Who discovered Zero? It is generally believed that Aryabhatta discovered Zero. But truly speaking, he did not. It is a wrong information that has been spread during the course of time for reasons unknown. One may ask how we can say that. It is because in the treatises written by him which is available today, there is no mention of discovery of Zero. Who then discovered Zero?
When we talk about discovery of Zero, we need to think what type of Zero are we referring to. For a layman, he would not be thinking beyond the meaning of Zero as ‘nothing’. But Zero degree Celsius doesn’t mean there is no temperature. Here, Zero is a place holder between positive and negative numbers. This notion of Zero must have come while negative numbers were first discussed by Brahmagupta (7th century Indian Mathematician) who was the first one to discuss about rules for operations with negative integers (12 centuries before the West started discussing about negative integers).
The Babylonians and Greeks had some notion of Zero but that was not the kind of Zero that we use today (refer the two links given at the end of this article). It was Pingala (300 BCE) the author of Chandass Shastra (one of the 6 Vedangas) who used Zero as a symbol for the first time in History. Pingala was also the originator of Binary Mathematics (Ref: ‘History of Mathematics in India from Vedic Period to Modern Times‘ online course on NPTEL, lecture on Pingala by Prof. M.D.Srinivas, Institute of Policy Studies). Pingala also dealt with Permutations and Combinations (2 millenniums before it was discussed in the West). It is very clear that for discussing such areas of Mathematics the knowledge of Zero is very important.
In other civilisations during the period of Pingala, Mathematics was not that advanced and they could manage without Zero. In fact, due to the lack of knowledge of Zero, they lagged behind in Mathematics.
So who discovered Zero? The answer is ‘we don’t know’. Looking at the kind of Mathematics that was dealt in India during the BCE period, it can be said that the use of Zero as not just a symbol, but as place holder and a symbol for nothing should have happened in India. But as Indians weren’t interested in accrediting their names to discoveries, it is not known who discovered Zero. As a matter of historical evidence, all that can be said is that it is in Pingala’s Chandass Shastra that we find the appearance of Zero as a symbol (as a digit) for the first time. In other civilisations, till that time zero was used only for absence of something. The development of decimal place value system should have developed somewhere between the period of Pingala (300 BCE) and Aryabhatta -1 (5th Cent CE) because by the time of Aryabhatta the decimal place value system was very much in use.
(Some interesting reads can be http://yaleglobal.yale.edu/about/zero.jsp
NPTEL Course on History of Indian Mathematics https://www.youtube.com/watch?v=p2WankcGP3Q&list=PLbMVogVj5nJThf31TNSQzuN7zqxe7HdRN)
Day 3: 8th Jan 2013
The day started with a few students’ presentations followed by the much awaited Final Quiz. Sixteen finalists from the written test paper on Mathematics (which happened the earlier day) were selected and formed into four teams. The rules were explained to all the team members by the Quiz Master – Sadagopan Rajesh, who happened to be a trainer for Olympiad kind of exams. The quiz was very well-organized and conducted by him. Once I started seeing the questions, I was glad that I wasn’t selected for the finals or else my team would have got almost zero contribution from me. The questions were very tough and the maximum time to think was 45 seconds where they got questions like – If you add 2 at the beginning and at the end of a natural number and subtract it from the original number, then you will get the difference as 2317. Find the number. Another question was – ‘n’ is a natural number such that 2^n divides 2013! (2013 factorial) exactly. What is the hightest power of ‘n’? And the answer was 2004, i.e., 2^2004 divided 2013! exactly. I had no clue how the kids got the answer in JUST 45 SECONDS!!!!!!!!!
Everyone enjoyed the quiz. The following session was Prof. Narasinga Rao Memorial Lecture to be given by a very renowned mathematician and Mathematics teacher, Prof. Chukka Ramaiah. Prof. Narasinga Rao was a very learned mathematics teacher who did research in Number Systems and made enormous contributions in the field of mathematics teaching. This 85 year-old man’s (Prof. Ramaiah) standing 90-minute talk moved the audience. He spoke in such a humble manner that it touched everybody’s heart. He emphasized both on Mathematics and how the attitude of teachers should be. He also shared how Mathematics is taught in countries like Singapore where they could teach concepts of Algebra in second standard. Their teaching techniques are far advanced while we still walk by the old road without much innovative ways of teaching. He also said that mathematical geniuses are coming out from small countries like North Korea, Vietnam, etc. but we, even though being such a big nation, are not able to nurture much mathematical talents. The primary reason what he said (which appealed to me very much) was that we teachers ourselves are not much motivated, then how can we motivate children? A teacher’s job is not a salaried labourer’s job. It is even more respected than a doctor or a soldier’s job. What struck to me was his words – ‘It is not the answer that is important, but the process’. How true! And today everybody is running after marks. A child is judged by the marks he gets and not the concepts he knows. He also remarked that all Guides and Digests should be banned by the Govt. because that is not the right way to learn. Learning should happen from thinking and not from copying down the solution. A child might come up with a better solution that what is taught to him. But how can thinking happen if the solutions are given readymade? Hence, we all should pressurize the Govt. to ban all such books which give out solutions readymade. He also spoke about how question papers are set these days. All the questions are taken from the textbook or from previous years’ question papers. What are we ultimately giving to our children? Is that really an exam? In one of the states in India, the Govt. had declared that the progress of a teacher will be judged by the scoring of his students. So, there came teachers who taught children ‘how to score in exams’ and not how to gain knowledge. How pathetic! How do we expect our country to progress if our prime focus is on getting marks instead of gaining knowledge?
