Sunday, October 23, 2011

Creating your own mathematics

Wanting to do something different for a change a few days ago, I amused myself with a little bit of mathematics. In our work, we are always trying to make the life of the teachers and children much better in the primary school classroom. Any mathematical activity which we undertake is with this objective.

I was playing around with a seemingly innocent problem of subtracting 169 from 637. However, I did it “my” way which was different from the “normal” and “accepted” method taught in school. Here goes:

100 + 100 + 100 + 100 + 31 + 37 = 468 (answer)

169 --- 269 --- 369 --- 469 --- 569 --- 600 --- 637

As you can see, we start with 169 and then progressively keep adding hundred. Why add hundred? Because it is easier to add hundred at a time. You are correcting an annual examination paper, and one of your children has come out with a working process with the answer as written above – what would your reaction be? Without being harsh on the teacher community, I would be tempted to say that this would invoke the much dreaded red cross against (and across) the answer. Not only that, the child would be pulled up with “This kind of an answer will not get you anywhere. No marks for this! Where is your working?” This would be followed by a “recapitulation” (torture) of the method:

637
- 169
-------------
468
-------------

Of course, you’ll have to remember how to “carry one”, “borrow one”, and “pay it back”.

Have you noticed how bus conductors deal with cash and return change? Suppose you have bought a ticket for Rs.5/-, and you hand over a Rs.50/- note. Many conductors, who have the habit being vocal about how they return the change, will be heard to say, “Ten, twenty, thirty, forty, fifty”. When “ten” is being said, the conductor actually gives you Rs.5/-, and then successively gives ten rupee notes till he comes to 50. In the process, you’ll have been given

5 + 10 + 10 + 10 + 10 = 45

Sabjiwalas use this method every minute. Where do the conductors and sabjiwalas have the time to “carry, borrow and pay it back?” Some of the sabjiwalas may not be even “numerate” in our definition. How are they able to manage?

Inspired with this discovery, I continued playing around with more problems. Consider the product: 17 X 14. This is how we normally do it in school:

17 X 14
----------
68
170
----------
238

There are other ways of doing this problem. One way would be:

(10 + 7) X 10 + 4)
= (10 X 10) + (10 X 4) + (7 X 10) + (7 X 4)
= 100 + 40 + 0 + 28
= 238

Can you think of other ways? How many of us wonder what takes place in the mind of the child when given a problem like

56 + 27. Answer?

There is something about the ease with which we use a ready made algorithm (an algorithm is a set of rules or procedures) whose logic we may not even understand, which stops us thinking about how we would approach a problem like 56 + 27. Mechanically, we follow the rule of “carrying over”. As teachers, we do not take the trouble to find out what happens in the mind of a child when it sees the problem. It may be more natural for many children to add 50 and 20 to get 70, and then add 7 and 6 to obtain 13. The final answer can then be got by adding 70 and 13. No wonder, we then let the children fall into a set of habits which, in the long run, close their minds to other possibilities. Our associations with particular procedures and our rigidity with particular symbols are so often tight that a child of eight may not know the answer to 7 multiplied by 5 but know straight away what 7 times 5 gives!

To subtract 169 from 639, it is surprising how many personal procedures (or algorithms) there are, and yet, we are often stuck with the method we have learned in school. Algorithms may help us to ease the problem of writing procedures in a symbolic way, but that is not the end in itself. Consider another personal procedure for the same problem (i.e., 637 – 169):

169:31
31 and 37 is 68
and there are 6-less-2 hundreds
468 (answer)

Here is a final example:
Since 637 = 100 + 100 +100 +100 + 100 +100 + 30 + 7, and
169 = 100 + 60 + 9, it follows that 637 – 169 can be written as

100 + 100 + 100 + 100 + 100 + 100 + 30 + 7
-- 100 + 60 + 9)
------------------------------------------------------------
0 + 40 + 91 + 100 + 100 + 100 + 30 + 7
------------------------------------------------------------
The reader is left to obtain the answer in whichever way is convenient. Notice how the above approach also helps in understanding the idea of the expanded notation. Notice also, that, in this case, “borrowing” as we would do it normally in a subtraction problem, is not required at all! After enough practice has been provided in the use of the expanded notation, the same problem could be now written as:

600 + 30 + 7
-- (100 + 60 + 9)
---------------------
468 + 0 + 0
---------------------

The process would be: Nine is greater than 7. Therefore we would need to borrow at least 2 from 30, and add it to 7. 9 – 9 is now equal to 0. By taking 2 from 30, we have made it 28. To subtract 60, we would need to take at least 32 from 600. Similarly, 60 – 60 is 0. Since we have removed 32 from 600, we are left with 568. 568 – 100 is therefore 468, the answer.

