Thursday, October 31, 2019

E-MARKETING STRATEGY Essay Example | Topics and Well Written Essays - 3000 words

E-MARKETING STRATEGY - Essay Example In a nutshell, the aim of this report is to analyse the fundamental issues related to e-marketing strategy and making recommendations to Michelle who is currently running an outlet Michelle’s Fancy Dress Costumes’ located in the Metro Centre in Gateshead. Correspondingly, it is recommended to create her website as the most important step for achieving the goal of e-marketing strategy. Moreover, extensive range of e-marketing strategy is suggested to her. Alongside, the use of Search Engine Optimisation (SEO) including placing of keywords on-site and off-site was recommended to her. The use of banner and affiliate adverting is also suggested for marketing of her products and services. In the backdrop of increasing effectiveness of direct marketing, the application of e-mail based permission marketing is also recommended to her. Owing to the increasing popularity and wide spread use of social media for marketing, Michelle is also recommended to use social networking sites like Facebook and Twitter in order to reach wide masses quickly and in cost effective manner. The evolution of internet has provided significant opportunities to businesses across the world. The continuous improvement in technology and greater access of internet to large populaces have offered marketers with an opportunity to market both directly and indirectly and conduct their business in an efficient manner. Currently, internet is being used by various business organisations for conducting online sales as well as marketing their respective products or services. Marketing through internet is becoming extremely popular among the marketers in the present day context. Marketing through internet is known by several names such as digital marketing, web marketing and internet marketing. However, e-marketing is the most commonly used term, which has gained considerable popularity amid the

Tuesday, October 29, 2019

Italian renaissance Research Paper Example | Topics and Well Written Essays - 1750 words

Italian renaissance - Research Paper Example Chronology accounts that the renaissance marked a drastic change in the society since artists, rulers, and other groups of elites intensified their activities towards improvement of the existing novices in different fields. Notably, Leonardo Da Vinci presumed an abundant role in propulsion of the renaissance activities since his works affected various disciplines that included art and painting, architecture, science, and engineering (Fisher, 2006). The following discussion shall evaluate Leonardo’s influence in the disciplines of history, art, and architecture in the ancient Italian society, and the prevalent influenced that activities continue to decipher in the modern society. Leonardo’s activities harness him to global recognition. History cites that 1452 was Da Vinci’s year of birth at an Italian city called Vinci. This intelligent artist concentrated on matters that influenced the entire world. Debatably, he had a passion to nature and science, factors that extensively influenced his artwork (Riding, 2006). The Italian renaissance period would not have been a success in the absence of artists such as Leonardo holding to the fact that he was the first person to paint a picture that bore a landscape (Pernis & Adams, 2006). This approach influenced a major change in artistic painting since all his predecessors started incorporating landscapes and shadows in their pieces of art. History indicates that his artwork bore abundance in delivering the desired perspective in each image. Through his paintings on Mona Lisa, the Last Supper, and the Flying Machine, Da Vinci was able to create knowledgeable to the human society such that his predecessors were able to contemplate how the world would be with a set of realism in the various aspects employed in the paintings (Austen, 2006). Further, Leonardo was able to influence his successors’ artistic designs including those of Ambrogio Lorenzetti who embarked in the profession and

Sunday, October 27, 2019

Vedic Mathematics Multiplication

Vedic Mathematics Multiplication 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: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena 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 Ãâ€" 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 Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, 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 12 (iv) Thus 92 = 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 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 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: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m 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 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya 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 dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 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 Ãâ€" 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 = x2 + 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 = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam 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: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat 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 2x2 + 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 22 + 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 VyavakalanÄ bhyÄ m 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 (x2 + 7x + 6) and x2 – 5x – 6 x2 + 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: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m 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 dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam 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: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena 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: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena 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 Ãâ€" 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 x3 + 6x2 + 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 x2 + 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 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 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 à ª5à ª, 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 brains 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 Ãâ€" 58, and the second 23 Ãâ€" 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 4s explanation for 1c), 32 Ãâ€" 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 Vedic Mathematics Multiplication Vedic Mathematics Multiplication 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: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena 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 Ãâ€" 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 Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, 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 12 (iv) Thus 92 = 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 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 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: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m 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 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya 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 dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 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 Ãâ€" 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 = x2 + 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 = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam 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: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat 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 2x2 + 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 22 + 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 VyavakalanÄ bhyÄ m 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 (x2 + 7x + 6) and x2 – 5x – 6 x2 + 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: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m 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 dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam 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: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena 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: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena 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 Ãâ€" 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 x3 + 6x2 + 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 x2 + 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 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 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 à ª5à ª, 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 brains 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 Ãâ€" 58, and the second 23 Ãâ€" 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 4s explanation for 1c), 32 Ãâ€" 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

