The Louisiana Purchase, The Oregon Treaty, And The...

There were many important events that helped to achieve the goal of Manifest Destiny. For example, the Louisiana Purchase, the Oregon Treaty, and the California Gold Rush all helped achieve this goal. All of these events had either increased the amount of land in the United States, or increased the population of people living in Western United States. One event that occurred during the time of Manifest Destiny was the Louisiana Purchase. In the early 1800s, President Thomas Jefferson wanted to get control over the Mississippi River, where many farmers traded. He sent negotiators to France to offer to buy New Orleans. At that time, New Orleans was the main trading port on the Mississippi, whoever controlled New Orleans controlled the Mississippi and the amount of trade on the river. Jefferson told the negotiators to offer France $2 million but he said to go as high as $10 million. Napoleon Bonaparte, emperor of France counter-offered and said that the United States should pay $15 million for all of The Louisiana Territory. Jefferson was happy and the U.S. accepted the offer. After the purchase, France’s finance minister, Charles de Talleyrand, sent a letter thanking Thomas Jefferson for purchasing the Louisiana Territory. He said, â€Å"I hope you see that both countries will benefit from this purchase. The United States h as increased in size tremendously. The French can now wage a more effective war with Great Britain.† (Document 2a). There is a map titled, â€Å"Major LandShow MoreRelatedExpansion of the United States during President Jefferson and President Polk535 Words   |  3 PagesUnited States. Great treaties and compromises signed by both will be forever recognized in history to the help of expanded our country. Jefferson conducted one of the largest territory gains in United States history with the Louisiana Purchase. Louisiana was France territory, originally from Spain; and Napoleon was already eager to sell because his empire was expanding and needed money, especially from the loss at Santo Domingo. France did not want Britain to take Louisiana because they controlledRead MoreManifest Destiny and Foreign Policy1135 Words   |  5 Pagesduring the 19th century. Westward expansion A.K.A. â€Å"Manifest destiny† led to America’s rapid acquisition of the old Mexican Southwest and the Oregon Territory that marked the fulfillment of President James K. Polks expansionistic campaign promises. Polk ran on only one platform -- westward expansion. He wanted to officially claim the southern part of Oregon Territory; annex the American Southwest from Mexico; and annex Texas. Thus, Polk’s desire for land would eventually cause a great deal of sufferingRead MoreManifest Destiny By James K. Polk1257 Words   |  6 Pageskilling of Indians who were in the way of the belief of Manifest Destiny. The president that followed through with the belief of Manifest Destiny the most is that of James K. Polk. His war with Mexico and strong stand against the British about the Oregon territory solidified Manifest Destiny. The Unites States of America and its government drilled into its citizens that they should spread the political idea of democracy for the common good of the people. The United States government has establishedRead MoreWestward Expansion Of The United States1017 Words   |  5 Pages it continuously proved to be aggressive, racist, and imperialistic. Enthusiasm over territorial expansion began in 1803 when Napoleon decided to offer the United States the entire Louisiana Territory and later escalated with the issues of Texas and Oregon. In the 1820s, the United States offered, twice, to purchase Texas from the Mexican government. However, it was not until 1824 that Mexico enacted a colonization law offering cheap land and a four-year exemption from taxes to any American willingRead MoreEssay on The Extension of Slavery Into the American West 1658 Words   |  7 Pageson the issues surrounding the possible extension of slavery into the following areas: the Missouri Territory and the Louisiana Purchase in general, Texas (annexation), the Oregon Territory, California (annexation), Nebraska (unorganized), and the Kansas Territory. The initial conflict over the extension of slavery westward focused on the territory acquired in the Louisiana Purchase in general and, in specific, application for statehood by the Missouri Territory. In 1819, there were 11 slave andRead MoreThe American West And American History Essay1724 Words   |  7 Pagesacross the Appalachian Mountains to explore, find resources, and find new settlements. Towards the 1800’s many white Americans from the east began to settle in the western part of the country â€Å"the Great Plain† after the United States got the Louisiana Purchase, by doing so they have to approach native Indians who have been living there for many years. President Thomas Jefferson commissioned the expedition, first set out the Lewis and Clark expedition to explore and chart the new territory in 1804Read More Western Expansion Essay4115 Words   |  17 Pagesfurther expansion of the frontier and many disappointed pioneers even backtracked from the west to the east. When the treaty of Paris was signed in 1783, the Americans had thought that they had enough land between the Atlantic coast and the Mississippi river. Yet in 1803, by the Louisiana Purchase, the area of the United States doubled and not long after, it was augmented by the half-purchase-half-conquest of Florida. By the end of 1820, as many as 6 states were created, east of Mississippi-Indiana (1816)Read MoreGive Me Liberty Chapter 13 Notes1842 Words   |  8 Pagesmoving west The Mormons’ Trek Went to modern-day Utah: founded by Joseph Smith (polygamy) National boundaries meant little to those who moved West The Mexican Frontier: New Mexico and California Mexico achieved independence from Spain in 1821, issue of slavery arose. Mexico could now trade w/ the US! California Californios - Mexican cattle ranchers By 1840, it was already linked commercially w/ the US The Texas Revolt Mexican gov, in order to develop the region, accepted an offer by MosesRead MoreThe American Civil War1765 Words   |  8 Pagesand he would not have gotten the annexation of Texas and also the lands that the US got in the Treaty of Guadalupe Hidalgo. The argument is that because of Clay and his stance on policies leading to the annexation of Texas that if Clay was chief executive, that Texas would not have been admitted into the Union. Clay also seemed to have this view also with the California territory as well as the Oregon Country. Clay had the vision that those areas were â€Å"not in the nation’s interest† to make themRead MoreImagine a land, untouched by modern civilization, its resources untapped, its plants grow wild and3300 Words   |  14 Pagesthe Americas. Christopher Columbus, being the most celebrated of the many explorers, had found many islands in what is now known as the Bahamas. The tribes of islanders he came across were nothing more than animals to him in his quest for power and gold. After reporting his findings, the news spread across Europe that there was uncharted land across the ocean. This land promised riches by the boatload, whose native people who were more than willing to give help. The native people were so easy to take

