a space of our own
TRANSCRIPT
December 15, 2012 Issue No.: 4
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....a space of our own
Yadgir District Institute
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CªÀjUÉ ®©ü¹zÀ CvÀÄå£ÀßvÀ ¥Àæ±À¹ÛUÀ¼ÀÄ: - 1948, ¥sɧÄæªÀj 28, J¥sï.Dgï.J¸ï ¥sÉ ÉÆà D¥sï gÁAiÀÄ¯ï ¸ÉƸÉÊn. - 1916, PÉÃA©æeï «±Àé«zÁå®AiÀÄ¢AzÀ ©.J ¥ÀzÀ«. - 1918, CPÉÆÖçgï 13, PÉÃA©æeï£À næ¤n PÁ¯Éf£À ¥sÉ ÉÆ.
EAUÉèAr£À ²ÃvÀ ªÁvÁªÀgÀtPÉÌ ¨ÉÃPÁzÀ ªÀÄÄAeÁUÀÈvÉ PÀæªÀÄUÀ¼À£ÀÄß ¤®ðQë¹zÀÝjAzÀ PÀëAiÀÄgÉÆÃUÀ ¦ÃrvÀgÁzÀ 33 ªÀµÀðzÀ gÁªÀiÁ£ÀÄd£ï 1920gÀ J¦æ¯ï 26gÀ ¸ÉÆêÀĪÁgÀzÀAzÀÄ PÉÆ£ÉAiÀÄĹgɼÉzÀgÀÄ.
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For more pictures from the Yadgir DI launch, please have a look at the attachment in the email
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INDEX
by Parimal Agnihotri (English) ............................ 4
2. by Sukanya Mohanti-Poem (Hindi) ................ 7
3. ¤gÁ¼À ªÀiË£À
Deep Silence by Zabeer Bavaji-Poem
(Kannada) ...................................................................... 7
5. J®èzÀPÀÆÌ MAzÀÄ ¯ÉPÁÌZÁgÀ«zÉ
Everything has it’s own calculation by
Devaraj Kodabala (Kannada) ............................. 9
by Peri (English) .......................................................11
8. ¯ÉPÁÌZÁgÀzÀ GvÀÛgÀ §AiÀĸÀĪÀ JgÀqÀÄ ¥Àæ±ÉßUÀ¼ÀÄ
“The two questions which expect
calculations” by Jayadevappa (Kannada) 17
9. Maths in Anisha’s World
by Shadakshari (English)................................... 17
11. £À£Àß ¯ÉPÁÌ(D)ZÁgÀ
12. UÀtÂvÀ«®èzÀ fêÀ£À ¸ÁzÀåªÉÃ?
Can life be Possible Without Maths?
By Shivakumar M D (Kannada) ...................... 21
13. UÀtÂvÀ PÀ°PÉAiÀÄ ¥ÀæQæAiÉÄ ºÉÃVgÀ ÉÃPÀÄ?”
How Learning Mathematics Process
14. November Month Highlights………………25
15. Cool Clicks by Anirudh………………………26
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Zero-God-Infinity: Story of many stories
Parimal Agnihotri In September 1887, two months before her child was due to be born, a nineteen-year-old Kumbakonam girl named Komalatammal traveled to Erode, her parental home, 150 miles upriver, to prepare for the birth of the child she carried. Erode, was located at the confluence of the Cauvery and one of its tributaries, the Bhavani, about 250 miles southwest of Chennai (erstwhile Madras). Not far from the river, in "the fort," as the town's original trading area was known, was the little house, on Teppukulam Street, that belonged to Komalatamrnal's father. It was here that a son was born to her and her husband Srinivasa, just after sunset on the ninth day of the Indian month of Margasirsha-or Thursday, December 22, 1887. On his eleventh day of life, again in accordance with tradition, the child was formally named, and a year almost to the day after his birth, Srinivasa Ramanujan Iyengar and his mother returned to Kumbakonam, where he would spend most of the next twenty years of his life. Ramanujan’s life is a story of many stories. The story of an inscrutable intellect and a simple heart. It is a story of the clash of cultures between India and the West; between the world of Sarangapani Sannidhi Street in Kumbakonam in South India, where Ramanujan grew up, and the glittering world of Cambridge; between the pristine proofs of the Western mathematical tradition and the mysterious powers of intuition with which Ramanujan dazzled East and West alike. It is a story of one man and his stubborn faith in his own abilities. In a way, is also a story about social and educational systems, and about how they matter, and how they can sometimes nurture talent and sometimes crush it. Is a story, too, about what you do with genius once you find it. Ramanujan was
brought to Cambridge by an English mathematician of aristocratic mien and peerless academic credentials, G. H. Hardy, to whom he had written for help. Hardy saw that Ramanujan was a rare flower, one not apt to tolerate being stuffed methodically full of all the mathematical knowledge he'd never acquired in India. Ramanujan was a man who grew up praying to stone deities; who for most of his life took counsel from a family goddess Namakkal, declaring it was she to whom his mathematical insights were owed; whose theorems would, at intellectually backbreaking cost, be proved true--yet leave mathematicians baffled that anyone could divine them in the first place. His life is also a story, then, about an uncommon and individual mind, and what its quirks may suggest about creativity, intuition, and intelligence. Ramanujan, who all his life believed in the Hindu gods and made the landscape of the Infinite, in, realms both mathematical and spiritual, his home. "An equation for me has no meaning," he once said, "unless it expresses a thought of God." Let me introduce some tit-bits from the life of unforgettable Indian mathematical prodigy, Ramanujan Childhood: Ramanujan's mother, Komalatammal, sang bhajans, or devotional songs, at a nearby temple. Half the proceeds from her group's performances went to the temple, the other half to the singers. With her husband earning only about twenty rupees per month, the five or ten she earned this way mattered; never would she miss a rehearsal. Yet now, in December 1889, she was missing them, four or five in a row. So one day, the head of the singing group showed up at Komalatammal's house to investigate.
