The gentleman who taught infinity

This is my first book, which is now under publication...wait for the book, whose story I have captured below. 

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The gentle man who taught infinity

            Nearly 30 years after they have parted ways, a student suddenly remembers his math teacher from school days. As he starts writing about him, he is surprised to discover that he remembers a lot of the interesting stuff he was taught. Gradually, as everything comes back, he ends up writing a full narrative. At the heart of this narrative is Channakeshava, the gentle teacher who took his students on a roller coaster ride of the world of mathematics. Though he worked in a regimented school, Channa often broke the boundaries of a restrictive school syllabus, sterile textbooks and mind numbing examinations.

In The gentle man who taught infinity, we journey with Channa and cross the Seven Bridges of Konigsberg, grapple with the intriguing Barber's Paradox, understand what took mathematicians nearly 350 years to solve Fermat's Last Theorem, appreciate why mathematics is beautiful and explore the paradoxes of infinity. Along the way, we are introduced to great mathematicians like Euclid, Bhaskaracharya, Cantor and Euler, among others.

Using storytelling to great effect, this remarkable teacher showed that mathematics is very much a human endeavour. It need not be the drudgery that we make of it, in our mad pursuit of marks and grades. Instead, the learning of math can be fun, meaningful and fulfilling for everyone. As the narrative unfolds, teacher and taught, subject and craft all get intertwined and result in a fascinating story.

 

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The CCE Conundrum

The entire enterprise of organized education, with its lakhs of schools, millions of teachers and thousands of department workers who constitute its bureaucracy at the national, state, district and sub district levels, who strive to make education happen in our country, can be said to be driven at different points in time and in various ways by a few fundamental questions. These questions actually flow from one to the other, and they are all interlinked. Each question can generate many different and debatable responses, with these debates often going back hundreds of years at least. There are no clear answers even today, and that is what makes the task of educating children that much more fascinating and complex. Everybody can have an opinion here. That is the best as well as the most frustrating part.

The questions

The first and obvious question is: Why should children go to school? This has to do with what we usually term as educational aims or more specifically, the aims of schooling. These aims can be many and often and they can be at cross purposes, if you will. Ultimately, the aims we choose are also a reflection of the kind of world or society we want to create. There is this belief that education has a role in the creation of that society, that schools have a social role. The Naxals, for instance, will have a conception of educational aims that will be very different from the aims a capitalist or industrialist would articulate. The two worlds are quite different – the collision of these worldviews is perhaps at the heart of the Maoist problem. That is a separate debate I don’t want to get into right now.

The second question is: What should they be doing there (in school)? How should they pass their time there? This usually boils down to the question of what children should learn, since schools claim that they teach children this and that – the passing on of knowledge is seen as central to what schools do. As part of this learning, what kinds of experiences would we want children to get at school? Again, the responses are bound to be divergent, depending on who is responding and under what conditions the response is being made. If a natural disaster strikes, for example, there are often shrill cries for including disaster preparedness in the curriculum. If there is a communal conflagration, then secular elements get emphasized. And so it goes. It is a long list. And it is controversial, as we have seen in the recent instance of the tamasha that has been generated over the inclusion of the Ambedkar cartoon in a textbook prepared by NCERT.

I can give another example. Some years back in Jammu and Kashmir, it was decided that English would become the medium of instruction in all schools from grade 1. The story that did the rounds was that the Education Minister was very upset one year with the 10^th standard exam results of the state. Someone suggested to him that this was because children were not getting an English medium education! And that is how an education policy was influenced. The point here is that what children should learn gets decided often in convoluted ways that seem to defy reason sometimes.  

Related to what is the question of how children should be taught. If getting good marks and cracking the exams is the idea, then everyone will apply their minds to beat the exam. There will be mock exams and the like, and the experts who have cracked it earlier will have tutorial sessions with the hopefuls. Bihar’s ‘Super 30’ is a classic example, where year after year children from disadvantaged backgrounds are coached to crack the entrance to the IIT. If ‘developing critical thinking’ is a key aim, then teaching methods as well as content will be different. Getting good grades in the exams may then become a peripheral concern. Of course, it can be argued that the two are not mutually exclusive. So the second question is the one that is concerned also with the methods of teaching and learning.

The third question is: How do we know if we are moving in the direction we want to go…? This is really at the heart of the assessment of the schooling experience. What should we do to find out if any child who is subjected to this system called school, is actually becoming what has been envisaged under question 1 above…? This is a complex one for it may be difficult to limit the response to the period of schooling. Given the vagaries of human nature, many things may happen after an individual has moved out of the institution of schooling. (Note here that we use the word school to indicate the various stages of formal, organized education.)

Question 3 is inextricably linked to question 1 – if this is what we want for our children from school, we need to do something with them which will move them in this direction and, further -- we have to develop a system which will enable us to find out if we are moving in the direction sought. These aspects are all intimately linked with each other, and cannot be seen in isolation.

I would like to mention here that the preparation of teachers is at the heart of the education enterprise and we will not be able to move an inch if teachers are not prepared enough to address the above questions in their day to day practice. And since we have not yet reached a stage where machines are teaching children (if one goes by what hard core proponents of IT sometimes have to say with regard to the potential they see in digital technologies for education, it would seem as there is no other alternative but to substitute teachers with machines), we may still consider the teacher challenge as the central one in education. 

The question of assessment

I will primarily address question 3 and the difficulties that come with it. Not because questions 1 and 2 are not important. They are, but at the moment, we are a bit obsessed with question 3, thanks to the emphasis on ‘Continuous and Comprehensive Evaluation’ (CCE) in the Right to Education Act. Perhaps very few of the many Education Acts enacted by countries across the world mention exactly how children should be taught, and how what they have learnt should be assessed. Ours does. I will come to this a little later.

Perhaps the easiest way of developing a system of finding out if children have moved in the directions we want them to go, (or, in more direct terms, to find out if they ‘have made the cut’) is to have exams. This is what we have been doing for a long, long time, with great effect. It feeds our collective anxiety and it has spawned a huge industry. Many years ago, when I was in school and later in the pre-university course (11^th and 12^th std.) we often heard that so and so college or school in Bangalore was the ‘best’, or provided ‘good education’. Then we would hear some name or the other of faculty members who were ‘fantastic’ and whose tuition classes if you attended would guarantee you good marks in the public exams and a passport to a better life.

It dawned on me much later that this entire discussion of best schools and colleges was more or less centred round how many ranks they obtained in public exams, and what percentage their ‘best students’ got in those exams. ‘Good education’ therefore got focused on questions two and three – for instance, what were the best methods of teaching which would lead to great marks? What kind of exams would ensure that children would make the cut? Now, exam performance is not the same as academic excellence, which is a much broader idea. Further, it certainly cannot be equated to ‘good education’, which can and does mean many more things. But a certain dominant interpretation of ‘good education’ can make it look as if exam performance is the main thing involved. Ultimately, every other consideration falls by the wayside. So it may not be uncommon to hear parents and teachers compare two schools and say one is better than the other since it churns out good results every year. It does not matter how those results are churned out and if children have understood (for instance) what they have studied -- or if they have inculcated along the way, a love for learning. They are treated much like the race horses which get whipped every now and then to run faster and faster and cross the finishing line. 

I must delve a little more on exams -- the system of exams with their marks can result in certain amusing and painful results and realities. When I was in 12^th standard with a science combination, we had two clear options – engineering or medicine. The IT industry had still not entered the scene in Bangalore in a big way in 1987 when I cleared my 12^th std. exams. The Call Centres and various other options that this thing called IT has spawned were still far away. So our teachers kept egging us on to do either engineering or medicine. It was made to look as if life would not be worth living if we didn’t become engineers or doctors. Since Karnataka those days (more so now) had a greater number of engineering colleges than colleges that made you doctors, the cut off percentage for engineering was lower than that of medicine. Those aspiring to become doctors had to study that wee bit harder and they had to get well above 95% or 97% if they had to secure a merit seat in a reasonably good medical college.

So, we had a system which decided who would build bridges and design machines and who would look after people’s lives, based just on a mischievous combination of their exam marks! No other attributes necessary to these professions were considered. And that is how I became an engineer, only to leave the profession some years later. The system has not changed a bit since those days.

