Teachers Day 2022

Like almost every teacher,  even I received - Teachers Day Greetings - from my students, their parents and even from some friends and teachers... 

Some things really touched me deeply.... And I felt to share those with you  

Photos Link: https://photos.app.goo.gl/saKbfao8vG38V4HAA

Description of Photos: 

(1)  Letter written in English to me by a student presently in class-12 science... We are associated since she was studying in class-5 marathi medium government school..

(2) Same girl made this chevda (snacks) for the first time on her own for me....  because I had probably told her once that I like this specific food item...

(3) Message typed in English and sent by a student who is presently studying in Second Year B.A.....We are associated since she was in class-9 marathi medium government school...

(4) Same girl has made & shared a short video of some memories / snaps, mostly of our celebrations during festival, picnic, get-together etc.

(5) Message sent by a person working as an IT professional since few years... We got associated when she was in her First Year Engineering in the college where I taught somewhere in 2009-10...

(6) Message sent by another student of her class / batch...

(7) Messages sent by my school teachers who have taught me when I was in middle / secondary year school..

(8) and (9) - Few of my government school students called to meet me in our class today, where we had some quality time together

(10) I could go and meet my school teacher who had taught me Maths when I was studying in Class-6....  Saw her after more than 25 years.... Luckily she could also see me doing some Math-Magic with her grand-daughter studying in Class-3... 

End of the Day:-

And I heard someone as if calling me from behind while I had started walking after boarding down the bus. I looked back. He said --- , "Your wallet fell down from your bag."

I checked my bag and found the zip of its front side open (may be while taking the ticket).. I ran towards him and thanked him deeply for his kind gesture...

Feeling grateful 

Thank you friends, for reading this message & checking the photos 

Rupesh Gesota

Demo-class to Teachers (on Fractions)

With children of migrant laborers working in IIT Gandhinagar campus, while a group of teachers sat around for observation (6-7 children of @ 13-15 yrs)

Children were asked – If there are 6 Rotis to be equally distributed among 4 of you, then how much will each get? They thought for a while and answered ‘Dedh’ (means 1 and half) , supported by reasoning. This solution was written on the board in picture and words. Similar representations were done on board for the next problem: 9 Rotis among 4 people, for which their answer was ‘Savaa Do’ (means 2 and a quarter). Now the challenge for them was to how to write these two fractions using numerals. Of the two numbers ‘one and half’ and ‘two and quarter’ they could write 1 and 2 , but how to write half and quarter.

So when the desire to know was created & expressed, they were given one proposal. I told them – I will tell you how to write half, but then you will have to find out how to write quarter. When they agreed to this condition, the notation was half was written on the board, but followed by a question – Why is half written as that way? And to this they reasoned – Because 1 Roti has been divided into 2 parts. So then they were asked how to denote Quarter, and they immediately suggested its correct notation. In fact they also figured out the notations for other smaller pieces like one-eighth (they called it as ‘aadhaa-paav’) and one-sixteenth. It was evident from their facial expressions that some of them had seen the pattern.

Now they were asked if half roti can be showed/ given in any other way. They were given paper plates to work and demonstrate. It took couple of mins and some rephrasing to understand this problem like if we cut the full roti (paper plate) in more than 2 pieces, and then can we make half.  And this was responded very well. They gave me 2 quarters. I kept one single half piece and two quarters next to each other, and asked them which one is half? And they said – Both. So then the problem posed was – there is one more name for this new representation of half. What it could be? After few seconds, this difficult to find answer was written on the board: 2/4 But then students were asked to reason about this and they did quite well with little help. Now they were asked to cut the plate into more pieces and make half and then express it symbolically. So they came up with the notations 3/6, 4/8 on their own with reasoning.  These fractions were written on the board side by side and they were asked to guess the next possible one... And one of them said 5/10.

Now they were asked which of the two is bigger 2/4 or 5/10, by writing one of these signs <, > , = between them. (I had to remind them of these signs they had used in their early school years like 43 > 17, 26 = 26, etc.)  And then one of them came forward and wrote ‘=’ between the two fractions 2/4 and 5/10 saying both are equal. They represent half. They also solved 1/2 is bigger than 1/8 with proper reasons (without any rules).

I now wanted them to see if they can find out the notation of another commonly used fraction ‘three-fourths’. So I asked them such question. How much Roti would one get if he is given half a roti followed by a quarter piece? This was backed by the use of paper plate (3 pieces of quarters). And they said it well – “pounaa’ (which means three-fourths).  When asked how would you write this using numerals, one of them said – 1/3, to which others agreed and one argued that its three pieces and one is taken away (because they saw three pieces of quarters, with one missing to form the whole).

