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### Abstract

Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method.

### Introduction

Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered.

Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West.

The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous.

If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves.

According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic.

Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable.

The Vedic system also provides for the solution of 'difficult' problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy.

The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils.

Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc.

But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible.

Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper.

### The First Sutra: Ekādhikena Pūrvena

The relevant Sutra reads Ekādhikena Pūrvena which rendered into English simply says "By one more than the previous one".

Its application and "modus operandi" are as follows.

(1) The last digit of the denominator in this case being 1 and the previous one being 1 "one more than the previous one" evidently means 2. Further the proposition 'by' (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one - line mental method.

A. First method

B. Second Method

This is the whole working. And the modus operandi is explained below.

Modus operandi chart is as follows:

(i) We put down 1 as the right-hand most digit 1

(ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit.

(iii) We multiply that 2 by 2 and put 4 down as the next previous digit.

(iv) We multiply that 4 by 2 and put it down thus 8 4 2 1

(v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 × 2 = 138 and so on).

(vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step.

(vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand.

(viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there.

Our chart now reads as follows:

The Second Sutra: Nikhilam Navataścaramam Daśatah

Now we proceed on to the next sutra "Nikhilam sutra" The sutra reads "Nikhilam Navataścaramam Daśatah", which literally translated means: all from 9 and the last from 10". We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication.

Suppose we have to multiply 9 by 7.

1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power.

Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin);

3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10.

4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts.

5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6

### The First Corollary

The first corollary naturally arising out of the Nikhilam Sutra reads in English "whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency".

This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear:

Suppose one wants to square 9, the following are the successive stages in our mental working.

(i) We would take up the nearest power of 10, i.e. 10 itself as our base.

(ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 1^{2}

(iv) Thus 9^{2} = 81

### The Second Corollary

The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of 'vulgar' fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context.

Its literal meaning is the same as before (i.e. by one more than the previous one") but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the "previous" one is 1. So one more than that is 2.

Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 × 2 and the right hand side is the vertical multiplication product i.e. 25 as usual.

Thus 152 = 1 × 2 / 25 = 2 / 25.

Now we proceed on to give the third corollary.

### The Third Corollary

Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines.

The procedure applicable in this case is therefore evidently as follows:

i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product.

The following example will make it clear:

The Third Sutra: Ūrdhva Tiryagbhyām

Ūrdhva Tiryagbhyām sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number.

The formula itself is very short and terse, consisting of only one compound word and means "vertically and cross-wise." The applications of this brief and terse sutra are manifold.

A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13.

(i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer;

(ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and

(iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 × 13 = 156.

The Fourth Sutra: Parāvartya Yojayet

The term Parāvartya Yojayet which means "Transpose and Apply." Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we don't wish to give this application to those polynomials.

However the four steps given by them in the polynomial division are given below: Divide x3 + 7x2 + 6x + 5 by x - 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient

x^{2} × –2 = –2x^{2} but we have 7x^{2} in the divident. This means that we have to get 9x^{2} more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x

As for the third term we already have –2 × 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient.

Q = x^{2} + 9x + 24

Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∴ Q = x^{2} + 9x + 24 and

R = 53.

### The Fifth Sutra: Sūnyam Samyasamuccaye

Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time.

Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator.

Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations.

Sixth meaning - With the same sense (total of the word - Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero.

Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time.

### The Sixth Sutra: Ānurūpye Śūnyamanyat

As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean "proportionately" and "the first by the first and the last by the last".

Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows:

i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x^{2} + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor.

ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus

Thus 2x2 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used.

### The Seventh Sutra: Sankalana Vyavakalanābhyām

Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples.

A concrete example will elucidate the process.

Suppose we have to find the HCF (Highest Common factor)

of (x^{2} + 7x + 6) and x^{2} – 5x – 6

x^{2} + 7x + 6 = (x + 1) (x + 6) and

x2 – 5x – 6 = (x + 1) ( x – 6)

the HCF is x + 1 but where the sutra is deployed is not clear.