Friday, October 25, 2019

Media Use of Stereotypes Essay -- Media Stereotypes Stereotyping

Media Use of Stereotypes We live in a world of technological innovation where mass media is a major part of us today. People make assumptions on what they hear. They do not try to analyze the situation to see who is right and who is wrong, and mass media is the main source of manipulating one's mind. The concept of propaganda has changed over time. Propagandists create ideas stereotypically through the use of propaganda and use media to promote it and target people's minds to have influence on their views towards a certain group of people. These ideas create negative or positive images in the intended audience's minds. However, it is notable that the information is only the one that is exemplified through media and therefore, can be wrong or changed than what reality is. Different stereotypes work well in propaganda with the help of propaganda tactics through the use of political campaigns and commercial advertising exposed by mass media. Many critics define propaganda differently; however, there is a general agreement that propaganda is concerned with influencing opinions (Cole, 1998). The word propaganda has many other synonyms such as big lie, persuasion, brainwashing, disinformation, etc. Propaganda is not a realistic portrayal of an issue; rather it is something that is changed to manipulate the intended public. Propaganda is the aim to change people's views about an issue and the way these aims are intended on the targeted audience is the way propaganda is presented, especially through mass media. Evidently, people strongly tend to select the media, which carry contents with which they already agree (Jackall, 1995). Media tends to produce what they know people will like to hear. On the other hand, people make ce... ...hat how much of the information they consume reflects reality and therefore, propagandists benefit by creating stereotypical views, which work well in portraying propaganda. WORK CITED "Definitions of Propaganda." Ed. Robert Cole. The Encyclopedia of Propaganda. 3 vols. New York: Sharpe Reference , 1998. "Do The Math." FIRST For Women On The Go 19 Apr. 2004: n. pag. Johnston, Carla B. Screened Out - How The Media Control Us & What We Can Do About It. Armonk: M. E. Sharpe, Inc, 2000. 23-24. LaRose, Robert, and Joseph Straubhaar. Media Now - Understanding Media, Culture and Technology. 4th ed. Belmont: Wadsworth Thomson Learning, 2004. 379. Propaganda. Ed. Robert Jackall. New York: New York UP, 1995. 89. Ryan, John, and William M. Wentworth. Media & Society - The Production of Culture in the Mass Media. Needham Heights: Allyn & Bacon, 1999. 52.

Thursday, October 24, 2019

Porters 5 Forces of the Retail Industry

Porters Five Forces of the Retail Industry I. Supplier Power The bargaining power of Suppliers is relatively low. There is a high competition between suppliers which means that their ability to raise prices or reduce quantity is very low. Suppliers include both domestic and international manufacturers and because many retail products are standardized, retailers have low switching costs which make the supplier power low.Larger retailers have power over their suppliers because they can threaten suppliers to change to a different suppliers which would significantly hurt the suppliers because of their great market share. Furthermore larger retailers can vertically integrate with suppliers they are having trouble cooperating with. II. Bargaining Power of Buyers The bargaining power of buyers is relatively low. This is because since there are so many customers, no one customer will have bargaining leverage. Therefore bargaining must be done in massive groups which are hard to organize.If c onsumers choose not to shop at a retail outlet they most likely miss out on value or price as well as convenience of shopping retail. III. Competitive Rivalry Competitive rivalry is medium to high. There are numerous competitors as well as many E-retailers that are entering the market rapidly. Several Rivals are highly dedicated to being industry leaders. Furthermore there are diverse approaches and differing goals between competitors. These are all factors that lead to a high force but because exit barriers are low.Therefore weak firms are more likely to leave the market which in turn, increases profits for remaining firms which weakens the power of competitive rivalry. IV. Threat of Substitutes Threat of substitutes is low because there are not many substitutes that offer low prices and convenience to consumers. The goal of retailers it provide a wide variety of products at one location and in many cases create a one stop shopping location which leaves little room for alternatives . V. Threat of New CompetitorsThreat of new competitors is low because customers are very loyal to existing brands and retail stores. The companies that are most likely to enter the retail market are grocery stores. However, it takes a lot of time and money to build a good brand image and then get consumers to you store. Because of this, new entrants will spending money on building a brand when establishing which leaves them less money that can be used to give themselves a competitive advantage in the market. Secondly strong distribution networks are required to keep a retail store stocked.Weak distribution networks result in more expense in moving goods around. Sources Nair, Sanel. â€Å"Walmart. † N. p. , n. d. Web. 23 Feb. 2013. . â€Å"Retail Industry – Five Forces Analysis. † N. p. , n. d. Web. 23 Feb. 2013. . Porter, M. E. (2000) What is Strategy? Harvard Business Review Retrieved February 5, 2012 from http://hbr. org/product/what-is-strategy/an/96608-PDF- ENG Porter, M. E. (1980) Competitive Strategy, Free Press, New York. â€Å"Porter’s Five Forces Analysis of Wal-Mart. † Write Academic, 12 Sept. 2012. Web. 23 Feb. 2013. .