Vedic Mathematics Multiplication

Sample details Pages: 30 Words: 9097 Downloads: 4 Date added: 2017/06/26 Category Statistics Essay Did you like this example? 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. Don’t waste time! Our writers will create an original "Vedic Mathematics Multiplication | Mathematics Dissertations" essay for you Create order 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 doesnt 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 Sankaracharyas invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: Ekdhikena Prvena The relevant Sutra reads Ekdhikena Prvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navatacaramam Daatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navatacaramam Daatah, 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 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: rdhva Tiryagbhym rdhva Tiryagbhym sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 13 = 156. The Fourth Sutra: Parvartya Yojayet The term Parvartya 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 2 = 22 but we have 72 in the divident. This means that we have to get 92 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have 2 9x = 18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = 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: Snyam 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: nurpye 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 22 + 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 Vyavakalanbhym 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: Puranpuranbhym Puranpuranbhym 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: Calan kalanbhym The term (Calan kalanbhym) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: Yvadnam Yvadnam 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: esnyankena Caramena The sutra esnyankena 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: Ekanynena Prvena The Ekanynena Prvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 62 + 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 3 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol 5, 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 boys and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 58, and the second 23 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure 1 for format), requiring students to multiply together two basic binomials, is given in Table 1. Question 2 was presented in each case with the binomials in the same line. Since the first questionnaire was administered immediately after the FOIL method was taught, most students employed this method, and only 8 (44.4%) correctly performed the expansion in the simplest example, question 2a). In the second questionnaire, where students were asked to use the vertically and Crosswise method, 10 (55.6%) were correct, but there was no statistically significant difference (c2 = 0.11, df = 1, ns). However, when the presentation format was changed from a single line to the grid format of question 3 (see Figure 1), the facility on 3a) (Multiply x + 3 and x + 4) improved to 14 correct answers (77.8%). Thus there was weak evidence (c2 = 2.92, df = 1, p = 0.1) of an improvement in students performance on this basic expansion of binomials in the second questionnaire, using the Vedic method with a grid presentation. However, the improvement was not sustained for question 3b) which had negative signs (see Table 2: c2 = 0.45), or for 2b), which involved 2x and negative signs. It seems that the arithmetic complexity caused problems, with only 3 students answering 2b) correctly on each test. The other results of question 3 on the second questionnaire also support the idea of improved understanding; Table 2 gives the results of the multiplications in this question. In this case question 3c) was the question of a standard corresponding to 2b) in the first test (2x 3 times x + 4), and yet the students did significantly better (50% correct) than they did on that question (c2 = 3.13, df = 1, p 0.05). It is also of interest that on question 2 in the first test all eight students who successfully applied the FOIL method wrote out all four terms and then added the two middle terms. No student wrote the final answer without the intermediate step of giving both middle terms. In contrast, when using the vertically and crosswise approach, all 14 students who were successful were able to write the answers straight down, as seen in the example in Figure 4. Furthermore, 5 students (12, 15, 16, 17, and 18) who were unable to multiply binomials using the standard method achieved success using the Vedic method. Figure 5 shows the corresponding working of student 18 on these questions. In test 1, following the FOIL method she was unable to multiply any terms together and even seemed to confuse the question with factorisation. However, with the Vedic method she makes a good attempt, correctly answering two parts. It might be argued that the improvement above was due to students spending more time learning how to perform such expansions, or that the previous FOIL learning took time to assimilate. However, it should be noted that the questions where the improvement occurred specifically involved use of the Vedic method, which, while it can be related to FOIL by an experienced mathematician, would no doubt look quite different to these students, since it is set out in a grid format rather than being performed in a single line. Furthermore, it was noted during the teaching episode that there was some resistance from the students to learning a second method when they already knew the FOIL method, and this could be expected to have a detrimental effect on performance. Factorisation of quadratic expressions Table 3 contains a summary of the results of question 3 from the first questionnaire and question 5 in the second, these being corresponding single line, traditional format factorisation questions (see Figure 1). Individually the results of these questions did not show any statistical difference between the performance before and after the Vedic method was introduced. For example, between questions 3a) and 