Extended Project

2010

To what extent should Maths teachers use Vedic Mathematics in teaching Maths to 11+ year old students?

Bhavik Patel

Blank Page

2

Abstract (223 words)

Many have forgotten the mathematicians who have spent the rest of their lives admiring the beauty of mathematics. The mathematics of today has helped modern society to deal with financial, economical and mechanical problems. Mathematics can be seen as the reason of Earths’ creation from the theory of relativity to the concept of quantum physics.

A number of ancient civilisations have taken into consideration the strength of mental mathematics, and have also considered whether these thoughts will stay intact in the future generation. As a result, many sources have been studied during the completion of this project. I have discovered that the past is never too old in the present and may also influence the future.

Society is facing many issues regarding the level of intelligence of young children in mathematics. Will the children of today as young as 11 years of age, be able to cope with mathematical problems in future of their life?

I will show why the methods of ancient mathematics should be used in the education system of today. These methods will strengthen the mental mathematical skills of young children and increase their confidence with the subject in the future. However, I will also go on to show why it should not be introduced to the education system.

The methods that will be shown will give an indication of how astonishing the minds of the past really were.

3

Introduction (829 words)

Learning mathematics from an early age has not been pleasurable for the majority, from simple fraction skills, to long division. Even the best of teachers are not fond of mathematics. According to a report by the Associated Content, the top 5 of the best taught subjects are Art, Foreign languages, ICT, Literature/Communication Arts and Philosophy.1

Let us be aware of the problems caused by mathematics within today’s society. Members of Parliament notify one in five children leave primary school with a poor grasp of maths, despite the fact that £2.3 billion are spent teaching the subject.2 During 2008, 30,000 children started secondary school with the maths skills of a seven-year-old.3 Part of the reason of this issue can be due to poor teaching skills and methods. Maths teachers do not improvise, or manipulate lessons to their own desire. Instead, lessons are heavily relied on textbooks and monotonous lessons. Children tend to obtain boredom from such lessons.

In addition, a fifth of teenagers leave school so illiterate and innumerate they lack the ability to deal with the challenges of everyday life. Consequently, it can be said that their maths skills are limited to little more than arithmetic.4

Furthermore, deficient maths skills of an adult cost Britain’s taxpayer up to £2.4 billion a year.5 A study conducted by the KPMG foundation estimates around seven million adults within the United Kingdom at best, have the maths skills of a nine-year-old. As a result, the learning potential of the people of the United Kingdom is lower, and have a high chance of being unemployed.6

Subsequently, the issues caused by deficient maths skills can influence a person’s future and capabilities. Lack of such skills relate to expensive educational needs provision, truancy,

1 http://www.associatedcontent.com/article/231288/the_top_five_school_subjects_with_the.html?cat=9

2 http://www.telegraph.co.uk/education/educationnews/5285397/Fifth-of-11-year-olds-with-poor-maths-skills-say-MPs.html - A secondary source information which the author obtained from Members of Parliament (MP’S)

3 http://www.telegraph.co.uk/education/educationnews/5285397/Fifth-of-11-year-olds-with-poor-maths-skills-say-MPs.html - A secondary source information which the author obtained from a special report.

4 http://www.guardian.co.uk/education/2010/may/07/poor-literacy-numeracy - secondary source information which the author obtained from a special report.

5 http://www.independent.co.uk/news/education/education-news/poor-maths-skills-cost-britain-16324bn-a-year-says-study-1225807.html - secondary source information which the author obtained from financial consultants.

6 http://www.independent.co.uk/news/education/education-news/poor-maths-skills-cost-britain-16324bn-a-year-says-study-1225807.html - secondary source information which the author obtained from a special report.

4

exclusion from school or even health risks.7 Such outcome can be avoided through a technique that does not require costly research and development. This technique can be referred to as Vedic Mathematics, which has been discovered from the Vedas.

Objectives and Rationale

This project will give me the opportunity to expose the simplicity and yet at the same time, the complexity of Vedic Mathematics. Introducing this unique mathematical system to the modern world will advance the mathematical skills of a young student.