### The Eight Sutra: Puranāpuranābhyām

Puranāpuranābhyām means "by the completion or not completion" of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasake'pi use of which is not mentioned in that section.

### The Ninth Sutra: Calanā kalanābhyām

The term (Calanā kalanābhyām) means differential calculus according to Jagadguru Sankaracharya.

### The Tenth Sutra: Yāvadūnam

Yāvadūnam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah.

### The Eleventh Sutra: Vyastisamastih Sutra

Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing.

### The Twelfth Sutra: Śesānyankena Caramena

The sutra Śesānyankena Caramena means "The remainders by the last digit". For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means "mere observation" He has given a few trivial examples for the same.

### The Thirteen Sutra: Sopantyadvayamantyam

The sutra Sopantyadvayamantyam means "the ultimate and twice the penultimate" which gives the answer immediately. No mention is made about the immediate subsutra.

The illustration given by them.

The proof of this is as follows.

The General Algebraic Proof is as follows.

Let d be the common difference

Canceling the factors A (A + d) of the denominators and d of

the numerators:

It is a pity that all samples given by the book form a special pattern.

### The Fourteenth Sutra: Ekanyūnena Pūrvena

The Ekanyūnena Pūrvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows.

For instance 43 × 9.

i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and

ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product

### The Fifthteen Sutra: Gunitasamuccayah

Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors.

Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x^{3} + 6x^{2} + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz.

Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 - 7 = 5. So, the quotient x^{2} + 5x + 6.

This is a very simple and easy but absolutely certain and effective process.

### The Sixteen Sutra :Gunakasamuccayah.

"It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product".

In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors).

For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80.

Similarly in the case of cubics, biquadratics etc. the same rule holds good.

For example (x + 1) (x + 2) (x + 3) = x3 + 6x2 + 11 x + 6 2 × 3 × 4 = 1 + 6 + 11 + 6 = 24.

Thus if and when some factors are known this rule helps us to fill in the gaps.

### Literature

Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey & MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor & Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation.

Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta & Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas.

The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan & Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them:

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. (Williams, 2002, p. 2).

Examples of the sutras are the "Vertically and Crosswise" sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here - see Figure 3), and the "All from nine and the last from ten" sutra that may be used in subtraction, vincula, multiplication and division.

Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, "A piece of cognitive structure that can be held in the focus of attention all at one time," and may include other ideas that can be immediately linked to it.

This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the

Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol ê5ê, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth & Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brain's holistic activity (Tall & Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure.

The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions.

### Methodology

The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school).

Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation.

The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised.

The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial

expression by a single value were revised, using, for example, expressions such as 5(x - 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a "guess and check" method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered.

Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer.

### Results

The first question (1a) in each questionnaire was a two-digit multiplication.

In the first, it was 37 × 58, and the second 23 × 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students' facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4's explanation for 1c), 32 × 69:

2 times 9 is 18 3 times 9 +

2 times 6 is 39 + carried 1 = 40

3 times 6 + carried 4 = 22

### Expansion of binomials

A summary of the results in the first of the algebra questions (Q2 - see Figure 1 for format), requiring students to multiply together two basic binomials, is given in Table 1. Question 2 was presented in each case with the binomials in the same line.

Since the first questionnaire was administered immediately after the FOIL method was taught, most students employed this method, and only 8 (44.4%) correctly performed the expansion in the simplest example, question 2a). In the second questionnaire, where students were asked to use the vertically and Crosswise method, 10 (55.6%) were correct, but there was no statistically significant difference (c2 = 0.11, df = 1, ns).