5a), c2 = 0.9, which is not significant. However, if the question parts are grouped together and the number of students correct on question 3 compared with those correct on question 5 then we find that there is a significant improvement (Q3 v Q5 c2 = 6.65, p 0.01) on the second questionnaire. For question 4 in each questionnaire the students were asked to supply the missing terms in two binomials that were multiplied together. Boxes were provided for the missing terms, which were in the binomials, the quadratic, or both (see Figure 1). While these are not standard straight factorisation questions they do require students to work backwards from the answer and thus display some conceptual knowledge of how to undo multiplication of binomials. There was no difference in facility on any of these matching questions (see Table 4), or on a comparison of the total number of correct scores between the two questionnaires (c2 = 0.39, ns). The students appeared to be able to do them equally well using either guess or check and decomposition or the Vedic approach. While the discussion above shows that the evidence for a better performance on individual questions following the teaching of the Vedic method was rarely present, a consideration of the students overall scores on the expansion and factorisation algebra questions did show a significantly better performance on the second test, (_x1 = 41.4%,_x2 = 51.5%, t = 2.66, p 0.05). Thus it appears that, overall, the Vedic approach may have contributed to student understanding of the methods, either by cementing in place the previous methods, or by complementing them. It is worth noting too that some students preferred to use the vertically and Crosswise method even when not directed to do so. For example, in question 2 of the second questionnaire students were simply asked to multiply two binomials together, with no method specified. In the event 4 students (4, 5, 16, and 18) chose to use the vertically and Crosswise approach, setting out their work in a grid. In addition, 3 of these students (15, 16 and 18) used it for the factorisation in question 5. An example of their work is shown in Figure 6. While they used the method with varying levels of success, it seems to have benefited both student 5, who answered no algebra questions correctly in test 1 (and 2 in test 2), and student 16 who used it for factorisation and went from 3 correct to 8 correct (see Figure 6). While student 18 preferred the Vedic method to expand and to factorise expressions, she used the traditional method to multiply numbers even when asked to multiply by the vertically and Crosswise method. This seems to suggest that she was comfortable using different methods in algebra from those employed in arithmetic. Discussion India has good reasons to be proud of a rich heritage in science, philosophy and culture in general, coming to us down the ages. In mathematics, which is my own area of specialization, the ancient Indians not only took great strides long before the Greek advent, which is a standard reference point in the Western historical perspective, but also enriched it for a long period making in particular some very fundamental contributions such as the place-value system for writing numbers as we have today, introduction of zero and so on. Further, the sustained development of mathematics in India in the post-Greek period was indirectly instrumental in the revival in Europe after its dark ages. Notwithstanding the enviable background, lack of adequate attention to academic pursuits over a prolonged period, occasioned by several factors, together with about two centuries of Macaulayan educational system, has unfortunately resulted, on the one hand, in a lack of awareness of our historical role in actual terms and, on the other, an empty sense of pride which is more of an emotional reaction to the colonial domination rather than an intellectual challenge. Together they provide a convenient ground for extremist and misguided elements in society to reconstruct history from nonexistent or concocted source material to whip up popular euphoria. That this anti-intellectual endeavour is counter-productive in the long run and, more important, harmful to our image as a mature society, is either not recognized or ignored in favour of short-term considerations. Along with the obvious need to accelerate the process of creating an awareness of our past achievements, on the strength of authentic information, a more urgent need has also arisen to confront and expose such baseless constructs before it is too late. This is not merely a question of setting the record straight. The motivated versions have a way of corrupting the intellectual processes in society and weakening their very foundations in the long run, which needs to be prevented at all costs. The so-called Vedic Mathematics is a case in point. A book by that name written by Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaja (Tirthaji, 1965) is at the centre of this pursuit, which has now acquired wide following; Tirthaji was the Shankaracharya of Govardhan Math, Puri, from 1925 until he passed away in 1960. The book was published posthumously, but he had been carrying out a campaign on the theme for a long time, apparently for several decades, by means of lectures, blackboard demonstrations, classes and so on. It has been known from the beginning that there is no evidence of the contents of the book being of Vedic origin; the Foreword to the book by the General Editor, Dr. A.S.Agrawala, and an account of the genesis of the work written by Manjula Trivedi, a disciple of the swamiji, makes this clear even before one gets to the text of the book. No one has come up with any positive evidence subsequently either. There has, however, been a persistent propaganda that the material is from the Vedas. In the face of a false sense of national pride associated with it and the neglect, on the part of the knowledgeable, in countering the propaganda, even educated and well meaning people have tended to accept it uncritically. The vested interests have also involved politicians in the propaganda process to gain state support. Several leaders have lent support to the Vedic Mathematics over the years, evidently in the belief of its being from ancient scriptures. In the current environment, when a label as ancient seems to carry considerable premium irrespective of its authenticity or merit, the purveyors would have it going easy. Large sums have been spent both by the Government and several private agencies to support this Vedic Mathematics, while authentic Vedic studies continue to be neglected. People, especially children, are encouraged to learn and spread the contents of the book, largely