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There she found, piled near the front door, leaves of the margosa tree; someone, it was plain to her, had smallpox. Stepping inside, she saw a small, dark figure lying atop a bed of margosa leaves. His mother, chanting all the while, dipped the leaves in water laced with ground turmeric, and gently scoured two-year-old Ramanujan's pox-ridden body-both to relieve the infernal itching and, South Indian herbalists believed, subdue the fever. Ramanujan would bear the scars of his childhood smallpox all his life. But he recovered, and in that was fortunate. For in Tanjore District, around the time he was growing up, a bad year for smallpox meant four thousand deaths. Fewer than one person in five was vaccinated. A cholera epidemic when Ramanujan was ten killed fifteen thousand people. Three or four children in every ten died before they'd lived a year. Ramanujan's family was a case study in the damning statistics. When he was a year and a half, his mother bore a son, Sadagopan. Three months later, Sadagopan was dead. When Ramanujan was almost four, in November 1891, a girl was born. By the following February, she, too, was dead. When Ramanujan was six and a half, his mother gave birth to yet another child, Seshan-who also died before the year was out. Much later, two brothers did survive-Lakshmi Narasimhan, born in 1898, when Ramanujan was ten, and Tirunarayanan, born when he was seventeen. But the death of his infant brothers and sister during those early years meant that he grew up with the solicitous regard and central position of an only child. After the death of his paternal grandfather, who had suffered from leprosy, Ramanujan, seven at the time, broke out in a bad case of itching and boils. But this was not the first hint of a temperament inclined to extreme and unexpected reactions to stress. Indeed, Ramanujan was a sensitive, stubborn, and-if a word more often
reserved for adults in their prime can be applied to a little boy-eccentric child. While yet an infant back in Erode, he wouldn't eat except at the temple. Later, in Kumbakonam, he'd take all the brass and copper vessels in the house and line them up from one wall to the other. If he didn't get what he wanted to eat, he was known to roll in the mud in frustration. For Ramanujan's first three years, he scarcely spoke. Perhaps, it is tempting to think, because he simply didn't choose to; he was an enormously self- willed child., It was common in those days for a young wife to shuttle back and forth between her husband's house and that of her parents, and Komalatammal, worried by her son's muteness, took Ramanujan to see her father, then living in Kanchipuram, near Chennai (erstwhile Mdras). There, at the urging of an elderly friend of her father's, Ramanujan began the ritual practice of Akshara Abhyasam: his hand, held and guided by his grandfather, was made to trace out Tamil characters in a thick bed of rice spread across the floor, as each character was spoken aloud. Soon fears of Ramanujan's dumbness were dispelled and he began to learn the 12 vowels, 18 consonants, and 216 combined consonant-vowel forms of the Tamil alphabet. On October I, 1892, the traditional opening day of school, known as Vijayadasami, he was enrolled, to the accompaniment of ancient Vedic chants, in the local pial school. A pial is the little porch in front of most South Indian houses; a pial school was just a teacher meeting there with half a dozen or so pupils.
But five-year-old Ramanujan, disliking the teacher, bristled at attending. Even as a child, he was so self-directed that, it was fair to say, unless he was ready to do something on his own, in his own time, he was scarcely capable of doing it at all; school for him often meant not keys to knowledge but shackles to throw off.
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Quiet and contemplative, Ramanujan was fond of asking questions like, Who was the first man in the world? Or, How far is it between clouds? He liked to be by himself, a tendency abetted by parents who, when friends called, discouraged him from going out to play; so he'd talk to them from the window overlooking the street. He lacked all interest in sports. And in a world where obesity was virtually unknown, where bones protruded from humans and animals alike, he was, first as a child and then for most of his life, fat. He used to say-whether as boast, joke, or lament remains unclear-that if he got into a fight with another boy he had only to fall on him to crush him to pieces. For about two years, Ramanujan was shuffled between schools. Beginning in March 1894, while still at his mother's parents' house in Kanchipuram, he briefly attended a school in which the language of instruction was not his native Tamil but the related but distinct Telugu. There, sometimes punished by having to sit with his arms folded in front of him and one finger turned up to his lips in silence, he would at times stalk out of class in a huff. In a dispute over a loan, his grandfather quit his job and left Kanchipuram. Ramanujan and his mother returned to Kumbakonam, where he enrolled in the Kangayan Primary School. But when his other grandfather died, Ramanujan was bounced back to his maternal grandparents, who by now were in Madras. There he so fiercely fought attending school that the family enlisted a local constable to scare him back to class. By mid-189S, after an
unhappy six months in Madras, Ramanujan was once more back in Kumbakonam. Why Hardy? Early on, I viewed a documentary about Ramanujan's life by the British filmmaker Christopher Sykes. Released by the BBC as Letters from an Indian Clerk, Sykes's film superbly distilled, into a single hour, something of the romance of Ramanujan's life. But watching it, I grew beguiled by G. H. Hardy, too. Hardy, it turned out, was the third English mathematician to whom Ramanujan had appealed; the other two declined to help. And Hardy did not just recognize Ramanujan's gifts; he went to great lengths to bring him to England, school him in the mathematics he had missed, and bring him to the attention of the world.
Was it sheer mathematical acumen? Probably not; the other two mathematicians were equally distinguished. There must have been other, less purely intellectual traits demanded of him-a special openness, perhaps, a willingness to disrupt his life and stake his reputation on someone he'd never seen. Hardy, was a bizarre and fascinating character-a cricket aficionado, a masterful prose stylist, a man blessed with gorgeous good looks who to his own eyes was so repulsively ugly he couldn't look at himself in the mirror. And this enfant terrible of English mathematics was, at the time he heard from Ramanujan, working a revolution on his field that would be felt for generations to come. One is, of course, moved to praise Hardy's ability to see genius in the tattered garb in which it was clothed, and to agree that the world was enriched as a result. But, it struck me, Hardy was enriched, too. His whole life was shaped by his time with Ramanujan, which he called "the one romantic incident in my life." The story of Ramanujan, then, is a story about two men, and what they meant to each other.