The second thing about exams, which some argue is very practical, is that they play the role of the perfect filter for the educational system. Not everyone who starts out in school stays long enough ends up doing the same things. Imagine if they did! There is the rural-urban divide, there is the boy-girl divide and then there is a divide depending on which community you come from and where you are located. The system then resembles a pyramid, and children keep routinely falling off it as they try and negotiate its steep and treacherous climb. Only a few get to the pinnacle. This, the exams achieve perfectly. Don’t just ask what happens to the ones who fall off the pyramid. They do not get the goodies that the others get. They get this message that they are second rate performers, relegated to some menial jobs. Knowing fully well that we have a system that produces inequality, we still go on saying that education is the fundamental right of every child -- as if by just saying it things will fall in place!            

CCE as Saviour

This limited approach has been the cause of some concern for many years – that we have the grandest of educational aims, enshrined in the various documents that we have routinely produced for more than a hundred years, but had the narrowest and harmful ways of ascertaining if we were moving in the directions we wanted to go. Of course, there are problems with the kinds of textbooks that the state produces, as well as the ways in which teachers are prepared. Along with this, no one had cracked the puzzle of assessment – how can we put in place a system which reflects the educational aims that we set out to achieve?

In the nineties, much before the struggles for instituting a law that recognizes education as a fundamental right in our country occurred, we started hearing about ‘Continuous and Comprehensive Evaluation’ (CCE). The usual lament was: ‘How can we just go on with a system that pronounces judgment on that one day and within those few hours…? We need to have a system that is fairer and which just doesn’t rely on marks at the end of the day…’ To add to this was the other refrain that the current system was not looking at the child as a ‘whole’ – exams, at best, tested short term memories whereas the child was much, much more than a person who was expected to give standard answers and repeat what was taught. So the terms ‘continuous and comprehensive’ came into being.

The idea behind the term continuous was that a child’s learning must be assessed continuously instead of only relying on term end tests and exams. This cumulative assessment would then provide a truer picture of the child’s abilities, difficulties etc. and would also provide the teacher useful reference points to intervene in case there were difficulties that needed to be addressed. The idea behind the term comprehensive was that we were assessing children only on the subject matter areas – math, language, science and so on. Would this provide a complete picture of the child’s abilities? Was education all about learning only these subjects? Obviously not! So went the arguments for an alternative or more comprehensive way of finding out where children were going. ‘Comprehensive’ was then understood in terms of expanding our understanding of what children could actually do, other than engaging with and learning the ‘core’ subject matter, other than just reading, writing and arithmetic – things we made them do.

I remember the jokes that usually did the rounds. ‘What’s the fuss anyway?’ Someone asked. ‘All this is important if teaching takes place in the first place!!’ Indeed, that was one concern which many were just not prepared to look at in the eye. Where was teaching happening? Survey after survey showed how little children learned, even on the most basic aspects of reading, writing and arithmetic (the three R’s, as we called it) even after 5-8 years of schooling. We talked about teacher absenteeism, non-teaching duties of the teacher and the poor quality of teacher preparation. What was the point of erecting a grand plan for evaluation with ill equipped teachers who could not even manage the basics? We seemed to have assumed that the basics were all in place, and that schools were now prepared to do this CCE. But there was also this argument that this examination paranoia was mostly applicable to urban areas fuelled by unhealthy competition.

Anyway, in workshops, meetings and many formal and informal interactions, we kept hearing this clarion call for CCE. It was seen as a part of what was broadly termed as ‘education reform’ – the set of policies and actions that were needed at all levels to improve the education that children got which we were all unhappy about. Many years passed, but CCE didn’t find the light of day. There may have been smaller experiments and pilots but these did not end up influencing day to day practice on a larger scale.

Meanwhile, document after document lamented our insensitive system of assessment which stifled all creativity and created distress instead. Much was written about it in 2005, the year that saw yet another National Curriculum Framework document being prepared. There were these ‘position’ papers – 21 in all, that elaborated the main document. One of them was entirely about examination reforms – how to make exams more appropriate for the ‘knowledge society’ of the 21^st century, and so on. The position paper on ‘Curriculum, Syllabus and Textbooks’ carried, in the end a very angry and powerful quote from the Scottish pilot turned educator, David Horsburgh thanks to whose efforts in a small school in Andhra Pradesh many states are now attempting to follow ‘Activity Based Learning’ based on his ideas (that is a different story altogether!):

‘…evaluation has been one of the most important forces in the gradual degeneration of all school education…the whole antiquated evaluation process, should as speedily as possible be hurled lock, stock and barrel out of the windows of our educational system in just the same way as the chamber pots were emptied in eighteenth century London…’          

CCE travelled some distance before it we saw it appear as part of law – the Right to Education (RTE) Act, in 2009. The Act stated in no uncertain terms that in order to find out what a child was learning, one had to do it continuously and in non-threatening ways and not just use term end tests and exams. You could talk to the child and get her involved in assessing what she had learnt, observe from a distance the various things that the child did and note them, provide descriptive feedback, give exercises of different kinds (useful things that we do very little about), or you could resort to some paper-pencil activity that was really engaging (again, something teachers don’t do much of).

All these methods could be employed together or separately – but they had to be done again and again, and not just once or twice in a year to brand the child in any which way. One had to build up some sort of a history to construct a more complete picture. And then, the whole process had to be ‘comprehensive’ – one had to look at the ‘whole’ child, something that we are woefully lacking in at the moment.

            It is interesting to see the initial scramble that results when anything becomes a law. Suddenly, something changes in the air. The RTE Act, which came into being after a tortuous journey, became the reference point of sorts after August 2009. In workshop after workshop, we started hearing statements like this one, which we started using in our various interactions as well: “This is not a scheme…it is a law. And do you know what will happen if this law is violated…? You can be punished -- Ye Dandaneeya hai!!” Further, we understood that the compulsion and responsibility of providing education is on the Government, and that Government alone is answerable -- as if it was not all these years!

It will be interesting to see how the government gets ‘punished’ as we go along. In any case, we have a poor track record of punishing or taking the government to task on the several promises it has made over all these decades to its people.  

While the Act has several provisions along with clear timelines and targets, parameters related to quality come into effect immediately. So we can start taking the government to task right away, and fill the court rooms with all sorts of pedagogical cases! There will be interesting court cases, I’m sure, if one takes the judicial route as the last resort to change the educational system. I’m not sure if this will alone work. And judges will then need to be educated about child centred education and the like. We can prepare primers for this.

The aspect of Continuous and Comprehensive Evaluation sits squarely within the notion of quality of education because this is what helps us find out, using various ways, the inherent abilities and potential of the child, and also if what we are doing in school is having the desired results in the directions we have set out. As I write this, state after state has, among other things, attempted to do something about CCE which was on the backburner for all these years. Workshops and trainings have been held galore and many states have developed manuals for teachers. The CCE lexicon sounds like this – formative and summative evaluation, scholastic and co-scholastic areas…rubrics and tools for this…phew! All of these are supposed to take care of CCE. It is now becoming a nice business and many organizations are beginning to become ‘experts’ who will deliver ‘CCE Products’ for our thankful consumption, both online and through workshops.

Assessment as a tool for reform…?

A question that has consistently bothered me is – how fundamentally different are these developments from what we have been up to so far? It is one thing to define or lay out the aspects that we would want to assess in a child. This wish list can be really long, and we can include all our fantasies of ‘good education’ in it. The issue, however, is whether the teacher is prepared to do it. From all accounts, the answer to this is in the negative. I do not wish to take stock of the results of efforts over the last twenty years to improve what happens inside the classrooms, but I do believe that the outlook is still gloomy. For one, if we map what children learn on even the most basic aspects – such as 3R’s – reading, writing and arithmetic, we find worrisome gaps. We know that children at the end of five years barely end up learning stuff that is equal to grade 1 or 2. Further, there are many studies that show just how un-child friendly, monotonous and inequitable our classrooms are, barring some exceptions. In such a scenario, one wonders if reform which uses assessment as a basis is going to make any difference. In fact, it may only end up causing more confusion.   

So far, CCE looks like old wine in a new bottle, which is what happens when we do tinkering. The earlier tests and term exams seem to have now morphed into this formative and summative evaluation business, which is now called the ‘CCE method’ of teaching! The spirit, therefore, has not changed. To add to this, teachers seem to be making effective use of the ‘no detention’ policy – that a child should never be detained in a classroom, at least till he reaches the eighth grade. The spirit behind this no detention policy was that the child should not be penalized for under achievement or no achievement. The argument was – why wouldn’t any child want to learn, provided we made learning interesting, engaging, meaningful, and provided all obstacles in the family, school and community, were removed? The problem was thus located outside the child. Fair enough. But teachers seem to have interpreted it quite differently, and this has resulted in ‘automatic’ promotion from one grade to the other – never mind if the child has learnt anything or not! In many ways, this is an excuse for not teaching which teachers can exploit.

