KIWI
10-20-2009, 08:14 PM
Being a maths dunce I have found this very interesting ...has any come across this before?
Mark Gaskell introduces an alternative system of calculation based on Vedic philosophy
At the Maharishi School in Lancashire we have developed a course on Vedic mathematics for key stage 3 that covers the national curriculum. The results have been impressive: maths lessons are much livelier and more fun, the children enjoy their work more and expectations of what is possible are very much higher. Academic performance has also greatly improved: the first class to complete the course managed to pass their GCSE a year early and all obtained an A grade.
Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago. Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.
The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: "by one more than the one before" and "all from nine and the last from 10". These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra "all from nine and the last from 10". Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.
1,000 - 564 = 436
1,000 - 5 6 4
subtract subtract subtract
from from from
9 9 10
¯ ¯ ¯
4 3 6
This can easily be extended to solve problems such as 3,000 minus 467. We simply reduce the first figure in 3,000 by one and then apply the sutra, to get the answer 2,533. We have had a lot of fun with this type of sum, particularly when dealing with money examples, such as £10 take away £2. 36. Many of the children have described how they have challenged their parents to races at home using many of the Vedic techniques - and won. This particular method can also be expanded into a general method, dealing with any subtraction sum.
The sutra "vertically and crosswise" has many uses. One very useful application is helping children who are having trouble with their tables above 5x5. For example 7x8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10.
7 x 8 = 56
7 3 (3 is the difference from base)
8 2
_________
A 7 3 starting at the left subtract
crosswise either 8-3 or
8 2 7-2 to get 5, the first figure
__________ of the answer
5
B 7 3 Multiply vertically
x to get 6 (3 x 2)
8 2
__________
5 6
The whole approach of Vedic maths is suitable for slow learners, as it is so simple and easy to use.
The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply 32 by 44. We multiply vertically 2x4=8. Then we multiply crosswise and add the two results: 3x4+4x2=20, so put down 0 and carry 2. Finally we multiply vertically 3x4=12 and add the carried 2 =14. Result: 1,408.
32 x 44 = 1,408
A 3 2 Starting from the right
x multiply vertically
4 4 2 x 4 = 8
B 3 2 Multiply crosswise
3 x 4 = 12 and 2 x 4 = 8
4 4 Add them together
_______
0 8 3 x 4 + 2 x 4 = 20
2 Put down 0 and carry 2
C 3 2 Finally multiply vertically
x 3 x 4 = 12 and add the
4 4 carried over 2 = 14
_______________
14 0 8
2
We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a "3 by 2" long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.
All the techniques produce one-line answers and most can be dealt with mentally, so calculators are not used until Year 10. The methods are either "special", in that they only apply under certain conditions, or general. This encourages flexibility and innovation on the part of the students.
Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below.
We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.
96 x 92 = 8,832
A 96 4 (4 is the difference from base)
92 8 (8 is the difference from base)
_____________
B 96 4 Subtract crosswise from the left
92 8 96 - 8 = 88 or 92 - 4 = 88
______________
88
C 96 4 Multiply vertically
x 4 x 8 = 32
92 8
____________
88 32
This works equally well for numbers above the base: 105x111=11,655. Here we add the differences. For 205x211=43,255, we double the first part of the answer, because 200 is 2x100.
We regularly practise the methods by having a mental test at the beginning of each lesson. With the introduction of a non-calculator paper at GCSE, Vedic maths offers methods that are simpler, more efficient and more readily acquired than conventional methods.
There is a unity and coherence in the system which is not found in conventional maths. It brings out the beauty and patterns in numbers and the world around us. The techniques are so simple they can be used when conventional methods would be cumbersome.
When the children learn about Pythagoras's theorem in Year 9 we do not use a calculator; squaring numbers and finding square roots (to several significant figures) is all performed with relative ease and reinforces the methods that they would have recently learned.
Mark Gaskell is head of maths at the Maharishi School in Lancashire
www.vedicmaths.org (http://www.vedicmaths.org) 'The Cosmic Computer'
by K Williams and M Gaskell, (also in an bridged
edition), Inspiration Books, 2 Oak Tree Court,
Skelmersdale, Lancs WN8 6SP. Tel: 01695 727 986.
Saturday school for primary teachers at
Manchester Metropolitan University on
October 7. See website.
