Monday, September 1, 2014

Talking Maths in the Bus with a young co-passenger

It was a public holiday – Ganesh Chaturthi. I thought that there wont be much traffic on this day, so I boarded the bus instead of train; but only to repent about this after some time..

Different options were then explored – playing with phone, reading the book, talking to person next to me, but nothing seemed to help that day.  Meanwhile, I found a 11-12 year old boy who I felt was going through the same trouble – killing the time. His mother was sitting 3-4 seats away from him. One could easily perceive his boredom/ uneasiness. I felt the urge to talk to him, but our physical separation was too much to allow this.  Our eyes met sometimes and later we even exchanged smiles but that’s all. However, after few stops, something interesting did happen.  The guy next to me stood up for his destination stop and this boy sprang up and came running to grab this window seat :)  Little did he know that this was a happy-news for me too! But soon I found him so deeply engrossed in watching the life outside the window that I decided to not disturb him (and his learning process) and I resorted to the gadget in my hand. While sifting through the images, an old puzzle caught my attention and I started thinking on improvising it.  And interestingly, the little head too had turned around and his eyes went straight onto my handset. After about 5-7 seconds, when I still found him ‘into’ the puzzle (image), the math-germ within me couldn’t be suppressed more.

“Hi”, I said.

And he looked up at me suddenly with a big & shy smile as if I had caught him doing something red-handed.


“Is it interesting?”

“What is it?”

“A Puzzle! Do you like puzzles?”

“Yes!” again with a big smile J

“Which class you are in?”  (What a silly question!)

“Seventh std.”

I handed-over the phone to him. “Ok! See if you can solve this puzzle then”



Without any hesitation, he grabbed the phone and then asked me for what exactly has to be done.

“What do you see in the image?”

“Cats, dogs and rabbits.”

“Ok. And what else?”

“They are standing on the weighing scale.”

“Ok. So?”

“But what has to be done?”

“You tell me. Anything related to weights?”

“Yes.. Their weights are given.”

 “Are the weights written everywhere?”

“Oh, haaaan, there is a question mark here,..... So do we need to find the weight read by the last scale?”

“What do you feel?”

And he now looks up at me with a confident smile (seems he was finally happy for being able to successfully trace out the question in that puzzle J )  and he further adds “But how to get this?”

“Well, this is what the puzzle is. How to solve this?”  :-)

“I don’t know!”

“Can we try to think from what is given and what is to be found?”

“Weights of animals are given at 3 places. And not given at the last place.”

“Ok.  Are all the places same?”   

“No. Weights of 2 animals are given everywhere. And we need to find the weight of 3 animals.”

“Good. So how will you find?”

“I don’t know.”

“Hmmm.... Let’s look at the first picture”

“Total weight of cat and rabbit is 10kg”

“Ok. So what do you feel what can be their weights?”

“6 and 4.”

“Ok. Who is 6kg and who is 4kg?”

“Cat is 6kg and rabbit is 4kg.”

“Fine. Then what about the dog?”

After 4-5 seconds, “16kg.”

“And how did you get that?”

Pointing at the second picture, “Because rabbit is 4kg, so dog is 16 kg.”

“Ok. Did you look at the 3rd picture?”

He looked there and after a while, “No...No... Not possible.”

“What is not possible?”

“6 and 4 is not possible.”

“Why?”

“Because then 6+16=22, and it is given that their total is 24.”

“Good. So what do we do then?”

“I don’t know. How to solve this?”

“Hmmm...  From where did you start last time?”

“First picture. 6 and 4.”

“Is anything else possible, if 6 and 4 does not work?”

After a while, “It can be 7 and 3 also”

“Right. But even 4 and 6 could be possible, don’t you feel so?“ 


(Why did I do/ ask this? Because I already knew that 7 and 3 is the right combination and I did not want him to arrive at the right solution so early (easily) ... (Why so? So that I can trouble/tease him more?) I leave this question open for you to think and respond to.)

So he tried for 4 and 6 on his own this time, checking all the 3 images (situations) and explained me with reasoning why this combination is also incorrect.

“So what do we do now?”

“7 and 3”

“Go ahead”  (he had anyways gone ahead w/o waiting for me ;)

“Yes, this is right!”

“What?”

“Cat is 7, rabbit is 3kg”

“Why?”

“Because then dog would be 17kg. And dog and cat would be 17+7=24kg and it is also given as 24kg. It is matching.”