Wednesday, October 23, 2019

Island of the Sequined Love Nun Chapter 49~50

49 The Bedside Manner of Cannibals Tuck slept through most of the day, then woke up with a pot of coffee over a spy novel. He looked at the words and his eyes moved down the pages for half an hour, but when he put it down he had no idea what he had read. His mind was torn by the thought of Beth Curtis showing up at his door. Whenever a guard crunched across the gravel compound, Tuck would go to the window to see if it was her. She wouldn't come here during the day, would she? He had promised Kimi that he would check on Sepie and meet him at the drinking circle, but now he was already a day late on the promise. What would happen if Beth Curtis came to his bungalow while he was out? She couldn't tell the doc, could she? What would her excuse be for coming here? Still, Tuck was beginning to think that the doc wasn't really the one running the show. He was merely skilled labor, and so, probably, was Tucker himself. Tuck looked at the pages of the spy novel, watched a little Malaysian television (today they were throwing spears at coconuts on top of a pole while the Asian stock market's tickers scrolled at the bottom of the screen in thin-colored bands), and waited for nightfall. When he could no longer see the guard's face across the compound, he made a great show of yawning and stretching in front of the window, then turned out the lights, built the dummy in his bed, and slipped out through the bottom of the shower. He took his usual path behind the clinic, then inched his way up on the far side and peeked around the front. Not ten feet away a guard stood by the door. He ducked quickly around the corner. There was no way into the clinic tonight. He could wait or even try to intimidate the guard, now that he knew they were afraid to shoot him. Of course, he wasn't sure they knew they were afraid to shoot him. What if Mato was the only one? He slid back down the side of the building and through the coconut grove to the beach. The swim had become like walking to the mailbox, and he was past the minefield in less than five minutes. As he rounded the curve of the beach, he saw a light and figures moving around it. The Shark men had brought a kerosene lamp to the drinking circle. How civilized. Some of the men acknowledged his presence as he moved into the circle, but the old chief only stared into the sand between his feet. There was a stack of magazines at his side. â€Å"What's going on, guys?† A panic made its way around the circle to land on Abo, who looked up and said, â€Å"Your friend is shot by the guards.† Tuck waited, but Abo looked away. Tuck jumped in front of Malink. â€Å"Chief, is he telling the truth? Did they shoot Kimi? Is he dead?† â€Å"Not dead,† Malink said, shaking his head. â€Å"Hurt very bad.† â€Å"Take me to him.† â€Å"He is at Sarapul's house.† â€Å"Right. I'll look it up in the guidebook later. Now take me to him.† Old Malink shook his head. â€Å"He going to die.† â€Å"Where is he shot?† â€Å"In the water by the minefield.† â€Å"No, numbnuts. Where on his body?† Malink held his hand to his side. â€Å"I say, ‘Take him to the Sorcerer,' but Sarapul say, ‘The Sorcerer shoot him.'† Malink then looked Tuck in the eye for the first time. His big brown face was a study in trouble. â€Å"Vincent send you. What do I do?† Tuck could sense a profound embarrassment in the old man. He had just admitted in front of the men in his tribe that he didn't have a clue. The loss of face was gnawing at him like a hungry sand crab. Tuck said, â€Å"Vincent is pleased with your decision, Malink. Now I must see Kimi.† One of the young Vincents stood up. Feeling very brave, he said, â€Å"I will take you.† Tuck grabbed his shoulder. â€Å"You're a good man. Lead on.† The young Vincent seemed to forget to breathe for a moment, as if Tuck had touched him on the shoulders with a sword and welcomed him to a seat at the Round Table, then he came to his senses and took off into the jungle. Tuck followed close behind, nearly clotheslining himself a couple of times on branches that the young Vincent ran right under. The coral gravel on the path tore at Tuck's feet as he ran. When they emerged from the jungle, Tuck could see a light coming out of Sarapul's hut, which Tuck recognized from his day in the cannibal tree. He turned to young Vincent, who was terrified. He had charged the dragon, but had made the mistake of stopping to think about it. â€Å"Kimi's with the cannibal?† Young Vincent nodded rapidly while bouncing from foot to foot, looking like he would wet himself any second. â€Å"Go on,† Tuck said. â€Å"Go tell Malink to come here. And have a drink. You're wigging out.