So, CCE has become continuous promotion with little or no learning.

Teachers have so far been adept at giving paper-pencil tests and scoring answer sheets using red ink. It gives them a sense of pleasure, of power, to be able to point out mistakes and slot the child. Even here, we can ask if understanding is assessed on these paper-pencil tests and exams. Currently, it is assessed in limited ways, since that is not the emphasis in teaching. Now we are saying that the teacher should move away and not be limited by this. We are saying that the nature of paper-pencil exercises and tests should change and assess whether the child has understood something or not. Secondly, we are saying that there is something beyond the so-called scholastic/subject matter areas – any child comes to school with a range of abilities – children can express and imagine in a variety of ways, and they may be good in certain kinds of physical activities and so on. These must be recognized (we use the term ‘co-scholastic’ to underline this) and must contribute to the overall assessment of the child. So, if a child draws a three headed elephant using his imagination, or if a child draws a parrot with red feathers and a green beak (which I have actually seen in a school in Chhattisgarh), what do we expect the teacher to do…? How does one evaluate this in CCE? What is the reference point? First of all, how does one evaluate expression, say, a drawing? I have a problem with this. We might at best say that the child can do such and such a thing, and make a record of his abilities, and celebrate these. It would be such a pity if a teacher sees these drawings and says, “But this is wrong! Such animals don’t exist! What have you done?”

Teachers, who have a very clear conception of ‘right’ and ‘wrong’ (and most teachers belong to this group), and who are schooled to believe that there is only one right answer, will take such children to task. They have not been prepared for this. Similarly, if children start asking questions which the teacher may not be able to answer, what is the teacher supposed to do under CCE? So far, we have shut them up. But are we prepared to let them ask?   

The more I think about CCE, the more I am inclined to believe that it is much more than evaluation. It is first of all about understanding the child  – where she comes from – her background, her family, her identity, her various abilities and the challenges that she faces as she negotiates school and its requirements. So let me add CCU – Continuous and Comprehensive Understanding, as a pre-requisite to CCE. Without this, can we say that our evaluations are going to be sound and helpful to the child? Secondly, is everything about evaluation? I suppose, no. Like I mentioned above, not everything that the child does needs to be judged on some grading scale. We may describe and narrate what the child does. This becomes a cumulative record of that child’s experiences and abilities as she passes through school. A well-kept record will read like a story of that child. It tells us a lot, actually.

What is currently happening in the name of CCE is sad. We have centrally developed manuals and formats which the teachers have to fill out routinely. There is very little of remedial teaching or support to address problems. Ideally, if we use this approach, we should be able to map out each child’s abilities, challenges and prepare plans to help the child overcome these challenges. We must also be in a position to truly appreciate each child as a person. But ‘filling the format’ has become the end rather than the means.

In order to look at the child differently, and in a more wholesome manner, the teacher’s conception of education, its purposes, its processes and its possible outcomes, has to undergo radical changes. This, I believe, has not taken place. All these years, we have focused more on techniques of teaching. There has been very little discussion on the ‘why’ and ‘what’ of education and how the teacher sees these things. Further, we have also not looked at the teacher as we would want the teacher to look at the child. Teachers, like children, need to be nurtured. Instead, we treat them with distrust at best.

To sum up, it seems pertinent to ask – how will you measure or evaluate something when in the first place you are not doing most of what is required to make that thing happen? We are trying to evaluate the outcomes of a teaching-learning process which is meant to be child centred but which is actually anything but that, which still requires children to learn by rote. Then we add a few things we need to measure under the so-called ‘co-scholastic’ domain without seriously asking whether we are doing them (sports, music, art and craft and the like) in the first place in our schools. These aspects, which should actually be central to the educational experience of every child, are relegated to mere formalities.   

You cannot just do something all of a sudden because some law or Act requires you to it. Presently, everyone is running around preparing CCE manuals and the like, hoping that these missing elements in a child’s education will somehow appear in the classroom once training on CCE is done. We need to dig deeper and ask why, in the first place we were not doing these things earlier. 

I believe that the route to reforming, changing, improving or transforming (call it what you want) classrooms cannot come alone from assessment. Something else needs to happen first in the way children are taught. This contains the seed for measuring what they have learnt. Assessment cannot be seen isolation, as we are seeing it now.

Which is why, I say that CCE as it exists now will remain at best another scratching on the surface. We have put the cart before the horse. Then we have missed the wood for the trees. They exist together and cannot be seen in separation.

Raipur

June 2012

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Enjoycation

          The word ‘education’ can have a variety of meanings depending on how you look at it and what you want from it. Thus, we hear the clichéd ‘Education is a contested terrain…’ statement often in discourses. The multiple meanings that we derive from education may or may not have anything to do with the etymology of the word -- which, from various accounts, comes from the 16^th century usage of the term ‘Educere’ in Latin, essentially meaning “bring out” or “lead forth.” Another Latin word, ‘Educatio’ means ‘breeding, bringing up, or rearing.’ What we do in the name of education is thus related to the etymology of the term, at least where it concerns the usage of the word in English.

Interestingly, the word ‘Siksha’ which we use to denote education (for instance, as in Sarva Siksha Abhiyan) has its origins in Sanskrit. Its meaning doesn’t quite come anywhere close to the meanings we ascribe to the educating of children and actually means the study of pronunciation of words and syllables through correct intonation. This is not central to the process of education now though it may have had critical importance during Vedic or pre-Vedic times especially when transmission of knowledge occurred orally. But I have seen discussions and debates about how important it is for children to correctly pronounce words. ‘Uccharan theek nahi karte ye bacche’ (note the emphasis on these children) is a constant refrain we hear from teachers. ‘Uccharan’ (pronunciation) is thus a frustrating area for some teachers, while others do not seem to be bothered too much about it.

Anyway, I think this is like missing the forest for the trees. Thankfully, in this huge Government of India flagship program called the Sarva Shiksha Abhiyan, we are not talking about universalization of correct pronunciation. Imagine if we did! We would have then spent much on establishing studio facilities everywhere and would have supplied headphones to every teacher and child to pronounce their way to success.

Further, I must add that this business of pronunciation (not ‘pronounciation’, remember!) is related to the dimension of Class, and how we use Class as a category for choosing who we relate to, and to what extent we relate to that person or persons. But this is a topic that needs separate discussion.

This is what fascinates me about education. It is a bit like searching for the Holy Grail, and it refuses to get tied down. There is no one formula that defines it, though there may be some underlying principles. Even here, there can be disagreement. Everyone -- educated or not, can have their opinion about it.  For some, education is like a passport to success – you get good grades in all your exams, land up a good job and fall into a certain slot in which you keep going all your life. This is the ‘reward’ that society gives you for studying hard. For others, like that great Brazilian educator Paulo Friere, education can lead to ‘critical consciousness’ in society – an interesting state in society in which a huge mass of (mostly poor) people start asking why they are that way, and why the rich stay rich. This realization, flash point or critical consciousness can then spur them into action like it did for instance in Naxalbari in West Bengal in May 1967, when poor peasants chose not to keep quiet about their exploitation at the hands of the landlords. Forty five years later, the ‘Naxals’ have been described as the biggest internal security threat by our Prime Minister. That’s another story which is unfolding.

Of course, one can ask – did the peasants develop that critical consciousness as a result of schooling? Very unlikely. But I am prepared to argue that they underwent a process that must have included, at various points in time, thinking about their condition and questioning it. We cannot also say if everyone thought in the same way. There must have been local leadership that goaded people to do something. Anger must have played its role. The point is not whether the poor peasants went to school. Friere believed that schools or organized learning in particular and education in general, whether it is inside or outside the boundary wall of the school, could result in critical enquiry leading to emancipation of oppressed people. He was able to demonstrate this through his work.  

Apart from yielding diverse meanings, what is equally interesting is the way the word education gets morphed to yield newer words with different meanings. In a sense, you can say that the word constantly gets bastardized. This is what I keep pointing out in conversations with friends. For instance, my daughter attended a preschool called ‘Jumbo Kids’ which stated in no uncertain terms that it provided ‘Kiducation’. This is what happens when kids get education.