Another typical way of solving this problem would be – starting from the left, i.e., from 600, we could begin by removing 100. We are now left with 500. But we see that 30 is lesser than 60, and 7 is lesser than 9. Therefore, how much would we need to borrow from 500 to subtract 60 and 9? First, take away 30, and add this to the 30 we already have. 60 – 60 is zero, and 500 – 30 is 470. We need to add 2 to 7, so that 9 is also cancelled in the same manner. Finally, we are left with 470 – 2, which is 468.

Notice how the expanded notation is used, and how this gradually gives way to the illustration and use of the place value. The procedure of “borrowing” is very clearly shown. You don’t always have to borrow 10 or 100. On the other hand, the amount that you need to borrow is flexible, and depends on what is required to be borrowed. Borrowing can also be done in many different ways. By admitting this idea, we are allowing enough scope for the child to think and explore to find out how to go about a particular problem. And each problem brings with it a new experience and challenge. Often, the tendency is to teach expanded notation, place value and operations on numbers separately. This piece meal approach prevents one from seeing the connections.

The methods explained so far do not destroy for me the other ways of subtracting that I know. Very often, the procedures we follow in our minds when doing a problem cannot be put on paper without making them to appear clumsy and chaotic to the reader. The above examples are sufficient to illustrate this. This does not mean that these methods are not correct, are ‘slow’, and therefore should not be followed. The only advantage of following the method learned in school is that it can be put down on paper without the need for elaboration. Secondly, these methods help us to compute quickly. This brings us to the next question…

“What is the best method?” I do not want to ask this question without counter-demanding, “For what purpose?” There is nothing sacred about a particular method. In fact, the popular perception which tremendously influences our attitudes as teachers and parents towards children is that:

Speed = Brilliance,
Slowness = Dullness

There are certain misplaced notions about what about the ‘qualities’ of a ‘good’ student of mathematics - the ability to compute fast, and the ability to handle big numbers. Shakuntala Devi is often referred to as a great mathematician (which she’s not!), because she can multiply two twelve digit numbers with ease, or obtain the square root of a ten digit number faster than the computer. Often, parents and teachers take pride in such skills that their children may have developed. Pray, what purpose will this serve to a child in an ordinary school classroom and later on in life? As adults, we can only pretend to understand the value of, say, 1 light year (the distance covered by light in one year, at the speed of 3,00,000 km/sec) which is 9460800000000 Km. Why should we torture our children then?

Mathematics is not just about how fast you can calculate, or your ability to play around with big numbers which may mean little to you in everyday life. It is not limited to the application of ready made, uniform procedures to the solution of problems. It is about cultivating the ability to create and explore paths which we can identify with. It is often said that in order to learn mathematics, one needs to create (re-create) it for oneself. The examples discussed so far clearly illustrate this. What we consider to be the “fundamental” or “basic” principles of mathematics at the school level have taken thousands of years to develop. It necessarily follows that we cannot force the learning pace with children. Yet, how easily frustrated we become when we see a “wrong” answer! The truth may be that this wrong answer represents a genuine exploration on the part of the child, a struggle to comprehend.

Most often, we do not let out children explore different ways to arrive at an answer with the argument that forming habits (in my words, the ability to mindlessly repeat) are a protection against the confusion that could take over if the mind began to charge off in too many directions. This uncertainty of not knowing what will happen makes us hold our cards close to our chests, and “protect” the interests of the child.

The truth is that, as parents and teachers, we would like our children to cultivate and perfect these skills and habits so that they can ‘do well’ in the examinations and score high marks. Remember, the competition is tough out there! But, in the name of this competition, are we not inhibiting the natural ways of learning in our children? You decide…

How can we have an environment where both experiences, i.e., formalized procedures and treatment of topics, are reconciled with exploration, imagination and the 'freeness' to think? While it is possible to go in all kinds of directions without necessarily having the ability to be able to compute fast, or be precise, this imagination would be useless without care in developing appropriate skills. On the other hand, these skills (of calculation, of being able to apply procedures, etc.) cannot be developed in isolation of the ability to be able to explore, imagine and think freely.

New Delhi
26th April
1998

Frozen Education -- a study of obstacles to children's education in Jammu and Kashmir

(The detailed study is not attached here. If you are interested in reading it, send me an e-mail. I will get back with a copy.)