This response was not new for me, as many other students have responded this way to three-fourths in the past. So now the teacher had two goals: 

(1) To help them discover that this is not the notation for ‘pounaa’

I knew that they had figured out the notations of other unit-fractions like half, etc. So strategy was to use this knowledge.  I asked them to draw the whole roti on the board, cut it into 2 equal pieces & write the notation of each piece. They did. Now repeat the similar process with cutting into 4 parts. They did. Now repeat the process with 3 parts. And they did this too. They drew the Y inside the circle and wrote 1/3 next to it.  Though the three parts were not precise but I choose to not correct their diagram at this moment because it did make sense to them.

So then I drew their attention to their previous guess/ solution for ‘pounaa’. And Wow!  They said that their previous guess/ solution was incorrect. So I cancelled it.

(2) To make them discover the correct notation of ‘pounaa’

I asked them – So what is the correct notation of this fraction ‘pounaa’? And they kept staring at the picture with wonder. Since they had figured out notations of different representations of halves few minutes back, so I thought to use their previous knowledge. I showed them the paper plate halves they had made & asked them what made them give those names to each of those. This conversation was enough for them to figure out the name of ‘pounaa’ (three-fourths) as 3/4.

And then finally, I wrote this notation of ‘pounaa’ next to its picture on the board J

Context : How this class shaped up? 

CCL team of IIT Gandhinagar ( https://ccl.iitgn.ac.in ) had invited me to interact with some KV and JNV school teachers who had come there for their residential training program. I was wondering as to what ‘sustainable’ could be done with teachers in just 2-3 hours, as I have now mostly restricted myself with long-term interventions with learners. So I thought of exposing them to reading & discussing of couple of thoughtful articles on maths education instead of doing some hands-on activities. 

My friend Swati https://azimpremjiuniversity.edu.in/people/swati-sircar suggested me to  demonstrate a session before reading a class-room story. She told me to do a session where few teachers can act / ask like children, but somehow I wasn’t much happier with having adults (teachers) as children (students) as I wondered if this conversation would be as natural/ spontaneous with simulated children :) Accidentally & luckily, just a day before my session with teachers, I found 3 children with their bags by the road-side, seemingly waiting for someone. On inquiring, they told me about their Nyasa School inside the campus. Then CCL team told me more about this social responsibility program run by IIT Gandinagar https://initaitives.iitgn.ac.in/nyasa/index.php 

So I couldn’t resist and found myself playing with these children, rather than attending the teachers training program J It was super-fun spending couple of hours with them (outdoor games & magic tricks).

While departing, I shared with few older children what was going on in my mind, if they would like to come for more playing (I meant playing maths now) the next day too (It was Sunday, a day-off for them). And they immediately & happily said YES! I guided them about the timings and venue. I got a bit worried the next day, when we (I and group of teachers) were ready for our session and they had not yet arrived. One of the CCL team members kindly waited outside the hall, looking for these children while I engaged teachers with some introductory fun activity. And soon, this bunch of children entered into the spacious hall full of teachers. I was thinking what must be going on in their heads while entering as they were not given any clue by me the previous day. So I oriented them about this, told them to relax & be as they were on the previous day ignoring other things / people around them. There was some hesitation in the beginning, but luckily the ice broke in few minutes and what emerged is as mentioned in the post above.

After the class, children left for outdoor play and teachers shared their observations & reflections. Some glimpses of this class and some bites of what teachers shared can be checked on links below. We then read and discussed an article that was published in this maths magazine. https://azimpremjiuniversity.edu.in/at-right-angles

-- Link of the article we read (classroom story having shades of Inquiry & Discovery:  https://tinyurl.com/3bvx36pn

-- Demonstrating a class with children : https://youtu.be/eIh4_OPVolM

-- Teachers sharing their observations: https://youtu.be/uEcshqAzBn

Thank you

Rupesh Gesota

www.rupeshgesota.weebly.com

Onset of out-station 5-day Maths Camps..

Sharing with you some thoughts and snaps of the lovely moments I experienced during my past couple of trips, to conduct 5-day summer Maths camps outside my hometown (Mumbai).

 

 

My immediate 2nd trip in May, after the 1st trip in April, became a possibility mainly because of the love and support of my friends & their families - Parin’s (at Bangalore) and Lalitha’s (at Tumkur) - who not only hosted me in their homes for 5 days each, but also opened up the doors of their hearts and homes to offer the access to learning (growing) opportunities to many unknown people in their cities. Lalitha even took me to a few government schools, some much away from her place so that those children too can get a flavor of Maths, which is generally missing in our modern racing systems.