There is a school called ‘Brighton Academy’ in Raipur which is on the way to the N. H Goel ‘World School’ which my daughter goes to. Interesting are the ways in which schools get named. Interesting too are their claims. The names of many schools contain the word ‘International’ nowadays. In my daughter’s case, her school’s claim is that it is ‘Global’. Do we mean international or global citizens as an outcome for children attending these schools? I’m not sure. Or, are they using international or global methods in teaching? Any school that has a swimming pool, air conditioned classrooms, dining spaces where five star caterers serve food, horse riding and golf, and exposure visits to other countries considers itself as a prime candidate for using ‘international’ or ‘global’ as part of its name. Even schools that do not have these facilities use these words in their names. Of course, to be fair, some schools try and use some innovative or progressive approaches to teaching children. So I will not entirely deny their claims.  

What is the message these schools want to send out to parents? Since I have not yet figured this out, I’ll just say that it’s a marketing gimmick. Just naming something does not change its nature. The very same schools will provide the most conventional education, as we have seen, far from what Friere wanted. Just as names do not change the nature of things fundamentally, appearances too do not contribute significantly to the educational experiences of children. You can have wi-fi enabled classrooms and web enabled systems where your child’s progress may be recorded, or horse rides for that matter but the teaching-learning can still be rote and stifle the child’s potential and creative impulse.

The Brighton Academy which actually looks like a corporate office from the front will supposedly produce ‘bright’ children, whatever that means. What takes the cake, however, is that this academy claims that they provide ‘Enjoycation’. This, I believe, is their version of a particular form or kind of education. Or education itself, in a new form -- you can take your pick.

I can hazard a guess and can imagine what the principal of the Brighton Academy would have to say: ‘Education should be enjoyable, you see. We should not burden children. That is why we use a new term: Enjoycation!’ How noble! This sounds a little like the term ‘Joyful learning’ which was bandied around mostly in the nineties. Also, in the nineties, we were also subjected to other terms, such as ‘child centered’ and ‘activity based’ education. On the whole, the idea was that education should become less burdensome on the child and actually should be more play and fun. Several slogans like ‘Khel Khel mein Siksha’ were coined. Then, there was that famous committee called the Yashpal Committee, which talked about the burden of non-comprehension and the weight of the school bag on the child’s delicate shoulders.     

I can understand the reaction to educational processes that are boring, do not allow the child to express, get involved, ask, do, understand, and all that. I can also understand the need to change the environment in schools which induces fear in the name of discipline. Yet, I feel that we did not work hard enough to understand the implications of these terms in practice. We are very good at writing documents, including that great sounding National Curriculum Framework of 2005 but pretty poor when it comes to these ideas in the reality of the Indian classroom.

In our half-hearted efforts, these terms often assume quite different meanings from their original intent. Thus, it was a common sight to see thousands of teachers across the country do a physical activity (like some action song, for instance) before the ‘actual teaching’ began – this was how activity based learning was interpreted. ‘Actual teaching’ which followed the activity, was business as usual. And then, in the thousands of workshops all over the country, we did discuss these terms but I suspect we did not really grapple with them deeply enough. For instance, take child centered – should we provide education based on the interests, needs, moods etc., of the child? Should we make it easy so that children do not have to struggle…? These and other questions require deeper engagement. Child centred does not just mean doing what the child likes to do.

My contention is that the whole business of enjoyment and fun and less boredom in education has the effect of diluting the educational process. What is the process of education without a struggle? The human mind has to struggle and apply all its powers to understand patterns in nature and in society. There is real enjoyment and fun in doing this. The ‘Aha!’ moment that a child experiences and the ‘Ahaa!’ that the teacher experiences on seeing the child thus, come as a result of great struggle which we are all capable of. This is a different view of Enjoycation. In any case, as we get along in life, there will be several instances where we will be forced to give up cherished notions, forced to move out of our neat little grooves, our comfort zones, reconsider our understanding of ourselves, people, relationships, society and the kinds of lives we would want to lead. Sometimes, it can be all very painful. All of these, which I believe are experiences of learning, leading to one’s education, do not come easily – one can actually run away from them if one wishes to, if one does not want to confront them head on. The enjoyment, however, consists in the savoring of the moment of discovery, insight and the individual liberation that these experiences bring.

I’m not sure what drives the thinking behind the Enjoycation that the Brighton Academy would like to give. I would beg to differ with them if they have coined the term merely as a marketing strategy. I suspect they have.

Raipur

May 2012

Channakeshava

Baldwins

My father would not have settled for anything other than a church managed school for my education. This school was for him an epitome of educational quality. The emphasis on instruction in English, seen by many as a passport to success in today’s world, and the ‘discipline’ that he thought was part of the ethos of every Christian educational institution, would help me go forward in life. I’m not sure what my mother or granny thought of this argument, or even if they did, if it made any difference to my father’s outlook. I remember heated discussions at home about the ‘excessive fees’ of Rs. 56/- per month in Baldwins in 1978, but my father was adamant in his belief. He was prepared to scrounge around from his meagre salary to give me what he thought was a good education.

The first school that I attended was St. Anne’s Convent on

Cunningham Road

, a couple of kilometres away from our cantonment residence in a much quieter and more beautiful Bangalore those days. An ageing rickshaw puller took a bunch of four or five of us every morning to this school, which was co-educational up to the fourth grade. I had some interesting experiences in this school about which I will write separately. When I reached grade 3, my father must have worried about my next school. Since he had set his sights on the well-known Baldwin Boys’ High School located near Johnson Market on

Hosur Road

, I was coached to clear the entrance test.

I joined the 98 year old Baldwins in 1978. But I would meet Channakeshava (Channa, as he was affectionately called), the mathematics teacher only in 1983 in the eighth grade. For reasons I could not understand, I had to repeat grade 3 despite passing the entrance test -- may be the chaps who gave me admission thought that my grade 3 pass certificate from St. Anne’s was not good enough. They made me lose a year, though.

Beginning

There is nothing unique about my mathematical experiences up to the seventh grade. We had sincere teachers, all ladies who went through their motions of teaching to the syllabus and preparing us for the exams. I don’t recall being excited about math. Mrs. Thomas, our class teacher in sixth who also taught us math, was perhaps the best of the lot – she was always pleasant, never lost her patience, and once told me that she could solve all the problems from the fat and complicated looking 10^th std. ICSE textbook written by someone called O. P Sinhal. My respect for her increased several notches after that even if I did not have the chance to verify what she claimed! That complex looking book which my seniors brought to school every day gave rise to both fascination and fear, and if someone told me she could solve everything in it, I could not help but only admire her for that.

I have memories of two ‘mathematical events’, both rooted in incomprehension, in class 6 and 7. The first was to do with the idea of inequalities. Since my father was confident of teaching me till things got mathematically tough around grade 7, he took it upon himself to teach me those vexed inequalities using signs like >, <, ≤, ≥ which appeared in between two or three numbers. Some thing was greater than or equal to something, while it was lesser than or equal to something else – and so it went! One evening, when much of it went overhead, my father asked me to do what he called Guddi-patam – ‘Mug it up!’ He said when he saw me struggling with some problems. In fact, that was his constant message throughout my school and college days, even when I was doing my engineering. If you couldn’t understand it, it had to be ‘mugged’ up only to be vomited in the exams. I couldn’t figure out the concept of inequalities despite his best efforts that evening. My reward for incomprehension was a hard slap! That was the only time he slapped me, for I could see that he was ashamed, apologetic and emotional about what he had done.

The second event was to happen in grade 7. I found the idea of Integers, those negative numbers that extended the other way from zero on the number line, difficult to understand. Till then, everything seemed fine in mathematics. One had to ‘solve’ problems and get the right answers. Or so I thought. Little did I realize then that the world of mathematics was full of strange numbers (rational, irrational, surreal, transcendental and the imaginary and complex, among many other such creatures) which were related to each other through even stranger relationships. Beautiful relationships, many mathematicians would say, as we will see a little later.

