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’?”

Playing Maths - "But these problems need brains! And our text-books have easy ones!"

Hello friends,

Firstly thanks a lot for taking out time in reading my long emails -- Math conversations. And more importantly, even sending your feedback/comments. I had requested this earlier too and I request/ repeat again, Plz share your reply with the group so that everyone can benefit and learn from your views/opinions and may even comment on the same, which can probably lead to an interesting discussion thread. After all, that's why we have created a group right? :-)

Secondly,I have also received requests to re-send the group invitation because the system generated emails had slipped into other folders like Promotion/Social etc and by the time you read it -- the invitation had expired. So here is the link that you can simply hit and your request to join the group will straightaway come to my inbox, which I will approve much before it expires.

https://groups.google.com/forum/?nomobile=true#!forum/maths-learners-group/join

Thirdly, I am so happy that within such a short span of time, more than 100 (116 till now) Mathsteachers of various schools and educators/ educationists have become part of this google-group. Hope to see more constructive discussions in the near future. If you know any Maths teachers who would like to be part of this Learners group, plz share this idea with them, and by this way, you get an opportunity to touch the lives of even their students! :)

New members/ recipients may plz visit our group page (above link) to get updated about/ learn from/ comment on any of  the previous stories/ posts/ documents shared in the group.

And yes, finally the main part --- I am glad to share with you all one of the Math conversations that I had with a student couple of weeks back. Hope you will see some value in this. And I would wait for/ love to listen to your feedback :-)

It was 9 and I had just reached home, when the door-bell rang again and I could hear the voice of my mom, “Yes, you may go inside!”

And I saw a boy dragging his feet towards me while carrying a wide smile and raised eye-brows. What was more surprising that he also carried a book in his hand! It’s been over 3 years I stay with them, but nothing such has ever happened. I mean, the only thing that the children of our building ask/expect from me, whenever they see me, is a ‘Game or Puzzle.’ But today – a Book! My childhood memories came alive and I saw myself in him. I was pretty sure that he had come for some clarification or guidance. I was happy to see these changes, but more than that – was Excited J

“I don’t understand this problem”, he says with a shy tone, handing over the book to me.

“Ok, what is in it that you don’t’ understand?”

“How to find the value of Circle minus Triangle? Which option is correct?”

“What do you feel, which option is correct?”

“I don’t know. Everything is given as addition”

“Ok. What information is given in the problem?”

He converted the written symbolic language into speech and then stopped for me.  I felt I figured out the cause and so tore out a page from my book and copied on it, the diagrams same as above, but with different numbers.

“Tell me, what do you feel what could be the values of Circle and Triangle here?”

“I don’t know.”

“Give a try. What can be their possible values?”

Playing Maths: "Hundred and Hundred and Hundred and Hundred and....."

If you remember, my previous story was about my visit to my sister's place where I face the ‘adventurous encounter’ with negative numbers with my 6-yr old nephew (Jash) and his 9-yr old cousin (my ‘new niece’ ;-)  

For those, who haven’t read that story, may find it here: 
http://rupeshgesota.blogspot.in/2014/09/it-was-day-of-rakhi.html

Soon I had to visit my sister’s place again, but this time for Ganpati darshan. And I had no idea that I will again collide with these two young mathematicians, so soon.  It was only after few minutes, that the same voice (but it was loud and insisting this time) fell into my ears --

“Maamaaa, today you have to teach me 7^th standard Maths!” And I was like........Bowled!

No doubt, as a Math teacher I am always alert, looking for such natural opportune moments to start theMath-talk, however to her, I gave a pat response – “What? Do I come here to teach you Maths?”

“Mama, plz teach na.....plzzzzzz...plzzzzzzzzzzzzzz...”

I was so much enjoying this desperation and was, in fact now, more desperate than her to kick off, but --- I again wore the devil’s hat -- “No, I am not going to teach you this time. I am sure you would have forgotten the 6^th std. Maths that I taught you last time.”

“No...No.. I remember everything.”

“Oh, Really?”

“Yes, you can ask me.”

“Theek hai, Tell me what is 50-20?”

After a pause, “You did not teach me this way. We did smaller minus bigger.”

With this proud feeling of achievement, I forge ahead – “Ok. I am happy you remember. So then tell me what is 20-50?”

“It’s Minus 30.... Simple!”

“Hmmm...That was easy. Now let me give you a difficult one. What is 17-30?”

She started moving her fingers and lips and watching this disappointed me a bit. You know, we had done a bit of Mental Math too that day, before signing off... But soon, I also found myself pacifying ‘how unjust of me to expect a 9-yr old to ‘remember, recall and use’ the strategies showed orally/casually to her in ‘just 10-15 minutes’, and that too after a gap of 20 days'. And while I was stoned into this looking-for-justification, a confident and loud voice brought me back into the moment.

“Minus 13.”

“Oh, good. And how did you do that?”

“30-17=13...... So 17-30= -13.”

“Hmm... I get it. But how did you get 30-17? I saw you counting fingers. Remember, we did something interesting last time – Mental Math?”

‘Oh, Yes...Wait...Wait.... I will do by that method now.”

And soon, after speaking to the wall and air for few seconds, she spoke to me and even reasoned out beautifully and stole my heart “We want to remove 17 from 30. So, we first remove 10 from 30. This leaves 20. And 20-7 = 13.”