To teach a child, many a times it calls for going down to his level. He gave a very good example which he used to teach the concept of Mathematical Limits when children found it difficult. He said, “A movie actress told the director that I will act in your movie only on one condition: Nobody should touch my body. And there was a kissing scene in the movie. So, the actor and the cameraman had to take the shot in such a way that both the actor’s and the actress’ faces came very near but the lips never touched. That is how a Limit in Mathematics is!” The audience burst into laughter. He continued, “Don’t think that why this old man is coming here and saying such things to the young audience. If you have to teach something to a teenager, try getting down to their level and see how fast they pick up the concept. I could see no better example to explain the concept of Limits to a 11th grader.
He also added that we should give the children an opportunity to come up with new questions. A person who is able to ‘spit questions’, he can have a Creative Thinking, which is very important in Mathematics. From the question that we pose to the students, they should be able to come up with more doubts and form more questions. One should also see to it that in a child a Critical Thinking is also inculcated. These qualities of Creative and Critical Thinking, can go a long way in Mathematics. He also requested bodies like AMTI to reach out to Govt. schools where the talents are not much recognized. All these students who had come for the National Seminar were from well-to-do schools and that’s not where the majority of students lie.
His talk was equally inspiring for the mixed age group in the audience and the applause after his talk lasted for a long time. One could see how much he was involved in his talk and how the words were coming out of his heart. He belonged to a rare class of teachers who could teach any student right from primary to University students. I saluted him from the bottom of my heart! I took the opportunity to follow him after the talk and present him a copy of my book – The Teacher Who Taught Us To Think that I had written about my Mathematics teacher. He was happy to know about the content of the book and gave a small pat on my face. It was a great feeling to have got even a touch of such a great teacher!
We all went to have lunch and came back to the hall for a ‘Short-Talk’ session where anyone could speak about the programme or anything on Mathematics, for 3-5 minutes. I took that opportunity to share couple of my thoughts. One – to have an Olympiad kind of test, like the one we had the previous day, for teachers so that we can equip ourselves with better thinking and facilitate better teaching to our students. Second – few students who presented papers were not trained properly. In the sense, they were just reading out from the PPT and some ones not even looking at the audience. The teachers could have given them rehearsals earlier so that they could feel more confident on the stage. But otherwise, the kids did a superb job which was beyond our expectations.
During the valedictory function, Prof. Mahadevan, one of the core Committee members of AMTI, expressed his sincere thanks to the Sri Prakash Synergy school for extending their support and appraisable hospitality. In fact, the teachers and students of the school were having holidays during this time of the year but all of them were volunteering for the event, with a BIG smile and lovingly they were doing their assigned jobs. Usually, employees of any organization start cribbing when they are called to work even for a weekend. Whereas these guys had taken atleast 5 days off their vacation to volunteer for the Conference. The children of the school were really sweet. They would greet every delegate with a Namaste and folded hands. That showed the culture they get from their school. The vision of the school was – Indian roots, International standards. Truly they were living upto that with respect to the values inculcated and infrastructure provided to the students.
After the valedictory, everybody started moving to their rooms or leaving to catch their trains. I had enough time since my train was at midnight. I was back in my room where I sat to pen down my thoughts while it was fresh in my mind.
This National conference of Association of Mathematics Teachers of India was overall a great experience for me. It gave me opportunities to meet a lot of teachers, retired teachers, mathematicians, talented students from different parts of the country and few researchers. It gave a forum to discuss various mathematical ideas with scholars on the subject. The exhibits and paper presentations by the students gave me more ideas of how the same thing can be organized in my town atleast in a small level. The students who came could also learn a lot of things from the Conference. There were a lot of books on Mathematics kept for sale. The reason I thought of penning down this experience is that atleast the people who read it might know some student, teacher or parent who could possibly expose their child to such events which are really great opportunities for them to learn. Let us be instruments in facilitating growth of children in whichever possible way.
Day 2: 7th Jan 2013
I was all excited since morning because in the morning we were going to have a test based on numerical and logical questions. It was after a long time that I would be appearing for a mathematical contest. The ones who would clear it would go into a final round of multimedia quiz. The quiz started at 9 am. There were 25 questions and 1 hour to solve it. 15 of the questions had negative marking. I was solving such type of questions after a long time and I could see how slow my processor was becoming. I could solve only 14 questions in an hour. My room-mate Shrivatsav and Varada, both solved 21 questions. I got only 9 questions correct and 5 wrong out of which one had a negative marking. So, my score should have been somewhere around 34.
