Thursday, May 14, 2015

22 - 9 is 11 or 13 ?

Hello friends,

Hope you must have got and read my previous email about the starting of Maths Teachers Study Group.  If not, then plz read it and also let me know if you are coming. I will send you the exact address, etc. I am so happy that 9 Teachers have expressed their interest! It is on Monday 18th May at Airoli, Navi-Mumbai. For more details, plz check that email. 

I remember I had mentioned in that email, about sharing one of the fantastic experiences I had while working with students on fractions that day, but then I thought of first sharing with you what happened Today!   :-)

Meanwhile, the word fractions remind me of one more interesting thing. A group of students were asked what is the half of 2/8 (two upon eight)  And it was interesting to listen to the response of few students. They said it is 1/4 (one upon four)  :-)

If possible, plz share your views on reading the conversation below. Will be happy to read.


We were solving a problem which encouraged her to make a list of all the possible numbers that would leave a reminder of 1 when divided by 3. And then, even hunt for any possible pattern in this list. So she wrote: 1, 4, 7, 10, 13, 16.....

After few seconds, I ask her,” Could you figure out anything?”

“Yes Sir. There is a common difference of 3 between any two adjacent numbers in the list. But I am wondering, why we got this difference of 3, because these numbers are not the multiples of 3.”

And I was like, Waaaaowww! What an interesting query! I paused for a while, to think as to how I should respond to this.

“So do you mean, they should not have the common difference of 3?”

“Yes. I have added one to every multiple of 3 to get these numbers.”

“Hmmm... Can you write down all the numbers from 0?”

Once she is done, I ask her to highlight the movement of multiples of 3 from the beginning. She draws the path. Now I further ask her to repeat the process for ‘her’ list i.e. for 1, 4, 7, 10....  She does that too. But after this, I choose to stay mum for a while. I wanted the bulb to glow, but what’s fun in it if the student gets it so easily, isn't it? J

So she asks me,” Sir, now what?”

“Means? Now it’s your job dear! I have guided you enough.”

She smiles at me and turns at the board. After few seconds, she again looks at me with a smile, but also with wrinkle on her forehead. “Sir, I am not getting.”

I realized that it’s the time to roll up my sleeves now.

‘Let’s say, you (Poonam) and Yogita are standing two meters apart. How will this distance change if both of you move ahead by the same amount?”

{While narrating, I also represent this situation pictorially on the board. But now I realize that ‘she’ could have been encouraged to draw the diagram}

She responded correctly that the distance between them would still remain the same; and she could also reason for this. I further probed her for the other case viz. she travelling more than Yogita. And she again responded correctly, along with the justification. So Now, I drew her attention to our original problem.

And she was like – “Yes Sir, I got it now.” with a wide smile J

“Please explain it to me then.”

“It was so simple sir. 4 is one more than 3 and similarly 7 is also one more than 6. Difference btwn 3 and 6 is three. And hence the difference btwn 4 and 7 will also be three.”

Good Job, isn't it? – I could make her observe, relate, think (reason), use logic and finally, Discover!  J

We had just settled down and suddenly something stroke me.

“You know what Poonam, we can apply this reasoning to solve the problems of subtraction, mentally?”

“How Sir?”

I stood up again, writing the problem on the board. ‘Tell me what is 22 - 9 ?”

She knows that resorting to fingers and even pen and paper disappoints her teacher and hence, she goes for Thinking!

“Sir, 22-9 is 13”

“How did you do this?”

“I kept the 2 of 20 aside. I know 20-9 = 11. Adding back the 2 to 11 gives 13.”

Interesting! I was happy that she remembers the strategies we had discussed. But this time, I wanted to show her more beautiful way – application of a concept (same difference) that we had just discussed and discovered. I could have explored the new strategy there itself, but I thought this problem probably was not so appropriate for stimulating to think for another (new) strategy, hence I posed another problem to her.

“Good. So tell me what 22-7 now is?”


“So quick?”

“Sir, we just now saw that 22-9 = 13. So 22-7 has to be equal to 15.”

What? She has used ‘logic’? Wow! But I wanted to pacify myself by ‘hearing’ it from her :-) So I tinkle her. “Please elaborate.”

“If we remove 9 from 22, we get 13. So if we remove 7 which is two less than 9, we will be left with two more than 13 i.e.15”

I was much happier now... Efforts paying back, you know? J

I too resorted to this first problem now. “So can you solve this problem 22-9 using another way?”