on the baseless premise of their being from the Vedas. With missionary zeal several devotees of this cause have striven to take the message around the world; not surprisingly, they have even met with some success in the West, not unlike some of the gurus and yogis peddling their own versions of Indian philosophy. Several people are also engaged in research in the new Vedic Mathematics. To top it all, when in the early nineties the Uttar Pradesh Government introduced Vedic Mathematics in school text books, the contents of the swamijis book were treated as if they were genuinely from the Vedas; this also naturally seems to have led them to include a list of the swamijis sutras on one of the opening pages (presumably for the students to learn them by heart and recite!) and to accord the swamiji a place of honour in the brief history of Indian mathematics described in the beginning of the textbook, together with a chart, which curiously has Srinivasa Ramanujans as the only other name from the twentieth century! For all their concern to inculcate a sense of national pride in children, those responsible for this have not cared for the simple fact that modern India has also produced several notable mathematicians and built a worthwhile edifice in mathematics (as also in many other areas). Harish Chandras work is held in great esteem all over the world and several leading seats of learning of our times pride themselves in having members pursuing his ideas; (see, for instance, Langlands, 1993). Even among those based in India, several like Syamdas Mukhopadhyay, Ganesh Prasad, B.N.Prasad, K.Anand Rau, T.Vijayaraghavan, S.S.Pillai, S.Minakshisundaram, Hansraj Gupta, K.G.Ramanathan, B.S.Madhava Rao, V.V.Narlikar, P.L.Bhatnagar and so on and also many living Indian mathematicians have carved a niche for themselves on the international mathematical scene (see Narasimhan, 1991). Ignoring all this while introducing the swamijis name in the brief history would inevitably create a warped perspective in childrens minds, favouring gimmickry rather than professional work. What does the swamijis Vedic Mathematics seek to do and what does it achieve? In his preface of the book, grandly titled A Descriptive Prefatory Note on the astounding Wonders of Ancient Indian Vedic Mathematics, the swamiji tells us that he strove from his childhood to study the Vedas critically to prove to ourselves (and to others) the correctness (or otherwise)of the derivational meaning of Veda that the Vedas should contain within themselves all the knowledge needed by the mankind relating not only to spiritual matters but also those usually described as purely secular, temporal or worldly; in other words, simply because of the meaning of the word Veda, everything that is worth knowing is expected to be contained in the vedas and the swamiji seeks to prove it to be the case! It may be worthwh ile to point out here that there would be room for starting such an enterprise with the word science! He also describes how the contemptuous or at best patronising attitude of Orientalists, Indologists and so on strengthened his determination to unravel the too-long-hidden mysteries of philosophy and science contained in ancient Indias Vedic lore, with the consequence that, after eight years of concentrated contemplation in forest solitude, we were at long last able to recover the long lost keys which alone could unlock the portals thereof. The mindset revealed in this can hardly be said to be suitable in scientific and objective inquiry or pursuit of knowledge, but perhaps one should not grudge it in someone from a totally different milieu, if the outcome is positive. One would have thought that with all the commitment and grit the author would have come up with at least a few new things which can be attributed to the Vedas, with solid evidence. This would have made a worthwhile contribution to our understanding of our heritage. Instead, all said and done there is only the authors certificate that we were agreeably astonished and intensely gratified to find that exceedingly though mathematical problems can be easily and readily solved with the help of these ultra-easy Vedic sutras (or mathematical aphorisms) contained in the Parishishta (the appendix portion) of the Atharva Veda in a few simple steps and by methods which can be conscientiously described as mere mental arithmetic (paragraph 9 in the preface). That passing reference to the Atharva Veda is all that is ever said by way of source material for the contents. The sutras, incidentally, which appeared later, scattered in the book, are short phrases of just about two to four words in Sanskrit, such as Ekadhikena Purvena or Anurupye Shunyam Anyat. (There are 16 of them and in addition there are 13 of what are called sub-sutras, similar in nature to the sutras). The mathematics of today concerns a great variety of objects beyond the high school level, involving various kinds of abstract objects generalising numbers, shapes, geometries, measures and so on and several combinations of such structures, various kinds of operations, often involving infinitely many entities; this is not the case only about the frontiers of mathematics but a whole lot of it, including many topics applied in physics, engineering, medicine, finance and various other subjects. Despite its entire pretentious verbiage page after page, the swamijis book offers nothing worthwhile in advanced mathematics whether concretely or by way of insight. Modern mathematics with its multitude of disciplines (group theory, topology, algebraic geometry, harmonic analysis, ergodic theory, combinatorial mathematics-to name just a few) would be a long way from the level of the swamijis book. There are occasionally reports of some researchers applying the swamijis Vedic Mathematics to advanced problems such as Keplers problem, but such work involves nothing more than tinkering superficially with the topic, in the manner of the swamijis treatment of calculus, and offers nothing of interest to professionals in the area. Even at the western teaching Vedic Mathematics deals only with a small part and, more importantly, there too it concerns itself with only one particular aspect, that of faster computation. One of the main aims of mathematics education even at the western teaching consists of developing familiarity with a variety of concepts and their significance. Not only does the approach of Vedic Mathematics not contributes anything towards this crucial objective, but in fact might work to its detriment, because of the undue emphasis laid on faster computation. The swamijis assertion 8 months (or 12 months) at an average rate of 2 or 3 hours per day should be enough for