Sunday, September 8, 2013

Reduction and Backtracking



In an earlier post we had shared how easy it was for Abhi to learn to write Hindi. We later realized that that story (and other stories of our children learning) could be read in a way completely different than intended by us. It could be read as a story of an exceptional child, doing something exceptional. Instead, our intention was to illustrate how easy it is for any child to learn to write, provided the child is motivated and the methodology sensible. This post continues in the same vein and we hope it will not be read in as a story of exceptionalism.
Aparna has always been fascinated with numbers and basic arithmetic, as we wrote earlier. Recently we were surprised when she demonstrated some sophisticated mathematical thinking that was not ever taught.
We occasionally go see Dev’s sister in a different part of Chennai and the bus ride is typically long. During one of these rides, Aparna wanted to do division that involved “big” numbers. So I asked her what 39 divided by 3 was. After working on it for a minute, she came up with the correct answer and was eager to share her algorithm with me. Here is the translated version of it:
“Let us set aside 9 and consider only the number 30 for now. So the question becomes ‘what is 30 divided by 3?’ 30 is a big number for me. So I am going to halve it and the question now is ‘what is 15 divided by 3?’ I know that there five 3’s in 15. Now I go back and use this information to answer the previous question – i.e. 30 divided by 3 must be twice as much as 15 divided by 3, which leaves me with the answer 10. Now I bring back the 9 which was left out earlier. I know that there are three 3’s in 9. So I add 3 to 10 and that is it!”

Weeks later I realized that what Aparna described to me, incidentally, is a problem solving strategy called “Decomposing and Recombining” by G. Polya in his famous work How to Solve It.

I believe that the enormous amount of time and freedom with which she has explored numbers on her own has helped Aparna build the number sense that is depicted here.
-- Hema