Throughout this dissertation, I will go on to show whether Vedic mathematics should earn a place within the education system. I will compare the modern mathematical techniques to the more ancient techniques of Vedic mathematics. Various questions will arise from this topic, such as accuracy of the system, whether the system is out of date or not, identifying flaws within the system and whether the system will actually take effect on young children, or not.

Literature Review

What are the Vedas?

The Vedas are ancient Indo-Aryan religious literature which are related with the Vedic civilisation, and are adherents of the Hindu religion. The Vedas are composed in Vedic Sanskrit which is the oldest language of the Indo-Iranian branch of the Indo-European family.8 The phrase, “Veda” when translated into English is “Knowledge”.9 The Vedas consist of four accepted principles of Samhitas, or Hymns. Three of the Samhitas are performances of sacrifice.10

The four Samhitas are as followed: Rigveda, Yajurveda, Samaveda and Atharvaveda. The Rigveda include hymns and prayers that are addressed to the pantheon of Gods and Goddesses.11 The Yajurveda, as well as the Samaveda consist of ritual formulas. The Samaveda

7 http://www.independent.co.uk/news/education/education-news/poor-maths-skills-cost-britain-16324bn-a-year-says-study-1225807.html - secondary source information which the author obtained from a special report.

8 http://www.astrovalley.com/vedas.php – a secondary source. Correct, reliable piece of information is required for the origin of the Vedas.

9 http://en.wikipedia.org/wiki/Vedas - A secondary source, may come in useful to discuss the definition of it, and relevance to Vedic mathematics.

10 http://en.wikipedia.org/wiki/Vedas – A secondary source, Radhakrishnan & Moore 1957, p. 3; Witzel, Michael, "Vedas and Upaniṣads", in: Flood 2003, p. 68; MacDonell 2004, p. 29-39; Sanskrit literature (2003) in Philip's Encyclopedia. Accessed 2007-08-09

5

is mostly a rearrangement of the Rigveda for musical rendering.12 The Atharvaveda consist of a collection of spells that are considered to have magical effects to praise a deity. The Vedas were compiled during the era of Krishna, a Hindu Deity, around 3500 B.C. 13 Even at that time, the Vedas were hardly understood hence, they are considered to be very ancient.

The compiler of the Vedas is known as Veda Vyasa (splitter of the Vedas).14 Veda Vyasa is also known as Vyasa Krishna Dwaipayana, this is due to his similarities with the great deity, Krishna. He is the twenty-eighth of the compiler of the Vedic knowledge.15

What is Vedic Mathematics?

Vedic mathematics was rediscovered between the years of 1911 and 1918 by Sri Bharati Krishna Tirthaji (1884-1960), who was a Hindu scholar and mathematician.16 According to Tirthaji, the world of mathematics is based on sixteen sūtras.17 Sūtra is a thread or a line that hold objects together; metaphorically it refers to an aphorism or method.18 An aphorism is an original thought spoken, or written in a memorable form, hence, the laws of mathematics are easily remembered through the sūtras.

The sūtras differ from the modern mathematical system. The sūtras were designed to match the way the mind naturally performs, this can be ideal for a student as they are guided to an

11 http://en.wikipedia.org/wiki/Vedas - http://en.wikipedia.org/wiki/Vedas – A secondary source, Radhakrishnan & Moore 1957, p. 3; Witzel, Michael, "Vedas and Upaniṣads", in: Flood 2003, p. 68; MacDonell 2004, p. 29-39; Sanskrit literature (2003) in Philip's Encyclopedia. Accessed 2007-08-09

12 http://en.wikipedia.org/wiki/Vedas – a secondary source. Correct, reliable piece of information is required for the origin of the Vedas.

13 http://en.wikipedia.org/wiki/Vedas – a secondary source. Correct, reliable piece of information is required for the origin of the Vedas.

14 http://www.astrovalley.com/vedas.php – a secondary source. Correct, reliable piece of information is required for the origin of the Vedas.

15 http://en.wikipedia.org/wiki/Vedas – a secondary source. Correct, reliable piece of information is required for the origin of the Vedas.

