INTRODUCTION TO VEDIC MATHEMATICS
Vedic (pronounced 'Vaydik') Mathematics is an ancient system of mathematics
originating in India in Vedic times. It was rediscovered between 1911
and 1918 by Sri Bharati Krsna Tirthaji (1884-1960) who studied the
ancient Sanskrit texts called the Vedas. The date these texts were
written is unknown but the content of the Vedas was passed on by an
oral tradition long before writing was invented. The Vedas are said
to cover every area and aspect of knowledge, including for example,
ethics, grammar, architecture, astronomy. The word 'Veda' literally
means 'knowledge'.
Since the publication of his book "Vedic Mathematics" (reference 1)
in 1965 interest in this system has been growing and some schools
now successfully teach it. Many teachers attending courses and talks
on Vedic Mathematics in recent years have expressed a strong interest
in teaching the Vedic system in their school, but not without a textbook.
This course has been written to meet this need. It is based on careful
study and research over the last 27 years and is being tested in various
schools.
The Vedic system is attractive to teachers and pupils because it has
many striking advantages over the mathematics currently taught. The
most significant of these are the coherence of the system, the easy
and simple Vedic methods, the emphasis on mental calculation, the
use of basic principles or Sutras, the use of a Unified Field chart,
its effectiveness over all ability ranges.
The Vedic system is extremely refined. The methods are simple and
complementary, so that for example 'long' division is a simple reversal
of the one-line multiplication process; similarly with squaring and
square roots.
MENTAL MATHEMATICS
Anyone familiar with the Vedic system will be aware of the remarkable
Vedic techniques: 'difficult' problems or huge sums which can be solved
immediately by the Vedic method. These striking and beautiful methods
are just a part of a complete system of mathematics which is far more
systematic than the modern 'system'. Children are inspired by these
delightful techniques and often ask "Why was I not shown this before?".
Vedic Mathematics manifests the coherent and unified structure of
mathematics and the methods are complementary, direct and easy.
The simplicity of Vedic Mathematics means that calculations can be
carried out mentally and this is very much encouraged in the Vedic
system. There are many advantages in using a mental system. Starting
with a short mental test, the class settle down at the beginning of
the lesson and focus on mathematics; they enjoy the challenge of using
only their own mind to solve problems; the questions in the test can
revise the previous lesson and lessons and introduce ideas to be used
in the present lesson. Although the system encourages mental calculation,
apart from the initial test at the beginning of the lesson its use
in the rest of the lesson is not insisted upon. It is important that
the pupil's study is fun and enjoyable and they should not be forced
to do what they find too difficult, though the able ones may enjoy
doing nearly everything in their head. There are many advantages in
a system which emphasises mental mathematics and these are discussed
in some detail in the introduction to reference 2.
CREATIVITY
Learning mathematics should be a delightful experience for all children
and they should all succeed in it. The Cosmic Computer course offers
a complete system of mental mathematics which can be taught in a holistic
way. The straightforward and beautifully interrelated Vedic methods
mean that mathematics can be done mentally, and this and the many
methods of solution which the Vedic system offers, encourages flexibility
and innovation. This in turn leads to the development of creativity
and intuition. The Vedic system does not insist on a purely analytic
approach as many modern teaching methods do. This makes a big difference
to the attitude which children have towards mathematics.
Being naturally creative students like to devise their own methods
of solution. The Vedic system seeks to cultivate intuition- having
a conscious proof or explanation of a method beforehand is not considered
essential in the Vedic methodology. Students are shown general methods
and also special methods which apply in special cases. This means
they do not rigidly have to follow a certain procedure but have a
choice and are encouraged to invent their own methods. Every problem
is unique and invites its own style of solution.
The Vedic system does not impose unnecessary restrictions- we can
for example work from left to right or right to left; with numbers
which are partly positive and partly negative; we can work 2 or more
figures at a time and can feel comfortable with long numbers.
These benefits of the Vedic system have been observed where it has
been used. Pupils become generally more focused, confident and intelligent.
THE SUTRAS OR WORD-FORMULAE
The Vedic system use a collection of sixteen Sutras (or formulae),
given in word form, and some sub-Sutras. These are listed in the front
of each book and express fundamental principles which run like threads
(the word "Sutra" means "thread") through the whole of mathematics,
unifying diverse topics. They are extremely useful in education for
this reason.
We use our mind in certain specific ways: we might extend an idea
or reverse it or compare or combine it with another. Each of these
types of mental activity is described by one of the Vedic Sutras:
they describe the ways in which the mind can work and so they tell
the student how to go about solving a problem. It is not necessary
for the teacher to be familiar with these. It is best not to stress
the Sutras- they become familiar after a while, and seem quite natural.
As an illustration:
an
equilateral
triangle appears here
with the three altitudes drawn in
asked how many triangles there are in this figure you can easily get
the answer. There are sixteen of these but did you notice how you
could mentally make the various shapes stand out- seeing some lines
and ignoring others, and then seeing different triangles- at will?