† Vincent nodded and ran off. Tuck approached the door slowly, creeping up until he could see the old man crouched over Kimi, trying to pour something into his mouth from a coconut cup. â€Å"Hey,† Tuck said, â€Å"how's he doing?† Sarapul looked around and gestured for Tuck to enter the house. Tuck had to bend to get through the low door, but once inside the ceiling opened to a fifteen-foot peak. Tuck knelt by Kimi. The navigator's eyes were closed, and even in the orange light of Sarapul's oil lamp, he looked pale. He was uncovered and a bandage was wrapped around his middle. â€Å"Did you do this?† Tuck asked Sarapul. The old cannibal nodded. â€Å"They shoot him in water. I pull him in.† â€Å"How many times?† Sarapu held up a long bent finger. â€Å"Both sides? Did it go through?† Tuck gestured with his fingers on either side of his hip. â€Å"Yes,† Sarapul said. â€Å"Let me see.† The old cannibal nodded and unwrapped Kimi's bandage. Tuck rolled the navigator gently on his side. Kimi groaned, but didn't wake. The bullet had hit him about two inches above the hip and about an inch in. It had passed right though, going in the size of a pencil and exiting the size of a quarter. Tuck was amazed that he hadn't bled to death. The old cannibal had done a good job. â€Å"Don't take him to the Sorcerer,† Sarapul said. â€Å"The Sorcerer will kill him. He is the only navigator.† The old cannibal was pleading while trying to remain fierce. A sob betrayed him. â€Å"He is my friend.† Tuck studied the wound to give the old cannibal a chance to gather himself. He couldn't remember any vital organs being in that area. But the wounds would have to be stiched shut. Tuck wasn't sure he had the stomach for it, but Sarapul was right. He couldn't take Kimi to Curtis. â€Å"Do you guys have anything you use to kill pain?† The cannibal looked at him quizzically. Tuck pinched him and he yelped. â€Å"Pain. Do you have anything to stop pain?† â€Å"Yes. Don't do that anymore.† â€Å"No, for Kimi.† Sarapul nodded and went out into the dark. He returned a few seconds later with a glass jug half-full of milky liquid. He handed it to Tuck. â€Å"Kava,† he said. â€Å"It make you no ouch.† Tuck uncapped the bottle and a smell like cooking cabbage assaulted his nostrils. He held his breath and took a big slug of the stuff, suppressed a gag, and swallowed. His mouth was immediately numb. â€Å"Wow, this ought to do it. I need a needle and some thread and some hot water. And some alcohol or peroxide if you have it.† Sarapul nodded. â€Å"I put Neosporin on him.† â€Å"You know about that? Why am I doing this?† Sarapul shrugged and left the house. Evidently, he didn't keep anything inside but his skinny old ass. Kimi moaned and Tuck rolled him over. The navigator's eyes fluttered open. â€Å"Boss, that dog fucker shot me.† â€Å"Curtis? The older white guy?† â€Å"No. Japanese dog fucker.† Kimi drew his finger across his scalp in a line and Tuck knew exactly who he meant. â€Å"What were you doing, Kimi? I told you that I'd check on Sepie and meet you.† Tuck felt a pleasant numbness moving into his limbs. This kava stuff would definitely do the trick. â€Å"You didn't come. I worry for her.† â€Å"I had to fly.† â€Å"Sarapul say those people very bad. You should come live here, boss.† â€Å"Be quiet. Drink this.† He held the jug to Kimi's lips and tipped it up. The navigator took a sip and Tuck let him rest before administering another dose. â€Å"That stuff nasty,† Kimi said. â€Å"I'm going to stitch you up.† The navigator's eyes went wide. He took the jug from Tuck and gulped from it until Tuck ripped it out of his hands. â€Å"It won't be that bad.† â€Å"Not for you.† Tuck grinned. â€Å"Haven't you heard? I've been sent here by Vincent.† â€Å"That what Sarapul say. He say he don't believe in Vincent until we come, but now he do.† â€Å"Really?† Sarapul came through the door with an armload of supplies. â€Å"I don't say that. This dog fucker lies.† Tuck shook his head. â€Å"You guys were made for each other.† Sarapul set down a sewing kit and a bottle of peroxide, then crouched over the navigator and looked up at Tuck. â€Å"Can you fix him?† Tuck grinned and grabbed the old cannibal by the cheek. â€Å"Yum,† Tuck said. â€Å"Sorry,† Sarapul said. â€Å"I'll fix him,† Tuck said. Silently he asked for help from Vincent. â€Å"I can't feel my arms,† Kimi said. â€Å"My legs, where are my legs? I'm dying.† Sarapul looked at Tuck. â€Å"Good,† he said. â€Å"More kava.† Tuck picked up the jug, now only a quarter full. â€Å"This is great stuff.