finishing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also foreign universities, is patently absurd and hopefully nobody takes it seriously, even among the activists in the area. It would work as a cruel joke if some people choose to make such a substitution in respect of their children. It is often claimed that Vedic Mathematics is well appreciated in other countries, and even taught in some schools in UK etc. In the normal course one would not have the means to examine such claims, especially since few details are generally supplied while making the claims. Recent years have brought about the development of powerful tools for verifying specifications of hardware and software systems. By now, major companies, such as Intel, IBM, and [emailprotected]/* */ Molecules have realized the impact and importance of such tools in their own design and implementation processes as a means of coping with the ever-increasing complexity of chip and computing tools. Protocols, networks, and distributed systems can generally not be described by code of some deterministic programming language. Such systems exhibit concurrent behavior and they are typically reactive in the sense that their behavior depends on what the environment can offer (e.g. Is the printer busy?). Computation tree logic (CTL) is currently one of the popular frameworks used in verifying properties of concurrent systems. We study its syntax and semantics, and use those insights to design an automated verification algorithm which takes a description of a system and specifications of expected behavior as input and checks whether that system meets those expectations. That algorithm is the foundation for a tool, the symbolic model verifier (SMV), which they use to evaluate some basic designs, e.g. simple elevator systems and a mutual exclusion protocol (3-4 weeks). Exposure to a labeling algorithm for finite-state verification illustrates depth-first backwards search in a directed graph; this search is recursive and the recursion is driven by the logical structure of the specified behavior, written as a CTL formula. The description and evaluation of small designs with the tool SMV makes students appreciate how such graphs can be modeled with a modular guarded-command language with non-deterministic assignment. The discussion of program logic contains the linear algorithm for computing minimal-sum sections of integer arrays as a case study. Finally, binary decision diagrams require algorithms that implement the familiar logical operations on such diagrams. Some of these algorithms illustrate dynamic programming at an accessible level. Conclusion In this study students were taught an appropriate Vedic sutra following teaching of the traditional FOIL method of multiplication of binomials, and the decomposition method for factorisation. We found that afterwards the students performed significantly better overall on these types of algebra questions, and specifically on the factorisations, and there was weak evidence of better results on expansion using a grid format. The reasons for the improvement are not easy to pinpoint since they appear in some areas and not in others. This seems to indicate that the value of the method may lie in what it adds to the students overall algebraic conceptions and knowledge of mathematical structure. Thus we have found no evidence that it should be seen as a replacement for the former approaches, but our results suggest it could rather be recommended as a useful adjunct, a complementary method. While this may take longer in terms of teaching time, the results indicate that possession of a range of strategies may have value above and beyond their individual benefit. References Barnard, T. Tall, D. O. (1997). Cognitive units, connections and mathematical proof. In E. Pekhonen (Ed.), Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 4148). Lahti, Finland. Bhatnagar, P. L. (1976). Some Vedic Sutras for arithmetical operations or a fresh look at arithmetic after learning some algebra. Mathematics Teacher (India) 11A, 85120. Booth, R. D. L. Thomas, M. O. J. (2000). Visualisation in mathematics learning: Arithmetic problem-solving and student difficulties. Journal of Mathematical Behavior, 18(2), 169190. Datta, B. B. Singh, A. N. (2001). History of Hindu Mathematics: A Source Book Parts I and II. New Delhi: Bharatiya Kala Prakashan. Joseph, G. G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics. Harmondsworth: Penguin. MacGregor, M. Stacey, K. (1999). A flying start to algebra. Teaching Children Mathematics, 6(2), 7885. Morrow, L. J. (1998). Wither algorithms? Mathematics educators express their views. In L. J. Morrow M. J. Kenney (Eds), The Teaching and Learning of Algorithms in School Mathematics (pp. 119), National Council of Teachers of Mathematics. Presmeg, N. C. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6(3), 4246. Shan, S. J. Bailey, P. (1991). Multiple Factors: Classroom Mathematics for Equality and Justice. Trentham Books. Stacey, K. MacGregor, M. (1997). Building foundations for algebra. Mathematics in the Middle School, 2, 253260. Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4, 295312. Tall, D. Thomas, M. (1991). Encouraging versatile thinking in algebra using the computer. Educational Studies in Mathematics, 22, 125147. Tirthaji, B. K. (2001). Vedic Mathematics. New Delhi: Motilal Banarasidass. Van Hiele, P. (2002). Similarities and differences between the theory of learning and teaching of Skemp and the van Hiele levels. In D. O. Tall M. O. J. Thomas (Eds), Intelligence, Learning and Understanding in Mathematics: A Tribute to Richard Skemp (pp. 2748). Flaxton, Queensland: Post Pressed. Warren, E. (2001). Algebraic understanding: The importance of learning in the early years. In H. Chick, K. Stacey, J. Vincent J. Vincent (Eds), The Future of the Teaching and Learning of Algebra, Proceedings of the 12th ICMI Study Conference, Vol. 2 (pp. 633646). Melbourne, Australia. Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal, 15(2), 122137. Williams, K. (2002). The cosmic computer course. Obtained from www.vedicmaths.com. Abhyankar, K.D. A rational approach to study ancient literature, Current science, 87 (Aug.2004) 415-416. Adams, E.S., and D.A. Farber. Beyond the Formalism Debate: Expert Reasoning, Fuzzy Logic and Complex Statutes, Vanderbilt Law Review, 52 (1999), 1243-1340. https://law.vanderbilt.edu/lawreview/vol525/adams.pdf Allen, J., S. Bhattacharya and F. Smarandache. Fuzziness and Funds allocation in Portfolio Optimization. https://lanl.arxiv.org/ftp/math/papers/0203/0203073. Anitha, V. Application of Bidirectional Associative Memory Model to Study Female Infanticide, Masters Dissertation, Guide: Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, Chennai, March 2000. Ashbacher, C. Introduction to Neutrosophic Logic, American Research Press, Rehoboth, 2002. https://www.gallup.unm.edu/~smarandache/IntrodNeutLogic. Axelord, R. (ed.) Structure of Decision: The Cognitive Maps of Political Elites, Princeton Univ. Press, New Jersey, 1976. Bibliography Discover Mathematics, Kenneth Williams Vertically and Crosswise, Kenneth Williams Vedic Mathematics, Bharati Krsna Tirthaji www.hinduism.co.za/vedic.htm https://vedicmathsindia.blogspot.com/search/label/vedic%20math%20research https://vedicmathsindia.blogspot.com/ https://vedicmathsindia.blogspot.com/2007/06/10-questions-to-kenneth-r-williams-on.html