 

Photos:

 

Bangalore Maths camp 2 

https://photos.app.goo.gl/ddwGPWtSQMor5gUR6

 

with govt school students at Tumkur

https://photos.app.goo.gl/Nysds6ZUQ18fzjWC8

 

with school maths teachers at Tumkur

https://photos.app.goo.gl/1bBZgUkjMuPyuwWKA

 

Similar generosity & care for the community was also expressed during my 1st trip in April, by the families of my 2 students with whom I had worked when they were in Mumbai, but then shifted to Bangalore. They talked around to form a group of parents for the maths program, and offered me their house to stay with them as long as I wished. And what better, I could even share my time doing fun maths with a few city children going through some physical & socio-economic challenges..

 

Bangalore Maths camp 1 (April)

https://photos.app.goo.gl/YrfHfHaSjoGh9Fpy8

 

Volunteering activities

https://photos.app.goo.gl/Qo78mdUoMQCMdogLA

 

It was nostalgic visiting those lanes, places, meeting friends and colleagues after more than a decade.. And all of this became much easier because both the families had immediately & graciously handed over the keys of their vehicles to me during my stay.. 

 

And finally, I cannot forget my friend Bijal who had instantly & voluntarily offered her younger daughter's train seat to me while travelling along to Bangalore from Mumbai, because I didn't have a confirmed seat..

.

What a blessing it is to be able to share what you are learning in your journey with people around and receive so much trust, support & wishes... There is so much love and goodness in this world, and there are times when one can feel it...

 

I have felt it more while reaching out to others because people do wish to take part in such positive initiatives everywhere which it why I firmly believe in the adage of *Vasudhaiva Kutumbakam*

 

Rupesh Gesota

www.rupeshgesota.weebly.com

Students make / extend the squaring trick :)

While playing with the square numbers, as to how can we find the square of numbers using the square of other numbers in various ways, we figured out the extension / modification of a well-known trick, something which was completely new for the teacher too :)

But before we go ahead, I would like to thank you all who read & even responded to my previous post with your lovely comments & thoughts. That was quite encouraging. In case you are the one who has not yet read the previous post, then I would recommend reading that first before this 

http://rupeshgesota.blogspot.com/2022/02/getting-into-algebra-through-arithmetic.html

Almost all students were aware of the 'trick' of finding the square of number ending with 5, thanks(?) to their teacher who had directly fed them this technique.

In case you are unaware of this technique then this is a good opportunity for you to figure out on your own. I have seen even few grade-4 students been able to do so, and pretty quickly :-))

Check the image given in the end of this post for your help !!

So after allowing them to impress me with this trick for few such numbers, I challenged them for the square of little different numbers like 48. And most of them, as I had expected or rather wanted, said:2064  (Do you get this, how they guessed this number?)

But when I asked them to verify their guess, they realized that its incorrect & they soon concluded (after trying for few other numbers like 29, 63, etc.) that the trick (quick method for finding the square of no. ending with 5), does not work with all the numbers.

So after this discussion / conclusion we had then moved to the other exploration (mentioned in detail in the previous blog post) and once we were done with that, one of the students told me that we can find the square of number ending with 6 by modifying the trick for the one ending with 5. And I was like highly. surprised with this claim. . Was she thinking over that one for this whole span?  And secondly I was also very curious now. Because I had never thought of / was unaware of this 'modification' till now.

She said, " We need to do some addition after applying the same method as that of 5."

She explained this with an example...

Let me share an image with you, allowing you to figure out what she did. Would suggest you to study this before you read the explanation below.

                                       

So yes.. This did intrigue me very much.... And hence we all tried our hands with various numbers. As you can see below: ..

                                        

Students also got excited looking at this method. And they quickly started trying numbers ending in 8 & 9 too... And their guesses to these did work. 

I hope some queries must have come to your mind by now :)

1) What about numbers ending with digits less than 5?

2) What is the explanation for this trick / method / algorithm ? [proof]

Well, these students did work on the 1st question and could crack it. However second question was just posed to them as of now so that they become aware of this possibility or rather necessity in Mathematics. 

In fact some of them became more curious to know the explanation now :)

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Some questions I wish to ask you:

1) Were you too aware of this particular trick (esp. the modification/ extension) ?

2) If yes, then nice.... If no, then what was your reaction to this one, and esp knowing that it got discovered by a student :) 

I would be happy to know your response to these questions and any other comments / thoughts on this post.

And yes, as mentioned in the beginning of this post, here is the image to help those who wish to find the trick for squaring the numbers ending with 5.