“Hmmm... . Tell me one thing. What if 7 and 3 too would have not worked?”

He thought and replied with confident smile – “We can try (8,2) (9,1) (5,5)”

“Good. So is the puzzle solved?”

After 5-6 seconds and with some finger counting, he replied, “ the answer is 27 kg.”

“How?”

He could explain this quite satisfactorily.

“Do you feel there can be another way to solve this problem?”

After a while, “I don’t know.”

“Think yaaar. You have already solved the problem in one way. So you have the capability to solve it in another way too.”

After seeing him struggle and not responding after 10-15 seconds, I crept in.  “ How many animals are there in each of the figures?”

“Two-two in all and three in the last”

“Ok. What if I bring down the two animals in the 1st picture and make them stand on the weighing scale of the 3rd picture?”, I asked him this while moving my hands from top (1st) to bottom  (3rd) picture.

“Weight will increase as there will be 4 animals now.”

“Can you describe the new picture clearly to me?”

“We have 1 dog, 2 cats, 1 rabbit and total weight is 34 kg.”

“Good. What do you feel what would I ask you now?”

“Get the animals of 2nd picture also down?”

“Wow! Can you?”

“So now we have 2 dogs, 2 cats, 2 rabbits and total weight is 54 kg.”

“Good. So?”

No response.

“What do we need to find?”

No response.

“What’s the question?”, while pointing at the last picture.

“Haaaan.... Find the weight of three animals together.”

“Hmmm.... So now?”

No response.

“ What have just found out now?”

“Weight of 2-2 animals is 54kg”

“So?”

“Yes... Yes... So weight of one-one animal will be half of that which is 54/2 = 27kg.”

“Good.  And is your answer right?”

“Yes, I got the same using the 1st method also.”

“So what do we learn from this.”

“There can be more than 1 method to solve the problem.”

“And can we compare the two methods?”

No response.

“I mean,  what was the approach in both methods, time taken, etc?”

“Yes, the 1st one was very easy, but we have to keep testing all combinations.”

“Do you mean, 2nd method was difficult?”

“No..Not difficult. But it did not click me so easily.”

“It will.. If you practice few more problems. But is there anything interesting about the 2nd method?”

“Yes, it’s faster and even smarter.”

‘Hmmm....”  (So happy to hear this!)

“Do you have one more puzzle?” And while saying this he pointed at my handset J

“Oh! You want more?”

“Yes!”

“Ok. Take this. And this is my favorite. I want Rs.100 note from you. But you need to give this to me using 10 notes in all without using Rs10 notes. You can mix the notes if you wish.”

And I instantly found him doing the number crunching. After about 1-2 minutes, he came up with the solution, but it had 11 notes instead of 10 L


“But good. You were very close. I think you will get it soon now.”

And he was again absorbed. What a visual delight it is to watch kids play with numbers J.  Almost in 1-2 minutes, he came up with the correct solution. I can still recollect how satisfied he looked with that sense of achievement.

“ You know there are at least 3 different and correct solutions to this problem.”

“That way, there can be many. It is impossible to find all.”

“Correct. But can we try for some more solutions, at least when we know that 2 more exist?”

And he happily continued for further explorations. After about a minute or so, “ It’s difficult. Can you give me another problem?”

“This itself is another problem for you. 2 more different solutions.”

“I will do that at home. But now Plz give me different problem.”

“Ok.”  And I was about to think and we saw all passengers around us preparing to get down. And then even the boy looked back while his mother signaled him to get down.

“Plz give me the problem fast. We have to get down now.”

“Oh.. I so appreciate your enthusiasm. But let’s get down now. I will give you the next problem when we meet next time and when you give me the solutions for this problem.”

“Arrey,,, but when and how will we meet next time?”

I was almost speechless by now and meanwhile, the mother had arrived by our seat and she intervened, “ I was watching from behind how you kept him so engaged. He was trying and enjoying all the time. Do you take Maths classes?”  ;)




PS: All the above conversations happened in Marathi. 

21 comments:

  1. On Wed, Sep 10, 2014 at 5:01 PM, Hariharan R. rhari911@gmail.com wrote:

    Loved it:-) I loved the fact that u only helped him to solve. U didnt teach:-)

    ReplyDelete
  2. On Wed, Sep 10, 2014 at 6:36 PM, K.B. Kartha k.b.kartha@gmail.com wrote:

    Very nice.
    You have done a polishing work to bring out the gem in the boy.
    Only you can do it!!!