19th May 2000 Times Education Supplement (Curriculum Special)
Mark Gaskell introduces an alternative system of calculation based on Vedic philosophy
At the Maharishi School in Lancashire we have developed a course on Vedic mathematics for key stage 3 that covers the national curriculum. The results have been impressive: maths lessons are much livelier and more fun, the children enjoy their work more and expectations of what is possible are very much higher. Academic performance has also greatly improved: the first class to complete the course managed to pass their GCSE a year early and all obtained an A grade.
Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago. Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.
The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: "by one more than the one before" and "all from nine and the last from 10". These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra "all from nine and the last from 10". Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.
1,000 - 564 = 436
1,000 - 5 6 4
subtract subtract subtract
from from from
9 9 10
¯ ¯ ¯
4 3 6
This can easily be extended to solve problems such as 3,000 minus 467. We simply reduce the first figure in 3,000 by one and then apply the sutra, to get the answer 2,533. We have had a lot of fun with this type of sum, particularly when dealing with money examples, such as £10 take away £2. 36. Many of the children have described how they have challenged their parents to races at home using many of the Vedic techniques - and won. This particular method can also be expanded into a general method, dealing with any subtraction sum.
The sutra "vertically and crosswise" has many uses. One very useful application is helping children who are having trouble with their tables above 5x5. For example 7x8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10.
7 x 8 = 56
7 3 (3 is the difference from base)
8 2
_________
A 7 3 starting at the left subtract
crosswise either 8-3 or
8 2 7-2 to get 5, the first figure
__________ of the answer
5
B 7 3 Multiply vertically
x to get 6 (3 x 2)
8 2
__________
5 6
The whole approach of Vedic maths is suitable for slow learners, as it is so simple and easy to use.
The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply 32 by 44. We multiply vertically 2x4=8. Then we multiply crosswise and add the two results: 3x4+4x2=20, so put down 0 and carry 2. Finally we multiply vertically 3x4=12 and add the carried 2 =14. Result: 1,408.
32 x 44 = 1,408
A 3 2 Starting from the right
x multiply vertically
4 4 2 x 4 = 8
B 3 2 Multiply crosswise
3 x 4 = 12 and 2 x 4 = 8
4 4 Add them together
_______
0 8 3 x 4 + 2 x 4 = 20
2 Put down 0 and carry 2
C 3 2 Finally multiply vertically
x 3 x 4 = 12 and add the
4 4 carried over 2 = 14
_______________
14 0 8
2
We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a "3 by 2" long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.
All the techniques produce one-line answers and most can be dealt with mentally, so calculators are not used until Year 10. The methods are either "special", in that they only apply under certain conditions, or general. This encourages flexibility and innovation on the part of the students.
Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below.
We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.
96 x 92 = 8,832
A 96 4 (4 is the difference from base)
92 8 (8 is the difference from base)
_____________
B 96 4 Subtract crosswise from the left
92 8 96 - 8 = 88 or 92 - 4 = 88
______________
88
C 96 4 Multiply vertically
x 4 x 8 = 32
92 8
____________
88 32
This works equally well for numbers above the base: 105x111=11,655. Here we add the differences. For 205x211=43,255, we double the first part of the answer, because 200 is 2x100.
We regularly practise the methods by having a mental test at the beginning of each lesson. With the introduction of a non-calculator paper at GCSE, Vedic maths offers methods that are simpler, more efficient and more readily acquired than conventional methods.
There is a unity and coherence in the system which is not found in conventional maths. It brings out the beauty and patterns in numbers and the world around us. The techniques are so simple they can be used when conventional methods would be cumbersome.
When the children learn about Pythagoras's theorem in Year 9 we do not use a calculator; squaring numbers and finding square roots (to several significant figures) is all performed with relative ease and reinforces the methods that they would have recently learned.
Mark Gaskell is head of maths at the Maharishi School in Lancashire
www.vedicmaths.org (http://www.vedicmaths.org) 'The Cosmic Computer'
by K Williams and M Gaskell, (also in an bridged
edition), Inspiration Books, 2 Oak Tree Court,
Skelmersdale, Lancs WN8 6SP. Tel: 01695 727 986.
Saturday school for primary teachers at
Manchester Metropolitan University on
October 7. See website.
19th May 2000 Times Education Supplement (Curriculum Special)