                                      

Thanks & Regards

Rupesh Gesota

www.rupeshgesota.weebly.com

PS: These session are with a bunch of government school students from disadvantaged economic background, as a part of maths enrichment program MENTOR run with them. More details can be found here: www.supportmentor.weebly.com

Getting them to Algebra via Arithmetic - 1

So by now, we (maths teachers) would most probably know many (or at least a few) ways to make students 'discover or arrive at' (& not simply teach / give away) the formula of Difference of Squares i.e. 

a^2 - b^2 = (a-b) (a+b) 

I too am aware of few interesting entry points to achieve this, but I was a bit surprised (at least in the beginning) when I realized midway during the 2nd session - while recording the 1st session's discoveries done by the students on the board - that this content can also be easily led to the above formula,.... and hence I decided to steer the car in that direction, though the plan was to take them to some other place :-))

So let me first share what happened in our 1st session:

We were playing with square numbers as in if they can find the square of a number using the square of its previous number. They did not yet have the knowledge of any of the expansion formulae like (a+b)^2  etc.

I came to know that they knew how to quickly find the squares of multiples of ten. So I asked them to first find out 21^2 and 31^2. They calculated using std. multiplication procedure, & then I wrote these on the board:

20^2 = 400

21^2 = 441 

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

30^2 = 900

31^2 = 961

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

I asked them if there is a way to find out the squares of 21 & 31 from those of 20 & 30 resp.?

It didn't take much time for some of them to quickly see the pattern.

"Yes, Add 20+21 to 400....."

This certainly surprised them and I gave them few more problems to play - use this pattern to find & verify. Luckily they were also aware of the 'trick' of quickly finding the square of a number ending with 5 (thanks to their teacher or youtube), So they happily & quickly calculated the squares of numbers ending with 1 and 6 quickly like 41, 76, etc.

So then came another question:

"Is it possible to find the square of 42 quickly?"

One of them said - "Sir, we can first find the square of 41 from that of 40 and then find out 42^2 from 41^2 using the same method.."

"Agreed.... But that's too long.... I need a short-cut... directly from 40^2..... Possible?"

And after some time, couple of them came up with this method:

"Add double the sum of 40 & 42 to 40^2"

Others verified this to be true. Some cheers. Some practice. And again another question:

"Now, how about 43^2 from 40^2 ?"

This time came a guess immediately - " Add triple the sum of 40 & 43 to 40^2..."

While this guess was followed by laughter of few, & some resorted to verification :)

And now the whole class was super excited by this emerging unexpected pattern..

When asked for 44^2 , ALL of them answered loudly & happily --

40^2 + 4 (40+44)

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

We did some more wandering around this zone , but with some deviation, about which I will probably share in the next post...& then we were time out for the day..

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

2nd session:

So today, I entered the class with a plan in my mind -- To lead them to the expansion formula (a+b)^2 through the previous session's exploration / discovery they did.

So we began with some revision:

1) I told them to find 11^2 from 10^2

They did.

2) Find 12^2 , 13^2  from 10^2 

They did these two quite quickly / easily.

3) Now they were told to find 27^2  from 10^2 

This took little more time, but then they figured out...

4) Now, time for another Change (in base)

Like 20^2 from 8^2, etc . You can check the image below for the progression.  One may also note that changes in representation are done - one step at a time and with Understanding, without any rules. 

Instance-1: Intentionally I replaced 12 by the expression 20-8 without brackets first. Few didn't find anything incorrect here, but with my pause, some could sense the flaw and they insisted for brackets with reasoning.

Instance-2: The term 7^2 was brought to the left of the equation without stating any rule like + term becomes - term when it crosses the = sign. The question asked to them was how to represent the addition statement using the subtraction statement.

Then, with appropriate explanation, it was time to generalize this arithmetic using variables but the words used were number-1 and number-2 along with their short-hand notation n1 and n2 instead of directly using 'a' and 'b' as found in most of the text-books. This made the transition to generalization (alphabetic representation) acceptable / easier for everyone to comprehend.

We decided to further short-hand this two-symbol & confusing variables (numbers) with one-symbol ones.

Though not all, but I could see some of them recall this formula that they had read / seen in their text-books & rejoice with Wonder and Joy !!

Yes, I am aware that this is not the derivation / proof of the formula, & we need to take them to that stage, but I think what is more important is that they could first arrive at this formula on their own with the knowledge/ tools that are accessible to them at their present level of ability.

Also, showing the visual representation of this formula to students is also in the mind. We might arrive at this formula using other ways too later :)

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

I will hopefully soon come back with the next post on -

a) some more interesting explorations that we did in our previous session (as mentioned in the middle of this post)

b) how students went towards & reached the destination that was thought of / planned by me (the teacher) initially.  (a+b)^2 using / from their explorations.

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

Meanwhile, I will be happy to hear your thoughts on this post / class and how would have you done in such case or how do you facilitate discovery of such formulae in your class.

Waiting for you :)

Thanks and Regards

Rupesh Gesota

wwww.rupeshgesota.weebly.com

PS: These session are with a bunch of government school students from disadvantaged economic background, as a part of maths enrichment program MENTOR run with them. More details can be found here: www.supportmentor.weebly.com