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I simply had to share a very interesting new approach to multiplication that one of my students came up with today. His name is Arjun, one of the bunch of 8th std. municipal school students with whom I work, voluntarily, for an hour or two after school.
Yesterday, we had stumbled upon a problem where we felt the need for multiplication as an intermediate step. I don’t remember the numbers except that it was a 2-digit number multiplied by a 1-digit number. The calculation was quite simple, yet the students were unable to do it mentally and wanted to use pen and paper. I insisted on mental maths and, after some discussion, the very concept of multiplication began to grow clearer to them. Gradually, they started solving these single and double digit problems mentally, through understanding rather than by using conventional multiplication procedures.
In our own school days perhaps most of us, myself included, would have used the conventional procedure, but probably without any real understanding. I shall not elaborate on understanding-based pedagogical approaches or strategies here, as I assume that most teachers, today, would be familiar with these; besides, an internet search would furnish many details.
As our two-hour session was drawing to a close, the students requested that I give them some on multiplication problems as an assignment, so that they could practice this new understanding-based approach, which had got them fascinated. I gave them about 10 problems.
Today, once our 10-minute maths warm-up was done, we started discussing the overnight assignment. It was fulfilling to see that students had solved the problems in more than one way. They had used both understanding and logic.
Example: to solve 38 x 5, a student had first worked out that 38 x 10 = 380 and then halved that figure to get 190, the required solution. There were many more such beautiful strategies, that left me pleasantly surprised.
Here, however, I specifically want to share the innovative approach discovered by Arjun to solve the problem: 29 x 8.
No doubt, his approach emerged from an understanding of the previous strategies, but the way he solved the problem was nevertheless remarkable.
He said, “Let’s consider (29 x 8) as (30 x 10), which yields the product 300.”
“But we could have also considered 30 x 8 (instead of 30 x 10). 30 x 10 yields us a surplus of 30 x 2 = 60.”
“Let us subtract this 60 from the 300. This which gives us 240, which is nothing but 30 x 8.”
“But we set out to calculate 29 x 8, and not 30 x 8. Thus we have a further surplus of 1 x 8, which is 8.”
“So we need to subtract this 8 from the 240, which gives us 232 – the correct solution to the given problem.”
I want to make an honest confession here. When Arjun described these steps for the first time, verbally and quickly, I could not follow his thinking right away. It was only when his classmate Suvarna joined him and re-framed the approach, a bit more slowly, that I was able to fathom this strategy.
I was simply blown away by the levels of understanding they had crossed. I was so impressed that I challenged them to solve another problem using the same approach, 38 x 7 =?
They did not even take a minute to apply the approach and deliver the solution. Check the picture below and see if you can follow their reasoning.
While they were enjoying themselves, and working on these problems using multiple approaches based on their conceptual understanding, various thoughts flew through my mind. My basic question to the teachers and parents reading this is: should students be encouraged to think independently in this way and to discover their own approaches to solving maths problems?
How can we guide students towards discovery? How do we learn to pose the right questions to them, so that they are able to discover and correct mistakes on their own? I have addressed some of these concerns in the various stories and conversations around maths in my blog.
Blog & photos: Rupesh Gesota, Mathematics Educator, Mumbai
Translation: samata.shiksha team