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The above two bits suggests us that, barring his Brahmin hood, if we compare Socio, Economic, Health and Education conditions, that of him and most of other children, of his time as well as subsequent times, even till today are same. Hence, plenty of Ramanujans born in India. But In last 125 years, after the Birth of a man who knew Zero, God and infinity better than anyone else, No one was brought to lime light as Ramanujan. In India it is not the scarcity of Ramanujans but surely it is of Hardys. Ramanujan’s life begs us to ask, how many Ramanujans dwell in India today unknown and unrecognized? Let us strive to be Hardys.
Ramanujan’s home on Sarangapani Street, Kumbakonam
Ramanujan (centre) with other scientists at Trinity College
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There is no one answer! Peri
Always it was a puzzle for me. My brother was very
good in all the work in our house and farm. He
could do all the farm work before me. I was not
able to figure when and how he learnt all these.
The puzzle is not about his learning these things,
but it was about the way he could not learn Maths
in school. To our great surprise we were called by
the head teacher and told that he is not attending
Maths classes for 2 years now. This boy who could
do a lot of things connected to our agriculture and
house hold chores which needed estimation,
calculation was doing very bad in school. This was
something our family was not able to digest.
Today, my brother can estimate the volume of
seeds needed to sow in one acre. He can calculate
the amount of paddy we will be getting this year
looking at the grains in the field. He can also very
accurately calculate storage space needed for the
grains looking at the crop in the field. In South
Canara we store rice in a well packed round
storage space created out of hay known as Mudi.
This is a very complicated shape that skilled people
pack 40 Kgs of rice in oval shaped hay packing
which has exactly 24 slices made of hay rope. Very
complicated calculations and estimation required
to get it right and my brother gets it always right,
to the extent that we do not see him calculating it
at all. My brother discontinued studies very soon
and he would always say that it was because of
Maths.
What went wrong we do not know? If he was good
in simple calculations which he still is, what went
wrong in school, we don’t know. This was the first
time I encountered the disconnect between every
day Maths and school Maths. But, later I saw this
disconnect everywhere and troubling a lot of
children mostly from rural families and more so
from poor families. So there is this disconnect
between the everyday Maths and the school
Maths. I can see children coming from rural areas
really suffering from this phobia of Maths. Why
rural areas, majority of our people are suffering
from this phobia.
If we look around the situation it is not very
different. In agricultural families the children are
engaged in a lot of activities which need a lot of
calculation and estimation skills. They learn /
acquire these skills without much difficulty. They
learn these skills in their play and simple matter of
fact simple activities. All the middle class and lower
middle families too need their children to work in
their occupation and very soon they get all the
skills they need. And I have seen how these same
children find it difficult to do simple Maths in
school.
Shashidhar a Maths teacher who enjoys and loves
Maths. I told him about how these children who
outside school are good at Maths usually fear
Maths in School and invariably fail in Maths. Shashi
told me that these things are fine and that it is true
that these children are doing some calculation and
are good at it. But, he said that this is not enough. I
told him if they can manage why unnecessarily
teach them something meaningless in the name of
Maths teaching? He said we are not teaching
children just calculation that is not the aim of
teaching mathematics to children. There is a lot
more and higher aim in Mathematics. We could not
discuss further on this since we had to leave.
12
Maths to children. I share with you some thoughts
I gathered through reading and discussion with
people who are engaged in Maths teaching.
The National Curriculum Framework (NCF), 2005
suggests the need for developing the ability for
Mathematisation in the child. It points out that the
aim of learning mathematics is not merely being
able to do quantitative calculations but also to
develop abilities in the child that would enable
her/him to redefine her/his relationship with the
World. The NCF-2005 also lays emphasis on
development in the children logical abilities as well
as abilities to comprehend space, spatial
transformations and develop the ability to visualise
both these. It recommends that mathematics
needs to slowly, move towards abstraction even
though it starts from concrete experiences and
models. The ability to generalise and perceive
patterns is an important step in being able to relate
to the abstract and logic governed nature of the
subject. (Hardy, Vidya Bhavan Publications).
What we are referring to here is the space for the
child to think, articulate, create and function
independently. Even with just a black board and
children sitting in their own seats, a lot of space for
thinking and for mental activity can be created.
This is what maths learning should be all about.
Unless we have this feeling a lot of materials, list of
activities, details of how to do activity etc., would
not help us overcome the major barriers to
children learning mathematics. (Hardy, Vidya
Bhavan Publications)
directions. I quote Morris Kline from his
bookMathematics for Liberal Arts:
study mathematics if we take a moment to
consider what mathematics is. Unfortunately the
answer cannot be given in a single sentence or a
single chapter. The subject has many facets or,
some might say, is Hydra-headed. One can look at
mathematics as a language, as a particular kind of
logical structure, as a body of knowledge about
number and space, as a series of methods for
deriving conclusions, as the essence of our
knowledge of the physical world, or merely as an
amusing intellectual activity.
What is the purpose of the study of mathematics in
a classical education? As one classical school states
it:
an ever-increasing sense of wonder and awe at the
profound way in which the world displays order,
pattern, and relation. Mathematics is studied not
because it is first useful and then beautiful, but
because it reveals the beautiful order inherent in
the cosmos. (from The Education Plan of St. Jerome
Classical School, Hyattsville, MD)
been to reveal and describe an order in the natural
world.
The body of knowledge and practice known as mathematics is derived from the contributions of thinkers throughout the ages and across the globe. It gives us a way to understand patterns, to quantify relationships, and to predict the future. Math helps us understand the world — and we use the world to understand math.