Shrivatsav and Varada both went for Olympiad-coaching in Mathematics and we could see the difference in their speed of problem-solving. It was fun watching them crack any given problem. The same was the case in the students from Kota (in Rajasthan) which happens to have the maximum number of IIT-entrance coaching centres. There the students would be admitted right from 6th or 7th standard onwards and they don’t attend schools at all. I was told that their institutes had tie-ups with schools which permitted the students to do away with regular schooling. The institutes would give them extra-coaching in the place of schools. Their normal coaching hours were around 8 hours a day. But these children are usually extremely bright and even a 6th standard child would know concepts of 8th or 9th standards. The students from Kota had come from Allen Career Institute. There were exhibits displayed by students in few classrooms. Most of these exhibits were charts. But these students from Kota had powerpoint presentations on various topics. One of those was on ‘Paradoxes’ which was explained by Ms. Astha Agarwal, a girl of 9th standard. Her presentation was very lively. She explained what a paradox is (which is usually very new and interesting for school children). She also respond well to my questions. Another was of Paavan Gaur of 10th standard from the same institute. He presented on geometrical representation on Arithmetic Progression, Geometric Progression and Harmonic Progression. He said that he is very much interested in physics and astro-physics. In my mind I was thinking, when I was on his age, I never knew that a subject by the name ‘Astro-physics’ even existed!
I asked both Astha and Paavan how it feels by not going to school. I felt that somewhere inside they missed the school. Paavan said that these institutes, even though they charge a good fee, are able to give them the best of teachers and best of teaching methodology, which should be ideally be undertaken by the Govt. But since our Govt is not able to provide such extensive training in schools for the really talented children, they have to rely on such coaching centres which give them training in depth in their subjects of interest. Some children learn the talk for paper presentations by-heart and fumble if something is asked to them. But these kids were really good with their concepts as well as their presentations. I had a nice time with them before I went for the afternoon tea break.
In the late afternoon, the host school, Sri Prakash Synergy School, had arranged buses for the delegates to visit a very old temple of Peddapuram. But I wasn’t in a mood to travel. I chose the time to sit alone and pen down something. During this time, I got some students who came from another school that belonged to Sri Prakash group of schools. These boys were really bright. They picked up whatever topics I explained to them. We discussed on different base systems and how numbers could be converted from one base to another base. I also gave them questions to think upon and give me the answers tomorrow. They did come up with the answers and ran towards me, when they saw me, to share their answers. One of them said, “Sir, we did not know these things. When you asked us such questions, we thought upon it and found out answers on our own. Thank you for asking such questions.” I smiled back.
Dinner that night was a buffet with a lot of varieties of cuisines. It was arranged in the school garden. Everyone could be seen enjoying the food and having casual conversations with others. I too took the opportunity to discuss few things with the people I came across.
Post dinner, we came to our room and Varada, Shrivatsav and I started discussing on logical solving of geometric problems. All of us posted questions for which we had to think of easier ways of solving other than the routine ways to solve the problems. Around midnight, we retired to bed.
…to be continued in the following post https://vinayrnair.wordpress.com/2013/01/09/experiences-at-national-conference-of-amti-2013-day-3
Day 1: 6th Jan 2013
Everyone in our room woke up around 6 am. Srivatsav, the boy of 11th standard, asked me to give him a hand in folding his bedsheet. I was taken aback when I saw him struggling to fold the bedsheet which was a bit bigger than the normal size. I showed him how he could fold it. However he still seemed to find it difficult.
After the shower, I got into a small conversation with Mr. Karmarkar and told him about Mr. Nair’s poem that I penned down the earlier day. He immediately recited a small poetry on fraction-addition in Marathi. I quickly penned down the beautiful composition which I was sure would come handy for a Maharashtrian kid. It was a surprise for me to have met the second mathematician-poet in such a short while.
Soon we headed for the canteen to have breakfast, following which; we proceeded to the lecture hall. The hall was full with around 200 delegates comprising equally of teachers and students. After the inaugural ceremony, the first session was Prof. Bhatnagar Memorial Lecture on ‘Applied Mathematics in solving problems in Aerodynamics’. Honestly speaking, I could follow only what was explained in the first ten minutes. After that was beyond my learning capacity. Yet it was a revelation for me to see so much of applied Mathematics in engineering.
What I understood was that aerodynamics dealt with the study of motion of air (or fluid) past bodies. For example, kite, boomerang, bird, aero plane, missile, etc. The speaker, Mr. Venkateshan Bhaskaran, said that basically the study of motion of flying objects involves four parts:
• The structure of the flying object
• The propulsion of the object
• Avionics – the mechanics involved in the flying object
• Aerodynamics – again divided into two sections:
o Given the shape of the body, what are the forces that will be faced by it, or
o What should be the structure of the body which has to be constructed to be carried in the given force?
