"2 rupees 20 paise for 100 grams" - Maths class or Ethics class?

Hello friends,

I did not know that I will be back with the new post so soon... But then, this summer has given me a wonderful opportunity to work with the children Daily. And hence, I am so lucky to be a part of (learn from) such stories daily. 

If some of you have been unable to read the earlier post, where Poonam is intentionally allowed to make a spate of mistakes in the multiplication and division algorithms, how she arrives at a (erroneous) conjecture by observing / analyzing some patterns in these mistakes, and finally how she spots her mistake to counter/ discard her own conjecture. A beautiful cycle of discovery that probably mathematicians and scientists would go through. 

This is the link to this story: 
http://rupeshgesota.blogspot.in/2015/06/we-can-divide-product-only-with-first.html

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It all started with just a casual inquiry to one of my students - "what do you feel how much does my bag weigh?"

From which the discussion gradually navigated to the fact that how some shopkeepers cheat the consumers by fixing the std. weights. Sachin claimed that he has seen the 'hole' on the other side of the standard 1kg weight so that no one can spot it. To which Sushma responded that in such cases, the vendor would earn more money.  This as seem did not at first augur well with Sachin because his opinion was stationed on the fact that consumer would get any quantity lesser than the weight on the other side of the balance.

It was interesting for me to learn that what is sometimes so obvious for one student (& the teacher) may not be so obvious for the other student. I allowed Sushma to explain her view and this quickly brought a smile on Sachin's face.

Sushma had already brought into this discussion her experience of purchasing 1-kg rice from the vendor. So I thought to just built upon it. I did not know that my next question would help me learn something so interesting from my students.

"If we assume that the vendor has removed the piece of weight worth 50 gms from the std. weight of 1 kg, then how much will he earn from the transaction of 1 kg of rice, sold at Rs.24 per kg?"

Of course, both the above values -- 50 gms and Rs.24 per kg were arrived at with the help of "students", they were not my (the teacher's) enforcement!  It was equally interesting to study their estimation sense and even help that develop using this real-life example) But I am not going deeper into those conversations this time. Why?  Well, Because I hardly get any acknowledgement from the readers for this effort :-)

Yes, coming back to the problem and how my students saw this and solved this.... Again, here too I am going very quick this time i.e I am not mentioning all the (beautiful) conversations that Really happened. 

Sachin finally could settle down with this argument that if 1 kg costs Rs.24, then half kg would cost Rs.12. So 50 grams would be worth Rs.2.20

It was interesting for me to see him arrive at the cost of 50 grams directly from half kg.  I would suggest you to pause for a while and answer this question. (of course, I will be much happier if you even reply to me with your answer) - 'What according to you must be the way that Sachin would have devised to arrive at Rs.2.20?'    I hope you will honestly pause and think before reading further :)

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.

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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.”

Starting the "Maths Teachers' Study Group" - at Airoli, Navi-Mumbai from 18th May

Hello friends,

It’s that time of the year again -- end of April and most of the teachers must be on the verge of finishing their academic responsibilities. Some of you might have already planned your holiday-trips, isn’t it? Hmmmm...  J

You know what? Today morning, I had a fantastic experience with my students, while working on fractions. And that’s when I decided, that I will share this story with you in the evening. So I sit down, write few lines, and all of a sudden, some other thing comes to my mind. I pause. I thought that the latter one is more important and even interesting. So, what next? I press the Ctrl+A, Ctrl+X, Ctrl+N, Ctrl+V. Yes, you guessed it right. The fraction story will come to you ‘after’ this broadcast in the next email J

A strange yet interesting thing happened last week (And I am so-so-so-happy that Finally, it did happen!) I got an email from couple of maths teachers. They wrote to me that this time, they want to make a better use of their month-long vacation - for their own professional growth.  (Waaaaaaaaow!) They wish to dive deeper into this mighty and beautiful ocean of mathematics.

Isn’t this really exciting? I really appreciate their decision – because they are embarking upon this learning journey, completely on their own, without any support (or push? :) from their respective schools. Excellent!! My dream to start a Maths Teachers’ Study Group has come to a reality, finally!!

So - to be very honest, I am personally much more excited than these couple of teachers now. To start working with them, to listen to them – about their classroom stories, their challenges, discussing about the most common maths mistakes that students make, analyze and learn from these mistakes, mathematics content and pedagogy related matters, investigation and inquiry-oriented mechanisms of teaching and learning mathematics, and of course, share with them my book-readings, experiences and even insights of senior global mathematics educationists -- their articles and host of educational videos, and ,.....and........and........ – there will be loads of FUN – like Playing mathematical games, solving ancient mathematics puzzles, and trying to unlock and appreciate the value embedded in these (often ignored) mathematically rich entities.... Oooooffff :-)  :-)

All this might remind some of you about my initiative of starting the maths-learners google group, and then even the compilation and circulation of "Math Energizer", isn't it? For the new folks in this group, this is the link that will navigate you to all the documents and maths stories that have been shared in this group till now: (you may esp. like to check out the oldest posts):  https://groups.google.com/forum/?nomobile=true#!forum/maths-learners-group/join

But then, I am sad that I am unavailable till 16^th May, because of some other prior-commitments. Hence, despite these teachers being ready to start immediately, we will be able to start only from 18^th May.