“But why 20-7?”

“Arrey, we have to remove 17 na? Not 10. So I removed 7 more.”

“Oh, ok...ok... I got it. Shall I give you one more problem?”

Playing Maths, again: "Yes! It is Minus 6."

It was the day of Rakhi. This time my sister was not going to visit our place as she had planned to host all the bhai-behens from her in-laws family, at her home.  Being aware about this situation, I reach her home on time – I mean, directly at lunch time, and with no surprise - only to be received by new faces with staring eyes but wide smiles. The confusion was evident.

“Hello Rupesh.... Why so late?........... He is my cousin”, came a happy familiar voice from somewhere inside and I then confidently (or still reluctantly?) dragged my feet, straight into bed-room and guess what? This space too was hijacked by few strangers, but this time – the very beautiful and adorable ones – Children!!

“Maaamaaaaa.....!” and here comes my bhaanjaa (Jash) running to me, and then pulling me into his new gang, while proudly showing his Rakhis and gifts he had given to his sisters.  The energy and enthusiasm quickly connects us and I was now Mama to all the bacchha-party there J Most of them were toddlers and hence watching them talk and play is a fantastic amusement, in itself and even a good learning for me, quite often. So I decide to lie down, to enjoy this free entertainment, while scrolling through my handset.

But soon I felt someone is whispering to me -- “Mama, do you have games in your phone?”

Jash knows by now that I don’t have games on my handset and so he never comes to me for that. So I was surprised hearing this voice. I turn around and find my ‘new niece’ sitting beside me; the quest for game was clearly visible in the tiny eyes and polite tone of this 9-year old, but sadly, I had to disappoint her with a No.  I also noticed that she was done with playing with other toddlers, who were now highly engrossed in ‘pakdaa-pakdee’ J

“Do you watch cartoons?”

“Mummy is not allowing to turn on TV”

“Why?”

“She insists, they want to talk and there is already too much noise at home.”

I so much wished to tell her that chit-chat of these adults was audible even at the staircase.  But then something strikes me all of a sudden and I modulate my voice.

“You know, I don’t have games. But I have another interesting thing in my phone.”

“What?”

“Guess. You already know it.”

She jumped back near me with eyes stuck on the screen. I open the calculator app and she shouts immediately --

“It’s calculator! I know that. “

I was kind of aware of this response, but still I was somehow confident of taking it ahead in some way to amuse her (or perhaps ‘us’? :-) How can kids be unaware of this interesting tool – calculator esp. when they are constantly navigating in their parents’ gadgets? ‘Interesting’ I say because I have realized, it is indeed interesting to the children of, at least, school-goers. (Do you know why?) I still remember how I and my friend enjoyed in taking the pocket calculators (even the fancy ones with beep sounds or flashing LEDs) in our schools, of course secretly. In fact, I was even once scolded my Mathsteacher for doing so.  Silly or funny it may sound, but I and my friend often sat on the last bench to playthe number crunching race ‘using calculator’ J And I, almost always used to lose!

Anyways, focus the camera on my ‘new niece’ now.

“Yes, I too knew that you would knew it.”

“You know, yesterday it was my maths test and I got 9/10”, while saying this she even grabbed the phone (calci) from my hand.

“Oh good, you just lost one mark.”

“I did a silly mistake.” I loved her ‘cursing’ expressions while she said this. But I wonder, if those were originally hers J

“Hmmm...What mistake?”

“I did a mistake in subtraction sum.”

(The rhyming ‘s’ sounded so sweet in her voice that I did not bother to correct her math-vocab at this point.)

“Can you try that on calculator now?”

She was already exploring the ‘new’ calculator. So I thought, probably it would be easy for her to work on this question/idea that I threw at her.

“I don’t remember the question now.”

“Oh ok. Was it a difficult problem?”

“It was with borrow.”

I guess some of you might have already identified yourself with her situation. Even I smiled on hearing this, and really wanted to tell her that – I understand dear. It’s not your problem. It’s not your mistake, if you are getting confused. But instead, I chose something else --

“Can I give you one problem?”

“Yes, but I will do that on calculator. Not with pen and paper.”

“Of course, yes!”

“You are allowed to use the machine!”

“Machine? This is not the machine. This is a calculator!”

And this reminded me of.... 3 idiots! Remember the humorous argument between Rancho and his teacher on his first engineering class.

“50-20”

“That’s so easy. I can even do this even orally. It’s 30”, and while saying this she pressed the numbers and showed me the answer. I could see, she was unhappy with this ‘easy’ question. In fact, I too ‘hate’ giving ‘easy’ questions to students, as much as ‘they hate’ solving ‘easy’ questions.

“Fine. Tell me what 5-2 is?”  (This one was, of course, for teasing her ;-)

“And now, she was furious -- I am not in 1^st standard. Give me difficult sum.”

“Oh..Ok. Check 2-5” (I swear, I don’t know how this slipped out of my mouth. Perhaps, because of her challenge?)

“What? 2-5 is not possible!” She said to me with confidence, that it is a wrong question.

“Well, can you check this on Calculator?”

Her typing was no-less than a real climax for me now. I don’t know what was she thinking, but I was much more anxious to see her response.

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 3^rd 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?“