16 http://www.hinduism.co.za/vedic.htm#What%20is%20Vedic%20Mathematics? – Secondary source. It’s important to provide correct information of who discovered Vedic mathematics, and where it came from.

17 http://www.hinduism.co.za/vedic.htm#What%20is%20Vedic%20Mathematics? – Secondary source. It’s important to provide correct information of who discovered Vedic mathematics, and where it came from.

18 http://en.wikipedia.org/wiki/Sūtra – secondary source, it’s important to be aware of the sūtras as they build up the Vedic system.

6

appropriate method of solution.19 The Vedic system is best known for its simplicity and coherence. When compared to the modern mathematical system, the modern system appears to have unrelated techniques that baffle the mind.

The Vedic Mathematic sūtras 20

The following list of sūtras is taken from the book Vedic Mathematics, written by the researcher himself, Sri Bharati Krishna Tirthaji. The main list of sūtras is:

1. By one more than the one before.

2. All from 9 and the last from 10.

3. Vertically and Cross-wise

4. Transpose and Apply

5. If the Samuccaya (on both sides of the equation, then) is the Same it is equal to Zero

6. If One is in Ratio the Other is Zero

7. By Addition and by Subtraction

8. By the Completion or Non-Completion

9. Differential Calculus

10. By the Deficiency

11. Specific and General

12. The Remainders by the Last Digit

13. The Ultimate and Twice the Penultimate

14. By One Less than the One Before

15. The Product of the Sum

16. All the Multipliers

In addition, the sūtras consist of sub-sūtras:

19 http://www.hinduism.co.za/vedic.htm#What%20is%20Vedic%20Mathematics? – Secondary source, Vedic mathematics, tirthaji. Important to be aware of the purpose of the sūtras.

20 http://www.hinduism.co.za/vedic.htm#What%20is%20Vedic%20Mathematics? – Secondary source, Vedic mathematics, tirthaji. The sūtras will be in relevance in the discussion as they are to be demonstrated.

7

1. Proportionately

2. The Remainder Remains Constant

3. The First by the First and the Last by the Last

4. For 7 the Multiplicand is 143

5. By Osculation

6. Lessen by the Deficiency

7. Whatever the Deficiency lessen by that amount andset up the Square of the Deficiency

8. Last Totalling 10

9. Only the Last Terms

10. The Sum of the Products

11. By Alternative Elimination and Retention

12. By Mere Observation

13. The Product of the Sum is the Sum of the Products

14. On the flag

Why the use of sūtras?

Tirthaji, in his book, Vedic mathematics, mentioned the special use of sūtras/aphorisms during the Vedic period: "In order to help the pupil memorize the material assimilated, they made it a general rule of practice to write even the most technical and abstruse textbooks in sutras or in verse (which is so much easier - even for the children - to memorize)... So from this standpoint, they used verse for lightening the burden and facilitating the work (by versifying scientific and even mathematical material in a readily assimilable form)!"

A former High Commissioner of India in the UK, Dr L M Singhvi, who approves the Vedic system states: "A single sutra would generally encompass a varied and wide range of particular applications and may be likened to a programmed chip of our computer age".

According to Clive Middleton of vedicmaths.org, the formulae of the Vedic system describes the way the mind naturally operates, and that is a great help in directing the student to the appropriate method of solution.

8

I will now demonstrate two of the sūtras from the Vedic system. This will give a partial view of the Vedic system and will allow us to realise whether or not it should be used within the education system.

The Second Sūtra (All from 9 and the last from 10)

The second sūtra is applied to the all-known multiplication. All students and teachers of the primary classes are expected to teach and learn the multiplication tables up to the number twelve. However, the Vedic system only requires you to be acquainted up to the five times tables. The Vedic system allows any child who is familiar with simple addition and subtraction skills to be able to manipulate any multiplication-table that involves large numbers. This allows the child to attain the answer of any multiplication easily, and immediately.

A worked example of the second sūtra will now be presented.