    ReplyDelete
  3. On Wed, Sep 10, 2014 at 7:16 PM, Sneha Titus sneha.titus@azimpremjifoundation.org wrote:

    Introduction to algebra on a bus journey? Fantastic!

    Sneha

    ReplyDelete
  4. On Wed, Sep 10, 2014 at 8:22 PM, Naresh Bhoir naresh@navnirmiti.org wrote:

    nice experience
    and good puzzle also.

    Regards,
    Naresh Bhoir

    ReplyDelete
  5. On Wed, Sep 10, 2014 at 9:06 PM, Sunanda Mane (Lend-A-Hand India) sunanda.mane@lend-a-hand-india.org wrote:

    Hi Rupesh,

    You made my day! (though I read this at the end of day, just now).
    Take a break for one day and visit Pune. Want your presence for some brainstorming.
    Best, Sunanda

    ReplyDelete
  6. On Wed, Sep 10, 2014 at 9:38 PM, Skrishnakumar skrishnakumar00@gmail.com wrote:

    Ur solution for the animals problem was quite interesting rupesh. Have u heard of lateral thinking concept and Edward de Bono.

    He talks about logical and lateral thinking methods and is famous for his lateral thinking books and puzzles. The boy's solution was logical while the one u gave was lateral !

    Regards
    Krishna

    ReplyDelete
  7. On Wed, Sep 10, 2014 at 11:33 PM, Chetna Mehrotra chetna@chetnamehrotra.com wrote:

    Wow..superb :)

    I liked the way ...step by step you triggered the child to a systematic approach..

    he cud add, divide.. without pen ,papers...

    I normally skip such puzzles,as i am not fond of Math, but this one held my attention.

    Thank you !

    Wishes..

    ReplyDelete
  8. On Thu, Sep 11, 2014 at 6:49 AM, 'vivek monteiro' via Maths-Learners-Group maths-learners-group@googlegroups.com wrote:

    Very sweet experience, and well written !

    Dr. Vivek

    ReplyDelete
  9. On Thu, Sep 11, 2014 at 8:21 AM, Ajay Dalmia drivechange@gmail.com wrote:

    Hello Rupesh,

    This is a great interaction. Thanks for sharing.

    ReplyDelete
  10. On Thu, Sep 11, 2014 at 9:11 AM, Bharat Saroopa bharatschool@gmail.com wrote:

    Good Morning

    Lovely

    ReplyDelete
  11. On Thu, Sep 11, 2014 at 10:32 AM, Shivani Ramchandani shivrekha9999@yahoo.in wrote:

    good one

    ReplyDelete
  12. On Thu, Sep 11, 2014 at 10:41 AM, Shabda Priya shabdapriya@gmail.com wrote:

    Very very interesting. ...

    ReplyDelete
  13. On Thu, Sep 11, 2014 at 10:44 PM, Subodh Kembhavi subkem@gmail.com wrote:

    Hi Rupesh,

    This documentation of your discussion with a student could serve as a nice lesson plan which many math teachers could try out in their classrooms.
    Thanks.

    - Subodh

    ReplyDelete
  14. On Sat, Sep 13, 2014 at 11:35 AM, Jeevan Mendonsa jeevanforlife@gmail.com wrote:

    Thanks Rupesh for sharing. it was interesting to see how u guided the boy in finding the answer for himself. as for the suggestion of asking him to consider other combination tho he had hit upon the right one, of course the reason was that he considers other possibilities too. the process is more important than the product. but here the role of the teacher becomes crucial. u wud agree with me that what is important here is to ensure that the students' interest is kept engaged. the intention should not be to stretch the puzzle just for the sake of it or trouble/tease the students. if the students lose interest, then it wud backfire. thus, the challenge should not be too difficult just as it should not be too easy. in this regard, if the teacher sees that the student's thinking ability is stretched beyond the limit, then s/he should give clues so that the students' interest and sense of excitement remains. striking the right balance in this regard is the challenge for the teacher.

    ReplyDelete
  15. On Sat, Sep 13, 2014 at 3:58 PM, Aarti Mulani aartimulani@gmail.com wrote:

    Yes, one needs to be extremely sensitive and observant so as to prevent the kid from reaching frustration level.A fine teacher would never allow the motivation to diminish.