A number of people and numerous conversations that we all had together are all responsible for this study. I will begin by mentioning Renu Singh from the Delhi office of Save the Children (SC), who first invited me to undertake the exercise of finding out the obstacles to the education of all children in Jammu and Kashmir. We began talking about it in the second half of 2009, but it was not until February 2010 that discussions on developing the tools for the study began. Sharif Bhat and Mufti Riyaz from SC’s office in Srinagar, and Neha Gandotra based out of Jammu were of tremendous help, particularly in identifying an enthusiastic group of young people who were willing co-learners and travelers and who did all the hard work of walking up and down the beautiful mountains, seeking to meet teachers, children, parents, and community leaders to understand what stops children from going to school. This list is long, but I must mention everyone -- Ufera, Soliha, Huzaifa, Umer, Shahid, Saleem, Wasim, Muzaffar, Tahir, Neelofar, Sharika, Rashida, Fayaz, Sheikh Ali, Mohd Ali, and Farrukh. I’m sure they enjoyed all the hard work and I hope they will eventually develop into the next generation of education activists that our society badly needs.

Bashir, Gulzar and Targez drove us across the seven districts and helped us unearth many a story whose hints we can only give in this document.

Save the Children’s local NGO Partners also deserve mention, especially for their support for the logistical arrangements in all the districts. The study would have been enriched further had they played a bigger role. Many thanks therefore to Yateem Tust, Yateem Foundation, Jay Kay Women’s Welfare Society (JKWWS), Modern Culture Club (MCC), and Kargil Development Project (KDP).

Representatives from the educational departments told us much and helped us develop insights regarding the way their departments function (well, in many cases, how they do not function!). While some were wary of us, many opened up in the hope that their voices would reach far and remove the constraints in educating all children. In the same vein, Renu Nanda from Jammu University, Javed Rahi from the Tribal Research and Cultural Foundation, J & K, Prof. Madhosh (retired from the Kashmir University), and Mohd Rafi, District Collector, Budgam, need to be thanked in particular for enriching us.

Finally, my deepest gratitude goes to the hundreds of parents, children and teachers who spent many hours with us, understanding patiently our incessant questions before responding to them, and allowing us to share their spaces at home, in the school and in their communities. Their spontaneous warmth and affection helped us to carry on. We learnt much in those hundreds of conversations, but I’m not sure if we have done enough justice in documenting the multitude of experiences in this document.

We have tried to map the kinds of obstacles that prevent children from going to school in the state of Jammu and Kashmir, enjoying school and learning something meaningful and relevant that they can use on their lives. These obstacles are quite complex, and our observations show that they are often rooted within the families, local communities and the functioning of the schooling system itself. They may arise due to the phenomenon of Gurbat (the Kashmiri word for poverty, which goes beyond income related poverty), or may still be affected due to civil unrest (as in the case of the Kashmir Valley, when many teachers happily do not attend school whenever there is a hartal). In many cases, it is in the nature of human behavior that the greatest obstacles arise – an insensitive parent, family member, an abusive teacher or a corrupt and complacent worker in the education department – all of them play significant roles in making or breaking the educational cycle of many a promising boy or girl. We have seen all these examples in the course of this study. I still remember interviewing this teacher in a Poonch village for over an hour. He was very articulate and gave me the impression that he was one of those hard working and conscientious teachers. Happy that the interview went off well, I then walked into the next room where children from that school were waiting for me to initiate a discussion. The first thing they said was “Jis teacher ke saath aap baath kar rahe the, woh ek mahiney ke baad school aa raha hai…mahiney mein ek ya do din aata hai bas…” This put me in deep thought. Isn’t this one of the biggest obstacles, I wondered? Coupled with this, when one heard stories of how corrupt the system of governance (including the educational system) has become, one wonders which obstacle is the biggest of them all – the family, the teacher, the education department, civil unrest, or political instability, among others?

Perhaps the answer lies in creating an environment where everyone concerned -- parent, teacher, politician, policy maker or administrator or child, is able to overcome one’s own limitations, develop deeper understanding and sensitivity to the child’s needs, and bring all the resources at his or her command to make the difference. The energy for this has to come from within the government, from that conscientious minority pegging away, trying to reach the poorest of the poor. Still, this may not be enough, and we will need to involve external agency to mobilize the constituencies to develop a mass of critical consciousness that can make a difference.