He explained how different kinds of wings of aero planes were invented and that we have thousands of designs of such wings. One just has to choose the right body to achieve his purpose of getting the optimum utilization of resources. He said that airlines have tough competition and research going on in their field so that they can choose the most effective design on the aircraft to suit their needs which will incur minimum costs as compared to their competitors. He explained various differential equations and other topics from calculus. The only few symbols that looked somewhat friendly to me in those equations were 0, +, = and π. But I thought, had we been taught the applications of calculus in college while we were learning to solve the equations, it would have given altogether a different meaning and approach for Mathematics. In college, we just solved equations without knowing the least where and how it’s going to be useful. But there was so much of use of calculus in applied Mathematics of engineering, aero dynamics, etc.
The second session was about π. As soon as I heard it, I was very happy because π happened to be one of my favourite topics. But my heart was saddened because the speaker didn’t talk about π in the way I understood it. He explained π using calculus. Yet, I wasn’t totally disheartened because he showed a series by ancient Indian Mathematician Madhavacharya, known as the Madhava-series which said:
1 – 1/3 + 11/5 – 1/7 + … + ((-1)^n)/2n+1 = π/4
As we learn in school that π comes in calculations related to circular shapes, here is a series given by a Mathematician who lived somewhere in the 13th or 14th Cent. CE, where he expresses the value of π/4 in the form of a series and without a circle!! So, π can exist even outside a circle!!
The presenter, Prof. Pandurangan, said that the above formula could be proved using Gregory-Leibnitz series (~1670 CE) which says:
π/4 = z – (z^3)/3 + (z^5)/5 – (z^7)/7 + ….
We just have to take the value of z as 1 in the above series.
It gave me immense joy when I learnt that this was proved by Madhava, an Indian mathematician three or four centuries before the mathematicians in the West discovered it.
The session was followed by lunch break. I took a short nap in my room after the lunch and went back to the lecture hall to attend the remaining sessions. On my way I saw a display of books for sale. And all of them were on Mathematics. Books have always been my weakness. I chose 6 books which dealt with magic squares, a book on Pi, e & i, and few other topics.
A teacher was presenting on a software called Geogebra which could be used to give more visual clarity in clearing concepts mainly in geometry. Constructions could be made very quickly using this software. Everyone enjoyed the presentation because the teacher made it really precise and clear. The next teacher presented upon ‘Exploring & Visualising Mathematics through Graphic Calculators’. She too did a great job in explaining. There is something called as a Graphic Calculator available in the market. In it we can draw graphs and plot the points, draw curves, and do almost anything with graphs. This gives the student an immediate picture of what an equation 4x + 3y = 24 would mean. A graphic calculator can also be downloaded from the internet. Her point was that if we are able to show the children the graphs of functions visually, then they would be able to understand the functions better than what they seem to understand merely in terms of x and y.
I didn’t quite enjoy the next two presentations by two teachers who almost emphasized that everything should be electronic and using technology. In their explanations they almost left no space for a teacher to think and set a question paper. Even that was suggested to be prepared from using a software. However, certain points were good and acceptable to me.
The following session was presentations by students. The students were divided into two classrooms and the audience could choose whichever classroom they wanted. I chose Classroom 1. The presentation skills of children in the classroom where I went into, were average even though they spoke of very nice concepts. It seems they were trained on the concepts more and less on presentation skills. Nevertheless, the kids did a great job on the stage. I was told that the students in Classroom 2 had done excellent presentations. My roommate, Shrivatsav, had his presentation in Classroom 2 of which I wasn’t aware and I missed it. His presentation was on methods to solve complex problems in simple ways using logic and observation.
I came to my room after the students’ session and didn’t go for the cultural programme that was arranged. I chose to take a quick nap for 20 minutes and started going through the notes I prepared and working on something. Soon it was time for dinner and I came back to the room immediately after dinner. I asked Shrivatsav how his presentation was and he said that it was good. I asked him what he feels about Calculus (which most of the students are scared of.) He said, “It’s easy. I like it.” I was surprised. This was the same boy who found Calculus a lot easier than folding his bedsheet. I had never imagined that easy and difficult things could be so relative!