Beauty

I reached the eighth grade in 1983, finally. It was Channakeshava’s first class with us in June that year and it had very little to do with our syllabus and textbooks. He could have so easily started off with the first chapter, and we would have plodded through, as usual. Instead, he wanted to begin by showing us ‘beauty’ in mathematics (This, I say in hindsight. It wasn’t so obvious then). So he posed the following question on the blackboard:

142857 X1?

This was easy. Next, he asked:

142857 X 2?

While we all got busy with the multiplication, he quickly wrote the answer – 285714. Then, he asked again:

142857 X 3?

428571 is the answer, which he had written down as if he knew it all along while we were busy multiplying! Then he went on to show what happens when 142857 is multiplied by 4, 5 and 6. Interesting stuff was emerging:

142857 X1 = 142857

142857 X 2 = 285714

142857 X 3 = 428571

142857 X4 = 571428

142857 X 5 = 714285

142857 X 6 = 857142

What we saw was a ‘cyclic permutation’ of the number’s original digits even as we multiplied it by 2, till 6. It was difficult to discern a pattern in the manner the digits got shifted -- sometimes one digit got shifted, sometimes two, and sometimes three but in all cases the ones getting shifted were consecutive digits. But what was startling was that it was the same number whose digits got shuffled around. Such numbers are called ‘cyclic numbers’, CK told us. And mathematics is full of such curiosities, he added, which can be studied by just about anyone. In conclusion, he asked: what happens when we multiply 142857 by 7? We wondered if the original number would get recycled again. But no! What we got instead was 999999! This was puzzling, indeed! While we were soaking in all in, CK just smiled.

That is how we were introduced to the beauty of mathematics in grade 8. I use the word beauty because it hits you, as a beautiful sunrise would – everything appears to be in place and it cannot get better than that moment. In mathematics, beauty lies in the patterns that unfold, like in the above example. At the risk of deviating from Channakeshava’s story, let me illustrate mathematical beauty through another example.

It is acknowledged that among the most beautiful relationships or patterns in mathematics is perhaps the Euler equation (after the great Swiss mathematician, Leonhard Euler) or identity e^i∏ = --1. To put it another way: e^i∏ + 1 = 0. Now each of these numbers – e, i, Pi and 0, are unique numbers which keep recurring in mathematics. Without getting into the details, for one can write pages and pages about each of these numbers, let’s just note that ‘e’ is what is called as an irrational number (simply put, a number that has a decimal/fractional portion that never ends and which can never be computed!) whose value is 2.7182818284…this ‘e’ is also the base for the natural logarithms.. There are different ways of arriving at e, and I don’t want to get into this discussion here. But I will just say that if you start operating the expression (1+1/n)ⁿ as ‘n’ gets bigger and bigger, you will approach ‘e’.  Try it out and see!   

But ‘i’ is another equally crazy number (called the imaginary number) which is the square root of minus 1 – written as √-1. This is like asking ‘What number when squared (multiplied by itself) gives – 1?’ This ‘i’ is neither 1 nor minus 1 and is therefore called imaginary!

Well, well, well! Why on earth does one need such numbers? You may very well ask. These numbers were invented to derive solutions to certain types of equations is all that I can say here. And then, you have the mysterious Pi, which has had a long and chequered history and can claim to really be the king of all numbers, sitting proud on a pedestal in the number world, and daring anyone to claim him! One would think that the ratio of the circumference of the circle and its diameter (that is, the number Pi as it is defined) would yield a decent number – after all, we can easily construct circles of definite sizes in terms of their diameters and radii. When people attempted to find this ratio centuries ago and in different cultures across the world, they found that it was a little more than 3 and so they approximated the ratio to 3. But that little portion beyond 3 kept nagging everyone and as the years progressed, it was discovered that that little portion is a never ending stream of decimal digits! And so, Pi (the symbol is ‘^∏’), as this ratio was called, looked something like 3.14159265…ad infinitum! You can draw the circle and its diameter but can never measure the ratio of its circumference to its diameter accurately! This was what stumped Pythagoras and quite shattered him – that there could be numbers whose actual values we might never know. Let me give another example – we can draw a right angled triangle whose sides are one unit each – we know from the Pythagorean Theorem that the length of the hypotenuse (the side opposite the angle that measures 90⁰) is √2, which is an irrational number. So there you are – you can draw this triangle but can never measure the length of its hypotenuse! It cannot get crazier than this!  

In the case of Pi, You can use mnemonics to remember the decimal places if you want (this is a little bit of consolation, anyway). Once, when I was invited as a math teacher to a creative writing class in English (talk of integrating the disciplines!), I wrote out the first 100 decimal places for Pi and asked the children to write a paragraph in such a manner that each word that was used would have as many letters as the value of that digit, in that sequence – for example, if you say, ‘May(3) I(1) have(4) a(1) large(5) container(9) of(2) coffee(6)?’ you can write out the first seven digits of Pi: 3.1415926. And when they followed this rule, the children wrote out some hilarious stuff which made us all roll with laughter for the next few days.

The other thing we did in the same school where I taught many years ago was that we (the children and I) prepared a ‘Pi tail’ using old newspapers – we made these newspaper strips, and used marker pens to write out the first 2500 decimal places of Pi – this resulted in a Pi tail some 850 feet long! On Science Day in 1994 in that school, we took out this paper tail and wound it round the school, starting from the library. Throughout the day, I saw children running around, following the tail – along the walls, inside the rooms, inside one of the toilets and then out all the way into the jungle gym, where the tail finally climbed a tree and went all the way up, and we left it dangling there. Finally, when a group of children who I was teaching about Pi, came to me and said they understood then why Pi was called irrational, I knew I had managed to convey a fundamental mathematical truth. I’m not sure if any of them thought about it beyond the science day.       

Now, let us visualise the Euler identity: (2.7182818284…)^{(3.14159265…) X (√-1)} + 1= 0

Note that ‘raising one number to a certain power’ (which is what we do in the above expression) simply means this – if you raise a number ‘a’ to the power ‘n’, it means you multiply ‘a’ by itself ‘n’ times. So if we say a³, it would look like this: a x a x a.

There is something very intricate and crazy happening with the Euler identity which we lay persons can only intuitively grasp. You have this irrational e which when raised to the product of another irrational (Pi) and the imaginary i, gives –1. Then you bring this –1 on the left side and that leaves us with zero on the right hand side. What beats me is how this happens. First of all, you have this e and ∏ whose decimal portions are never ending or recurring as even the most powerful supercomputers of our age have found. If one can find a way of adding them, it might result in another irrational number. Ditto, if one is subtracted from the other. If they are multiplied by each other, we might end up with yet another irrational number. But if e is raised to the power of the product of i and ∏, then the strangest of things occurs – the infinite strings of digits just disappears, leaving us with –1! And when this is added to +1 as we see in the Euler identity, we are left with nothing!

How can this be...? What is happening…and what is the role that ‘i’ plays in this? Euler, the great genius he was, discovered this profound relationship more than 300 years ago. How did he come to it understand it, and what would he have said about it to a layman he would have met on the street? The Euler identity tells us something fundamental about how the world of numbers is structured and ordered. You can visualise this in terms of a number line and try and locate all these numbers there.  

It struck me that the Euler identity in many ways is like the alchemy (or so I thought) I once saw in the chemistry lab of our school. Our chemistry teacher was demonstrating how different acids react with each other. So there was this colourful liquid which was mixed with another colourful liquid and Voila! The mixture became colourless! There was a collective gasp in the room, and since fairly large glass containers were used to pour out these liquids, the entire exercise looked spectacular. While we were all aware of the chemicals involved and the chemical reaction itself, there was no discussion on what made the coloured liquids transparent when they came into contact with each other. Obviously, it had to do something with the change in the structure at the molecular level, and the way the new molecules interacted with light. But these aspects were probably complex to be discussed at that stage in that chemistry class. We did not know enough of quantum physics or chemistry to answer that question.

The Euler identity is similar in terms of mathematical alchemy and results in zero, the midpoint on the number line, the mathematical nothingness. I was explaining this to a friend recently and he said his hair stood on end when he thought long and hard about what was happening. I’m sure that any mathematician worth his or her salt might be able to explain away or prove this phenomenon (I use the word phenomenon quite deliberately here) theoretically. There is fun, however, in keeping it a bit mysterious and struggling with the infinity of digits that appear and disappear in the problem, much like the colours we saw disappearing in the lab. I’m tempted to quote Benjamin Pierce the American mathematician below:

“Gentlemen, that (e^i∏ + 1 = 0) is surely true, but it is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore, we know it must be the truth.” 

