Tuesday, November 11, 2014

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?”


“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’?”


“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?”


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


“Do you know Quarter?”


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”  :-)


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?"  :-)


  1. On Tue, Nov 11, 2014 at 10:48 AM, Shahnaz Patel shahnaz.patel@fountainheadschools.org wrote:

    Wow! This is just wonderful. I am in IBPYP school and we emphasize on teaching everything through inquiry. We always look for examples of how good inquiry looks like, for teachers to see, understand and try out. And this one is surely one of them. I will share this with other teachers too.

  2. On Tue, Nov 11, 2014 at 11:09 AM, Anil anil@kdpaccountants.com wrote:

    Dear Rupesh- your exploration will not go waste-pl. preserve them systematically in your data bank so that one day you can come out with very useful publication beneficial to all those who are interested.

  3. On Tue, Nov 11, 2014 at 11:18 AM, Rushikesh Kirtikar wrote:

    That was a good reading again.
    But why didn't you ask her the meaning of 1/2 when she said half initially? Did she said 1/2 (for half) just because she might have learnt it somewhere in school or so?

  4. On Tue, Nov 11, 2014 at 12:08 PM, diptisurve76 . diptisurve76@gmail.com wrote:

    Its just awesome we have really tried with our kids .Even we had the same experience.And from that day we have made it a point when you start a new topic especially for higher grades have a recap of the basic knowledge of the topic
    Ms Dipti

  5. On Wed, Nov 19, 2014 at 6:58 PM, Aarti Mulani aartimulani@gmail.com wrote:

    Thanks for this lovely share, Rupesh. Totally in confirmation in what I would do also for the topic.
    In fact, this model can be used for the next level, to introduce the concept of equivalent fractions. This enables the student to discover and relate to the fact- multiply the numerator and denominator by the same number to obtain the set of equivalent fractions.

    Warm regards,
    Sent from my iPhone

  6. On Wed, Nov 19, 2014 at 8:04 PM, Bharat Karmarkar wrote:

    Very true. Students must be motivated for decision-making while they learn, because school education is the empowerment of students to lead successful life. Success depends on decision-making and commitment.

  7. On Wed, Nov 12, 2014 at 11:54 AM, Archana Natraj anatraj@gmail.com wrote:

    Hi Rupesh,

    Love the way you teach..Also love that you did not "teach her". really appreciate you shareing this in a group as it often gives me things to try out at home.

    Since my daughter Janani i in Class 4 co incidently doing equivalent fractions..

    I found that instead of saying 1 upon 2 and numerator/denominator etc, in the outset we say one OUT OF 2 PIECE.

    Cadbury chocolate is a great example two. See how quickly they will realise that 1/8 is way smaller than 1/2 and that 4/8 is also 1/2

    In school they also do this with ribbons. .. Start with 10 ribbons of same length. Then cut all except one in to 2 pieces ,3, 4 etc...
    Then see how many bits are required to make the same length


  8. On Tue, Nov 11, 2014 at 8:20 AM, Saji K sajimol.k@gmail.com wrote:

    Dear Rupesh....

    At the outset thank you for persisting...not only with the children but also also with the adults. Adults (like me :) with all their "knowledge" are so much more tiresome to deal with...but I find that because you persist you slowly chip away years of resistance and give space to think...so thank you once again :)

    It was lovely reading this share about the girl and her eventual comprehension ...what is laudable is your admission of instinctively wanting to give her the solution and you choosing to persist with her till she "gets it"...bravo...for fully trusting that "she will get it"!

    Growing up I wish I had teachers like you....teachers who trusted and believed in our capacity and capability...Im sure the world would have been a different place.Today our children too are administered " daily doses of how to do everything" and then we wonder how they are turning into doers rather than thinkers or creators.

    I am glad and grateful for the sacrifice, patience and consistency you show towards all your students...whether above or below 18...:)

    Kudos and Bravo again Rupesh Sir!

    (PS...I find that once kids leave your class they are ravenous almost like they've come out of a couple of hours of active sports....they are muching and crunching in the car...I realised all the thinking in your class uses up their sugar and here obviously its the brain thats playing heavy duty sports ;) ...its a pleasure to see them that way ;)

    Cheers and Rgds

  9. On Tue, Nov 11, 2014 at 11:00 AM, Yatri wrote:

    Awesome explanation and motivation given by you :)

    I am still confused how do you plan to conduct such workshops for my kids. Is it done on regular basis or parents have to be involved ... Pls make it more clear

    Dr Yatri thacker

  10. On Wed, Nov 12, 2014 at 3:56 AM, Dr. NandKumar Jadhav wrote:

    Dear Rupesh,

    Thanks for sharing !!!

    You are really doing a wonderful work.


    Dr nandu

  11. On Wed, Nov 19, 2014 at 8:26 PM, Skrishnakumar wrote:

    Sajimol's feedback is very nicely presented and sums up what parents and students feel about ur teaching methods ! Great work Rupesh !



    On Thu, Nov 20, 2014 at 9:58 AM, 'priya wrote:
    Keep up the good work, Rupesh. It is always interesting to read your stories though we may not have something to say in reply. Your contribution towards a brighter future generation is commendable.

    Priya J Parab


  12. On Thu, Nov 20, 2014 at 10:16 AM, Sidharth Telgote wrote:

    Thats true Rupesh. Its interesting to read your articles, we may not reply, may be because we dont hv much to reply.
    I personally try to incorporate a few of your thought processes when I interact with my daughter's teachers.

    Pl keep educating us.



    On Thu, Nov 20, 2014 at 10:42 AM, Nishigandha Lokhande nishigandha.lokhande@gmail.com wrote:

    I second that ..Rupesh Sir, Whtever u r doing is really great job... I personally use some of ur tricks at home with kids..I am also interested in attending any of ur session if it is in mumbai. Requesting you to keep sending such beautiful mails. Thank you.


  13. On Tue, Nov 11, 2014 at 9:05 AM, nitieprasad prasad nitieprasad@gmail.com wrote:

    Great work Rupesh. Pleasure to read...
    Faculty Adviser, NITIE Center for Student Enterprises, NCSE, NITIE, Mumbai - 87.