Emma Goldman Anarchist, Feminist, Writer, Lecturer

Emma Goldman is known as a rebel,  an anarchist, an ardent proponent of birth control and free speech, a feminist, a lecturer, and a  writer. Born on June 27, 1869, she became known as Red Emma both for her heritage and her political involvement. Emma Goldman died on May 14, 1940. Early Life Emma Goldman was born in what is now Lithuania but was then controlled by Russia, in a Jewish ghetto that was largely German Jewish in culture. Her father, Abraham Goldman, married Taube Zodokoff.  She had two older half-sisters (her mothers children) and two younger brothers.  The family ran an inn which was used by the Russian military for training soldiers. Emma Goldman  was sent when she was seven to Kà ¶nigsberg to attend private school and live with relatives. When her family followed, she transferred to a private school.   When Emma Goldman was twelve, she and the family moved to St. Petersburg.  She left school, though she worked on self-education, and went to work to help support the family.  She eventually became involved with university radicals and looked to historical women rebels as role models. Activism in America Under suppression of radical politics by the government, and family pressure to marry, Emma Goldman left for America in 1885 with her half-sister Helen Zodokoff, where they lived with their older sister who had emigrated earlier.  She began working in the textile industry in Rochester, New York. In 1886 Emma married a fellow worker, Jacob Kersner.  They divorced in 1889, but since Kersner was a citizen, that marriage was the basis for Goldmans later claims to be a citizen. Emma Goldman moved in 1889 to New York where she quickly became active in the anarchist movement.  Inspired by the events in Chicago in 1886, which she had followed from Rochester, she joined with fellow anarchist  Alexander Berkman in a plot to end the Homestead Steel Strike by assassinating the industrialist Henry Clay Frick.  The plot failed to kill Frick, and Berkman went to jail for 14 years.  Emma Goldmans name was widely known as the New York World depicted her as the real brains behind the attempt. The 1893 panic, with a stock market crash and massive unemployment, led to a public rally in Union Square in August.  Goldman spoke there, and she was arrested for inciting a riot.  While she was in jail, Nellie Bly interviewed her.  When she got out of prison from that charge, in 1895, she went to Europe to study medicine.   She was back in America in 1901, suspected of participating in a plot to assassinate President William McKinley.  The only evidence that could be found against her was that the actual assassin attended a speech Goldman gave.  The assassination resulted in the 1902 Aliens Act, classifying promoting criminal anarchy as a felony.  In 1903, Goldman was among those who founded the Free Speech League to promote free speech and free assembly rights, and to oppose the Aliens Act. She was the editor and publisher of  Mother Earth  magazine from 1906 until 1917.  This journal promoted a cooperative commonwealth in America, rather than a government, and opposed repression. Emma Goldman became one of the most outspoken and well-known American radicals, lecturing, and writing on anarchism, womens rights, and other political topics. She also wrote and lectured on new drama, drawing out the social messages of Ibsen, Strindberg, Shaw, and others. Emma Goldman served prison and jail terms for such activities as advising the unemployed to take bread if their pleas for food were not answered, for giving information in a lecture on birth control, and for opposing military conscription. In 1908 she was deprived of her citizenship. In 1917, with her  long-time associate  Alexander Berkman, Emma Goldman was convicted of conspiracy against the draft laws, and sentenced to years in prison and fined $10,000. In 1919 Emma Goldman, along with Alexander Berkman and 247 others who had been targeted in the Red Scare after World War I, emigrated to Russia on the Buford. But Emma Goldmans libertarian socialism led to her Disillusionment in Russia, as the title of her 1923 work says it. She lived in Europe, obtained British citizenship by marrying the Welshman James Colton, and traveled through many nations giving lectures. Without  citizenship, Emma Goldman was prohibited, except for a brief stay in 1934, from entering the United States. She spent her final years aiding the anti-Franco forces in Spain through lecturing and fund-raising. Succumbing to a stroke and its effects, she died in Canada in 1940 and was buried in Chicago, near the graves of the Haymarket anarchists.

The History of Organ Donation and Transplantation

Organ donation is the surgical removal of organs or a tissue of one person to be transplanted to another person for the purpose of replacing a failed organ damaged by disease or injury. Organs and tissues that can be transplanted are liver, kidneys, pancreas, heart, lungs, intestines, cornea, middle ear, skin, bone, bone marrow, heart valves, and connective tissues. Everyone regardless of age can consider themselves as potential donors. After one dies, he is evaluated if he is suited for organ donation based on their medical history and their age as determined by the Organ Procurement Agency (Cleveland Clinic). The main advantage of this medical surgery is that it is conceived for the purpose of saving people’s lives – one organ can†¦show more content†¦The transplantation that took 5 hours and a half hours which made Richard survive for eight years, was the stepping stone for thousands of transplants done after this operation which ultimately led surgeons to perform transplants of other organs. Dr. Joseph Murray who performed the transplantation on the twins and the lead surgeon at the Brigham Hospital in Boston won a Nobel Prize in 1990 (Shelly, 2010). 