You could spend your entire life unravelling these relationships, like mathematicians did. What was even more fascinating, as I realized much later in my explorations of physics, were the intimate connections these abstract relationships had with the ordering of the natural world. Surely, there is something fascinating here and I’m not sure if even the best mathematicians have understood why this is so.

But human relationships are far more complicated, I think!

Channakeshava

CK had cast a spell with his cyclic number example. And there was more to come that year as well as in grades 9 and 10! There wasn’t much exploration of numbers after that session on cyclic numbers -- it had to be business as usual in a school that prided itself on exam results. But there was a definite difference in the way we were taught mathematics. First, CK came across as very calm and self-assured both as a human being and teacher. We couldn’t get close to him, and there was no question of getting friendly with him – he ensured that there was always some distance between us and him – fair enough, I think, given that he may have been in his late forties or early fifties then. But when it came to learning math with him, it was sheer fun. With his partly bald head and bespectacled face and his immaculate suit, he looked quite the archetypal mathematician himself!

When I look back on those three years that he taught us, I realized how grounded he was in the knowledge of the subject itself. Years later, I realized that he was indeed an avid explorer of the mathematical world. This, I now believe, is a pre-requisite for every teacher of the subject – a teacher of mathematics cannot limit his or her teaching to some sharing of procedures to solve problems but share the real excitement that comes from exploring patterns and relationships in the mathematical world. And he must do this in a manner that children can understand. While access to good reference materials would certainly help, the fundamental requirement is curiosity, and that is sadly in short supply among most teachers – how can one convey excitement to the child when one is not generally curious…? 

There was this unmistakable gleam in CK’s eyes whenever he conveyed something fundamental or profound. When he emphasized some deeper aspect of the topic at hand, his mouth would open a bit wider than usual and he would pull his lips backward to make a point -- like I remember when we first heard the Latin phrase ‘Quod Erat Demonstrandum’ when we were grappling with proof in geometry. Q.E.D, literally meaning ‘What was required to be proved’ is what one says when one has demonstrated a proof. A proof is a kind of achievement in the mathematical world through which one demonstrates underlying patterns and relationships between numbers, spaces and the like.

Everyone was attentive in Channakeshava’s class, from the first to the last bench because we knew that we would end up learning something.

The other unique element of his teaching was his impeccable running handwriting and use of blackboard space. I have not come across another teacher who used the blackboard as effectively and as beautifully as he did. His handwriting was a visual treat, not flamboyant but clear and pleasing to the eye. The sense of proportion he had when using the blackboard was just amazing. He would clearly demarcate areas on the blackboard – for instance, for drawing diagrams, for arguing, and writing down the steps of the analysis, and for rough work or calculations.

Proof

That year, we encountered the idea of the theorem for the first time in geometry. We had earlier learnt that most common property of triangles – that the sum of the three angles of a triangle always adds up to 180⁰ (Well, I later realized that it doesn’t, always, and depends very much on what kind of surface you draw the triangle on – this kind of thinking has led to the development of different kinds of geometries, such as Riemannian and Lobachevskian geometry, for instance.). The geometry we have studied in school is called Euclidean, after Euclid, the great Greek geometer who lived more than 2000 years ago. His brand of geometry – Euclidean geometry is what you do on flat surfaces.

‘How do you know this is true?’ CK asked as a matter of fact. ‘Measure and see and you will get 180⁰!’ we said.

‘How many triangles should I draw and measure?’ This question stumped us a bit and I remember that we didn’t agree on any one number. In fact, any number would have been arbitrary – 10, 50, 100, 1000…? ‘In any case’, CK said, what if the 1001^st triangle’s angles do not add up to 180⁰?’ There was no response from the class.

‘For this reason, we have to prove that no matter what, the angles of a triangle add up to 180⁰.’ And so we went about proving this elementary theorem and learnt along the way that the word ‘theorem’ is nothing but a statement claiming such and such a thing, which has a proof that is generated using deductive reasoning – this is something like saying ‘If A, then B’. Now, all these theorems in geometry that one encounters in school (and usually breaks one’s head against) rest on certain foundational statements called axioms. In his Elements, Euclid had proposed five axioms on which the whole of his Euclidean geometry rests! I had difficulty in accepting these axioms as ‘self-evident truths’ which did not need proving but from which all of Euclidean geometry flowed. Later, one realised that there had to be certain starting points anyway without which, one couldn’t even move forward.

I’m recounting all of this because CK shared this fascinating history with us. Imagine -- we were learning something that was thought of two millennia ago and put down in a book! In the coming months and years, we would prove many theorems, in geometry and algebra and trigonometry, and in algebraic geometry. I enjoyed the study of theorems and their proofs only intermittently and kept wishing that CK would tell us more and more stories instead. There were days when the proof of a theorem would just appear effortlessly and then there were days when one would struggle. Channa would go on, as enthusiastically as ever, adorning blackboard space with these eternal theorems and their mind bending logical proofs!

Years later, when I read that great book Men of Mathematics by E. T Bell (a feminist would have questioned the title itself!), it struck me when he said that the Euclidean method that insisted on proof may have actually hampered the development of mathematics by at least two thousand years. On the other hand, if mathematicians had followed the unfettered thinking of Archimedes, one of the three greatest mathematicians ever (along with Newton and Gauss, and, I would add, our own Ramanujan) that Bell identifies, the age of modern mathematics (and with it, science) could have occurred two millennia earlier. Anyway, that’s a discussion for another day. I wonder what position Channa would have taken. Would he have taken Euclid’s side or would he have plumped for Archimedes…?

Famous problems we discussed

As we journeyed with Channa, we were always treated to snippets from the fascinating history of mathematics. Some years later, when I took to teaching mathematics at The Valley School in Bangalore, I realized how well he had employed the history of mathematics as an effective tool to make the subject absorbing. In fact, I realized later that this was the cornerstone of his teaching and was perhaps one of the most effective ways in which one could fire the imagination of the learner. That there were hundreds of stories of mathematical discovery, many of them rooted in daily life problems and that mathematicians were as human as anyone else, hadn’t occurred to most of us who by then had reconciled to our highly developed math phobias.

The Seven Bridges of Konigsberg, the Barber’s Paradox, Fermat’s problem and the Four Colour Problem are among the most interesting and intricate problems of mathematics which have defied the best minds for generations. Attempts at their resolutions have given rise to entire branches of mathematics. CK told us those fascinating stories. I remember them to this day. I passed them on to my students in the best manner I could.  

First, the ‘Seven Bridges of Konigsberg’ -- I must share this.

I’m not sure which grade it was, may be 9^th or 10^th, around the time when we were supposed to learn the idea of matrices. Any lesser teacher would have plunged headlong into the subject and would have introduced matrices as ‘an array of numbers…arranged horizontally and vertically…and these are the rules for their addition and multiplication…’ This would have been followed by boring problem solving from the exercises in the textbook. With CK, that was not to be. He simply had to get to the root of the idea, to the bottom of the matter, and share the excitement of his explorations with his students. And that is how we were treated to the 300 year old ‘Seven Bridges of Konigsberg’ problem. I do not remember the connections that were made (or mentioned) between the Konigsberg problem and the idea of matrices in mathematics, but I do remember we spent a couple of periods discussing this puzzle that originated in daily life and informed the development of new areas of mathematics such as graph theory, which finds wide applications.

Briefly, the Seven Bridges problem originated in the town of Konigsberg (founded in 1254 A.D) in Prussia (Kaliningrad in modern day Russia). The town itself was made of four land masses that were connected to each other and the mainland by a network of seven bridges built on the Pregel River which ran through Konigsberg. The seven bridges (not all of them exist now) were named ‘Blacksmith, Connecting, Green, Merchant, Wooden, High’ and ‘Honey’, and the problem, which was attacked (actually, negatively resolved) by the great Euler in 1736 was to walk through the entire town of Konigsberg in such a manner that one would have to cross each bridge once and only once and finish at the starting point. The story goes that when people of Konigsberg took their leisurely walks on Sundays and used the bridges to reach different points in the town it occurred to them that they could actually generate a puzzle based on the bridges. Thus, the above problem was posed.