13
The world is interconnected. Everyday math shows these connections and possibilities. The earlier young learners can put these skills to practice, the more likely we will remain an innovation society and economy.
Any contexts used in math should add to an understanding of the math, as well as the world. To do that, teachers should stay focused on teaching good, sound, rigorous and appropriate math content and use contextual examples that work for students.
Math is often studied as a pure science, but is typically applied to other disciplines, extending well beyond physics and engineering. For instance, studying exponential growth and decay (the rate at which things grow and die) within the context of population growth, the spread of disease, or water contamination, is meaningful. It not only gives students a real-world context in which to use the math, but helps them understand global phenomena – they may hear about a disease spreading in India, but can’t make the connection without understanding how fast something like cholera can spread in a dense population. In a similar vein, a study of statistics and probability is key to understanding many of the events of the world, and is usually reserved for students at a higher level of math, if it gets any study in high school at all. But many world events and phenomena are unpredictable and can only be described using statistical models, so a globally focused math program needs to consider including statistics. Probability and statistics can be used to estimate death tolls from natural disasters, such as earthquakes and tsunamis; the amount of aid that might be necessary to help in the aftermath; and the number of people who would be displaced.
It’s important, though, to only include examples that are relevant to the math and help students make sense of the world. By the time a student graduates high school, he or she should be able to use mathematical tools and procedures to explore
problems and opportunities in the world, and use
mathematical models to make and defend conclusions and actions. (Understanding the World Through Math, Asia Society)
The NCF 2005 speaks of developing
Mathematization in students. So what does the
Mathematization mean? Rajni Dewedi helped me
to understand this. She says that Mathematization
is a process. This means developing an ability to
think in abstract. It is more organizing and
generalizing the essence of experience: seeing
patterns in them and interpreting in relation to
other context and concepts.
than what we today do in the name of Maths
teaching. We will stop looking for answers and
right answers which according to us is only one!
We will concentrate more on understanding the
problem and looking for different ways of solving
the problem. Hence we will not have only one
answer- the answer! But, we will have many ways
of solving the problem and many answers to this
question.
need teachers who can identify the skills and
knowledge with which our children come to school.
They need to build on this knowledge of the
children. Once the children see that what they
have is knowledge and is respected by others
especially by teacher’s community, they will
develop a lot of confidence in themselves and a
strong self-esteem. Once, this is established then,
our teacher’s job would be to slowly take our
children to the realm of abstract thinking. It is
very clear that we need to help our children think
in abstract. Use a lot of imagination. Help them
14
think out of box to solve problems, see what is
emerging, see the pattern, construct their own
meaning in things which otherwise looks
meaningless. Predict with some evidence, with
some base. Hence, Maths is not only for solving the
immediate problem of day to day living but more
so to improve the quality of our life and living. It is
to prepare the child to think, observe closely and
gain insights in the process and enjoy life. This is
the process of preparing the child for better living.
15
Discussion with Govt. Functionaries
16
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CwèAzÀ M§â ¸ÁºÉç §jÛzÀÝ. CªÀ£ÀÄ ¨Á¼À NzÉÆÌAqÀ §ÄzÀéAvÀ. ºÁzÁåVz ÀÝ F zÀqÀØ §gÉÆà ¸Á»Ã§£Àß ¤°è¹, “¸ÁºÉçæ, £ÀAzÉÆAzï PÀéöåZÀÑ£ï Lw, ºÉý ºÉÆÃVæ” CAzÀ.
“K£À¥Á CzÀÄ, ºÉüÀÄ ªÀÄvÉÛ” CAzÀ ¸ÁºÉç.
“£Á®ÄÌ PÀtÄÚ, MAzÀÄ PÉÆÃqÀÄ, JgÀqÀÄ PÁ®Ä EgÉÆà UÉÆêÀiÁvÉ LvÀAvÀ°æ , CzÀÄ J¯ÉèöÊw, £ÉÆÃrzÁæ?”
“CAiÉÆåà ºÀÄZÁÑ, CAvÁ ºÀ ÀÄ«gÉÆÃzÀÄ §zÁÝ?” CAzÁ ¸Á»Ã§.
“ºÀAUÁzÉæ £Á vÉÆÃgÀ¸ÉÛä. MAzÀÄ ¥ÀPÀë £Á£ÀÄ ¸ÉÆÃvÉæ 2 PÉÆqÉÛä. ¤Ãªï ¸ÉÆÃvÉæ 5 PÉÆqÉâÃPÀÄ” CAzÀ.
“DAiÀÄÄÛ” CAzÀ ¸Á»Ã§.
ºÁUÁzÉæ, CªÀæ°è AiÀiÁgÀÄ UɯÁÛgÉ, AiÀiÁgÀÄ ¸ÉÆïÁÛgÉ? ¯ÉPÁÌZÁgÀzÀ GvÀÛgÀ ¨ÉÃPÀÄ. AiÀiÁgÀÄ PÉÆrÛÃj? ÀÆPÀÛ §ºÀĪÀiÁ£À GAlÄ. C®èzÉ, MAzÀÄ §ºÀĪÀiÁ£À UÉzÀݪÀjUÉ ªÀÄvÉÆÛAzÀÄ GavÀ!!
Maths in Anisha’s world
Shadakshari T S
while when I visited home recently.
He updated most of the things
happened all around his school,
friends, Elder brother Nakshatra, Family, Farm,
livestock etc.
Later he came closer and started showing lot of
affection by rolling all over me and kissing my
cheeks. He slowly opened up and asked me to
promise for one thing. I said “no I will not if I don’t
know the reason?” But he kept forcing me to touch
any one of his two fingers. I touched one finger and
he jumped so happily and shouted joyously and
told me “You should buy me a Bicycle “Now” that’s
18
fingers!!!!”
I said OK and asked him “But where is the money?
