Math Coach: Demo-class to Teachers (on Fractions)

With children of migrant laborers working in IIT Gandhinagar campus, while a group of teachers sat around for observation (6-7 children of @ 13-15 yrs)

Children were asked – If there are 6 Rotis to be equally distributed among 4 of you, then how much will each get? They thought for a while and answered ‘Dedh’ (means 1 and half) , supported by reasoning. This solution was written on the board in picture and words. Similar representations were done on board for the next problem: 9 Rotis among 4 people, for which their answer was ‘Savaa Do’ (means 2 and a quarter). Now the challenge for them was to how to write these two fractions using numerals. Of the two numbers ‘one and half’ and ‘two and quarter’ they could write 1 and 2 , but how to write half and quarter.

So when the desire to know was created & expressed, they were given one proposal. I told them – I will tell you how to write half, but then you will have to find out how to write quarter. When they agreed to this condition, the notation was half was written on the board, but followed by a question – Why is half written as that way? And to this they reasoned – Because 1 Roti has been divided into 2 parts. So then they were asked how to denote Quarter, and they immediately suggested its correct notation. In fact they also figured out the notations for other smaller pieces like one-eighth (they called it as ‘aadhaa-paav’) and one-sixteenth. It was evident from their facial expressions that some of them had seen the pattern.

Now they were asked if half roti can be showed/ given in any other way. They were given paper plates to work and demonstrate. It took couple of mins and some rephrasing to understand this problem like if we cut the full roti (paper plate) in more than 2 pieces, and then can we make half. And this was responded very well. They gave me 2 quarters. I kept one single half piece and two quarters next to each other, and asked them which one is half? And they said – Both. So then the problem posed was – there is one more name for this new representation of half. What it could be? After few seconds, this difficult to find answer was written on the board: 2/4 But then students were asked to reason about this and they did quite well with little help. Now they were asked to cut the plate into more pieces and make half and then express it symbolically. So they came up with the notations 3/6, 4/8 on their own with reasoning. These fractions were written on the board side by side and they were asked to guess the next possible one... And one of them said 5/10.

Now they were asked which of the two is bigger 2/4 or 5/10, by writing one of these signs <, > , = between them. (I had to remind them of these signs they had used in their early school years like 43 > 17, 26 = 26, etc.) And then one of them came forward and wrote ‘=’ between the two fractions 2/4 and 5/10 saying both are equal. They represent half. They also solved 1/2 is bigger than 1/8 with proper reasons (without any rules).

I now wanted them to see if they can find out the notation of another commonly used fraction ‘three-fourths’. So I asked them such question. How much Roti would one get if he is given half a roti followed by a quarter piece? This was backed by the use of paper plate (3 pieces of quarters). And they said it well – “pounaa’ (which means three-fourths). When asked how would you write this using numerals, one of them said – 1/3, to which others agreed and one argued that its three pieces and one is taken away (because they saw three pieces of quarters, with one missing to form the whole).

This response was not new for me, as many other students have responded this way to three-fourths in the past. So now the teacher had two goals:

(1) To help them discover that this is not the notation for ‘pounaa’

I knew that they had figured out the notations of other unit-fractions like half, etc. So strategy was to use this knowledge. I asked them to draw the whole roti on the board, cut it into 2 equal pieces & write the notation of each piece. They did. Now repeat the similar process with cutting into 4 parts. They did. Now repeat the process with 3 parts. And they did this too. They drew the Y inside the circle and wrote 1/3 next to it. Though the three parts were not precise but I choose to not correct their diagram at this moment because it did make sense to them.

So then I drew their attention to their previous guess/ solution for ‘pounaa’. And Wow! They said that their previous guess/ solution was incorrect. So I cancelled it.

(2) To make them discover the correct notation of ‘pounaa’

I asked them – So what is the correct notation of this fraction ‘pounaa’? And they kept staring at the picture with wonder. Since they had figured out notations of different representations of halves few minutes back, so I thought to use their previous knowledge. I showed them the paper plate halves they had made & asked them what made them give those names to each of those. This conversation was enough for them to figure out the name of ‘pounaa’ (three-fourths) as 3/4.

And then finally, I wrote this notation of ‘pounaa’ next to its picture on the board J

Context : How this class shaped up?

CCL team of IIT Gandhinagar ( https://ccl.iitgn.ac.in ) had invited me to interact with some KV and JNV school teachers who had come there for their residential training program. I was wondering as to what ‘sustainable’ could be done with teachers in just 2-3 hours, as I have now mostly restricted myself with long-term interventions with learners. So I thought of exposing them to reading & discussing of couple of thoughtful articles on maths education instead of doing some hands-on activities.

My friend Swati https://azimpremjiuniversity.edu.in/people/swati-sircar suggested me to demonstrate a session before reading a class-room story. She told me to do a session where few teachers can act / ask like children, but somehow I wasn’t much happier with having adults (teachers) as children (students) as I wondered if this conversation would be as natural/ spontaneous with simulated children :) Accidentally & luckily, just a day before my session with teachers, I found 3 children with their bags by the road-side, seemingly waiting for someone. On inquiring, they told me about their Nyasa School inside the campus. Then CCL team told me more about this social responsibility program run by IIT Gandinagar https://initaitives.iitgn.ac.in/nyasa/index.php

So I couldn’t resist and found myself playing with these children, rather than attending the teachers training program J It was super-fun spending couple of hours with them (outdoor games & magic tricks).

While departing, I shared with few older children what was going on in my mind, if they would like to come for more playing (I meant playing maths now) the next day too (It was Sunday, a day-off for them). And they immediately & happily said YES! I guided them about the timings and venue. I got a bit worried the next day, when we (I and group of teachers) were ready for our session and they had not yet arrived. One of the CCL team members kindly waited outside the hall, looking for these children while I engaged teachers with some introductory fun activity. And soon, this bunch of children entered into the spacious hall full of teachers. I was thinking what must be going on in their heads while entering as they were not given any clue by me the previous day. So I oriented them about this, told them to relax & be as they were on the previous day ignoring other things / people around them. There was some hesitation in the beginning, but luckily the ice broke in few minutes and what emerged is as mentioned in the post above.

After the class, children left for outdoor play and teachers shared their observations & reflections. Some glimpses of this class and some bites of what teachers shared can be checked on links below. We then read and discussed an article that was published in this maths magazine. https://azimpremjiuniversity.edu.in/at-right-angles

-- Link of the article we read (classroom story having shades of Inquiry & Discovery: https://tinyurl.com/3bvx36pn

-- Demonstrating a class with children : https://youtu.be/eIh4_OPVolM