Mathematics is good for the soul, getting things right enlivens a sense of truth, efforts to understand automatically purifies desires.
– Iris Murdoch
I am Rupesh Gesota, an engineer-turned-Maths-educator. I love to see the sparkles of understanding in the eyes of my students, and know that I was part of that process. I enjoy doing Maths with children and sharing that passion with parents and teachers.
After rather successfully ‘failing’, with a spate of private schools and children from rich families, I now work closely with a group of middle-school students from municipal schools, who hail from challenged socio-economic backgrounds. It has been a thrilling experience to re-learn with them, helped by the “safe” space in which we explore Maths together. We solve puzzles, play games, watch videos, and read books that take us on journeys beyond the syllabus. We believe that Maths is not about speed, but depth, and we learn mathematical concepts primarily through interesting problems. Students are first encouraged to think of solutions independently, based on their previous knowledge, and then to share their approach with partners – followed by a collective, whole class discussion that might lead to the evolution of a new technique or rule.
The class is centred on “Why?”, where the views of peers are heard, and challenged when necessary on logical grounds or through counter-examples. Students are motivated to use any approach that makes sense to them, but they must know – or know how to find out – the reason behind any rule or formula used.
A few examples:
Simplify: -6 – (-7)
Usually, students would solve this by opening the brackets, using few rules i.e. they would solve in the following way:
– 6 – (-7)
= – 6 + 7 …… minus x minus = plus
= + 1 ….if one number is positive & other is negative, then perform subtraction with sign of the answer same as the bigger number.
But students in our class could finally solve this problem in these two ways:
a) -6 – (-7) = (?) This is same as (-7) + (?) = (-6). Now it’s easy to see that +1 must be added to (-7) to give a lower negative number (-6).
b) (-6) and (-7) are consecutive numbers. So their difference is +1 or -1. In this case, the smaller number is subtracted from bigger number. So, their difference should be +1.
However, these two ways of solving the problem were not arrived at promptly. In fact, the students’ first guess was -1.
Once they had solved the problem correctly, an inquiry was carried out to understand their reasoning processes and mathematical insights.
Solve 3/8 ÷ 1/2 (three eighths divided by one half)
Again, students are generally taught the “rule” for dividing fractions, which is to flip the second fraction and multiply it with the first fraction.
This procedure gives the correct answer, but students do not have an understanding of why they are doing what they are doing. Further, they can neither estimate nor justify the answer obtained. But when the same problem was posed to a group of students who had been taught to understand division and fractions, then this was what happened.
Two different solutions were offered. Some said it was 3/4, while others said it was 3/16. I was certain that none of them had used the “flip and multiply” rule.
Why did some students think the answer was 3/16? They had calculated half of 3/8, which is 3/16. When the other group was asked for their rationale for arriving at 3/4, they explained it thus:
The problem required them to find out the number of halves in 3/8. They first calculated the number of halves in 1/8, by converting the division problem into one of multiplication — how many times half would make 1/8?
The answer was 1/4, because 1/4 x 1/2 = 1/8.
To get to 1/4, they reasoned that
1/2 x 1 = 1/2 (half taken one time gives half)
So 1/2 x 1/2 = 1/4 (half taken half times or half of half)
So 1/2 x 1/4 = 1/8 (half taken quarter times, or a quarter of half)
So now, if there are 1/4 halves in 1/8, how many halves would be there in 3/8?
Since 3/8 is 3 x 1/8, it would have 3 times as many halves, or 3 x 1/4 = 3/4
It could be argued that this is a longer method compared to the flip and multiply approach. That may well be so, but if understanding the concept is considered more important than rote-learning a procedure, then the former is definitely the winner. Further, if we look at it closely and carefully, we can “see” the flip and multiply happening in the former method as well.
Recently, I gave a puzzle to a group of 6th std students that was basically a set of three simultaneous equations. They solved the puzzle in a manner that might have been a little difficult for a conventional Maths teacher to arrive at. The students’ approach is detailed here:
It was not easy at first to inculcate such different approaches to doing and learning Maths, because their Maths classes in school had focused only on the conventional, quick answer-getting procedures, but the attempt to transform an instruction-driven classroom into an inquiry-driven learning space is well worth the effort.
How can students be guided towards discovery by asking the right questions, so that they are able to discover and correct mistakes on their own? Some answers may be found in the various stories and Maths conversations recorded in my blog.
Blog & photos: Rupesh Gesota, Maths Educator, Mumbai
Translation and editing: samata.shiksha team