2.2.3. Later Attempts and Discoveries Blood and Skin. In 18th century, a German surgeon, Karl Thiersh, introduced an epidermal skin grafting called â€Å"Thiersh Graft† which is a specific grafting method using the epidermis and a portion of the dermis. This advancement in skin transplantation is still used today. Aside from skin grafts, blood transplantation was also gaining interest during this time. English gynaecologist, James Blundell was the first doctor to ever invent a machine dedicated to blood transplantation. In 1900, Paul Ehrlich discovered the potential of erythrocytes in creating haemolytic antibodies in blood transfusions and his discovery was thoroughly explained by Karl Landsteiner (Shelly, 2010). By 20th century, Alexis Carrel experimented with dogs and discovered a way to effectively anastomose blood vessels though this doesn’t bypass graft rejections and transfusions caused by antibodies. Total body irradiation was found to supress the immune system to maintain the transplantation however it also produced bone marrow aplasia which resultedShow MoreRelatedAn Analysis Of Organ Donation Essay1509 Words   |  7 PagesAn Analysis of Organ Donation Flashback to when you were sixteen years old. Young, naive, and about to be ruling the streets with your very own Driver’s License. You passed your written and physical driving exams, but before you are able to get your â€Å"right of passage†, you must indicate whether or not you are willing to donate your organs in the case of your death. But how does one know which box to check? It is your responsibility to educate yourself in the matter because ignorance is not alwaysRead MoreOrgan Donation and Transplantation 982 Words   |  4 PagesOrgan and tissue donation is life-saving and life transforming medical process wherein organs and tissues were removed from a donor and transplant them to a recipient who is very ill from organ failure. It is said that one organ can save up to 10 people and may improve the lives of thousands more (Australian Red Cross Blood Service, 2011). Most of the donated organs and tissues came from people who already died but in some cases, a l iving person can donate organs such as kidneys, heart, liver, pancreasRead MoreBioprinting Human Organs: The Past, Present, And Future.1488 Words   |  6 Pages Bioprinting Human Organs: The Past, Present, and Future Written by: Emmitt Mikkelson, Alexander Turnbull and John Wesley Table of Contents: I. Introduction II. History of Organ Transplants III. Development of Bioprinting IV. Current Bioprinting Processes V. Bioprinting Human Organs for Transplantation VI. Insurance Coverage for Organ Transplants VII. Ethical Considerations and Alternative Ideas VIII. The Future of Bioprinting IX. Conclusion Read MoreInformative Speech Organ Donation and Transplant Essay1042 Words   |  5 Pageshours someone dies waiting for an organ transplant. 18 people will die each day waiting for an organ. One organ donor can save up to 8 lives. . THE NEED IS REAL In Jan 2006 I began to lose my eyesight. A year later I became a candidate for cornea tissue transplant. I am a cornea tissue transplant recipient. As a result I felt is necessary to inform you about the history and facts on organ donation and transplantation. C. Audience Adaptation – Organ transplantation represents a unique partnershipRead MoreOrgan Allocation Case Study : Correctional Healthcare1147 Words   |  5 PagesOrgan Allocation Case Study: Correctional Healthcare Healthcare workers and the ethics board make tough decisions that impact the patient’s future, specifically related to organ allocation. Organ transplantation is extremely important in order to save lives, prolong survival, and increase the quality of life (Beyar, 2017). Each year the number of people on the waiting list continues to rise at an alarming rate. According to the U.S. Department of Health and Human Services, organ donation statisticsRead MoreOrgan Donation Essay1194 Words   |  5 PagesOrgan Donation Today in the United States there are thousands of people currently waiting for some type of transplant. If one were to ask a group of people if they have friends or family who have either had a transplant or are waiting for one, one would find that most people know at least one person who has had a transplant or is waiting for one. Transplantation is a great advance in modern medicine. The need for organ donors is much larger than the number of people who sign up to donateRead MoreAn Emergency Room On A Calm Day1424 Words   |  6 Pagesexplain to the parents that the medical team discovered that Karly carried an organ donor card in her wallet. As a registered organ donor, Karly, in the unpropitious possibility of her death, elected to donate all of her surviving organs to those in need. Ultimately, she wanted one last chance to save someone’s life and change his world. Two situations could arise out of this predicament: Karly’s parents could deny organ donati on or they could allow Karly’s sacrifice save someone else’s life. UltimatelyRead MoreImportance Of Organ Donation1094 Words   |  5 Pagesof them die because of the lack of donor organs? Still don’t care? What if among all the persons there is someone whom you know? I suppose now you care. I will introduce you the myths about organ donation, real facts and solutions. Let me introduce you first the definition of organ donation and some details of the history of donation. Organ donation is the process of providing an organ, organs, or partial organ to transplant into one or more people. Organ donors can be deceased or living. AccordingRead MoreOrgan Donation Is The Surgical Removal Of Organs Or A Tissue Transplant?1430 Words   |  6 PagesOrgan donation is the surgical removal of organs or a tissue transplanted to another for the purpose of exchanging a failed organ injured by disease or injury. Organs and tissues that can be transferred are liver, kidneys, pancreas, heart, lungs, intestines, cornea, middle ear, skin, bone, bone marrow, heart valves, and connective tissues. Each person regardless of age can consider themselves a prospective donors. Before one dies, he/she is assessed to see if they might be appropriate for organ donationRead MoreOrgan Donation And The Ethical Implications1441 Words   |  6 Pagesbecoming an organ donor can save lives and improve the quality of life of the recipient. When an individual is in need of an organ transplant, it is typically known that they are in need of an organ that facilitates a restoration of physiological functioning and will often mean the difference between life and death. A vast majority of individuals are apprehensive about what happens to their bodies after their death. In this paper we will go through the origins and history of organ donation, the process

Medicare Fraud - 647 Words