† â€Å"I'm dying,† Kimi said. Tuck rolled the navigator over on his side. â€Å"Kimi, did I tell you I saw Roberto?† â€Å"See, I didn't eat him,† Sarapul said. â€Å"Where?† Kimi asked. â€Å"He came to my house. He talked to me.† â€Å"You lie. He only speak Filipino.† â€Å"He learned English. Can you feel that?† â€Å"Feel what? I am dying?† â€Å"Good,† Tuck said and he laid his first stitch. â€Å"What Roberto say? He mad at me?† â€Å"No, he said you're dying.† â€Å"I'm dying, I'm dying,† Kimi wailed. â€Å"Just kidding. He didn't say that. He said you're probably dying.† Tuck kept Kimi talking, and before long the navigator was so convinced of his approaching death he didn't notice that Tucker Case, self-taught incompetent, had completely stitched and dressed his wounds. 50 Don Quixote at the Miniature Golf Course He was sleeping, dreaming of flying, but not in a plane. He was soaring over the warm Pacific above a pod of hump-back whales. He swooped in close to the waves and one of the whales breached, winked at him with a football-sized eye, and said, â€Å"You da man.† Then the whale smiled and blew the dream all to hell, for while Tuck knew himself to indeed â€Å"be da man† and while he didn't mind being told so, he also knew that whales couldn't smile and that bit of illogic above all the others broke the dream's back. He woke up. There was music playing in his bungalow. â€Å"Dance with me, Tucker,† she said. â€Å"Dance with me in the moonlight.† The smooth muted horns of â€Å"Moonlight Serenade† filled the room from a portable boom box on his coffee table. Beth Curtis, wearing a sequined evening gown and high-heeled sandals, danced an imaginary partner around the room. â€Å"Oh, dance with me, Tucker. Please.† She glided over to the bed and held her hand out to him. He gave her the coconut man's head, rolled over, and ducked under the sheet. â€Å"Go away. I'm tired and you're insane.† She sat on the bed with a bounce. â€Å"You old stick in the mud.† A pouty voice now. â€Å"You never want to have any romance.† Tuck feigned sleep. Pretty well, he thought. â€Å"I brought champagne and candles. And I made cookies.† This is me sleeping, Tuck thought. This is exactly how I behave when I sleep. â€Å"I twisted up a joint of skunky green bud the size of your dick.† â€Å"I hope you got help carrying it,† he said, still under the covers. â€Å"I rolled it on the inside of my thigh the way the women in Cuba roll cigars.† â€Å"Don't tell me how you licked the paper.† She slapped him on the bottom. â€Å"Come on, dance with me.† He rolled over and pulled the sheet off his face. â€Å"You're not going to go away, are you?† â€Å"Not until you dance with me and have some champagne.† Tuck looked at his watch. â€Å"It's five in the morning.† â€Å"Haven't you ever danced till dawn?† â€Å"Not vertically.† â€Å"Oh, you nasty boy.† Coy now, as if anything short of being caught at genocide could make her blush. The song changed to something slow and oily that Tuck didn't recognize. â€Å"This is such a good song. Let's dance.† She swooned. She actually swooned. Swooning, Tuck noticed, looked very much like an asthma attack wheezed in slow motion. A rooster crowed, and seven thousand six hundred and fifty-two roosters responded in turn. â€Å"Beth, it's morning. Please go home.† â€Å"Then you're not going to dance with me?† â€Å"No.† â€Å"All right, I guess we'll skip the dancing, but I want you to know that I'm very disappointed.† She stood up, pulled the evening gown over her head, and dropped it to the floor. The sequins sizzled against the floor like a dying rattlesnake. She wore only stockings underneath. Tuck said, â€Å"I don't think this is such a good idea,† but there was no conviction in his voice and she pushed him back on the bed. Tuck was staring up at the ceiling, his arm pinned under her neck, silently mouthing his mantra, â€Å"After this, I will not bone the crazy woman. After this, I will not bone the crazy woman. After†¦Ã¢â‚¬  Boy, how many times had he said that? Maybe things were getting better, though. In the past it had always been â€Å"I will not get drunk and bone the crazy woman.† He had been only sleepy this time. He tried to worm his arm out from under her, then used the â€Å"old snuggle method.† He rolled into her for a hug and when she responded with a sleepy moan and tried to kiss him, the space under her neck opened up and he was free. It worked as well on murdering bitch goddesses as it did on Mary Jean ladies. Better even, Beth didn't wear near as much hair spray, which can slow a guy down. God, I'm good. He rolled out of bed and crept into the bathroom. While he peed, he softly chanted, â€Å"Yo, after this, I will not bone the crazy woman.