2011

Kuni

I

My son liked this story so much that I had to tell it to him many times over, Ditto, with my daughter. I still tell them Kuni’s story if I do not have any other story to tell. Little did I realise that my childhood tryst with Kuni would mean a lot to my children. At some point, as I realised how much they liked this story, I decided to write it for children. That day is not far off.

The year was 1978 – thirty three years ago! ‘Thirty three years ago…’ sounds nice when one tells a story, especially to those much younger than you. It feels as if you have accumulated so much experience! Anyway, let me get on with the story. My granny’s ancestral home in Bengalooru’s Cantonment, off Queen’s Road, was about to be sold for rupees seventy nine thousand. There are conflicting versions of how it happened. My parents always maintained that my uncles – three of them – were very keen on getting that house sold. Nobody wanted to stay together, and they had their own lives. So the question of keeping that property for everyone to live together one fine day didn’t arise. Often, they talked to my granny about selling the house and getting their share of the little booty. Left with no other option, the old woman gave in. Since she was staying with us, it was agreed that my father would get a substantial share of the proceeds.

I don’t remember the many details of what it took to sell our home, but there is this vivid picture of a discussion the entire family had in our small living room with a chap called Samiullah who finally bought the house. I was allowed to sit on a wooden stool somewhere in between two adults as they all haggled over the final price. The other piece of memory is that of my father and uncles counting seventy nine thousand rupees! I marvelled at their speed, as I always do when I stand at the bank counter sometimes to see money getting counted. No cheques, no electronic transfers. We didn’t even imagine that there could be something called electronic money those days. I’m pretty sure that my father didn’t have a bank account. He brought home his meagre few hundred rupees salary in cash and handed it to my granny who managed the household with my mother. Life was simple those days, and I didn’t even realise as a child that I missed out on anything.

Years later, when I could understand life’s nuances better, I was told by my older cousins and some aunts that my father had committed a Himalayan real estate blunder. I was told that that the Muniswamy Road house was worth crores! Had it been sold later, everybody would have become ‘lakhpatis’, at least – so went the argument. I had several such conversations and everyone seemed to point fingers at my father for his lack of foresight. My parents always thought otherwise and said ‘Look, they needed money, that’s why they sat on your granny’s head and made her agree.’ These arguments mattered little when I was young.

Anyway, we left 14, Muniswamy Road one day in 1978 from my childhood home, never to go back again. It was the first time that I moved from one home to another. My father called my childhood home an ‘outhouse’ meaning that all the rooms of the house were built in a line at one end of the plot of land leaving a large compound space in which we had the Tulsi katte, several flowering plants, a huge coconut tree (which my mother said was planted on the day I was born) and a big tree that gave the fragrant Sampangi flowers which were in great demand in the Shivaji Nagar market nearby. My friends and I played and pranced around in the large compound. I remember tri cycling, playing hide and seek, watering the coconut tree and climbing the Sampangi Mara. There are other memories, such as pissing out by standing at the room window holding its railings, when it was raining. Bahadur, our tenant who had rented one room at the corner of the plot, ran past and, when he saw this parabolic stream coming out of the window ducked underneath and carried on!

II

We shifted southwards to a place called Jayanagar 9th block. For one year, my father said, we would have to stay in a rented place while our new house was getting built in a new residential area called J. P Nagar. The roads were just getting laid out, and my mother would often walk from 9th block to J. P Nagar 2nd phase to supervise construction. Later, our house would appear as a white speck from the end of 9th block Jayanagar. It was only the second house in all of J. P Nagar!

Our rented home in 9th block was very small and I remember the difficulty I had adjusting in this space after 14, Muniswamy Road. Anyway, I quickly made many friends in the neighbourhood and spent most of my time on the streets of 9th block, playing lagori, marbles, flying kites, spinning the tops and playing cricket – all on the street. I keep telling my son that these games have more or less disappeared from our cities nowadays. I can vouch that kite flying in Bangalore has all but disappeared. Kids spend a lot of time playing virtual games on the cell or computer. And then there are these malls, which have mushroomed all over the city, where you pay through your nose to satisfy your child’s desire to play. Gone is that spontaneity which we enjoyed as children.