Let us start with the multiplication of single-digit numbers. Note, that this can be achieved even with weak multiplication skills. All that is needed are the simple skills of addition and subtraction.

In order to achieve an answer, deficiencies/complements will be used. A deficiency/complement is what relates a number to unity. The unity is expressed as 1 or 10 or 100 or 1000, etc. I.e. for the numbers relating to 10:

The deficiency of 9 is 1. The deficiency of 8 is 2. The deficiency of 7 is 3 The deficiency of 6 is 4, etc.

The sūtra (all from 9 and the last from 10) relates any number back to unity, or 1. This is achieved by calculating what must be added to the number to make it up for the next base of 10 above. Alternatively, that power of 10 which is closest to the numbers that are being multiplied.

The fourth sūtra (Transpose and apply)

I have fourth sūtra, transpose and apply consists of a special-case formula. This formula can be used for the division of very large numbers, and very small numbers however; I will be using this formula to solve simultaneous equations. The formula is as follows:

9

(B*R) – (C*Q)(B*P) – (A*Q)

X=

6-48-2

Compliments

6-48-2 /

Where (B*R) and (C*Q) are cross-multiplications, from the centre on the right. Likewise, (B*P) and (A*Q) are also cross multiplications, from the centre on the left.

In the same way, (C*P) and (A*R) are cross-multiplications. While the denominator is the same as before.

Discussion (3904 Words)

As we are now aware of two sūtras, let us put them into context. I will demonstrate the sūtras as mathematical examples and will express how straightforward it simply is. The sūtras will be compared with the modern mathematical system/techniques. The literature reviewed is known to be rediscovered from the Vedas, and therefore based on the Vedas itself. A range of sources have been analysed to investigate the mathematical techniques within the sūtras.

The Second Sūtra (All from 9 and the last from 10)

Example (1)

In this example, the number 7 will be multiplied by the number 8.

Step 1: The base of this calculation will be 10 as it is the nearest unity to the numbers being multiplied. Place the two multipliers on top of each other on the left-hand-side.

Step 2: Subtract each of the multipliers from the base to obtain the complements. Write the answer on the right-hand-side with a connecting minus sign (-) between them. This shows that the complements are less than the number ten.

Step 3: The answer will now have two parts, one on the left-hand-side and one on the right. A line can be drawn between them to separate the two parts.

10

68

(C*P) – (A*R)(B*P) – (A*Q)

Y=

Right-hand-side of the answer

6-48-24/8Left-hand-side of the answer

Step 4: The left-hand-side of the answer can be achieved through cross-subtraction. (6-2=4), or (8-4=4); either of them give the same answer. The left-hand-side can also be achieved through two other methods:

1) Subtract the base 10 from the sum of the two numbers being multiplied. (6+8=14) therefore, (14-10=4). Answer is the same.

2) Subtract the sum of the complements from the base 10. (4+2=6) therefore, (10-6=4). Note that you get the same answer.

Step 5: The right-hand-side of the answer can now be obtained through this step:

Multiply the two compliments together, in this case 4 and 2. (4x2=8) Therefore, the right-hand-side of the answer is 8

As a result, the answer to 6 multiplied by 8 is 48. Both the left-hand-side and right have been formed into the actual answer of the question.

Multiplication using a unity/base of 100

In this example, we will be multiplying two-digit numbers together. As mentioned earlier, in mathematics, the unity is expressed as 1 or 10 or 100 or 1000, etc. I will now demonstrate multiplication using a unity/base of 100. This method is ideal for multiplying numbers close to 100, easily. In this case, there will be two complement digits for each number because a base of 10 is being used.

For example, when multiplying 95 by 96, the compliment of 95 is 05 and the compliment of 96 is 04. The allocation of the complements was achieved by using a method that is stated within the name of this sūtra itself, all from 9 and the last from 10.

Compliment of 95: subtract left-hand-side of the number from 9, (9-9=0). Subtract right-hand-side of the number from 10, (10-5=5). Therefore, the compliment of 95 is 05.

Compliment of 96: subtract left-hand-side of the number from 9, (9-9=0). Subtract right-hand-side of the number from 10, (10-6=4). Therefore, the compliment of 96 is 04.