She thought for a while. “Yes, Instead of subtracting 9, I will subtract 10 from 22. This would leave 12. Removing that one gives 11.”

And I was now experiencing a mixed feeling. On one side, I was very impressed by her second strategy, but at the same time, I was a bit disappointed as she has erred up at one point. But I ensured that my body language doesn't hint her about the goof-up, I ask her, “Oh, It’s 11? And I thought its 13, as said by you sometime back.”

This simulated her to think again. “No no sir, it is 13.”

“What? But you just said it is 11.”

“Sir, Wait... Let me think!”

I was enjoying seeing her confused and struggled. Because, I knew that this struggle is Must! If I steal her struggle, then I will steal her learning.

And she bounced back. “Sir, I am confused. I am getting 13 by one method. And 11 by another method. How can this be possible?”

“Hmmm..So what does that mean Poonam?”

“It means that I have done some mistake in one of the methods. But I am unable to figure out any mistake in both the methods. Both seem to be perfect.”

“Ok. Both 'seem' to be correct. But what do you feel, can both be so?”

“No.. How can one subtraction problem have two different answers?”

“I am happy that you realize this.”

“Siiiiiiiiiiiiiiiiiiiiiir, plz help me....”, she insists with a pleading tone. I was actually surprised to see her melt. Because I have seen her persevering in much difficult problems.

“Do you ‘really’ want me to find out your error?”

“No sir. I will find out my own.” Her instant confident determination and willingness to persevere catapulted me back, to cloud seven J

“But can you give me a small hint?"

“Ok.” And she thought that her appeal has yielded. But it seems she forgot that I am such a tough nut to crack when it comes to revealing hints and giving answers. J

“Let’s write both the approaches so that we can observe and compare.”

She writes it down. I notice she has goofed up while expressing her mathematical thinking in the written form (she has equated the two unequal expressions 20-9 and 11+2). But as per thought process, she was in perfect synchronization and hence I decided to NOT point out to her any other mistake at this juncture (tackle only one thing at a time).

She stares at me (waiting for me (the hint)).

While pointing at the second approach, I ask her the reason for 'subtracting one' from 12.”

“Because I had to remove only 9 from 22. But instead, I removed 10. So I will have to further subtract one from the answer because 9 is one less than 10.”

“Really? Will we have to compensate that way?” And.......I couldn’t resist my gentle laughter at this moment and this was enough to make her more impatient.

“Poonam, there is one easier way to figure out which answer is correct. Take stones and ......”

“Sir, I am not in class-1. I will do it mentally itself and I will figure out the error.”

“I am happy with your attitude, but remember, sometimes there is nothing wrong in going back to the basics.”

She looked at me without any expression.  I guess she was trying to digest my advice J

“Ok fine, tell me what 12-9 is?”


“And how did you do this?”

She almost jumped. “Sir, 13 is the correct answer. It cannot be 11.”

“And why so?”

“Because if 12-9 = 3, then 22-9 has to be equal to 13.”

Waaaaowww --- Another pat on the back!!

“Well, now you just need to figure out the error in 2nd approach. Why you are getting 11 there?”

“Yes, I will think about this at home today itself and tell you for sure by tomorrow.”

“I will eagerly wait for your response, Poonam”

We rose for the national anthem and they left.... Meanwhile I started thinking -- And what about the new strategy of subtraction that I wanted to share with her?  J  


  1. On Fri, May 15, 2015 at 9:45 AM, Anjali wrote:

    Dear friends,
    That was extremely inspiring, Rupeshji. I am sharing a link with you where you can find a MOOC program being held by an organization,TESSI, which is producing open educational Resources that can be used to make teaching more child centred. If you can spare time please enrol for it.
    Of course I hope that Rupeshji is successful in arranging his get together too. In fact if it is at the same time you could do it together.
    I hope you benefit through this.
    Anjali Gupte

  2. On Sat, May 16, 2015 at 10:57 PM, Sameer Khadilkar wrote:

    ​Rupesh - I'm not one of the parents of the children you are working with presently, but definitely a well-wisher. Am writing back just to assure that it makes a lot of sense (to me) to share the detailed interaction the way you did in your example - so that (i) the parents understand whats happening (how the children are reasoning) and (ii) can spot or suggest something that you might have missed

    this approach of interaction with the child seems wonderful. I usually try to do the same with my child (and not just for maths), and see the value in not giving the formula/method/approach/solution..but just coax them to figure it out. The only thing is that sometimes the child comes with a question and *is* looking for a quick answer... and not the "you figure it out response" in which case I have to give in (assuming I know the answer of course :)

    one other thing I notice (in my interactions) is that once the "answer/solution" is accepted by the teacher/parent, the child tends to accept it without verifying. Am not sure if the teacher/parent/guide purposely accepting wrong answers and letting them figure out later that it was incorrect - will help break the blind acceptance habit?

    cheers and best wishes..


