Playing Maths: "It is Half... No, it is 1 Upon 6... No, it is Half... No,it is...."

Few days back, I was assessing a bunch of 6th & 7th class students in a school – asking them some practical questions (in a friendly way) to gauge their conceptual understanding.  Having done this exercise with a host of schools across the spectrum, I have a decent amount of variety of responses of students by now. And I am so thrilled by the quality of learning that happens by analyzing this type of first-hand experiences that now I wish, some day, to systematically compile and publish/post it in a form such that it would be accessible/useful to all the interested learners. I don’t know when this will happen, but I do share and seek observations while interacting with the schools/teachers/parents during the workshops/study-circles.  And then, a thought crossed my mind as to why not, meanwhile, share this information in our group? 

So here I go –   (while waiting for your observations/ comments/ views) -

The student is shown the following pieces spread on one of the desks.

​​After giving her around a minute to play with these pieces on her own, I drew her attention to the full circular piece. 

​​“What shape is this?”

“Circle”

“Hmm.... Do you eat something of this shape at home?”

“Yes... Chapati...papad”

“Right! So let’s say this is your one-full chapatti”, while handing over that piece to her.

Now I take another piece in my hand and ask her,” So how much chapati would this be?”

​“Half” (super fast response!)

“Ok. Can you write ‘half’ using numerals?” 

She writes: 1/2

“Good. Is there any other way of reading this notation?”

“One upon two.”

“OK.” I replace this piece with another and continue “Ok. So what about this piece then?”

​​​

She compared it with Half and said, “It’s one upon four”

 “Ok. Let’s write this using numerals?”

She writes:  1/4

“And why this ’4’?”

Silence!

“If you want, you may work out with these pieces and figure out the reason.”

And then it was a pleasure watching her find out the right pieces from the mess, then assemble these four pieces into one full circle. She tells me,” See. Four pieces make one circle.”

​“Ok, you mean to say – Four such pieces make one whole?”

“Yes.”

“Is there was any other way of naming this piece, as you said Half is same as One upon two?

Silence.

“Do you know Quarter?”

Silence.

I gave her a new piece, asking for its name.

​She looked at the previous piece & its notation ( 1/4 ) alongside, thought for a while and - “This is one upon three.”

I smiled in my mind. But by now, I have learned to 'not react' to the students’ incorrect responses --

“I see. But why one upon three?” while laying more emphasis on the last word.

“Because it is smaller than this piece”, she explained, while showing me the previous (bigger) piece ( 1/4  )

“Would you like to verify your response, the way you did before, by assembling?”

“No... I am sure!” 

I waited for few seconds, wanting for her to ‘do’, but she stood still waiting for me (the next instruction).

“Ok fine. Can you write its notation?”

I kept the piece down and she writes 1/3 below it.  I gave her a new piece now – “So what about this?”

​

She looks at the set-up created below, and finally says – “This is one upon five.”

“Oh. And why so?”

She again points at the quarter-sized piece and explains her reasoning to me. “This piece is bigger than this piece.”

“Can we assemble and check?”

“Not needed.”

“Fine. Let’s write its notation too.”

She writes: 1/5

I realized that there was enough Masala to take the movie towards Climax now. So I finally showed her the piece, which she had recognized as ‘Half’ in the very beginning.

“So what is the name of the piece?’

​​Instantly – “One upon six!”

“Ok. And why so?”

"Because it is bigger than this piece." (while comparing it with the previous smaller piece – i.e.‘her’ one upon five)

“Oh... I am confused now. Because some time back, I think you told me that the name of this piece is.................................”

Silence! I looked at her. And soon, even she looked at me puzzled, but with a Smile! (I LOVE such scenes!)

“What happened? Why are you laughing?”

“No...No...This is not one-upon-six. It is Half!”

“Oh, is it? But just now you said that it is one-upon-six? And you seemed pretty confident about your reasoning too!”

She looks down......then up.....and then down again....And our smiles don’t seem to stop! :-)

After a while, I intervene -- “So what is the name of this piece? Half or one-upon-six?”

She again laughs, and recoils back to “one upon six”  :-)

“Sure?”

Silence! And her head was again down.

Seeing her so desperate for the correct solution, I felt a strong urge to lead her to the discovery... give her back the piece of one-third and ask her ‘So then, if that is your half, then what according to you, would be this piece?’ I was quite sure that she will beautifully pick up, but then there were reasons for not doing so at that time, in that way.

However, I was also very sure that she had enjoyed this interaction, and confusion. In fact my experiences with students tell me that they love being confused (challenged) this way and even to come out of such confusions on their own. So to finally leave her with a lesson (thinking-assignment, actually), I thought of picturing this entire sequence on the desk. My plan of thinking aloud worked, she got motivated and I was so happy that she joined me, to correctly correct the mistakes that I was intentionally making in portraying our sequence of discussion. But I stopped when I had to to write ‘One upon six’ and so, I asked her “Where do I write?”

And she again smiled.

​I thanked her and waved her Bi..... But she pauses for a while and asks me -- "When will you come again?"  :-)