11

6-48-24/

Left-hand-side of the answer

9793

97-0393-07

Compliments

97-0393-0790/

97-0393-0790/21

Here, I have subtracted the last digit of both numbers from 10, and subtracted the remaining digit from 8. Hence, the name of this sūtra, all from 9 and the last from 10. This method was used in example (1), but it was best for it to be explained when using the base of 100. This method is easier to understand when taking the base as 100.

As we now know how to achieve a compliment, I will demonstrate how to answer a question involving two-digit numbers.

Example (2)

In this example, the number 97 will be multiplied by the number 93

Step 1: The base of this calculation will be 100 as it is the nearest unity to the numbers being multiplied. Place the two multipliers on top of each other.

Step 2: use the all from 9 and the last from 10 technique to calculate the compliment. Subtract left-hand-side of the number from 9, (9-9=0). Subtract right-hand-side of the number from 10, (10-7=3). Therefore, the compliment of 97 is 03.

Subtract left-hand-side of the number from 9, (9-9=0). Subtract right-hand-side of the number from 10, (10-3=7). Therefore, the compliment of 93 is 07.

Step 3: The left-hand-side of the answer can be achieved through cross-subtraction. (97-07=90), or (93-03=90); either of them give the same answer.

Step 4: Multiply the two compliments together, in this case 03 and 07. (03x07=21) Therefore, the right-hand-side of the answer is 21. The LHS and the RHS of the answer have been calculated therefore, the answer for 97x93 is equal to 9021.

Multiplication using a unity/base of 1000

In this example, the base of 1000 will be used. Therefore, we will be multiplying three-digit numbers. This can be achieved easily by using the all from 9 and the last from 10 technique. In this case, the compliment of the two numbers being multiplied will have three-digits; this is because a base of 1000 is being used.

12

697997

697-303997-003

Compliments

697-303997-003694/

697-303997-003694/909

For example, when multiplying 598 by 991, the compliment of 598 is 402 and the compliment of 991 is 009. As mentioned earlier, the allocation of the complements was achieved by using a method that is stated within the name of this sūtra itself, all from 9 and the last from 10.

Compliment of 598: Subtract left-hand-side of the number from 9, (9-5=4). Subtract the digit in the middle also from 9, (9-9=0). Subtract right-hand-side of the number from 10, (10-8=2). Therefore, the compliment of 598 is 402.

Compliment of 991: Subtract left-hand-side of the number from 9, (9-9=0). Subtract the digit in the middle also from 9, (9-9=0), Subtract right-hand-side of the number from 10, (10-1=9). Therefore, the compliment of 991 is 009.

As shown above, I have used the all from 9 but the last from 10 technique. We now know how to achieve a compliment therefore; I will demonstrate how to answer a question involving three-digit numbers.

Example (3)

Let us multiply the numbers 697 and 997.

Step 1: Place the two numbers being multiplied on top of each other.

Step 2: Use the all from 9 and the last from 10 technique to calculate the compliment. Subtract left-hand-side of the number from 9, (9-6=3). Subtract the digit in the middle also from 9, (9-9=0) Subtract right-hand-side of the number from 10, (10-7=3). Therefore, the compliment of 697 is 303.

Subtract left-hand-side of the number from 9, (9-9=0). Subtract the digit in the middle also from 9, (9-9=0) Subtract right-hand-side of the number from 10, (10-37=3). Therefore, the compliment of 997 is 003.

Step 3: The left-hand-side of the answer can be achieved through cross-subtraction. (697-003=694), or (997-303=694); either of them give the same answer.

Step 4: Multiply the two compliments together, in this case 303 and 007. (303x003=909) Therefore, the right-hand-side of the answer is 909. The LHS

13

and the RHS of the answer have been calculated therefore, the answer for 697x997 is equal to 694909.