So friends, here’s an Invitation to All of you.....

If you or any of your colleagues would be interested to embark upon this thrilling-learning journey of mathematics, you are Most Welcome!! Write an email to me about your interest and participation in this "Maths Teachers’ Study Group" and you are IN! So simple, isn’t it? J

But what about the Fees, Time/ Duration, Frequency of sessions/meetings, Number of hours? 

Well, nothing is decided on these 'trivial' matters - as of now. We will work out all these details later, as the group evolves (when I say evolve, I do not mean by quantity - but by wisdom :-). We, the interested teachers and me, believe that all these details are important, but still are secondary and hence, can be easily worked out later, esp. by a person or group who is really hungry to learn!

I can personally relate to this fact because some time back, I used to travel for around 6 hours by ST buses, on the first Sunday of every month, to be a part of similar Teachers’ Study Group based at Wada Taluka of Thane district. To reach there, I had to first travel by a train, then change two buses and then again an auto. J Yes, the journey drained me out completely, but the peer-teachers and the discussions in the Study Group recharged and refreshed me, much more!

The only thing we are sure about now is that We Are Beginning – without worrying too much about the nitty-grittes. We are also not much concerned now about the destination and the challenges that we may face in this journey.

We are meeting on 18^th of May at Airoli, Navi-Mumbai at sharp 10 am – to Begin! 

I am eagerly waiting to hear those precious three words from you --  "I am IN"  :)   I will then share with you the necessary guidelines and documents. 

Imp: I am not taking any space on rent and all... So the group size will (unfortunately) be limited by the size of the room.

I never use this phrase - First come First Serve -- but sadly, it will be applied this time :-)

PS: I am also happy to share with you all an article that was recently published on Divya Bhaskar, a marathi newspaper. Though some statements and figures do need some correction, however I am thankful to the reporter for sparing few minutes to listen to me and even the editor for to share my story, work and vision with people around. People often wonder about my switch from engineering to teaching school maths. And it has been Real challenge for me to answer this question to anyone (esp. strangers) in even half an hour... So, I guess, this 2-minute quarter-page article will turn out be a good handy beginner to quench their curiosity :)

This is the link:  http://divyamarathi.bhaskar.com/news-srh/MAH-MUM-positive-monday-news-mission-for-math-4976021-NOR.html

The beautiful strategy he Discovered to multiply two numbers, mentally.....

Hello friends,

Surprised? that I am back so early ?   :))

Thanks for patiently reading and sending your views on the previous story shared by me. If you haven't been able to read that yet, then you may just hit this link

https://groups.google.com/d/msg/maths-learners-group/8itYsl_8N6Q/J0l_L-WY2kEJ

(It was about how beautifully a visually challenged student had solved a problem in one of the non-conventional ways :)

I remember I had talked to you about the deal that I will post a story in response to your 2 stories, but am sorry -- I couldn't resist my urge of sharing with you all a very interesting approach of doing multiplication, as discussed by one of my students today. His name is Arjun and he is part of the bunch of 8th std. students of a municipal school with whom I have recently started working with, voluntarily after their school hours. 

Yesterday, we had stumbled upon a problem where we felt the need for doing the multiplication as a part of its intermediate step. I don't remember the numbers but it was a 2-digit no. multiplied by 1-digit number. The numbers were quite easy, still the students were unable to do i mentally and wanted to resort to pen-paper. I insisted for mental maths and after some discussions, the team could Discover the meaning of multiplication and gradually, they even started doing these 1-d x 2-d problems Mentally, and with Understanding and Not by using the conventional standard procedure of multiplication (something that I and perhaps most of us would have studied/used, and probably even without understanding, in our school days :)        

{I am not elaborating these strategies in this post as I assume most of the teachers must be aware of these understanding-based approaches.... one can even google out) 

It was the end of the 2-hour session and so they requested me to give them some problems on multiplication as assignment, so that they can practice this new approach (Understanding-based) which they had started loving a lot. So I gave them about 10 problems.