At night I saw that Varada was sitting on his bed, feeling sleepy but not wanting to sleep. I had seen him playing with a Rubik’s cube all the while. The cube was almost worn out. I could imagine how many lakhs of times he must have rolled his fingers on it to get it into that state. Suddenly I observed that the cube was perfectly arranged with single colours on all sides. I asked him if he actually did that. He smiled and replied, “Yes.” “How much time do you usually take to complete it?” I asked. He said, “Less than a minute.” I said, “WHAT? What is your record time?” “37 seconds” was his reply. I had seen the genius in his eyes the very moment I saw him the first day. But he was not happy when he said this answer. He said, “That’s not a good timing. I’m still the second in my class. There’s a boy who does it in 25 seconds in my class.” I was like ‘WOW’ because I never knew that record timings were so fast. He said that the world record timing is of a man who could do any cube in 5 seconds (however badly they are jumbled.) Then I watched him arrange a cube. He held the cube very firmly in his hands and moved the cube with his index and ring fingers of both hands. Yes, he used just one finger to get the rotations done. That saved a lot time of his as compared to beginners who use one the whole of their palms to rotate the cube. Of course, he did that through practice. So I asked him how much time he took to achieve this skill. I was thinking that he would say some time with respect to years or many months. He said, “One and a half months back I started training myself. And I practiced for 3-4 hours a day. I somehow wanted to crack it. Cracking the whole cube took me two full days. The rest of the time was spent in studying the algorithms and formulae for getting it done faster.” This was a new class for me. I never knew such things like algorithm and formulae existed for solving a Rubik’s cube. He continued, “There’s a lady who has given many formulae for solving different situations in Rubik’s puzzle. The number of formulae goes beyond 150 or 250. The formulae go something like this – Right-inverted, Up, Right, Right, Up-inverted….” He went on saying such words. He said, in a particular situation, we have to think which formula to choose. If we are able to do that correctly then we can solve a cube in less than 30 seconds. “But have you by-hearted all the formulae?” I asked. He replied, “No. I prefer to do it using algorithms and pattern-observation. I had my friends who guided me to get out of some situations where I would get stuck up sometimes in a puzzle. And I was passionate about this to such an extent that I couldn’t get myself out of it.” And I could see that in him. All the while he was talking to me, he would solve the cubes and he would be hardly looking at it. I was imagining the capacity of a human brain and how much of it we were actually utilizing!
We retired to bed by 11:30 pm.
…to be continued in the following post https://vinayrnair.wordpress.com/2013/01/09/experiences-at-national-conference-of-amti-2013-day-2
Day 0: 5th Jan 2013
I was all set to attend the National Seminar of Association of Mathematics Teachers of India (AMTI) going to be held at Sri Prakash Synergy School, Peddapuram near Samalkot Junction which is some 3 hours train journey from Vishakapattanam (Vizag, Andhra Pradesh, South India). I boarded the train from Mumbai on 4th afternoon. My wife had packed my favourite curd rice for dinner and the following day’s lunch. Luckily, my co-passengers were busy talking amongst themselves and (luckily) I didn’t seem to be an interesting person to them. That gave me enough time and room for me to do my work.
I spent my time in the train on some official emails initially and then I watched a few videos on Mathematical Thinking that I had downloaded from www.courera.org where one can register for online courses offered by Professors from various Universities across the world at absolutely zero cost. I had found this course really interesting because it said that it was not on solving mathematical problems but developing that ability to solve mathematical problems; in other words, to develop the mathematical thinking. Professor Keith Delvin from Stanford University was the Course Instructor and was taking the course very interestingly. So, I spent time on the side-upper berth to watch the videos and came down occasionally to recharge my laptop. The quiet journey and simple food made the journey very comfortable.
As I reached Samalkot Junction, couple of teachers from the host school (Sri Prakash Synergy School) along with two students of the school had come to receive delegates who were arriving at the railway station. There were other delegates and Members of AMTI who had arrived at the station during that time. We were escorted by the students to the school in auto rickshaws. In my autorickshaw was a Mathematics trainer, Mr. Bharat Karmarkar (in his fifties), from Nigri (in Pune) who took classes to improve conceptual clarity in Mathematics both online and through home visits. Since we had the subject of common interest and since I had all the patience in the world to listen to such people, we began the discussion even before we exchanged our names. He said that his objective was to make the concepts clear rather than scoring marks or getting homework done from children. I was happy to see such a teacher after a long time. He also conducted annual excursions for children where they get to learn a lot of things practically through their visits. I was totally impressed by his ways of educating.
In no time we reached the school and were again greeted by the student-volunteers and their teachers. The school campus was big and their hospitality was really appreciable. Few students greeted us at the reception with a smile and Namaste. They applied kumkum (turmeric powder mixed with lime which gives it a red colour, often used in Hindu occasions) on our foreheads and sandalwood at the back of our palms. It was so nice to see the humility with which they greeted us. One of them asked me to keep our luggage at a place and proceed for the registration counter where we were to fill up a form and proceed to our rooms. Another student-volunteer took us to our room. Everything was so well organized that we didn’t face any hurdles at all in getting the initial procedures done.
In my room, there were four beds. One of my roommates was none other than Mr. Karmarkar and two others were Varada and Srivatsav, boys of class 9th (from P.S. Senior Secondary School, Chennai) and 12th (from Padma Seshadri Bala Bhavan School (popularly known as PSBB), Chennai) who had come to give presentations in the Seminar. The boys were really smart and energetic.