However, when the presentation format was changed from a single line to the grid format of question 3 (see Figure 1), the facility on 3a) (Multiply x + 3 and x + 4) improved to 14 correct answers (77.8%). Thus there was weak evidence (c2 = 2.92, df = 1, p = 0.1) of an improvement in students' performance on this basic expansion of binomials in the second questionnaire, using the Vedic method with a grid presentation. However, the improvement was not sustained for question 3b) which had negative signs (see Table 2: c2 = 0.45), or for 2b), which involved 2x and negative signs. It seems that the arithmetic complexity caused problems, with only 3 students answering 2b) correctly on each test.

The other results of question 3 on the second questionnaire also support the idea of improved understanding; Table 2 gives the results of the multiplications in this question. In this case question 3c) was the question of a standard corresponding to 2b) in the first test (2x - 3 times x + 4), and yet the students did significantly better (50% correct) than they did on that question (c2 = 3.13, df = 1, p < 0.05).

It is also of interest that on question 2 in the first test all eight students who successfully applied the FOIL method wrote out all four terms and then added the two middle terms. No student wrote the final answer without the intermediate step of giving both middle terms. In contrast, when using the vertically and crosswise approach, all 14 students who were successful were able to write the answers straight down, as seen in the example in Figure 4.

Furthermore, 5 students (12, 15, 16, 17, and 18) who were unable to multiply binomials using the standard method achieved success using the Vedic method. Figure 5 shows the corresponding working of student 18 on these questions. In test 1, following the FOIL method she was unable to multiply any terms together and even seemed to confuse the question with factorisation.

However, with the Vedic method she makes a good attempt, correctly answering two parts.

It might be argued that the improvement above was due to students spending more time learning how to perform such expansions, or that the previous FOIL learning took time to assimilate. However, it should be noted that the questions where the improvement occurred specifically involved use of the Vedic method, which, while it can be related to FOIL by an experienced mathematician, would no doubt look quite different to these students, since it is set out in a grid format rather than being performed in a single line.

Furthermore, it was noted during the teaching episode that there was some resistance from the students to learning a second method when they already knew the FOIL method, and this could be expected to have a detrimental effect on performance.

### Factorisation of quadratic expressions

Table 3 contains a summary of the results of question 3 from the first questionnaire and question 5 in the second, these being corresponding single line, traditional format factorisation questions (see Figure 1). Individually the results of these questions did not show any statistical difference between the performance before and after the Vedic method was introduced. For example, between questions 3a) and 5a), c2 = 0.9, which is not significant.

However, if the question parts are grouped together and the number of students correct on question 3 compared with those correct on question 5 then we find that there is a significant improvement (Q3 v Q5 c2 = 6.65, p < 0.01) on the second questionnaire.

For question 4 in each questionnaire the students were asked to supply the missing terms in two binomials that were multiplied together. Boxes were provided for the missing terms, which were in the binomials, the quadratic, or both (see Figure 1). While these are not "standard" straight factorisation questions they do require students to work backwards from the answer and thus display some conceptual knowledge of how to "undo" multiplication of binomials. There was no difference in facility on any of these matching questions (see Table 4), or on a comparison of the total number of correct scores between the two questionnaires (c2 = 0.39, ns). The students appeared to be able to do them equally well using either guess or check and decomposition or the Vedic approach.

While the discussion above shows that the evidence for a better performance on individual questions following the teaching of the Vedic method was rarely present, a consideration of the students' overall scores on the expansion and factorisation algebra questions did show a significantly better performance on the second test, (_x1 = 41.4%,_x2 = 51.5%, t = 2.66, p < 0.05).

Thus it appears that, overall, the Vedic approach may have contributed to student understanding of the methods, either by cementing in place the previous methods, or by complementing them.

It is worth noting too that some students preferred to use the vertically and Crosswise method even when not directed to do so. For example, in question 2 of the second questionnaire students were simply asked to multiply two binomials together, with no method specified. In the event 4 students (4, 5, 16, and 18) chose to use the vertically and Crosswise approach, setting out their work in a grid. In addition, 3 of these students (15, 16 and 18) used it for the factorisation in question 5. An example of their work is shown in Figure 6.