Who pays for it?
from my Savings!!! In my savings box you need to
put ONLY the remaining balance money.”
“OK, but what is the total cost of your bicycle you
are dreaming of now?” I asked.
He said “APPA It’s JUST 15 fifteen THOUSAND
ONLY…!!!!. My friend told me at school”
My home was full of laugher. He was slightly
surprised as why all of us were laughing!!!!
Then I spoke to him about the difference between
Hundred and thousand in his own world.
(According to him 16 is GREATER than 15 but not
the Thousand against Hundred..…!!!!)
He is still in his 1st standard…and learning his
numbers…. For his life
I too keep counting on HIM...!!! For my rest of the life..!!!
Conceptual understanding
Rudresh
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CªÀjUÉ ®©ü¹zÀ CvÀÄå£ÀßvÀ ¥Àæ±À¹ÛUÀ¼ÀÄ: - 1948, ¥sɧÄæªÀj 28, J¥sï.Dgï.J¸ï ¥sÉ ÉÆà D¥sï gÁAiÀÄ¯ï ¸ÉƸÉÊn. - 1916, PÉÃA©æeï «±Àé«zÁå®AiÀÄ¢AzÀ ©.J ¥ÀzÀ«. - 1918, CPÉÆÖçgï 13, PÉÃA©æeï£À næ¤n PÁ¯Éf£À ¥sÉ ÉÆ.
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For more pictures from the Yadgir DI launch, please have a look at the attachment in the email
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INDEX
by Parimal Agnihotri (English) ............................ 4
2. by Sukanya Mohanti-Poem (Hindi) ................ 7
3. ¤gÁ¼À ªÀiË£À
Deep Silence by Zabeer Bavaji-Poem
(Kannada) ...................................................................... 7
5. J®èzÀPÀÆÌ MAzÀÄ ¯ÉPÁÌZÁgÀ«zÉ
Everything has it’s own calculation by
Devaraj Kodabala (Kannada) ............................. 9
by Peri (English) .......................................................11
8. ¯ÉPÁÌZÁgÀzÀ GvÀÛgÀ §AiÀĸÀĪÀ JgÀqÀÄ ¥Àæ±ÉßUÀ¼ÀÄ
“The two questions which expect
calculations” by Jayadevappa (Kannada) 17
9. Maths in Anisha’s World
by Shadakshari (English)................................... 17
11. £À£Àß ¯ÉPÁÌ(D)ZÁgÀ
12. UÀtÂvÀ«®èzÀ fêÀ£À ¸ÁzÀåªÉÃ?
Can life be Possible Without Maths?
By Shivakumar M D (Kannada) ...................... 21
13. UÀtÂvÀ PÀ°PÉAiÀÄ ¥ÀæQæAiÉÄ ºÉÃVgÀ ÉÃPÀÄ?”
How Learning Mathematics Process
14. November Month Highlights………………25
15. Cool Clicks by Anirudh………………………26
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Zero-God-Infinity: Story of many stories
Parimal Agnihotri In September 1887, two months before her child was due to be born, a nineteen-year-old Kumbakonam girl named Komalatammal traveled to Erode, her parental home, 150 miles upriver, to prepare for the birth of the child she carried. Erode, was located at the confluence of the Cauvery and one of its tributaries, the Bhavani, about 250 miles southwest of Chennai (erstwhile Madras). Not far from the river, in "the fort," as the town's original trading area was known, was the little house, on Teppukulam Street, that belonged to Komalatamrnal's father. It was here that a son was born to her and her husband Srinivasa, just after sunset on the ninth day of the Indian month of Margasirsha-or Thursday, December 22, 1887. On his eleventh day of life, again in accordance with tradition, the child was formally named, and a year almost to the day after his birth, Srinivasa Ramanujan Iyengar and his mother returned to Kumbakonam, where he would spend most of the next twenty years of his life. Ramanujan’s life is a story of many stories. The story of an inscrutable intellect and a simple heart. It is a story of the clash of cultures between India and the West; between the world of Sarangapani Sannidhi Street in Kumbakonam in South India, where Ramanujan grew up, and the glittering world of Cambridge; between the pristine proofs of the Western mathematical tradition and the mysterious powers of intuition with which Ramanujan dazzled East and West alike. It is a story of one man and his stubborn faith in his own abilities. In a way, is also a story about social and educational systems, and about how they matter, and how they can sometimes nurture talent and sometimes crush it. Is a story, too, about what you do with genius once you find it. Ramanujan was
brought to Cambridge by an English mathematician of aristocratic mien and peerless academic credentials, G. H. Hardy, to whom he had written for help. Hardy saw that Ramanujan was a rare flower, one not apt to tolerate being stuffed methodically full of all the mathematical knowledge he'd never acquired in India. Ramanujan was a man who grew up praying to stone deities; who for most of his life took counsel from a family goddess Namakkal, declaring it was she to whom his mathematical insights were owed; whose theorems would, at intellectually backbreaking cost, be proved true--yet leave mathematicians baffled that anyone could divine them in the first place. His life is also a story, then, about an uncommon and individual mind, and what its quirks may suggest about creativity, intuition, and intelligence. Ramanujan, who all his life believed in the Hindu gods and made the landscape of the Infinite, in, realms both mathematical and spiritual, his home. "An equation for me has no meaning," he once said, "unless it expresses a thought of God." Let me introduce some tit-bits from the life of unforgettable Indian mathematical prodigy, Ramanujan Childhood: Ramanujan's mother, Komalatammal, sang bhajans, or devotional songs, at a nearby temple. Half the proceeds from her group's performances went to the temple, the other half to the singers. With her husband earning only about twenty rupees per month, the five or ten she earned this way mattered; never would she miss a rehearsal. Yet now, in December 1889, she was missing them, four or five in a row. So one day, the head of the singing group showed up at Komalatammal's house to investigate.