Consequently, I have demonstrated the second sūtra as clearly as possible. This sūtra is ideal for calculating the multiplication of large numbers quickly, efficiently, easily and most importantly, it can be done mentally. Frequent practice of this sūtra will improve the strength of mental mathematics in multiplication. This is great for children as young as 11 or even younger to improve their mental maths. It will help them become confident in mathematics and may act as a base for their future study.

Comparison with the traditional (modern day) method

The traditional (modern day) method of multiplication involves memory, but not calculation. Young children are familiar with their times-tables because they remember the answers, not by calculating them. The Vedic system involves memory as well as calculation. Young children will be able to abide by the Vedic system easily. This is because, it will cause the child to remember the all from 9 but the last from 10 technique and manipulate it on any number for any calculation. This involves more brain activity which is beneficial for the child this is because, an increase in the stimulation of the brain will improve the way the child manipulates any kind of information. Hence, the child will perform well at school.

The fourth sūtra (Transpose and apply)

Example (4)

I will now solve this equation by using the transpose and apply formula:

To solve for X, start with the y-coefficients and the independent terms and cross-multiply rightwards. In other words, start from the top row and multiply across by the lower one, and vice versa. Link the two cross-products with a minus (-) sign. This gives us our numerator i.e. [(1x2) – (-4 x5)] = 22

To calculate the denominator, start with the x-coefficients instead and cross-multiply with the independent terms, leftwards. Remember to link the two cross-products with a minus (-) sign. i.e. [(1x3) – (2 x -4)] = 11

Therefore, the equation should look like this:

14

2x + y = 53x – 4y = 2

To solve for Y, start with the independent terms on the top row and cross-multiply with the x-coefficients on the lower row. This will give the numerator. i.e. [(5x3) – (2x2)] = 11

The denominator for Y is equal to the denominator of X therefore, the denominator is 11.

As a result, the solution of this simultaneous equation is:

Consequently, the transpose and apply formula allows anybody to solve simultaneous equations. It is clearly a quick and easy method. This method does require strong multiplication skills. All that is needed is simple addition, subtraction and division skills.

Comparison with the traditional (modern day) method

Using the traditional method will involve substitution and changing the subject of the equation. Many steps will need to be met to achieve the answer and therefore, there is a high chance of making a mistake. Whereas, if the transpose and apply formula were to be used, the chances of making a mistake are low.

Why Vedic mathematics SHOULD be introduced to the education system?

The Effects of Vedic Mathematics on children

15

[(1x2) – (-4 x5)][(1x3) – (2 x -4)]X =

X = 2211

X = 2

[(5x3) – (2x2)] [(1x3) – (2 x -4)]Y =

Y = 1111

Y= 1

X = 2 Y= 1

Now that we have an indication of two of the sūtras, let us discuss the effects of its method on children. A few years ago, a school based in London, St James’ School, and other schools, began to teach the Vedic system to children which seems to achieve noteworthy success. At the school in Skelmersdale, Lancashire, UK, a full course called "The Cosmic Computer" was written and tested on 11 to 14 year old children. According to Mahesh Yogi, "The sutras of Vedic

Mathematics are the software for the cosmic computer that runs this universe."

There are many advantages of using a flexible, mental system. Students will be able to devise their own methods as they are not limited to the one ‘correct’ method. As a result, there will be creative, confident and intelligent children. Research is being carried out in many areas including the effects of learning Vedic Maths on children. Research has proved that Vedic Mathematics has many benefits:

Is simple and flexible in nature

Can perform calculations at a rapid speed

Gives accurate results

Saves valuable time in competitions and examinations

Develops mental abilities

Enhances intuition and logic

Provides an extra competitive edge

Increases confidence and intelligence levels

Optimizes performance

Encourages creativity and innovation

The followers of the Vedic system argue that Vedic maths is far more systematic, coherent and unified than the modern system. It is a mental tool for calculation that encourages the development and use of intuition and innovation, while giving the student a lot of flexibility, fun and satisfaction. Therefore, it's direct and easy to implement in schools.

