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
Neat ! So Aparna is not only a alphabet scientist but also a number scientist !! It is so very wonderful they get to be all different kind of scientists they really are by birth...What a wonderful story Hema..Thanks for sharing..
ReplyDeleteReally nice to hear this Hema... Innovative thinking there...
ReplyDeleteThis is a wonderful story hema. Children are natural problem solvers and every child has different strategies for solving any problem (not just math problems). When we ask them to focus on only one algorithmic strategy for problem solving, we do them a disservice. And instead of developing a flexible relationship with numbers children become afraid of math! Btw, you may want to explore the work done by a group called cognitively guided instruction from madison. They basically suggest that anyone who works with children on mathematics would be better allowing and facilitating children to develop and fine tune their strategies. I will be happy to share some resources I have from them. Polya of course is a classic.
ReplyDeleteHi there!
DeleteThanks for your comment. I would like to know about the resources you have mentioned above.
Thanks so very much!
Nice, and I agree with the analysis. I too remember starting to split numbers and do them in the same way you've described, when I was growing. It came about from a lot of time staring aimlessly at number charts, finding patterns, or messing with the calculator. I guess I hated memorizing so much that I was constantly looking for ways to compute from something simple.
ReplyDeleteI suggest to introduce the girl to logarithms, when the right occasion comes up (say, news of an earthquake in the world).
ReplyDeleteSlowly, adapting the example given for division(like doing 30/6 by finding how many "6" added together are in 30), then the next step is: "how many 10 multiplied together are in 100, or 1000? (or how many 2 multiplied are in 8?)".
Next time there is an earthquake in the world (force 4, 5, 6...), she will be able to understand the difference between one with force 5 and one with force 6. It is fantastic for the child understand something that most adult people understand vaguely ("hey, I heard that a 6 quake is a hell lot stronger than a 5").
And it is all so simple, almost... childish.