Kuni quickly became a dear friend. At that time, I didn’t know that ‘Kuni’ in Kannada actually meant dog! This only dawned several years later. There she was, simply called ‘dog’ by all of us. For a dog that spent most of her time on the streets, she looked quite big and brown, almost like an Alsatian. I discovered from my friends that she had quite a few children who, like her, inhabited 9th block’s lanes. They all looked quite different. She must have had several lovers! I would feel strange but curious and excited nonetheless at the sight of her mating with another dog -- the famous ‘doggie position’, as I learnt some years later! Other dogs would wait to mount her, in heat, often snarling at each other and mounting her even if she was stuck with another dog. Some of my friends threw stones at dogs who had coupled. They enjoyed doing this and didn’t think much about the hurt they inflicted on other living beings. Children can be violent, too. I wondered why dogs had to get stuck that way…

Once I fed Kuni with some chapathis, she expected some titbit or the other from me all the time. My mother and granny readily obliged and we usually fed her food that we ate – rice, dal, chapathi, biscuits. Pretty soon, she started spending a lot of time in the vicinity of our new home. To escape the heat, she would just jump over the gate, come in and lie down in front of our door. The small plot of land on which we lived was partitioned into three separate houses, two of which were rented out. Subbamma, our landlord, lived in one of these houses opposite ours. She didn’t seem to mind Kuni.

As the days passed, Kuni started waiting for me to return from school. I usually walked home from the 9th block bus stand and as I entered the last stretch every day, I would often find my mother standing at the gate, waiting for me. Seeing me, she would look back and say something to Kuni, who often waited inside. Kuni would then charge across a very short stretch of compound, leap over the gate and come charging towards me, her tail wagging madly, joy written all over her face. Our evening meeting usually had the same script – she would stop just in time to avoid a collision and then would kneel in front, straightening her front legs, wagging her tail, waiting for me to initiate some affection. I would go close to her, pat her head, and ask her how she was. She would then pounce on me, and I would hold her forelegs as she walked on the other two, and that’s how we often covered some distance. Then she would run away, only to return after a few minutes. I fed her the biscuits I was given along with a glass of milk.

Every evening, as my friends and I played on the streets, Kuni hung around. Sometimes, she would go away for an hour two, but always came home for dinner. She would stay the night in the compound. Winters in Bangalore were quite nippy in the late seventies. I still remember how misty it would get in the mornings and how our skins would crack during those months. We all liked to ‘smoke’ in the mist then! To make Kuni comfortable at night, we spread out a gunny bag in front of our door. My parents and granny would not let Kuni in. I often tried very hard to cover her with another gunny bag, but she would usually come out of it and was not comfortable being covered.

Once, our neighbours had a social function and there were many people who had come for lunch. It was some festival and I remember being home that day. Lunch was spread out on banana leaves, which were all later neatly rolled out and thrown outside a little away from our house. There was no public dustbin. It was all left to the cows to eat the banana leaves. Before the cows came, Kuni and her friends appeared and rushed madly towards the banana leaves, hoping to find some morsels of food. There were many dogs fighting for very little food. Kuni was getting side lined by the other dogs and I didn’t like it one bit. I scared away the other dogs with a stick, and this gave Kuni a chance to eat some food all by herself. The other dogs kept growling, but they did not dare to come near me. I think Kuni became more close to me after that – at least, that’s what I thought then.

One day, as I walked towards 9th block bus stand to board my school bus, Kuni started trailing me, much to my surprise. She had not done this before. I kept asking her to go back but she wouldn’t listen. So she walked behind me, all the way till the 9th block bus stand. The school bus was ready and would leave in a few minutes. I got in, and Kuni followed! She didn’t want to let me go. What I did, I tried to tell her to get down and when that didn’t work she had to be pushed out by the bus conductor. I kept worrying about her all day and was relieved to find her fine in the evening.

As the days passed, Kuni and I only got closer. She would spend most of her time at our place, or somewhere nearby. She went with me whenever I set out of the house, to play or visit a friend’s place. On my part, I looked after her as best as I could. I became friends with her children too.

Finally, she didn’t come home in the morning over a weekend. I remember going out and looking for her. I enquired about her with a few friends, but they didn’t have an answer. So we started searching in the lanes of 9th block. ‘Go to the main road, there’s a dog lying there’, said one of our neighbours. We found Kuni lying inert in the middle of the main road, the one that connects the Bannerghatta road with Kanakapura road. Why is she lying down in the middle of the road? I thought. She would be run over by the Gaadis. I didn’t understand, till my friend pointed out to blood beneath her head. With a heavy heart, I realised that she had been run over by a callous driver. I couldn’t believe it. ‘Can’t we take her to a doctor?’ Nobody answered.

I came home crying. I looked back -- Kuni was surrounded by her children, all with sad looks on their faces, their tails down, sniffing her on that fateful main road.

Raipur
October 2011