So today, when we were done with our 10-minute math warm-up, we started discussing the assignment problems. It was so fulfilling to notice that each of these students had solved each of these problems in more than 1 way, by Understanding and at times even with the application of Logic. For example:  For solving 38x5, they first worked out 38x10 which equals 380 and this was halved to get 190 as the required solution. And there were many more beautiful strategies.....which can easily drop the jaws of many of us ;)

But in this post, I want to specifically share an innovative approach figured out by Arjun to the problem: 29x8  (of course, his approach is actually an off-shoot of the understanding of the previous strategies, but the way he has remarkably applied this strategy to solve this problem -- this is something that blew off my mind !!  )

He says:

Lets consider (29 x 8) as (30 x 10)  which yields the product 300. 

But we could have also considered (29 x 8) as (30 x 8) rather than (30 x 10)   ... thus we have taken a surplus of (30 x 2) i.e. 60

So lets subtract this 60 from 300 which gives us 240.......which is nothing but (30 x 8). 

But now, we wanted to calculate (29 x 8) and not (30 x 8)....... Thus we have further taken the surplus of (1 x 8) which is 8

So we need to subtract this 8 from 240 which gives 232, the solution to the given problem. 

I want to make an honest confession, that when he verbally and quickly described these steps at the first time, it went off my head straight-away. It was only when his peer, Suvarna, joined him and re-framed this approach, a bit slowly, I could fathom this strategy...

I was just taken away by the levels of understanding they had transcended to. I was so impressed that I challenged them to solve another problem using the same approach :))

They did not even take a minute to shoot off the approach and solution. Check the snap attached and see if you can comprehend their understanding. 

But while they were enjoying and working on these problems using various multiple approaches (based on their conceptual understanding), thoughts flew by my mind... to ask the teachers and parents in this group ---

Should students be encouraged this way to think independently and discover their own approaches to solve the problems?

"Sir, plz aur ek problem do naaaa......." :-)

Hello friends,

How have you all been doing???   :)

Yes, I know its been looooong, and there is no post on this group, isn't it?  Yet, I am little happy that its not only me, but some of you too who feel the same.... as I have been hearing from you via messages and emails about the silence in this zone......I feel very happy when some teachers tell me that they are missing my Long Math-conversations. :)

But to be honest, this has also been one of the demotivating reasons for me to write in this group....

@ Friends, I do work with schools, teachers and students. But I am not an amazing regular full-time school maths teacher like you.... Though I have created this group, but I really wanted You all to make the best of it and take it ahead.  ...like the community owned  By the teachers, For the teachers! ...so that there is no dearth of a helping hand or an ear, in case any ones needs it... a platform at your finger-tips to connect with teachers from other schools, learn some interesting stuff from peers across the boundaries, write to them and discuss with them about your their struggles, listen to their experiences, failures, innovative practices, and success-stories of many maths classrooms...

I remember how overwhelmingly most of you had responded to the idea of this google-group and even to the previous maths-conversations posted by me... And believe me, I have been really wanting to share so many more such beautiful math-stories with all of you, that I have accumulated in past few months with amazing number of middle school students based in urban-private, municipal as well as rural schools..... But.....   :( 

So folks, let's have a deal....Really?  :)  For every two classroom stories shared by anyone in this group, I too will post one Story.. How about it?   :)

"There is no difference between living and learning... it is impossible and misleading and harmful to think of them as being separate.” ~ John Holt

So here is one math-story from my side... Actually, this one is a very short piece of a beautiful maths class I had with a bunch of Visually Challenged students... I really want to share all our class-room stories..... Every session with this lovely bunch has been an enrichment session for me, enabling me to become a better maths teacher.... I hope I will blog all these conversations one day.....but for now, let me awaken this group with this booster :)

The learners (me and my students) -- all of us were sitting on the floor, in a circle.... I think they were around 5-6 of them (class-6-8)

While thinking on a particular math problem, we arrived at a situation where I wanted Juilee and Prasad to have equal amount of money, given that they had different amount of money in the beginning (Juilee had Rs.2,500 and Prasad had Rs.3,250) Also, they are not allowed to discard or borrow any money from outside.

I noticed that they were struggling to get through this problem and hence after about 10-minutes, I decided to break my silence... by posing a simpler problem to them.... I made Prasad's amount relatively easier... it was rounded to 3,500.... ..Now it was not so difficult for one of them to arrive at the solution...

"Sir, I will give my 500 to Juilie and both of us will 3,000."

"Hmmm.... So let's switch back to our original problem then.....2,500 and 3,250"

"Sir.... this one is very difficult."

"Really? Let's give a try now.. I don't think, it should be difficult now :)  "

Again....... a silence......and a struggle...... 

Meanwhile, one more student -- Faizaan -- joined us and this problem was posed to him too... He too is visually impaired, and I am completely awe-struck by the remarkable mathematical aptitude of this guy. This 15-16 yr old chap has just blown me off every time, with his super computation and concentration abilities... 

"Faizaan.... Sir will not tell us the answer......you plz help us.", complained Prasad with a frustrating tone :)

Feedback of Parents

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?”