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There she found, piled near the front door, leaves of the margosa tree; someone, it was plain to her, had smallpox. Stepping inside, she saw a small, dark figure lying atop a bed of margosa leaves. His mother, chanting all the while, dipped the leaves in water laced with ground turmeric, and gently scoured two-year-old Ramanujan's pox-ridden body-both to relieve the infernal itching and, South Indian herbalists believed, subdue the fever. Ramanujan would bear the scars of his childhood smallpox all his life. But he recovered, and in that was fortunate. For in Tanjore District, around the time he was growing up, a bad year for smallpox meant four thousand deaths. Fewer than one person in five was vaccinated. A cholera epidemic when Ramanujan was ten killed fifteen thousand people. Three or four children in every ten died before they'd lived a year. Ramanujan's family was a case study in the damning statistics. When he was a year and a half, his mother bore a son, Sadagopan. Three months later, Sadagopan was dead. When Ramanujan was almost four, in November 1891, a girl was born. By the following February, she, too, was dead. When Ramanujan was six and a half, his mother gave birth to yet another child, Seshan-who also died before the year was out. Much later, two brothers did survive-Lakshmi Narasimhan, born in 1898, when Ramanujan was ten, and Tirunarayanan, born when he was seventeen. But the death of his infant brothers and sister during those early years meant that he grew up with the solicitous regard and central position of an only child. After the death of his paternal grandfather, who had suffered from leprosy, Ramanujan, seven at the time, broke out in a bad case of itching and boils. But this was not the first hint of a temperament inclined to extreme and unexpected reactions to stress. Indeed, Ramanujan was a sensitive, stubborn, and-if a word more often
reserved for adults in their prime can be applied to a little boy-eccentric child. While yet an infant back in Erode, he wouldn't eat except at the temple. Later, in Kumbakonam, he'd take all the brass and copper vessels in the house and line them up from one wall to the other. If he didn't get what he wanted to eat, he was known to roll in the mud in frustration. For Ramanujan's first three years, he scarcely spoke. Perhaps, it is tempting to think, because he simply didn't choose to; he was an enormously self- willed child., It was common in those days for a young wife to shuttle back and forth between her husband's house and that of her parents, and Komalatammal, worried by her son's muteness, took Ramanujan to see her father, then living in Kanchipuram, near Chennai (erstwhile Mdras). There, at the urging of an elderly friend of her father's, Ramanujan began the ritual practice of Akshara Abhyasam: his hand, held and guided by his grandfather, was made to trace out Tamil characters in a thick bed of rice spread across the floor, as each character was spoken aloud. Soon fears of Ramanujan's dumbness were dispelled and he began to learn the 12 vowels, 18 consonants, and 216 combined consonant-vowel forms of the Tamil alphabet. On October I, 1892, the traditional opening day of school, known as Vijayadasami, he was enrolled, to the accompaniment of ancient Vedic chants, in the local pial school. A pial is the little porch in front of most South Indian houses; a pial school was just a teacher meeting there with half a dozen or so pupils.
But five-year-old Ramanujan, disliking the teacher, bristled at attending. Even as a child, he was so self-directed that, it was fair to say, unless he was ready to do something on his own, in his own time, he was scarcely capable of doing it at all; school for him often meant not keys to knowledge but shackles to throw off.
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Quiet and contemplative, Ramanujan was fond of asking questions like, Who was the first man in the world? Or, How far is it between clouds? He liked to be by himself, a tendency abetted by parents who, when friends called, discouraged him from going out to play; so he'd talk to them from the window overlooking the street. He lacked all interest in sports. And in a world where obesity was virtually unknown, where bones protruded from humans and animals alike, he was, first as a child and then for most of his life, fat. He used to say-whether as boast, joke, or lament remains unclear-that if he got into a fight with another boy he had only to fall on him to crush him to pieces. For about two years, Ramanujan was shuffled between schools. Beginning in March 1894, while still at his mother's parents' house in Kanchipuram, he briefly attended a school in which the language of instruction was not his native Tamil but the related but distinct Telugu. There, sometimes punished by having to sit with his arms folded in front of him and one finger turned up to his lips in silence, he would at times stalk out of class in a huff. In a dispute over a loan, his grandfather quit his job and left Kanchipuram. Ramanujan and his mother returned to Kumbakonam, where he enrolled in the Kangayan Primary School. But when his other grandfather died, Ramanujan was bounced back to his maternal grandparents, who by now were in Madras. There he so fiercely fought attending school that the family enlisted a local constable to scare him back to class. By mid-189S, after an
unhappy six months in Madras, Ramanujan was once more back in Kumbakonam. Why Hardy? Early on, I viewed a documentary about Ramanujan's life by the British filmmaker Christopher Sykes. Released by the BBC as Letters from an Indian Clerk, Sykes's film superbly distilled, into a single hour, something of the romance of Ramanujan's life. But watching it, I grew beguiled by G. H. Hardy, too. Hardy, it turned out, was the third English mathematician to whom Ramanujan had appealed; the other two declined to help. And Hardy did not just recognize Ramanujan's gifts; he went to great lengths to bring him to England, school him in the mathematics he had missed, and bring him to the attention of the world.
Was it sheer mathematical acumen? Probably not; the other two mathematicians were equally distinguished. There must have been other, less purely intellectual traits demanded of him-a special openness, perhaps, a willingness to disrupt his life and stake his reputation on someone he'd never seen. Hardy, was a bizarre and fascinating character-a cricket aficionado, a masterful prose stylist, a man blessed with gorgeous good looks who to his own eyes was so repulsively ugly he couldn't look at himself in the mirror. And this enfant terrible of English mathematics was, at the time he heard from Ramanujan, working a revolution on his field that would be felt for generations to come. One is, of course, moved to praise Hardy's ability to see genius in the tattered garb in which it was clothed, and to agree that the world was enriched as a result. But, it struck me, Hardy was enriched, too. His whole life was shaped by his time with Ramanujan, which he called "the one romantic incident in my life." The story of Ramanujan, then, is a story about two men, and what they meant to each other.