As we had reached the entrance of the school, Mr. Karmarkar met a friend of his – Mr. Prabhakaran Nair, a retired school teacher from Calicut, Kerala. They were delighted to see each other after a couple of years. Mr. Karmarkar told him that there’s another Nair with him (which was me) and then we greeted each other.
Me & Mr. Karmarkar were in room no. 104 and Mr. Nair was in 102. First thing I look for when I enter a new room is the charging point. And I was surprised to see that there wasn’t any! At last I found a point in the small dressing area before the rest rooms. But the plug had some loose contact and hence it wasn’t working properly. Finally we found two plug points outside our rooms (in the verandah of the floor) for the whole floor which had ten four-bedded rooms. Luckily Mr. Karmarkar had carried an extension cord (a tip for those who travel to such places where you never know how many plug points you might get.) I got my mobile charged for some time and we all went to the canteen for dinner.
After dinner I got an opportunity to talk to Mr. Nair and we both walked into his room continuing the conversation. His teaching experiences fascinated me a lot. Mr. Nair, to my utter surprise, was 70 years old. He didn’t look more than 60 to me. He was full of glee and energy while he was talking. He was a school teacher and also a mathematician-poet (a very rare combination we find these days.) He said that while he used to go to take classes for the differently-abled children, he found these poems handy to teach them concepts of Mathematics. One poem was such that a boy saw a flock of parrots flying in the sky and he asks one of those parrots how many you are in number. The parrot replies that ‘us + us + half of us + one-fourth of us + me, will make 100. Now you calculate how many we are in number.’ Saying this, the parrots fly away. The boy isn’t able to calculate and asks his grandmother. The old lady immediately gives the answer as 36. The answer forms a simple equation as x + x + ½ x + ¼ x + 1 = 100. Solving it further we get 2 ¾ x = 99, i.e. 11/4 x = 99. Thus x = 36. The boy gets a question that if the number of parrots were different, what would be the answer in the place of 100? He goes exploring and discovers an Arithmetic Progression 12, 23, 34, 45, and so on. Thus, in one poem, Mr. Nair explained simple equations and arithmetic progression. I’m sure if a student gets something like this, then undoubtedly he would find a flavor to Mathematics. It reminded me of all the great Indian Mathematicians like Bhaskaracharya who wrote down Mathematical questions in poetry and gave it some spice thus encouraging the students to get to the answers.
The poem in Malayalam was really beautiful; not as dry as I explained it above. He shared a couple of more poems. One of the other poems he shared was that of a boy who asks his father, ‘why are there only two types – even and odd numbers?’ (This ‘why’ is not a usual question from a school student.) His father somehow gives some explanation and tries to cover up his own ignorance. Then the boy corrects his father and tells him that in ‘even number’ of circles, each circle can be paired with another circle. But that’s not the case with ‘odd number’ of circles. In case of odd numbers, one circle will stay unpaired. When you combine two odd numbers, say 5 and 7, then the circles which were left out alone in 5 and 7, will be paired with each other. That is why when we add two odds we get an even number.
Mr. Nair also shared some ideas on how we can induce some thinking ability in children and ask them questions in general to make them think on ideas and encourage them to explore more. It was wonderful discussing with him till 9:45pm. I wished him good night and came out of his room to pen down this experience. In a few minutes another gentleman, Prof. Govinda Reddy from Nandyal (a place in Andhra Pradesh) came to me. I had met him last year at a National Mathematics Seminar at Rashtriya Samskrita Vidyapeetham, Tirupathy (Tirupathy is a place known for the famous Balaji temple in Andhra Pradesh.) In our earlier visit I couldn’t interact with him much. But this time, it was compensated. We sat till 12:45 am and finally I had to send him to his room to take rest.
Prof. Govinda Reddy was an old retired Mathematics teacher from Nandyal. He seemed to me to be in his 70’s (I was astonished to hear from Mr. Karmarkar that Prof. Reddy was 81!) He was tall, lean, dark and soft-spoken. He has his personal library which comprises of around 1000 books on Mathematics. He collects books on Mathematics from different sources. He subscribes some foreign magazines like Crux (a Canadian publication) and Putnam (a USA publication for Mathematics Olympiads.) Sometimes his students gift him some books but most of the times he purchases from his pension. He started the Nandyal Association of Mathematics Teachers 10 years back where he organizes Teachers’ Training Programmes, Contests on Mathematics for students and various other programmes. He showed me one of the question papers for the 10th standard in one of such contests. It had 6 questions and the students were given three hours to solve the paper. The levels of questions were very high but it would definitely bring out the best of upcoming mathematicians into light. He said that the cash prizes of some contests went as high as Rs.10,000/-. I was astounded to hear this and asked him from where does he get funded for such activities in a small town like Nandyal? He smiled and said, “I get a pension of Rs.27,000. My monthly expenditure comes to 7,000. My house rent is Rs. 6,000. The books that I buy amount to Rs.1,000 per month. After some miscellaneous expenses whatever is remaining, I save it for such purposes. At the end of a year, whatever amount is saved is spent for such events.” And he kept smiling.