What an introduction this was to the topic of matrices! We spent two classes on the Konigsberg problem, trying to draw our myriad routes across the seven bridges only to be stumped in the end. And then, CK told us that Euler stated that the Konigsberg problem could never be solved. The actual proof came much later, in 1873 by one Carl Hierholzer. But the thinking that went into Euler’s reasoning spawned an entire new branch of mathematics called Graph Theory which was in turn intimately connected with the idea of Matrices. It also preceded the development of another branch of mathematics called Topology, Channa said. He did not of course go into details and enlighten us about these connections, nor did we have the time to actually see how Euler resolved the problem (the proof is not so difficult to understand, as I discovered later, but during Euler’s time, it must have been breaking news!). When I look back today, it amazes me to even think that a math teacher could have thought so much in depth, to present just one of the myriad topics we studied at school. How much he must have read and reflected, before presenting us the problem, and linking it to what we had to study as part of our course!

       

Likewise, we had a fascinating discussion about the Barber’s Paradox, when Channa generally talked about paradoxes in mathematics. At that stage, I remember that we were beginning to discuss the idea of Sets. Channa walked in one day and asked if we knew what a paradox was. When there was silence in the room, he went on to explain that a paradox in mathematics occurs when we encounter a statement that contains ideas or thoughts that are conflicting (in any case, this is a simplistic way of understanding paradoxes and we will not get into a deeper discussion here). The Barber’s Paradox, of which there are many variations and which is also called a paradox of ‘self reference’, perfectly illustrates this. I remember the loud arguments and counter arguments in class when this paradox, first proposed by the mathematician and philosopher Bertrand Russell, was discussed. This is how it goes.

Suppose there is a village which has a barber who shaves only those who do not shave themselves, and no one else…the question is: who shaves him (the barber)? It looks simple at first sight but when you grapple with it, you get tied in knots. Now, if the barber shaves himself, he actually mustn’t, since he does not shave those men who shave themselves. However, if he does not shave himself, then he must, since he shaves those who do not shave themselves! So, we encounter contradictions in both cases. In the class as the paradox kept getting discussed, I actually remember imagining an unshaven barber whose beard kept growing and growing infinitely (actually, beards don’t grow that way!). Anyway, we had fun with this paradox. CK then mentioned that this paradox actually exposed a contradiction at the heart of set theory. In simple terms, this would mean that there is a statement ‘S’ such that ‘S’ and its negation (not S) are both true. Such inconsistencies would make the foundations of mathematics very shaky, since we would then have no basis for trusting any mathematical proof (remember, we discussed the angles of a triangle theorem and its proof earlier, where we remembered Channa’s insisting that proof must be solid and robust, no matter what kind of triangle one considered?).   

Just to illustrate paradoxes that are like the Barber’s Paradox, reflect on what is famously known as the ‘Liar Paradox’ (which, I remember, we also discussed in CK’s class) below:

“All Cretans are liars.” (attributed to Epimenides the Cretan) 

Like the Barber’s paradox, the above statement results in contradictions. Think about it.

Again when I look back, I wonder and marvel at the depth of CK’s understanding of the subject, which he used so effortlessly to help us appreciate the pillars around which mathematical thought has been built over the millennia. I remember the fascination, and it still gives me goose bumps when I recall those few classes which enabled us a deeper glimpse of mathematical reality. It is only when one loves the subject and when one cares enough for the learner when teaching it, that will make you take the trouble to go the root of that discipline, and expose it to the learner. This can bring real joy. Joyful learning, unfortunately, is a much misused term in the lexicon of education – instead of connoting that joy can arise from struggle and insight as well, we are talking of making things easier for the child through fun and play. This trivializes the struggle and achievement that is part of the process of learning.

            I’m not sure about the context in which the ‘Four Colour Problem’ (FCP) was presented and discussed by CK. May be it had to do with the Konigsberg problem, before we got on with matrices. Maybe CK did say that it informed the development of new areas in mathematics like Graph Theory, Topology etc. The FCP was, till the 90’s a great unsolved problem of mathematics (originating in map making and cartography) which stumped the best brains. Finally, supercomputers had to be called in (for the first time, literally, to establish a major theorem) to process the huge amounts of data required to establish the proof way back in the 70’s. Even then, there was debate within the mathematical community – would this constitute proof? Did this not sound like an experimental proof which the natural sciences use routinely? Remember, in mathematics we are all used to deductive proof, Euclidean style! Anyway, a typical statement of the FCP might look like this:

“Given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the regions of the map so that no two adjacent regions have the same colour.”

To put things simply, it is like saying that one does not need more than four colours to colour a map such that adjacent countries or regions do not have the same colour. While mathematicians were able to show/prove the case with five colours, the four colour problem stubbornly resisted a solution for well over a century. Interesting, isn’t it? One can never be sure which area of human activity can actually spawn a new area of knowledge which people keep pursuing even hundreds of years later!

And finally, who can forget Fermat’s last theorem? We were discussing the theorem of Pythagoras, that the square of the hypotenuse of a right angled triangle is equal to the sum of the squares of the other two sides: a² + b² = c², where ‘a’ and ‘b’ are the two sides of the right angled triangle and ‘c’ is the hypotenuse. Every high school kid knows this, but few teachers would take the discussions forward beyond stating the theorem and one of its proofs (I have heard that there are approx. 370 ways of proving the above result!). CK went on where most teachers wouldn’t tread, and we were treated to the then 348 year old Fermat’s ‘Last Theorem’, which states that no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two. This, the French amateur mathematician Pierre de Fermat had asserted way back in 1637 in his famous notebook on the margins. As we can see above, the Pythagorean case is a special case of Fermat’s Theorem. To quote some examples, 3² + 4² = 5²; 12² + 5² = 13², and there are various ways in which we can keep generating these ‘Pythagorean Triples’. Channa treated us to a history of this problem and said, rather sombrely, that the Fermat problem was one of the famous all time unresolved problems of mathematics. ‘Maybe one of you will solve it one day!’ he said, with a twinkle in his eye.

The mid 80’s were exciting times for Fermat’s Last Theorem (FLT). Countless mathematicians had by then grappled with it, and proofs had been presented for specific cases, and for ‘n’ running into several million. But there was no general proof. Little did we realize then (in 1985) that Andrew Wiles, the British mathematician was very close to deciding that he would spend the next seven or eight years in his attic in complete secrecy to crack the problem. Finally, in 1994, when the problem was declared as solved, I’m not sure if Channakeshava was still teaching at Baldwin’s to make this announcement to students of the 9^th or 10^th grades. Wiles’ story is fascinating and if I talk about it here, we will move away from CK. There are popular books written on FLT which explain the story of the struggle behind its proof.                   

As I mentioned at the beginning, we were treated to these mathematical gems by CK for three years. I have mentioned only a few, but I must mention my consternation when we encountered the number Pi (^∏), which belonged to a class of numbers called ‘Irrational’ numbers – simply, as I have mentioned earlier, numbers that didn’t behave ‘rationally’, whose fractional parts went on and on and on in a never ending stream of decimal digits. We also discussed about the mathematics of the infinite, though I do not remember any specific instance that caught my attention. There was one argument about whether the number of sand particles on a beach was finite or infinite. It was also interesting to note that there were ‘different orders of infinity’. These days, I have revived my interest in the mathematics of the infinite.  

History of mathematics

These forays into mathematics were fascinating, no doubt, for a fortunate few. That mathematics even had a history like this was beyond our imagination as students, used as we were to rote learning methods by and large. None of my other teachers exhibited this in-depth understanding of the discipline that Channa did. He would tempt you by showing little by little, the intriguing life worlds of mathematics. After a while, we would be on to our usual exercises from the text (the more boring part!) but the stories remained with us.        

Another topic that caught my attention in school was the discovery about Indian mathematicians. When we discussed Pythagoras’s theorem and did quadratic equations in algebra, Channa posed questions that Indian mathematicians attempted more than a thousand years ago. I clearly remember CK discussing the 12th century mathematician Bhaskaracharya II. In fact, what was fascinating was that he had picked up a problem from Bhaskara’s Lilavati. This is Bhaskara’s treatise on mathematics (written when he was in his thirties), and as the fascinating story goes, is dedicated to his daughter Lilavati. If I remember correctly, Channa picked up the following problem from Lilavati:

“A bamboo 18 cubits high was broken by the wind. Its top touched the ground 6 cubits from the root. Tell the lengths of segments of the bamboo.”   