    ReplyDelete
  16. On Sat, Sep 13, 2014 at 6:46 PM, Sangeeta Gulati sangeetagulati92@gmail.com wrote:

    Dear Rupesh ,

    You are quite a story teller ! Pleasure to read your experience.
    It is great you used the opportunity to ignite a spark in a young fellow. I am sure he will remember his interaction with you for a long time and the lessons he learned will help him as well.
    Best wishes to you.

    Sangeeta Gulati

    ReplyDelete
  17. On Fri, Sep 19, 2014 at 8:45 AM, 'S.N.Gananath' activemaths@yahoo.com [ActiveMaths] wrote:

    Wonderful. Really enjoyed it.

    Regards,

    Gananath

    ReplyDelete
  18. On Sat, Sep 20, 2014 at 3:55 PM, Rushikesh Kirtikar rushikesh.kirtikar@gmail.com wrote:

    I like all of it. I like the way you pitch in maths without the child actually knowing. And that is very important I guess. Math is a tool and it has to be used as a tool to solve problem rather thinking math itself a problem, haha! And you do it always I know.

    Actually, even I have the same question for you, that you had for me. In fact for all of us who are trying to improve the education. How can we spread these initiatives to more children?

    ReplyDelete
  19. On Sat, Sep 27, 2014 at 8:28 AM, Arvind Gupta arvindtoys@gmail.com wrote:

    Dear Rupesh,

    Very interesting article. Thanks for sharing.

    in peace

    arvind gupta

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    On Sat, Sep 27, 2014 at 11:20 AM, Vinay Somani vinay@findstone.com wrote:
    wonderful!
    thanks for sharing!

    -------------------------------------------------

    On Sat, Oct 4, 2014 at 9:35 PM, K. Subramaniam (Ravi) subra@hbcse.tifr.res.in wrote:

    Dear Rupesh,

    I read this account which you had posted to Active maths group. It's nice to read about your experience, both with the puzzle and the calculator. Do keep posting!

    with regards,

    - Ravi

    -------------------------------------------

    On Thu, Sep 18, 2014 at 9:57 AM, Harish Amur harishamur@gmail.com [ActiveMaths] wrote:

    Amazing! Thanks for sharing it. It helps to know that these things still happen in this mad world!

    Regards,
    Harish

    ----------------------------------------------------

    On Fri, Sep 19, 2014 at 11:25 AM, Amrita Patwardhan amritapatwardhan@gmail.com [ActiveMaths] wrote:

    Yes, thanks for sharing. Enjoyed reading and look forward to sharing the puzzle with my 8 year old!

    --------------------------------------------------------------

    On Fri, Sep 19, 2014 at 3:50 PM, Sundaram Subbharatnam sundaram48@yahoo.com [ActiveMaths] wrote:

    The best part was your concluding message!
    The entire key to learning is motivation!
    THanks for sharing the incidence.

    Sundaram

    -------------------------------------------------------

    On Sat, Sep 20, 2014 at 6:32 AM, Kaleswara Rao rao.kalesh@gmail.com [ActiveMaths] wrote:

    Very interesting. Constructivism is properly adopted. Thanks for this ,which I share in 7 th std new maths textbk state level mrp orientation Karnataka on22 sept. Kaleswara rao.N,Bangalore.

    -----------------------------------------------------

    ReplyDelete
  20. On Tue, May 12, 2015 at 10:25 AM, Shilpa B Roy shilpabroy@yahoo.com wrote:

    Dear Mr. Rupesh,

    Thank you so much for sharing such delightful experiences. My son Samarth is in Gundecha and I can just imagine him getting all challenged with such puzzles. I stopped my work at office and got engrossed in your puzzels:) How I wish my son could take flight into this world of Maths with you !

    Just wanted to add that I really appreciate your mails and your passion for the subject. This is the key to excellence in any field and this is what i want my kids to discover for themselves.

    Best Regards,
    Shilpa B. Roy

    ReplyDelete
  21. On Wed, Jun 10, 2015 at 1:52 PM, Anjali Chipalkatti anjalichip@gmail.com wrote:

    Dear Rupesh,

    When I read the above email, I realised it is as good as a lesson plan and I strongly suggest you should write more such LPs.

    I also watched the footage you shared with me ( discussion on 4 men wearing caps) and it is extremely interesting. Working with children is so much enjoyable than teachers...

    You don't need my appreciation, now that so many have already appreciated, but let me say I really loved the way you engage students in building inquiry abilities through maths.

    cheers
    anjali

    ReplyDelete