While they used the method with varying levels of success, it seems to have benefited both student 5, who answered no algebra questions correctly in test 1 (and 2 in test 2), and student 16 who used it for factorisation and went from 3 correct to 8 correct (see Figure 6). While student 18 preferred the Vedic method to expand and to factorise expressions, she used the traditional method to multiply numbers even when asked to multiply by the vertically and Crosswise method. This seems to suggest that she was comfortable using different methods in algebra from those employed in arithmetic.

### Discussion

India has good reasons to be proud of a rich heritage in science, philosophy and culture in general, coming to us down the ages. In mathematics, which is my own area of specialization, the ancient Indians not only took great strides long before the Greek advent, which is a standard reference point in the Western historical perspective, but also enriched it for a long period making in particular some very fundamental contributions such as the place-value system for writing numbers as we have today, introduction of zero and so on.

Further, the sustained development of mathematics in India in the post-Greek period was indirectly instrumental in the revival in Europe after "its dark ages".

Notwithstanding the enviable background, lack of adequate attention to academic pursuits over a prolonged period, occasioned by several factors, together with about two centuries of Macaulayan educational system, has unfortunately resulted, on the one hand, in a lack of awareness of our historical role in actual terms and, on the other, an empty sense of pride which is more of an emotional reaction to the colonial domination rather than an intellectual challenge. Together they provide a convenient ground for extremist and misguided elements in society to "reconstruct history" from nonexistent or concocted source material to whip up popular euphoria.

That this anti-intellectual endeavour is counter-productive in the long run and, more important, harmful to our image as a mature society, is either not recognized or ignored in favour of short-term considerations.

Along with the obvious need to accelerate the process of creating an awareness of our past achievements, on the strength of authentic information, a more urgent need has also arisen to confront and expose such baseless constructs before it is too late. This is not merely a question of setting the record straight. The motivated versions have a way of corrupting the intellectual processes in society and weakening their very foundations in the long run, which needs to be prevented at all costs. The so-called "Vedic Mathematics" is a case in point. A book by that name written by Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaja (Tirthaji, 1965) is at the centre of this pursuit, which has now acquired wide following; Tirthaji was the Shankaracharya of Govardhan Math, Puri, from 1925 until he passed away in 1960.

The book was published posthumously, but he had been carrying out a campaign on the theme for a long time, apparently for several decades, by means of lectures, blackboard demonstrations, classes and so on. It has been known from the beginning that there is no evidence of the contents of the book being of Vedic origin; the Foreword to the book by the General Editor, Dr. A.S.Agrawala, and an account of the genesis of the work written by Manjula Trivedi, a disciple of the swamiji, makes this clear even before one gets to the text of the book. No one has come up with any positive evidence subsequently either.

There has, however, been a persistent propaganda that the material is from the Vedas. In the face of a false sense of national pride associated with it and the neglect, on the part of the knowledgeable, in countering the propaganda, even educated and well meaning people have tended to accept it uncritically. The vested interests have also involved politicians in the propaganda process to gain state support. Several leaders have lent support to the "Vedic Mathematics" over the years, evidently in the belief of its being from ancient scriptures. In the current environment, when a label as ancient seems to carry considerable premium irrespective of its authenticity or merit, the purveyors would have it going easy.

Large sums have been spent both by the Government and several private agencies to support this "Vedic Mathematics", while authentic Vedic studies continue to be neglected. People, especially children, are encouraged to learn and spread the contents of the book, largely on the baseless premise of their being from the Vedas. With missionary zeal several "devotees" of this cause have striven to take the "message" around the world; not surprisingly, they have even met with some success in the West, not unlike some of the gurus and yogis peddling their own versions of "Indian philosophy". Several people are also engaged in "research" in the new "Vedic Mathematics."