Its Growing Popularity

Interest in Vedic maths is growing in the field of education where maths teachers are looking for a new and better approach to the subject. Students of the IIT (Indian Institute of

16

9-17-36/3

Technology) are said to be using this ancient technique for quick calculations. Dr. Murli Manohar Joshi, Indian Minister for Science & Technology, argued the significance of Vedic maths, while pointing out the important contributions of ancient Indian mathematicians, such as Aryabhatta, who laid the foundations of algebra, Baudhayan, the great geometer, and Medhatithi and Madhyatithi, the saint duo, who formulated the basic framework for numerals.

Why Vedic mathematics SHOULD NOT be introduced to the education system?

Issues arising from Vedic Mathematics

There are numerous amounts of issues that arise from Vedic mathematics. Let us first discuss the issue of the origin of the system.

According to Sri Bharati Krishna Tirthaji, the Vedic system was rediscovered between the years

of 1911 and 1918. The most commonly criticised aspect of Vedic mathematics is whether the system is Vedic or not. This is mainly because, it is claimed that none of the sūtras can be found within any Vedic literature. Some propose that the sūtras have nothing to do with the Vedas at all and Tirthaji composed the sūtras himself. Consequently, it is discussed whether the system deserves the name ‘Vedic’ or ‘mathematics’. This is because, according to critics, the language style does not seem Vedic. The General Editor of the book himself, Dr. A.S. Agrawala, has mentioned within the foreword to the book that the sūtras have nothing to do with the Vedas. He also mentions that the sūtras are not to be found within any Vedic literature.

It is also said whether the Vedic system is too old to be introduced at this stage in time. The arithmetic of the Vedic system can be performed on a computer, as well as a calculator. This makes the system slightly irrelevant in the modern world. Tirthaji’s methods are just rules that make mathematics seem like a bunch of tricks which are easy to put into action, but difficult to understand.

For example, the multiplication of 9 and 7. Line them along with their difference from the base 10.

You obtain the answer in the following fashion: the unit’s digit is the two differences multiplied together, –1 x –3 = 3 and the other digit 6 is just either of the diagonals added together, that is, 9–3 = 7–1 = 6. This method can be extended too much larger numbers. It is a neat trick, but it does not make multiplication easier to fathom, quite

the contrary.

On the other hand, there are several people who approve Vedic mathematics. It has been explained that textual references should not the basis for evaluating the ‘Vedicity’ of the mathematics. Some propose that Vedic mathematics is unique when compared with other

17

5-53-7

scientific work as it is not pragmatically worked out, but is based on a direct revelation, or an “intuitional visualisation” of fundamental mathematical truths.

The compilers of the Vedas, the much respected Rishis, were considered to be “approached reverentially” by God almighty. In the same way, Tirthaji has been described as having the same “reverential approach”. Some say that Tirthaji may not have found the sūtras within the Vedas, but instead he received them spiritually just as the rishis did, which should validate them as being Vedic. In addition, there is further confusion with the use of the word “Vedic”. When translated into English, the Word “Veda” is “knowledge”. From this, it is possible that the words “Vedic mathematics”, is referring to the fact that the sūtras represent all the knowledge of mathematics, hence the name, “Vedic mathematics”. It is not clear whether the use of the word is to represent “all knowledge” of mathematics or the Vedic texts itself.

Flaws within the Vedic system

I will now discuss the flaws that have been encountered from observing the sūtras. Let us discuss the flaws within the second sūtra, all from 9 but the last form 10.

The examples that were demonstrated for this sūtra involved multiplying numbers above the number 5. However, from carrying out my own calculations, this sūtra only applies to numbers above the number 5.

For example, let us multiply together two numbers that are less than the number 5. I.e. 5 multiplied by 3. I am still going to use the method of the sūtra.

Step 1: I will place the number on top of each other.

Step 2: I will now subtract the two numbers being multiplied from the nearest base, 10, to achieve the compliments. This is because; the nearest unity to the numbers being multiplied is 10.

Step 3: The answer will now have two parts, one on the left-hand-side and one on the right. A line can be drawn between them to separate the two parts.

18

53

5-53-7 /

Step 4: I will now use the one of the other three methods to achieve the left-hand-side of the answer; this can be achieved through cross-subtraction. (5 – 7 = -2), or (3 – 5 = -2); either of them give the same answer.