† It had taken on a rap cadence and he was feeling very hip along with the usual self-loathing. His scars made him think of Kimi's wound, and suddenly he was angry. He padded naked back to the bed and jostled the sleeping icon. â€Å"Get up, Beth. Go home.† And someone pounded on the door. â€Å"Mr. Case, tee time in five.† Tuck clamped his hand over Beth's mouth, lifted her by her head in a single sweeping move from the bed to the bathroom, where he released her and shut the door. Fred Astaire, had he been a terrorist, would have been proud of the move. Tuck grabbed his pants off the floor, which is where he kept them, pulled them on, and answered the door. Sebastian Curtis had a driver slung over his shoulder. â€Å"You might want to put on a shirt, Mr. Case. You can get burned, even this early.† â€Å"Right,† Tuck said. He was looking at the caddie. Today Stripe carried the clubs. The guard sneered at him. Tuck smiled back. Stripe, like Mato before him, was doing caddie duty unarmed. Time to play a little round for the navigator, he thought. He winked at Stripe. â€Å"I'll be right there.† Tuck closed the door and went to the bathroom to tell Beth to wait until he'd gone before coming out, but when he opened the door, she was gone. â€Å"Did you know that over ninety percent of all the endangered species are on islands?† the doctor said. â€Å"Nope,† Tuck said. He picked his ball up and put it on the rubberized mat, then turned to Stripe. â€Å"Dopey, give me a five iron.† They were on the fourth hole and had crisscrossed the compound pretending to play golf for an hour. Tuck swung and skidded the ball fifty yards across the gravel. â€Å"Heads up, Bashful,† Tuck said as he threw the club back to Stripe. â€Å"Islands are like evolutionary pressure cookers. New species pop up faster and go extinct more quickly. It works the same way with religions.† â€Å"No kidding, Doc?† They still had fifty yards to get to where Sebastian's first shot lay. Tuck had hit three times. â€Å"The cargo cults have all the same events associated with the great reli-gions: a period of oppression, the rise of a Messiah, a new order, the promise of an endless time of peace and prosperity. But instead of devel-oping over centuries like Christianity or Buddhism, it happens in just a few years. It's fascinating, like being able to see the hands of the clock move right before your eyes and be a part of it.† â€Å"So you must totally get off when daylight savings time comes around.† â€Å"It was just a metaphor, Mr. Case.† â€Å"Call me Tuck.† They had reached Tuck's ball and he placed it on the Astro Turf mat. â€Å"Sneezy, give me the driver.† Sebastian cleared his throat. â€Å"That looks more like a nine iron to me. You've only got fifty yards to the pin.† â€Å"Trust me, Doc. I need a driver for this one.† Stripe snickered and handed him the driver. Tuck examined it, one of the large-headed alloy models that had become so popular in the States – all metal. Tuck grinned at Stripe. â€Å"So, Doc, I guess you shitcanned the Meth-odist thing to watch the clock spin.† Tuck lined up the shot and took a practice swing. The club whooshed through the air. â€Å"Have you ever had faith in anything, Mr. Case?† Tuck took another practice swing. â€Å"Me? Faith? Nope.† â€Å"Not even your own abilities?† â€Å"Nope.† Tuck made a show of lining up the shot again and making sure his hips were loose. â€Å"Then you shouldn't make jokes about it.† â€Å"Right,† Tuck said. He tensed and put his entire weight behind the club, but instead of hitting the ball, he swung it around like a baseball bat, slamming the head into Stripe's cheek, shattering the bone with a sickening thwack. The guard's feet went out from under him and he landed with a crunch in the coral. â€Å"Christ!† Sebastian yelled. He grabbed the club and wrenched it from Tuck's grasp. â€Å"What in the hell are you doing?† Tuck didn't answer. He bent over the guard until he was only inches from his face and whispered, â€Å"Fore, motherfucker.† A second later Tuck heard a mechanical click and the guard who had been tending the pin had an Uzi pressed to his ear. Sebastian Curtis was bent over Stripe, pulling his eyes open to see if his pupils would contract. â€Å"Take Mr. Case to his bungalow and stay with him. Send two men with a stretcher and find Beth. Tell her to – † Curtis suddenly realized that the guard was only getting about a third of what he said. â€Å"Bring my wife.† â€Å"I'll get back to you on that faith thing, Doc,† Tuck said.