I was speechless! What kind of dedication he was showing towards his subject. How different are people. Some places we even find teachers who hesitate even to give a prize of a small toffee to their students and here is a teacher who spends his entire pension to bring up the upcoming talents in Mathematics. Here was a man, sitting before me in a plain white shirt and a white dhoti, whose hand shivers while he holds something, and is in his early 80’s, at such a stage in life what he is doing such remarkable things which the Govt. or other institutes should be taking up.
He showed me a book which was going to be released the next day. It was about brief life sketches of many mathematicians across the globe along with their photographs so that children can remember their faces when they hear the mathematicians’ names. He had also brought along with him some 132 flexes on the life of Srinivasa Ramanujan which explained the life of Srinivasa Ramanujan with photographs at various stages of his life. I couldn’t help myself wondering as to how much effort Prof. Reddy was putting in the field of Mathematics even at this age!
Finally, we retired to bed at almost 1 am.
… continued in the following post https://vinayrnair.wordpress.com/2013/01/09/experiences-at-national-conference-of-amti-2013-day-1/
DID YOU KNOW that Greek and Romans were idol worshippers and Geometry developed to a great extent in their country. Arabs were not idol worshippers and Algebra developed in their country. India was a combination of both philosophies where some people follow idol worship and some didn’t. Could that be one of the reasons why both Algebra and Geometry developed in India?
The ancient texts of Mathematics, Sulba Sutras, contain geometrical constructions pertaining to religious sacrificial rituals. When there was a need, the branch of geometry developed. Temples & rituals can be seen in Greek & Roman history which might have forced them to develop geometry in their country.
Be it a coincidence or a fact, it is surely a good point to think upon. Do religious faiths factor the development of mathematics in a culture?
If we ask somebody what ‘Psychology’ is, they will say it’s the study of the Psyche. Similarly, Biology is the study of living organisms. But if we ask someone, even a Mathematics Professor, to define ‘Mathematics’, he m…ay not to be able to define it clearly. One may say, it’s the science of numbers or shapes. However, that is not the exact definition.
The definition of Mathematics is beautifully said in the Shastras. The Shastras say, everything has got 24 qualities. E.g. a mango. It has got a form (Rupa), Taste (Rasa), Smell (Gandha), a different feel when you touch (Sparsha), and so on. The fifth quality is Number (Sankhya). There is ONE mango. So this number, ONE, is considered as a quality of the mango. The other qualities are magnitude, separateness, conjunction, disjunction, remoteness, proximity, weight, fluidity, viscidity, sound, intellect, pleasure, pain, desire, aversion, volition, merit, demerit and tendency. No field of modern sciences can give such an elaborate explanation on qualities like the above which is mentioned in the Nyaya Shastra.
The Sanskrit verses in the form of a question and answer form is very interesting.
Q: गणितम् किम्? (What is Mathematics?)
A: गुणितम् शक्यम् गणितम् (That which enables calculation is Mathematics)
Q: केन? (How)
A: संख्या या. (With numbers)
Q: संख्या का? (What is a number?)
A: Sankhya is a Guna present in the object.
संयक ख्यायते इति संख्या – That which enables ‘Systematic-Thinking’ is Samkhya
Mathematicians define mathematics as “Systematic Thinking”! Mathematics is not mere computation; it is something which enables a person to think systematically. This was the definition of Mathematics given in Vedic literature which gives the clear picture of Mathematics and why one has to learn it.
Maths is one of the oldest subjects that we have today. Oldest because, even the caveman used it. But who taught him? When birds fly, they calculate speed, time & distance. A cheetah, while chasing its prey, runs in such a way that it covers the minimum distance to catch. Who taught them Maths? The only possible answer is – they discovered it on their own. Every living organism has a Maths program in it. They use it knowingly or unknowingly as per their requirements. In the subject of Vedic Maths, we see how Maths can be discovered from our ‘within’.
Vedic Maths is about observation. All we need to do to learn a particular technique in Vedic Maths is – To keep looking at it. After some time, we see a pattern that keeps repeating.
Do you see any pattern repeating in the questions?
Ans a) 1225 Ans b) 2016 Ans c) 5621 Ans d) 7224
Now keep looking at the first two digits & last two digits of every answer, you will see a pattern.
Scientists say, left side of our brain receives data & information, analyses it and gives the result. The right side of the brain observes patterns, designs, etc. Creativity & Intuition can be developed only if the right side of our brain is used more. When any subject is ‘taught’, only the left brain is developed. But when it is ‘discovered’, our brain develops. As a result, we just don’t become good at Maths but also improve on other Creative areas.
