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Sunday, June 28, 2009

Introduction To Vedic Mathematics

I have talked about my attempts to teach mathematics to my children in an earlier post. Because of constant coaching from my wife and I, their level of mathematical knowledge is at a point where they can probably sail through middle school and the early part of high school without learning anything more. They know how to solve simultaneous equations in 2 variables, find slopes and intercepts of linear equations, apply the pythagoras theorem, find the volumes, surface areas, and plane areas of most regular 2-dimensional and 3-dimensional shapes, etc., etc.

Some of it is still dependent on the mood they are in. Sometimes, they apply themselves and can then solve any problem they have been introduced to earlier but stringing together what they know to proceed from problem to solution. At other times, they will get lazy and claim that they "don't get" something even though I know they know how to do it. That is when things get interesting as I lose my patience and berate them for being lazy and unwilling to think through something.

The next task I have been handed by my wife is to make them good at mental computation. They are good at basic mental computation like addition, subtraction and multiplication because they have been to classes where they teach the use of the Japanese Abacus. But my wife is not satisfied with the speed they can achieve using the abacus. Essentially, the use of abacus for doing arithmetic is a brute-force method that has its limitations.

So, my wife did some research and has asked me to teach them Vedic Mathematics. The subject intrigued me enough that I started doing some basic research before I even begin my teaching duties. This is what I have uncovered so far.

Vedic mathematics is a system of mathematics consisting of a list of 16 basic sūtras, or aphorisms. They were presented by a Hindu scholar and mathematician, Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja, during the early part of the 20th century.

Tirthaji claims that he found the sūtras after years of studying the Vedas, a set of sacred ancient Hindu texts. The calculation strategies provided by Vedic mathematics are creative and useful, and can be applied in a number of ways to calculation methods in arithmetic and algebra, most notably within the education system.

The word “veda” has two basic meanings. The first, a literal translation of the Sanskrit word, is “knowledge”. The second, and most common meaning of the word, refers to the sacred ancient literature of Hinduism, the Vedas, a collection of hymns, poetry and Hindu ceremonial formulae. Believed to be one of the oldest human written records, the Vedas date back over 4000 years. Traditionally, they were passed down orally and adapted from generation to generation by sacred sages called rishis, before eventually emerging written in Vedic, an ancient form of Sanskrit.

The Vedas are divided into four main sections: the Rig-veda, Sama-veda, Yajur-veda and the Atharva-veda, known collectively as the Samhitas. The first three, the Rig-veda, Sama-veda, and Yajur-veda are basically ritual handbooks that were used by priests during the Vedic period (1500–500 BCE). Vedic mathematics is apparently part of the fourth Veda, Atharva-veda, which is distinct from the others in several ways. First, unlike the religious focus of the other Vedas, the Atharva-veda contains hymns, spells and magical incantations for personal and domestic use. Also, the Atharva-veda, which was written later than the other Vedas, was not always considered authoritative, but only became so after being accepted by the Brahmans, the highest order of Hindu priests. Collectively, the Vedas include information about a huge range of subjects, spanning religion, medicine, architecture, astronomy, etc.

It is a well-known and accepted fact that ancient Indian Vedic civilizations were known for being skilled in geometry, algebra and computational mathematics complex enough to incorporate things like irrational numbers. Important contributions to early mathematics were made by Indian scholars like Aryabhatta, Brahmagupta, and Bhaskara II. Indian mathematicians made early contributions to the study of the decimal number system, zero, negative numbers, arithmetic, and algebra. In addition, trigonometry was well-developed and understood in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

Almost all ancient Indian mathematics literature is composed completely in verse; there was a tradition of composing terse sūtras, like those of Vedic mathematics, to ensure that information would be preserved even if written records were damaged or lost. Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered as important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form.

Mathematicians of ancient and early medieval India were almost all Sanskrit pandits, who were trained in Sanskrit language and literature, and possessed a common stock of knowledge in grammar, exegesis and logic. Memorization of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorization and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia.

Tremendous amounts of energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorization of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jatā-pātha (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order. The recitation thus proceeded as:
word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...

In another form of recitation, dhvaja-pātha (literally "flag recitation") a sequence of N words were recited (and memorized) by pairing the first two and last two words and then proceeding as:
word1word2, word(N-1)wordN; word2word3, word(N-3)word(N-2); ...; word(N-1)wordN, word1word2;

The most complex form of recitation, ghana-pātha (literally "dense recitation"), took the form:
word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ...