Tuesday, October 22, 2019

The Brain Computer Interface Psychology Essay Essay Example

The Brain Computer Interface Psychology Essay Essay Example The Brain Computer Interface Psychology Essay Essay The Brain Computer Interface Psychology Essay Essay A Brain-computer interface ( BCI ) is a communicating channel linking the encephalon to a computing machine or another electronic device.BCI represents a direct interface between the encephalon and a computing machine or any other system. BCI is a wide construct and comprehends any communicating between the encephalon and a machine in both waies: efficaciously opening a wholly new communicating channel without the usage of any peripheral nervous system or musculuss. In rule this communicating is thought to be two manner. But present twenty-four hours BCI is chiefly concentrating on communicating from the encephalon to the computing machine. To pass on in the other way, inputting information in to the encephalon, more thorough cognition is required refering the operation of the encephalon. From here on the focal point is on communicating straight from the encephalon to the computing machine. Typically, the intellectual cerebral mantle is the country of involvement in the encephalon computing machine interface. The intellectual cerebral mantle is the country of the encephalon responsible for playing a cardinal function in memory, attending, perceptual consciousness, thought, linguistic communication, consciousness and motor map. The manner these BCI s work is proficient but can by and large works as follows. The electrode is placed in the country of the encephalon responsible for the coveted motor map. These electrode recognize encephalon moving ridges, that step the minute differences in electromotive forces across active nerve cells, and construe this as a signal. But in this construct we traveling to utilize the encephalon moving ridges through radio EEG. These signals are stored and so synthesized utilizing assorted complex transforms and run through a plan, typically something like Matlab or C++ . It is good known that the encephalon is an electrochemical organ ; research workers have peculated that a to the full working encephalon can bring forth every bit much as 10 Watts of electrical power. Other more conservative research workers calculate that if all 10 billion interconnected nervus cells discharged at one clip that a individual electrode placed on the human scalp would enter something like five millionths to 50 millionths of a V. Whenever artefacts are detected the affected part of the signal can be rejected. This can be a valid pre-processing measure and does non hold to be a job. However the job with canceling a specific piece of informations is that it can ensue in unusual anomalousnesss where the two pieces are connected. Second, EEG information in general is comparatively scarce. For that ground a better attack is to take the artefact from the EEG information. This goes one measure further than artifact rejection. For practical intents in an online system, it is unwanted to throw off every signal that is affected with an artefact. Recovering the signal with 100 % rightness is impossible ; it is merely unknown what the information would hold looked like without for case the oculus wink. For offline systems this is less critical, since it does non count if some bids are lost. In the online instance nevertheless, the user demands that every bid that is issued to the system is recognized and executed. The user does nt desire to maintain seeking infinitely for a good test. At first utilizing ICA algorithm extract Independent constituents ( ICs ) of each test so GA select the best and related ICs among the hole ICs. The proposed attack to the usage of GAs for Artifact remotion involves encoding a set of vitamin D, ICs as a binary twine of 500 elements, in which a 0 in the twine indicates that the corresponding IC is to be omitted, and a 1 that it is to be included. This coding strategy represents the presence or absence of a peculiar Intelligence community from the IC infinite. The length of chromosome equal to