To top it all, when in the early nineties the Uttar Pradesh Government introduced "Vedic Mathematics" in school text books, the contents of the swamiji's book were treated as if they were genuinely from the Vedas; this also naturally seems to have led them to include a list of the swamiji's sutras on one of the opening pages (presumably for the students to learn them by heart and recite!) and to accord the swamiji a place of honour in the "brief history of Indian mathematics" described in the beginning of the textbook, together with a chart, which curiously has Srinivasa Ramanujan's as the only other name from the twentieth century! For all their concern to inculcate a sense of national pride in children, those responsible for this have not cared for the simple fact that modern India has also produced several notable mathematicians and built a worthwhile edifice in mathematics (as also in many other areas).

Harish Chandra's work is held in great esteem all over the world and several leading seats of learning of our times pride themselves in having members pursuing his ideas; (see, for instance, Langlands, 1993). Even among those based in India, several like Syamdas Mukhopadhyay, Ganesh Prasad, B.N.Prasad, K.Anand Rau, T.Vijayaraghavan, S.S.Pillai, S.Minakshisundaram, Hansraj Gupta, K.G.Ramanathan, B.S.Madhava Rao, V.V.Narlikar, P.L.Bhatnagar and so on and also many living Indian mathematicians have carved a niche for themselves on the international mathematical scene (see Narasimhan, 1991).

Ignoring all this while introducing the swamiji's name in the "brief history" would inevitably create a warped perspective in children's minds, favouring gimmickry rather than professional work. What does the swamiji's "Vedic Mathematics" seek to do and what does it achieve? In his preface of the book, grandly titled" A Descriptive Prefatory Note on the astounding Wonders of Ancient Indian Vedic Mathematics," the swamiji tells us that he strove from his childhood to study the Vedas critically "to prove to ourselves (and to others) the correctness (or otherwise)"of the "derivational meaning" of "Veda" that the" Vedas should contain within themselves all the knowledge needed by the mankind relating not only to spiritual matters but also those usually described as purely 'secular', 'temporal' or 'worldly'; in other words, simply because of the meaning of the word 'Veda', everything that is worth knowing is expected to be contained in the vedas and the swamiji seeks to prove it to be the case! It may be worthwhile to point out here that there would be room for starting such an enterprise with the word 'science'!

He also describes how the "contemptuous or at best patronising " attitude of Orientalists, Indologists and so on strengthened his determination to unravel the too-long-hidden mysteries of philosophy and science contained in ancient India's Vedic lore, with the consequence that, "after eight years of concentrated contemplation in forest solitude, we were at long last able to recover the long lost keys which alone could unlock the portals thereof."

The mindset revealed in this can hardly be said to be suitable in scientific and objective inquiry or pursuit of knowledge, but perhaps one should not grudge it in someone from a totally different milieu, if the outcome is positive. One would have thought that with all the commitment and grit the author would have come up with at least a few new things which can be attributed to the Vedas, with solid evidence.

This would have made a worthwhile contribution to our understanding of our heritage. Instead, all said and done there is only the author's certificate that "we were agreeably astonished and intensely gratified to find that exceedingly though mathematical problems can be easily and readily solved with the help of these ultra-easy Vedic sutras (or mathematical aphorisms) contained in the Parishishta (the appendix portion) of the Atharva Veda in a few simple steps and by methods which can be conscientiously described as mere 'mental arithmetic' "(paragraph 9 in the preface).

That passing reference to the Atharva Veda is all that is ever said by way of source material for the contents. The sutras, incidentally, which appeared later, scattered in the book, are short phrases of just about two to four words in Sanskrit, such as Ekadhikena Purvena or Anurupye Shunyam Anyat. (There are 16 of them and in addition there are 13 of what are called sub-sutras, similar in nature to the sutras).

The mathematics of today concerns a great variety of objects beyond the high school level, involving various kinds of abstract objects generalising numbers, shapes, geometries, measures and so on and several combinations of such structures, various kinds of operations, often involving infinitely many entities; this is not the case only about the frontiers of mathematics but a whole lot of it, including many topics applied in physics, engineering, medicine, finance and various other subjects.