  3. On Thu, May 14, 2015 at 11:53 PM, Prasad Madkaikar wrote:

    It was wonderful to read this experience of yours. I would love to know some basic approaches that we as parents should keep in mind while we embark on our journey with our child into the world of mathematics.
    We have a 2 year old by the way.

  4. On Sat, May 16, 2015 at 11:41 AM, purvi shah wrote:

    Dear Rupesh,

    I have been reading your posts for a while and enjoying the same.

    When i read this particular post it resonated a lot with me because the subtraction strategy that you discussed i had recently taught my 8 year old and he has been doing basic mental subtractions faster because of this.

    Now the query i have for you is this:

    My son goes to an alternate school ( waldorf school) he has a great liking for math and i wanted to encourage it outside of school as you maybe aware that in waldorf schools they start late so he will only be going to the second grade in June 2015.

    Therefore i started him with abacus, the sir comes home and we have only had just one class. The first class itself disappointed me because it was very dogmatic and i can be very picky on some things. ANd therefore wanted yr counsel.

    I have no friends who have tried this ( from alternate schools that is) tons of friends who have recommended this whose kids go to regular schools. Do you have any personal experience with this and would you recommend the same. My only objective thru this class really is getting his mental math stronger. And since i spend a lot of time with him on english, GK etc i felt that someone else doing it with him might be better. But i don;t want to takeaway the joy of learning and exploration away from a curious child.

  5. On Sun, May 17, 2015 at 11:48 PM, Prasad Madkaikar wrote:

    Hi Rupesh,
    The aspects that made me feel wonderful about the experience are below:

    1. A non judgemental approach by a teachers towards the learner's willingness to experiment,fail and persist.
    2. Desire to adopt flexible techniques based on learners needs
    3. Making learning a pull driven activity than a push driven one.


    Prasad Madkaikar

  6. On Thu, May 14, 2015 at 11:57 PM, Mona Dalmia wrote:

    Dear Rupesh,

    Thanks for sharing your interaction with all of us. I can feel your excitement as I have gone through this myself so many times, It is indeed heart warming to witness a child approaching the 'AHA' moment.

    Pat on your back for engaging in self-reflection too. Your intuition might be right that if she had drawn the first time, perhaps she could have thought about the problem just then. I am reminded of what Reuven Feuerstein has written, "I formulated a need to look not for capacity but for modifiability". A good example of mediated learning experience.

    Also kuddos to you for your patience and your ability to hold Poonam's hand to success. Looking forward to our weekly sessions at DCRC.

    Many regards

  7. On Sat, May 16, 2015 at 3:21 PM, ekta wrote:

    Wow!!! Really enjoyed reading your conversation, Rupesh.... :) Must tell you, you capture even the finer nuances of the conversation, eg. perseverance, confidence, etc... beautifully.... It's not just maths they are learning, but life skills - trusting themselves, finding their own mistakes, persisting, analysing, corelating, finding multiple methods, crosschecking..... wow wow wow... i'm still a big work-in-progress when it comes to these skills... :) I hope as I try to imbibe your philosophy of doing maths with my daughter, I will also learn... :)

  8. On Thu, May 14, 2015 at 11:30 PM, anita sharma wrote:

    Dear Rupesh ji

    I was going through the videos and the experience posted by you. It is really amazing to acquaint your work. I am working on a project with prof from Delhi Univ & Deptt of Sc.& Technology and we are in process of preparing a resource book for teachers teaching class VI and does not have access of resources and training.One chapter of the book is devoted to the success stories of teachers faced challenges of the classroom.Your experience can give us useful input regarding this.Kindly share your story in around 500 words so that it can reach to all and can be instrumental in producing many more passionate and committed teachers like you.

  9. On Tue, May 19, 2015 at 3:09 PM, anita sharma wrote:

    Thankyou Rupesh ji for your quick response and my apologies for delayed response.I have gone through the brief of your journey o far and highly impressed.A very important issue raised in the end about how a teachers can lead a child in direction of thinking mathematically.
    I would like to use your narration of the conversation between you and child,if you permit in the resource book.