pictures of pits on the sand. They even drew the channels which connect the pits, for passage of water. Then, they solved the problem!

Rahul cannot recite multiplication tables yet. But he is able to work with numbers if the problem is connected to his surroundings. Now, the challenge for us is to help Rahul progress from his somewhat flexible strategy to the abstract standard algorithm of multiplication. We have read research papers which state that children use their own flexible methods to solve mathematical problems. We are now seeing it for real. Kishor and I have worked out a plan for Rahul. Let us see how he responds to it.