That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the rig veda (ca. 1500 BCE), as a single text, without any variant readings. Similar methods were used for memorizing mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (ca. 500 BCE).

The most notable application of Vedic mathematics is in education. Vedic mathematical strategies may prove to be a useful resource for teachers and students, who may find elements of it easier and more accessible to teach and learn than conventional mathematics. In particular, these strategies may be an invaluable resource to students that already struggle with mathematics, and could benefit from alternative approaches.

One attempt at incorporating Vedic mathematics into education was made by Mark Gaskell, the head of mathematics at the Maharishi School, Lancashire, England. The school has developed a Vedic mathematics curriculum equivalent to the national one with impressive results. According to Gaskell, the alternative curriculum has resulted in livelier classes, greater student enjoyment and understanding, and improved academic performance. In fact, the first set of students to complete the course were each able to not only pass, but achieve over 80%, on the General Certificate of Secondary Education, a proficiency test taken by all secondary school British students, a year earlier than their peers in the regular curriculum.

I have also learnt a few tricks myself based on the research I have done and the books I have started reading. I am not a big fan of pure arithmetic in general. After all, calculators are common-place nowadays and they are getting more and more powerful. But, still there is something to be said for being able to do some mental computations from time to time. One may not have access to a calculator just when you need one. You don't want to be stuck not even able to do some intelligent approximations just because you don't have access to working electronic brains at that moment. That is what your organic brains are there for, and I do believe that learning to do mental computations may not only come in handy for accomplishing some useful things, but also to keep your brain sharp. The scientific evidence is unassailable that the more you exercise your brain with mental gymnastics like the ones involved in mental computations, the less likely you are to develop degenerative brain diseases like Alzheimer's disease. And the value of impressing friends and co-workers by beating them to the punch by telling them the answer before they have had time to even enter the problem into their calculators is priceless!

Once you understand a particular vedic method and learn how to do it in your mind, you can also amaze friends and others with your mental gymnastics. It is just uncanny how the techniques work: when you perform it in front of friends, it is as if you are either performing magic or you are a human calculator. I have even been accused of cheating by some people I have demonstrated these techniques to, though, of course, they just could not figure out how exactly I was cheating!

I think the best way to learn something is to absorb the methodology and then try to repeat it to others in my own words. That will demonstrate that I understand the methodology completely and will help me in teaching my kids what I have learnt. So, I will try to explain these mathematical techniques in my own words in future posts about Vedic Mathematics so that I am sure of the methodology before I sit my kids down and drill them on it. And these posts will also serve as a written record of the methodologies for future reference so that I don't have to hunt around for the original source as I try to relearn something I may have forgotten.

In the meantime, if you are curious, you can get a taste of Vedic Mathematics by going to the official website of the Vedic Mathematics Academy. They have tutorials on some basic methodologies that should be easy to master for most people. They also sell books and DVD's on the subject if you are interested in pursuing it further.


Sath said...

very informative. thank you for putting all together and easily understandable form. i am a great fan of vedic mathematics. i like to explore more on vedic maths have some cds, dvd's.
my email
anyinforformation on vm please help me

Blogannath said...
This comment has been removed by the author.
Robert said...


I have looked at Vedic math now the last two years. Being an Northeast american 1952 born and schooled as an engineer, of course, I find the techniques very facinating. I only wish this country would wake up and embrace what other cultutres have to offer. Great website. Robert.

Blogannath said...

Thank you very much, Robert. I am hoping to introduce these techniques to more people through this blog so that they can make use of it, and spread its utility through word of mouth to more and more people after experiencing the benefits first-hand. Please pass the word around, and thank you once again.

Anonymous said...

My compliments!
A clear and understandable explanation.

Blogannath said...

Thank you. I hope the actual lessons are equally clear and understandable.

Michelle said...

Hi I am a math teacher in the Philippines and wanting to make a research in helping students learn through Vedic Math. If I may I want to quote you in my paper. Yours is one of the most comprehensive introduction to Vedic Mathematics I have read. I am enlightened myself.

Btw, I am very amazed about the lesson of complement of 10. I recognize that it is used in some of the techniques I've encountered but I have no idea it was one of the basics. It emphasizes how much basic is not to be underestimated.

Thank you very much.

Blogannath said...

Thank you, Michelle. Please feel free to quote from these lessons, but please include a link to the blog post where possible.

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