IC infinite dimensions. Then the selected ICs used as input informations for classifiers. This paper used the fittingness map shown below to unite the two footings: Like other communicating and control systems, BCI s have inputs, end products, and interlingual rendition algorithms that convert the former to the latter. BCI operation depends on the interaction of two adaptative accountants, the user s encephalon, which produces the input ( i.e. , the electrophysiological activity measured by the BCI system ) and the system itself, which translates that activity into end product ( i.e. , specific bids that act on the external universe ) . Successful BCI operation requires that the user get and keep a new accomplishment, a accomplishment that consists non of musculus control but instead of control of EEG or single-unit activity.

Monday, October 21, 2019

Understanding Expressive Roles and Task Roles

Understanding Expressive Roles and Task Roles Expressive roles and task roles, also known as instrumental roles, describe two ways of participating in social relationships. People in expressive roles tend to pay attention to how everyone is getting along, managing conflict, soothing hurt feelings, encouraging good humor, and take care of things that contribute to one’s feelings within the social group. People in task roles, on the other hand, pay more attention to achieving whatever goals are important to the social group, like earning money to provide resources for survival, for example. Sociologists believe that both roles are required for small social groups to function properly  and that each provides a form of leadership: functional and social. Parsonss Domestic Division of Labor How sociologists understand expressive roles and task roles today is rooted in Talcott Parsons development of them as concepts within his formulation of the domestic division of labor. Parsons was a mid-century American sociologist, and his theory of the domestic division of labor reflects gender role biases that proliferated at that time and that are often considered traditional, though theres scant factual evidence to back up this assumption. Parsons is known for popularizing the structural functionalist perspective within sociology, and his description of expressive and task roles fits within that framework. In his view, assuming heteronormative and patriarchally organized nuclear family unit, Parsons framed the man/husband as fulfilling  the instrumental role by working outside the home to provide the money required to support the family. The father, in this sense, is instrumental or task-oriented he accomplishes a specific task (earning money) that is required for the family unit to function. In this model, the woman/wife plays a complementary expressive role by serving as the caregiver for the family. In this role, she is responsible for the primary socialization of the children  and provides morale and cohesion for the group through emotional support and social instruction. A Broader Understanding and Application Parsons conceptualization of expressive and task roles was limited by stereotypical ideas about gender, heterosexual relationships, and unrealistic expectations for family organization and structure, however, freed of these ideological constraints, these concepts have value and are usefully applied to understanding social groups today. If you think about your own life and relationships, you can probably see that some people clearly embrace the expectations of either expressive or task roles, while others might do both. You might even notice that you and others around you seem to move between these different roles depending on where they are, what they are doing, and who they are doing it with. People can be seen to be playing these roles in all small social groups, not just families. This can be observed within friend groups, households that are not composed of family members, sports teams or clubs, and even among colleagues in a workplace setting. Regardless of the setting, one will see people of all genders playing both roles at various times. Updated  by Nicki Lisa Cole, Ph.D.