Despite its entire pretentious verbiage page after page, the swamiji's book offers nothing worthwhile in advanced mathematics whether concretely or by way of insight. Modern mathematics with its multitude of disciplines (group theory, topology, algebraic geometry, harmonic analysis, ergodic theory, combinatorial mathematics-to name just a few) would be a long way from the level of the swamiji's book. There are occasionally reports of some "researchers" applying the swamiji's "Vedic Mathematics" to advanced problems such as Kepler's problem, but such work involves nothing more than tinkering superficially with the topic, in the manner of the swamiji's treatment of calculus, and offers nothing of interest to professionals in the area.

Even at the western teaching “Vedic Mathematics" deals only with a small part and, more importantly, there too it concerns itself with only one particular aspect, that of faster computation.

One of the main aims of mathematics education even at the western teaching consists of developing familiarity with a variety of concepts and their significance. Not only does the approach of “Vedic Mathematics” not contributes anything towards this crucial objective, but in fact might work to its detriment, because of the undue emphasis laid on faster computation.

The swamiji's assertion "8 months (or 12 months) at an average rate of 2 or 3 hours per day should be enough for finishing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also foreign universities," is patently absurd and hopefully nobody takes it seriously, even among the activists in the area. It would work as a cruel joke if some people choose to make such a substitution in respect of their children.

It is often claimed that "Vedic Mathematics" is well appreciated in other countries, and even taught in some schools in UK etc. In the normal course one would not have the means to examine such claims, especially since few details are generally supplied while making the claims.

Recent years have brought about the development of powerful tools for verifying specifications of hardware and software systems. By now, major companies, such as Intel, IBM, and Folding@Home-Protein Molecules have realized the impact and importance of such tools in their own design and implementation processes as a means of coping with the ever-increasing complexity of chip and computing tools.

Protocols, networks, and distributed systems can generally not be described by code of some deterministic programming language. Such systems exhibit concurrent behavior and they are typically reactive in the sense that their behavior depends on what the environment can offer (e.g. “Is the printer busy?”). Computation tree logic (CTL) is currently one of the popular frameworks used in verifying properties of concurrent systems. We study its syntax and semantics, and use those insights to design an automated verification algorithm which takes a description of a system and specifications of expected behavior as input and checks whether that system meets those expectations. That algorithm is the foundation for a tool, the symbolic model verifier (SMV), which they use to evaluate some basic designs, e.g. simple elevator systems and a mutual exclusion protocol (3-4 weeks).

Exposure to a labeling algorithm for finite-state verification illustrates depth-first backwards search in a directed graph; this search is recursive and the recursion is driven by the logical structure of the specified behavior, written as a CTL formula. The description and evaluation of small designs with the tool SMV makes students appreciate how such graphs can be modeled with a modular guarded-command language with non-deterministic assignment. The discussion of program logic contains the linear algorithm for computing minimal-sum sections of integer arrays as a case study. Finally, binary decision diagrams require algorithms that implement the familiar logical operations on such diagrams. Some of these algorithms illustrate dynamic programming at an accessible level.

### Conclusion

In this study students were taught an appropriate Vedic sutra following teaching of the traditional FOIL method of multiplication of binomials, and the decomposition method for factorisation. We found that afterwards the students performed significantly better overall on these types of algebra questions, and specifically on the factorisations, and there was weak evidence of better results on expansion using a grid format.

The reasons for the improvement are not easy to pinpoint since they appear in some areas and not in others. This seems to indicate that the value of the method may lie in what it adds to the students' overall algebraic conceptions and knowledge of mathematical structure. Thus we have found no evidence that it should be seen as a replacement for the former approaches, but our results suggest it could rather be recommended as a useful adjunct, a complementary method. While this may take longer in terms of teaching time, the results indicate that possession of a range of strategies may have value above and beyond their individual benefit.

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