With repeated observation of patters in calculations, our brain automatically starts performing them without us even realizing of such an event happening inside our brain. It’s like how we see in cricket. When the bowler releases the ball, how much time does the batsman get to decide how to hit it? Just a fraction of a second. And within that time he calculates the speed, spin / swing of the ball, his footwork, gap between fielders, force and angle at which he should hit the ball, and much more. Such is the capacity of our brain.
We can make our brains sharp by learning this wonderful subject – Vedic Maths; where Learning is Discovering!
(as appeared in Tapovan Prasad magazine in September 2010 issue)
Bharat has been the motherland of many great souls whose contributions have helped the whole of mankind. One among them was Swami Bharati Krishna Tirthaji who recreated a system of Mathematics that is popularly known today, as ‘Vedic Mathematics’.
Born to highly learned and pious parents in 1884 at Tirunelveli in Madras Presidency, Venkatraman was an exceptionally brilliant boy. He always stood first in his class for all the subjects. At the age of sixteen, he was awarded the title ‘Saraswati’ for his proficiency in Sanskrit. He was deeply influenced by his Sanskrit Guru Sri Vedam Venkatrai Shastri whom he remembered with deepest love, reverence and gratitude. When he was twenty, he passed M.A. of American College of Sciences, New York (from Bombay Centre) in seven different subjects (His subjects included Sanskrit, Philosophy, English, Mathematics, History and Science) simultaneously securing the highest honours in all. He was proficient in fourteen languages.
Prof. Venkatraman served as Principal of National College, Rajmahendri for a short period until his thirst for spiritual knowledge pulled him to Swami Satchidananda Sivabhinava Nrisimha Bharati at Sringeri. After eight years of extensive study of the scriptures, Prof. Venkatraman was initiated into the Holy order of Sanyasa as Swami Bharati Krishna Tirthaji. After few years, Jagadguru Shankaracharya Sri Madhusudan Tirtha’s (of Govardhan Math) health took a severe turn and Swami Bharati Krishna Tirthaji had to take up his position at Puri Govardhan Peeth.
During his life, Swamiji travelled throughout India. He was a spiritual dynamo. Vedic Mathematics was one of his epoch-making contributions to the world. During his Vedic studies at Sringeri (1911-1918), as a result of his intense tapas in the forests of Sringeri, Swamiji unraveled the hidden meanings of certain Sutras (aphorisms or word formulae).
Vedic Mathematics is based on Sixteen sutras and Thirteen upasutras; which Swamiji says, he ‘re-discovered’ from the Vedas. One may not find these sutras as it is in the Vedas, but the essence or the root is from the Vedas. Such was the greatness of Swamiji. He could have easily claimed the ownership of the New System of Mathematics, and nobody could have doubted it. But, like any other great saint of India; he gave the credit to the Vedas.
Swamiji had written sixteen volumes on Vedic Mathematics (one volume on each sutra). But before it could be printed, it was irretrievably lost. Everyone grieved over the great loss except Swamiji, for he said he could rewrite them all recollecting from his memory. In one and a half month’s time, he wrote one introductory volume. However, the toll that had taken on his failing health, on account of his rigorous work for almost four decades (and losing eye-sight due to cataract), did not allow him write more. And in February 1960, the Mahatma left his mortal frame.
Swamiji’s introductory Volume of Vedic Mathematics itself covers Addition, Subtraction, Multiplication, Division, Squares, Square Roots, Cubes, Cube Roots, Factorisation, Simple & Quadratic Equations, H.C.F, L.C.M., Decimals, Fractions, Analytical Conics, and much more. The remaining fifteen volumes had higher levels of Mathematics based on the same sixteen Sutras and thirteen upa-Sutras. Never in the past has anyone made such a discovery by encapsulating such a vast subject like Mathematics in just a few Word Formulae.
When one hears the word ‘Sutras’ in a language like Sanskrit, one need not think that he should have knowledge in Sanskrit to understand the subject. It is so lucid that children of any caliber can learn it with ease. Unlike the Mathematics that is taught in school which has only ‘one way’ to do a particular calculation, Vedic Mathematics teaches different ways to solve the same problem. Perhaps, this was Swamiji’s way to connect Mathematics to God – Different ways to attain the same ‘Truth’!
Using Vedic Mathematics, even a ten year old can calculate 999 x 991 or 20111 ÷ 975 in less than five seconds, and that too in a method of his own choice. This may look like an exaggeration but it is the Truth. More than ‘Techniques’ this subject facilitates ‘Thinking’, which today’s educational system in India unfortunately does not allow. Another sad thing is that such a great text has not been introduced in Indian Universities. But it has been accepted as a part of syllabus in some schools of England. Today, there are many books available on Vedic Mathematics (based on the introductory volume of Vedic Mathematics), many of them written by Western Mathematicians.
Many school students are afraid of Mathematics and they have no option but to learn it. Using Vedic Mathematics we can enjoy the subject and experiment & discover different methods. Imagine how much time we could save doing calculations if we learn Vedic Mathematics!