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The above two bits suggests us that, barring his Brahmin hood, if we compare Socio, Economic, Health and Education conditions, that of him and most of other children, of his time as well as subsequent times, even till today are same. Hence, plenty of Ramanujans born in India. But In last 125 years, after the Birth of a man who knew Zero, God and infinity better than anyone else, No one was brought to lime light as Ramanujan. In India it is not the scarcity of Ramanujans but surely it is of Hardys. Ramanujan’s life begs us to ask, how many Ramanujans dwell in India today unknown and unrecognized? Let us strive to be Hardys.
Ramanujan’s home on Sarangapani Street, Kumbakonam
Ramanujan (centre) with other scientists at Trinity College
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There is no one answer! Peri
Always it was a puzzle for me. My brother was very
good in all the work in our house and farm. He
could do all the farm work before me. I was not
able to figure when and how he learnt all these.
The puzzle is not about his learning these things,
but it was about the way he could not learn Maths
in school. To our great surprise we were called by
the head teacher and told that he is not attending
Maths classes for 2 years now. This boy who could
do a lot of things connected to our agriculture and
house hold chores which needed estimation,
calculation was doing very bad in school. This was
something our family was not able to digest.
Today, my brother can estimate the volume of
seeds needed to sow in one acre. He can calculate
the amount of paddy we will be getting this year
looking at the grains in the field. He can also very
accurately calculate storage space needed for the
grains looking at the crop in the field. In South
Canara we store rice in a well packed round
storage space created out of hay known as Mudi.
This is a very complicated shape that skilled people
pack 40 Kgs of rice in oval shaped hay packing
which has exactly 24 slices made of hay rope. Very
complicated calculations and estimation required
to get it right and my brother gets it always right,
to the extent that we do not see him calculating it
at all. My brother discontinued studies very soon
and he would always say that it was because of
Maths.
What went wrong we do not know? If he was good
in simple calculations which he still is, what went
wrong in school, we don’t know. This was the first
time I encountered the disconnect between every
day Maths and school Maths. But, later I saw this
disconnect everywhere and troubling a lot of
children mostly from rural families and more so
from poor families. So there is this disconnect
between the everyday Maths and the school
Maths. I can see children coming from rural areas
really suffering from this phobia of Maths. Why
rural areas, majority of our people are suffering
from this phobia.
If we look around the situation it is not very
different. In agricultural families the children are
engaged in a lot of activities which need a lot of
calculation and estimation skills. They learn /
acquire these skills without much difficulty. They
learn these skills in their play and simple matter of
fact simple activities. All the middle class and lower
middle families too need their children to work in
their occupation and very soon they get all the
skills they need. And I have seen how these same
children find it difficult to do simple Maths in
school.
Shashidhar a Maths teacher who enjoys and loves
Maths. I told him about how these children who
outside school are good at Maths usually fear
Maths in School and invariably fail in Maths. Shashi
told me that these things are fine and that it is true
that these children are doing some calculation and
are good at it. But, he said that this is not enough. I
told him if they can manage why unnecessarily
teach them something meaningless in the name of
Maths teaching? He said we are not teaching
children just calculation that is not the aim of
teaching mathematics to children. There is a lot
more and higher aim in Mathematics. We could not
discuss further on this since we had to leave.
12
Maths to children. I share with you some thoughts
I gathered through reading and discussion with
people who are engaged in Maths teaching.
The National Curriculum Framework (NCF), 2005
suggests the need for developing the ability for
Mathematisation in the child. It points out that the
aim of learning mathematics is not merely being
able to do quantitative calculations but also to
develop abilities in the child that would enable
her/him to redefine her/his relationship with the
World. The NCF-2005 also lays emphasis on
development in the children logical abilities as well
as abilities to comprehend space, spatial
transformations and develop the ability to visualise
both these. It recommends that mathematics
needs to slowly, move towards abstraction even
though it starts from concrete experiences and
models. The ability to generalise and perceive
patterns is an important step in being able to relate
to the abstract and logic governed nature of the
subject. (Hardy, Vidya Bhavan Publications).
What we are referring to here is the space for the
child to think, articulate, create and function
independently. Even with just a black board and
children sitting in their own seats, a lot of space for
thinking and for mental activity can be created.
This is what maths learning should be all about.
Unless we have this feeling a lot of materials, list of
activities, details of how to do activity etc., would
not help us overcome the major barriers to
children learning mathematics. (Hardy, Vidya
Bhavan Publications)
directions. I quote Morris Kline from his
bookMathematics for Liberal Arts:
study mathematics if we take a moment to
consider what mathematics is. Unfortunately the
answer cannot be given in a single sentence or a
single chapter. The subject has many facets or,
some might say, is Hydra-headed. One can look at
mathematics as a language, as a particular kind of
logical structure, as a body of knowledge about
number and space, as a series of methods for
deriving conclusions, as the essence of our
knowledge of the physical world, or merely as an
amusing intellectual activity.
What is the purpose of the study of mathematics in
a classical education? As one classical school states
it:
an ever-increasing sense of wonder and awe at the
profound way in which the world displays order,
pattern, and relation. Mathematics is studied not
because it is first useful and then beautiful, but
because it reveals the beautiful order inherent in
the cosmos. (from The Education Plan of St. Jerome
Classical School, Hyattsville, MD)
been to reveal and describe an order in the natural
world.
The body of knowledge and practice known as mathematics is derived from the contributions of thinkers throughout the ages and across the globe. It gives us a way to understand patterns, to quantify relationships, and to predict the future. Math helps us understand the world — and we use the world to understand math.
13
The world is interconnected. Everyday math shows these connections and possibilities. The earlier young learners can put these skills to practice, the more likely we will remain an innovation society and economy.