The Vedic formula By Alternate Elimination and Retention describes
this attribute of our mind and has many more obviously mathematical
applications also (in eliminating first x and then y when solving
a pair of simultaneous equations, for example).
The universe appears lawful, at every level, and nature is able to
respond instantly to every situation that occurs everywhere in the
universe. Nature appears to employ a Cosmic Computer which governs
all action with the utmost precision.
THE UNIFIED FIELD CHART
The Unified Field chart (invented by Maharishi Mahesh Yogi) is given
at the beginning of each Teacher's Guide and shows the whole subject
and how it develops sequentially. This means the student can always
easily see where they are in relation to the whole. Other subjects-
chemistry, computer science for example- all have their own Unified
Field chart showing their structure and development, and any part
of a Unified Field chart can itself be expanded into a Unified Field
chart.
The term "Unified Field" comes from modern Physics and refers to the
unified field of all the various forces of nature. This ultimate level
of existence corresponds in subjective terms to the self. So the Unified
Field is at the basis of creation just as your self is the observer
of creation, and therefore any structure, like Mathematics, must unfold
from here- from unity into diversity.
As the structure develops from the base the terms get more and more
concrete as the three branches of Arithmetic, Algebra and Geometry
develop. The first level above the Unified Field: zero, equality,
empty space, gives a quality of that branch which also has the infinite,
abstract quality of the Unified Field. Next, unit, symbol, point give
a more concrete base for the development of the three branches. Each
of these, with three further concepts allow the development of numbers
and number systems, algebraic expressions and geometrical forms. These
develop further into all the various areas of Arithmetic, Algebra
and Geometry. In some cases, graphs for example, a topic is a mixture
of two or all of the three branches. The large box shows all the various
topics studied in the Cosmic Computer books and so the student always
knows where they are in relation to the whole. The contents of this
box change from book to book, as the topics studied change. The Transcendental
Meditation referred to on the chart is a simple technique for bringing
the mind to a quiet state where it can consciously experience the
Unified Field.
The extent to which this chart is used is left to the discretion of
the teacher.
THE COURSE
This course follows the requirements of the National Curriculum for
England and Wales and at present three books are available.
The course is intended to cover the first three years of a secondary
school course (that is starting with 11 year old pupils), but may
be suitable for other classes too. Some of the material is very suitable
for children aged from 8 years.
The textbook is intended to supplement the course, rather than being
used for individual study and the course is intended to be delivered
as whole class teaching. The text is however self-explanatory and
so the pupil who misses some lessons in school would still be able
to continue. The reason for writing the book in this way is more for
the benefit of the teacher- who will probably be unfamiliar with the
Vedic system.
STRUCTURE OF THE COURSE
Lessons begin with a mental arithmetic test of 10 questions (included
in this Guide). Answers are given at the end of the test and any difficulties
are dealt with. The teacher then introduces the lesson for that day.
This Teacher's Guide gives, in the Notes, any points of special interest
for each chapter and other suggestions. (Some additional material
will also be found there for Book 1, on geometry.) There are plenty
of Extension Sheets for more able or fast pupils and many of these
are very appropriate for the whole class to work on. Revision Tests
are also contained in the Teacher's Guide and these revise the previous
4 or 5 chapters. There are also worksheets, games etc.
There is also an Answer Book which contains answers to all the exercises
and other numbered questions that come up in the text. The Answer
Book is intended to be made available to the class to check their
work.
Reference 3 is a useful addition to the course as it contains many
variations and extensions not covered in these books
THE MENTAL TESTS
These tests (also included in this Guide) are carefully structured
so that they
a) introduce new ideas,
b) develop ideas encountered earlier,
c) revise work from the previous and earlier lessons (in this way
pupils do not forget
what they have learnt and establish it more clearly),
d) introduce or revise ideas needed in the current lesson.
Underlined questions in the tests, and diagrams, are intended to be
written up on the board.
The tests are marked immediately at the end- they can mark their own
or exchange with a neighbour. They could record their marks.
Any misunderstandings and errors can be sorted out when they are marked-
a quick explanation for each answer can also be given.
The tests can also be diagnostic- the books could be taken in to see
who got what right or wrong, and it is useful to ask at some stage
of the marking 'who got this right?' (which of course tells you who
didn't).
The tests should not be too slow: repeat each question but decline
to say it a third time unless for a good reason, then go immediately
on to the next question. Occasionally a test can be speeded up (with
warning). The tests, and answers, should not take long. Hints or reminders
can be given if a question draws a blank response.
The tests are also intended to be flexible: questions can be changed
if they are too easy or too hard or if extra revision of some topic
is needed (but the aim would be for most pupils to get most of the
questions right). It should also be easy to improvise extra tests
if there are not enough, even if it means giving an earlier test with
the numbers altered. In some of the later tests working out could
be allowed for question 10.
References:
1. Tirthaji B.K. (1965) Vedic Mathematics, Motilal Banarsidass
2. Williams K.R. (1991) The Natural Calculator, Vedic Mathematics
Research Group
3. Williams K.R. (1984) Discover Vedic Mathematics, Vedic Mathematics
Research Group