It takes the simple application of the Pythagorean Theorem to find out that the lengths of segments of the broken bamboo are 8 and 10 cubits respectively. While we all enjoyed doing this problem and the class was greatly enlivened as a result, the fascinating part for me was the travelling back in time that we did and the realization at the time that there were Indian mathematicians going back a thousand years and beyond who had worked on what we were learning in school. While we had only touched upon the ‘great’ discoveries of Indian scientists and mathematicians in the history class, CK’s treatment of the subject brought things alive, and history was no longer restricted to boring dates and events and the mugging up of occurrences of the past. He would have been a very good history teacher as well! Incidentally, the above problem also appears in the history of Chinese mathematics and is known as the Kou Ku theorem. It appears prominently in the 13^th century text known as Hsiang Chieh Chiu Chang Suan Fa Tsuan Lei.     

In 1993, when I took to teaching at the Valley School in Bangalore after I had had enough of manufacturing tractors, I learnt something more about the history of mathematics and took off from where we had left with CK. A senior colleague of mine, who was also interested in understanding how mathematics was produced across cultures, came to know that someone called George Gheverghese Joseph, a scholar from the Manchester University was in town. Joseph had researched extensively the ‘Non-European Roots’ of mathematics. We went to meet him and invite him to a lecture at the school. To our delight, he readily agreed. His lecture was gripping, and covered a vast canvas. He showed how ‘Eurocentric’ the entire enterprise of mathematics and science was – indeed, this is what we are taught in schools to this day – that Europe was the centre of global mathematical and scientific development since the days of antiquity. Such a blinkered view ignores the fact that other ancient cultures also did a lot of mathematics, often predating the discoveries of Europe by at least hundreds of years. A lot of this took place during Europe’s great slumber, the Dark Ages. In this context and in particular, Joseph talked about his pet research project -- the discovery of mathematics of the ‘Kerala School’ which flourished between the 14^th and 16^th centuries most notably through the work of mathematicians such as Madhava of Sangamagrama and Nilakantha of Tirur, in Kerala.

Research on this alternative history of mathematics has conclusively shown that the work of the Kerala School predates the discovery of that great mathematical tool, the Calculus, by at least two centuries! While Newton and Leibnitz, generally acknowledged as the founders of the Calculus must be given their due for combining a range of disparate ideas into a coherent discipline of the Calculus, the discoveries of the Indian, Chinese and Arab mathematicians cannot be disregarded. In fact, as Joseph pointed out, we cannot overlook the transmission of mathematical ideas from Egypt, Babylon, China and India through the Arab world to Europe. Research in this area has thrown up compelling evidence that this mathematical transmission, right from the days of Pythagoras of antiquity, had informed the development of European mathematics (Pythagoras knew that the Egyptians knew his theorem, though he also knew they hadn’t proved it).                  

            Anyway, the seed of curiosity that CK had sown in the mid eighties was further explored by me as a teacher, thanks to that chance encounter with Joseph in the mid nineties! Faithfully, I shared these exciting discoveries with the children I taught. Also, I ended up buying his book Crest of the Peacock, in which Joseph elaborates the theme of mathematics as it was done and discovered outside Europe, starting with the mathematics of the ‘Ishango bone’ (actually, a lunar calendar) from the mountains of Central Equatorial Africa 20000 years ago! It is a fascinating read and will certainly open your eyes. I’m sure CK knew about the alternative perspective on global mathematical development, but we didn’t discuss it in school.  

The Valley School

            I graduated from Baldwins in 1985 and became an engineer by 1991. After a two year boring stint in the industry in which I was a production engineer in a company that manufactured tractors, I decided to become a teacher. One reason most certainly was that I wanted to keep on learning, and I wanted to share the joy that comes with discovery and insight with children. Also, I was angry with our insipid educational system consisting mostly of de-motivated teachers most of whom were just doing their jobs. In that surge of idealism that I felt as a young man, I wanted to change the world by becoming a teacher.

              I attribute my reasonable success as a mathematics teacher to the fact that I learnt from what CK did with us in school. I made it a point, for instance, to discuss the history of mathematics in my classes. The Valley School library offered me good resources. I dabbled in ‘Vedic mathematics’ (VM), demonstrated some of its methods, got children excited and got involved in the debates around its veracity as a system of mathematical system. Joseph does talk about Vedic mathematics in his book but there are questions about whether Vedic mathematics actually originated from the Vedas, or, as I said, if the system is robust enough to be called mathematics. I remember attending a workshop organized on VM by the local RSS Shakha in Chamarajapet in Bangalore. We were trying to solve cubic equations, and the ‘magical shortcuts’ of VM were on full view. But when we slightly changed the coefficients of ‘x’ in a cubic equation, the Vedic methods failed. Anyway, the children lapped up whatever I could teach.

            Wherever possible, I took children on a historical trajectory. We solved ancient problems based on the Pythagorean Theorem, in Trigonometry, Analytical Geometry, Logarithms and we widely debated paradoxes in mathematics. I included many of these discussions in worksheets and even managed to give problems based on these discussions in the various test papers that I set. Further, most of what I did with the blackboard as a teaching aid was from what I learnt from observing CK.

            There were several ‘Aha!’ moments that the children and I experienced. I still remember how awestruck they all were (class IX students) when I showed them, through a small table I had prepared, how the idea of a logarithm actually works and simplifies the operations of multiplication and division, by converting them to the easier processes of addition and subtraction, respectively. Imagine, instead, if I had begun by saying: ‘If a^x = y, then ‘x’ is called the logarithm of ‘y’ to base ‘a’! This is why children run away from mathematics. Well, we talked about the historical context in which logarithms were invented, and then I told them the story of the Scotsman John Napier who is generally credited with the invention of the logarithm. It is interesting that Napier indulged in mathematics as a hobby.    

As you can see, CK’s legacy lived on in my life as a when I became a teacher. It still does, nearly three decades later. See, that’s what a good math teacher can do to you!

Math teacher

            I met CK at his residence again in March 2000, just before my marriage. He was happy to see me and was curious to know what I was doing. We discussed the strengths and ills of our education system and he listened quietly to the many things I shared about my work in education. He had retired from Baldwins by then, but went to school everyday to leave his grandson. He couldn’t make it to my marriage reception and I learnt later that he was not keeping well that day. 

            For me, CK will always remain the model math teacher. I have not seen a better math teacher since. He perfectly straddled the discipline of mathematics and its teaching – he was its passionate and eternal student and at the same time he loved teaching the subject. It is widely recognized in educational theory that the teacher’s knowledge consists of both subject matter knowledge and the knowledge of teaching (pedagogic content knowledge) among other aspects of knowledge that are needed to become a good teacher (if one views the teacher’s problem as a knowledge problem, that is). In a class of nearly 50 students and in a school that was traditional at best, CK struck a great balance. If I remember his teaching nearly thirty years later, it is because what he taught was internalized – forever! The teaching of procedures in mathematics (procedural knowledge) is a small part of mathematics education. It is the understanding of deeper conceptual underpinnings that is important. If one develops insights here, these insights are likely to stay throughout one’s life. And CK was constantly chipping away at the deeper and mysterious structures of the mathematical world, inviting us to explore and savour its beauty.

            As I write this, I have taken easy recourse to the internet to check if what I remember from Channa’s classes is correct. The World Wide Web is a place where everything we discussed in school with Channa is easily available, in print, in videos and in all sorts of forms. You can access all the fascinating stuff and it will keep you occupied for a lifetime. At the same time, the internet can dumb you down – it provides many things on a platter and you are tempted to copy paste from the myriad notes and articles and call them your own! No wonder then, that there is software developed to detect plagiarism. But in the eighties when CK taught us, the ‘www’ was not even in our wildest fantasies of the future. He must have therefore read a lot, sitting in libraries and looking for his favourite books in bookshops. When some of us got a glimpse of his private library, we knew he was a serious reader of mathematics and science. His study was filled with hundreds of books then.

            I cannot help but reflect upon the status of the teacher now and wonder what it would mean to take a leaf or two from CK’s book. Can training make great teachers? Or, are teachers made even before that? I would tend to believe that one should love teaching and I’m not sure if this can alone come from teacher preparation. At the same time, I’m not discounting this preparation, both before and after, one becomes a teacher. This learning is an eternal journey. But it saddens me to see how teachers are treated within the educational system. We have created a system where we see the teacher as a contract worker who is often paid a pittance, who is made to do all kinds of things other than teaching and whose support systems are non-existent. Yet, we expect a lot from the teacher and somehow expect the teacher to be ‘different’ from the rest of the human species. How then can we expect children to glimpse that wonderful world that CK showed us, that I continue to see and marvel to this day?    

Giri

Raipur

March 2012