Any contexts used in math should add to an understanding of the math, as well as the world. To do that, teachers should stay focused on teaching good, sound, rigorous and appropriate math content and use contextual examples that work for students.
Math is often studied as a pure science, but is typically applied to other disciplines, extending well beyond physics and engineering. For instance, studying exponential growth and decay (the rate at which things grow and die) within the context of population growth, the spread of disease, or water contamination, is meaningful. It not only gives students a real-world context in which to use the math, but helps them understand global phenomena – they may hear about a disease spreading in India, but can’t make the connection without understanding how fast something like cholera can spread in a dense population. In a similar vein, a study of statistics and probability is key to understanding many of the events of the world, and is usually reserved for students at a higher level of math, if it gets any study in high school at all. But many world events and phenomena are unpredictable and can only be described using statistical models, so a globally focused math program needs to consider including statistics. Probability and statistics can be used to estimate death tolls from natural disasters, such as earthquakes and tsunamis; the amount of aid that might be necessary to help in the aftermath; and the number of people who would be displaced.
It’s important, though, to only include examples that are relevant to the math and help students make sense of the world. By the time a student graduates high school, he or she should be able to use mathematical tools and procedures to explore
problems and opportunities in the world, and use
mathematical models to make and defend conclusions and actions. (Understanding the World Through Math, Asia Society)
The NCF 2005 speaks of developing
Mathematization in students. So what does the
Mathematization mean? Rajni Dewedi helped me
to understand this. She says that Mathematization
is a process. This means developing an ability to
think in abstract. It is more organizing and
generalizing the essence of experience: seeing
patterns in them and interpreting in relation to
other context and concepts.
than what we today do in the name of Maths
teaching. We will stop looking for answers and
right answers which according to us is only one!
We will concentrate more on understanding the
problem and looking for different ways of solving
the problem. Hence we will not have only one
answer- the answer! But, we will have many ways
of solving the problem and many answers to this
question.
need teachers who can identify the skills and
knowledge with which our children come to school.
They need to build on this knowledge of the
children. Once the children see that what they
have is knowledge and is respected by others
especially by teacher’s community, they will
develop a lot of confidence in themselves and a
strong self-esteem. Once, this is established then,
our teacher’s job would be to slowly take our
children to the realm of abstract thinking. It is
very clear that we need to help our children think
in abstract. Use a lot of imagination. Help them
14
think out of box to solve problems, see what is
emerging, see the pattern, construct their own
meaning in things which otherwise looks
meaningless. Predict with some evidence, with
some base. Hence, Maths is not only for solving the
immediate problem of day to day living but more
so to improve the quality of our life and living. It is
to prepare the child to think, observe closely and
gain insights in the process and enjoy life. This is
the process of preparing the child for better living.
15
Discussion with Govt. Functionaries
16
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“K£À¥Á CzÀÄ, ºÉüÀÄ ªÀÄvÉÛ” CAzÀ ¸ÁºÉç.
“£Á®ÄÌ PÀtÄÚ, MAzÀÄ PÉÆÃqÀÄ, JgÀqÀÄ PÁ®Ä EgÉÆà UÉÆêÀiÁvÉ LvÀAvÀ°æ , CzÀÄ J¯ÉèöÊw, £ÉÆÃrzÁæ?”
“CAiÉÆåà ºÀÄZÁÑ, CAvÁ ºÀ ÀÄ«gÉÆÃzÀÄ §zÁÝ?” CAzÁ ¸Á»Ã§.
“ºÀAUÁzÉæ £Á vÉÆÃgÀ¸ÉÛä. MAzÀÄ ¥ÀPÀë £Á£ÀÄ ¸ÉÆÃvÉæ 2 PÉÆqÉÛä. ¤Ãªï ¸ÉÆÃvÉæ 5 PÉÆqÉâÃPÀÄ” CAzÀ.
“DAiÀÄÄÛ” CAzÀ ¸Á»Ã§.
ºÁUÁzÉæ, CªÀæ°è AiÀiÁgÀÄ UɯÁÛgÉ, AiÀiÁgÀÄ ¸ÉÆïÁÛgÉ? ¯ÉPÁÌZÁgÀzÀ GvÀÛgÀ ¨ÉÃPÀÄ. AiÀiÁgÀÄ PÉÆrÛÃj? ÀÆPÀÛ §ºÀĪÀiÁ£À GAlÄ. C®èzÉ, MAzÀÄ §ºÀĪÀiÁ£À UÉzÀݪÀjUÉ ªÀÄvÉÆÛAzÀÄ GavÀ!!
Maths in Anisha’s world
Shadakshari T S
while when I visited home recently.
He updated most of the things
happened all around his school,
friends, Elder brother Nakshatra, Family, Farm,
livestock etc.
Later he came closer and started showing lot of
affection by rolling all over me and kissing my
cheeks. He slowly opened up and asked me to
promise for one thing. I said “no I will not if I don’t
know the reason?” But he kept forcing me to touch
any one of his two fingers. I touched one finger and
he jumped so happily and shouted joyously and
told me “You should buy me a Bicycle “Now” that’s
18
fingers!!!!”
I said OK and asked him “But where is the money?
Who pays for it?
from my Savings!!! In my savings box you need to
put ONLY the remaining balance money.”
“OK, but what is the total cost of your bicycle you
are dreaming of now?” I asked.
He said “APPA It’s JUST 15 fifteen THOUSAND
ONLY…!!!!. My friend told me at school”
My home was full of laugher. He was slightly
surprised as why all of us were laughing!!!!
Then I spoke to him about the difference between
Hundred and thousand in his own world.
(According to him 16 is GREATER than 15 but not
the Thousand against Hundred..…!!!!)
He is still in his 1st standard…and learning his
numbers…. For his life
I too keep counting on HIM...!!! For my rest of the life..!!!
Conceptual understanding
Rudresh
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