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
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 10th 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 ei∏ = --1. To put it another way: ei∏ + 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 (ei∏ + 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
‘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 1001st 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
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 Euclid
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
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 9th or 10th, 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 9th or 10th 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 13th 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 Manchester University 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 14th and 16th 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 Newton 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 Bangalore
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 ax = 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