Step 5: Now, I am able to find the right-hand-side of the answer by multiplying the two compliments together. (5 x 7 = 35)

Consequently, the answer to 5 multiplied by 3, by using the sūtra, is equal to -235. Evidently, this answer is no where near the actual answer, which is equal to 15. The flaw is noticeable from step 4 where the left-hand-side of the answer is -2; from this point, it is obvious that the final answer is wrong.

Conclusion

From the observation of the sūtras, it can be said that there are an equal number of arguments from both points of view. However, I believe that Vedic mathematics should be part of the education system. Multiplication-tables seem to be a difficulty for young children. However, by following the Vedic system, children will be able to calculate multiplications easily, just as demonstrated earlier.

The Vedic system does not require a teacher to have any qualifications relating to maths in order to teach it. It is a quick, efficient and cheap way to teach mathematics to young children. The Vedic system will save the UK from wasting £2.3 billion pounds on the current, conventional method. The sūtras were designed to naturally work along with the brain, which will make it easier for young children to learn mathematics and even remember them. As a result, there will be creative, confident and intelligent children.

In addition, the discussion has shown that the Vedic system is not perfect, and includes flaws. It is also not sure whether the name given to the system is correct and whether it deserves to include the word “Vedic”. Also, one of the sūtras that were demonstrated included flaws. This raises an issue on whether the Vedic system is reliable, and whether it is even a mathematical system. However, the criticisms for the Vedic system do not out-weigh the approvals for it. I

19

5-5 3-7-2 /

5-5 3-7-2 /35

have proved that the Vedic system is an unique system with sufficient evidence and step-by-sep demonstrations.

Limitations

During the continuation of this project, there have been a number of occasions where I was not able to present every aspect of my research; this is due to lack of time and the limitation of quantity. Suppose I did have a sufficient amount time and quantity , I would have liked to explore, demonstrate and analyse more sūtras. This would have provided me with more evidence as to why the Vedic system should be within the education system.

Other limitations include lack of resources. The internet is overflowing with information more than we can ever imagine. However, locating ancient texts such as Vedic mathematics can be difficult due to its exclusivity. Also, the language style is different from the language style of today therefore, implanting it and understanding the context can be difficult.

Development of skills

I have learnt a number of new skills from the completion of this project. It has given me the chance to look further into the world of Vedic mathematics and manipulate it into different aspects. I have experienced a heap of independent learning and research, which will be very beneficial for me during my years in University.

One key aspect that I have learnt from this project, is that the values of ancient civilisation are never too old to be reintroduced. Those values will always stay intact and be known as the base of today’s values.

Bibliography

The source of information has been split into two parts: Documents and Websites.

Documents:

Bharati Krishna Tirtha's Vedic mathematics, English edition, 1965.

The Power of Vedic Maths, Atul Gupta, 2004.

Vedic Mathematics for Schools, Book 1, James T. Glover.

20

Websites :

http://www.telegraph.co.uk/education/educationnews/5285397/Fifth-of-11-year-olds-with-poor-maths-skills-say-MPs.html

http://www.guardian.co.uk/education/2010/may/07/poor-literacy-numeracy

http://www.independent.co.uk/news/education/education-news/poor-maths-and-literacy-results-2063062.html

http://www.independent.co.uk/news/education/education-news/poor-maths-skills-cost-britain-16324bn-a-year-says-study-1225807.html

http://www.guardian.co.uk/education/2010/apr/27/numeracy-skills-adults

http://www.timesonline.co.uk/tol/life_and_style/education/school_league_tables/article6738359.ece

http://www.openthemagazine.com/article/art-culture/the-fraud-of-vedic-maths

http://www.gayatrimata.org/Themagicofvedicmath.htm

http://www.associatedcontent.com/article/231288/the_top_five_school_subjects_with_the.html?cat=9

http://mastermindvedicmaths.com/blog_vedic/?p=6

http://www.astrovalley.com/vedas.php

http://en.wikipedia.org/wiki/Vedas

21