Medicare fraud is becoming a huge problem in today’s society. Medicare is a health insurance program for personnel paid by taxes the American population contributes to for personnel 65 years or older. When a health care provider, health suppliers, and private health companies deliberately bill Medicare for supplies or services that were not given is considered Medicare Fraud. To include, when a person uses another person’s Medicare card to receive health care for which the person does not qualify for. An individual, company, or a group can commit a Medicare fraud scheme. Medicare Fraud Scheme A physician, office manager for the physician’s medical practice, and five owners of health care agencies were arrested for charges related to the†¦show more content†¦Medicare Fraud Strike Force is a multi-agency team of federal, state and local investigators designed to combat Medicare fraud through the use of Medicare data analysis techniques and an increased focus on communityShow MoreRelatedMedicare Fraud Essay988 Words   |  4 PagesDepartment of Justice work to reduce healthcare fraud and investigate dishonest providers and suppliers. The Health Care Fraud Prevention and Enforcement Action Team recouped almost 3 billion in fraud, this year alone. Also, aggressive strategies exist to eliminate Medicare prescription fraud. Patients abusing or selling painkillers received by visiting several doctors and obtaining multiple prescriptions costs Medicare millions annually. Fraud affects everyone, preventing it requires governmentRead MoreFraud Case Study : Medic are Claims1852 Words   |  8 Pages Fraud Case Study: Medicare Claims Carolann Stanek University of Mary Fraud Case Study: Medicare Claims False claims are a parasite to the American health care system resulting in overall higher health care costs. The Department of Justice reported recovering $1.9 billion dollars in fiscal year 2015 from fraudulent and false claims in health care (Department of Justice, 2015). In 2011, fraud and abuse were estimated to add $98 billion to federal spending for Medicare and Medicaid (FurrowRead MoreMedicare Fraud2905 Words   |  12 PagesMedicare Fraud: The History, Incidence, Costs and Institutional Remedies John H Everett Wayland Baptist Medicare Fraud: The History, Incidence, Costs and Institutional Remedies What is Medicare fraud? (FA, 2011, p. 1) states â€Å"Medicare fraud happens when Medicare is billed for services or supplies you never got. Medicare fraud costs Medicare a lot of money each year.† What is Medicare abuse? (FA, 2011, p. 1) defines this as â€Å"Abuse occurs when doctors or suppliers don’tRead MoreThe Senate Committee On Health And Human Services1493 Words   |  6 PagesAlso, Charge VII: to monitor the implementation of laws addressed by the committee and make recommendations for improvement of any legislation in the sectors of Medicaid long-term care, Dental Board Reforms, Cancer Prevention and Research Institute, fraud, waste and abuse reductions of Medicaid and other health and human services programs in Texas (Governor, 2014). Other duties include Charge II: monitoring the implementation of programs to improve mental health and substance abuse services and makeRead MoreMedicare : Medicare And Medicaid1426 Words   |  6 PagesWeek 2 D B Billing 2 Compare and contrast Medicare and Medicaid; including funding sources, fraud and/or abuse, and eligibility requirements for recipients Both Medicare and Medicaid are administered through a division of Health and Human Services called the Centers for Medicare and Medicaid Services (CMS). CMS’s primary role is to monitor contractors and state agency to ensure the proper administration of Medicare and Medicaid. CMS establishes policies for the provider reimbursements, researchesRead MoreThe Impact Of Data Mining On The Healthcare Industry1451 Words   |  6 PagesData mining is used in various forms by different agencies. Detecting fraud and abuse is one of the benefits of the use of data mining. The healthcare industry is big and one of the biggest payers is CMS. However, detecting fraud and abuse in healthcare claims is crucial because billions of money is being wasted in unnecessary care. Data mining is defined as the process of data selection and studying and building models using massive data stores to disclose previously unidentified patterns in databasesRead MoreHealthcare Claims and Processing1600 Words   |  7 PagesA current LCD for the regional Medicare intermediary (Michigan - Region V) is shown in the example below. This LCD is for Erythropoiesis Stimulating Agents, L25211. The LCD is active and became effective on 12/1/2007 with an date of 11/01/2013 for the 10/22/2013 revision (cms.gov, 2014b). Question 8: Report on the Health Insurance Portability And Accountability Act (HIPAA) and its impact on healthcare claims processing. The Health Insurance Portability and Accountability Act (HIPAA) of 1996Read MoreSocial Security and Medicare Seeing Through Loopholes Essay1549 Words   |  7 Pages People receiving Social Security and Medicare need to prepare for drastic changes. Social Security and Medicare in unity has been around since 1965. President Lyndon B. Johnson decided to help the elderly pay for expensive medical necessities from doctor visits to medicine. President Johnson knew that elderly individuals would have less income and pay more for medical necessities than younger individuals. He made Social Security and Medicare a number one priority during his presidency, unaware yearsRead MoreThe Role Of Governmental Agencies On The Health Care Industry1271 Words   |  6 Pagesservices, perform studies, manage training, and technical assistance. Health Care Fraud Prevention and Enforcement Action Team (HEAT) The Department of Health and Human Services (HHS) and Department of Justice (DOJ) created the Health Care Fraud Prevention and Enforcement Action Team (HEAT). The (HEAT) collect information through the states to help avoid losing, fraud, and exploitation in health care programs such as Medicare and Medicaid. Chase on persons and associations who abuse the system which itRead MoreDelivering Health Care in America1726 Words   |  7 PagesCHAPTER 9 Medicare Enrollment â€Å"Ringing in the New Year with New Health Care Benefits† Some of the most important provisions of the Affordable Care Act will take effect in 2011. Starting this year, the vast majority of people with Medicare will see several new benefits, including free annual wellness check-ups, recommended preventive services without cost-sharing and discounts on prescription drugs in the Medicare Part D â€Å"donut hole† (Sebelius, Health Care, 2011). In addition, new rules will hold

Health Essay Research Paper Health free essay sample

Health Essay, Research Paper Health Often we take our wellness, or the absence of unwellness, disease, or hurt for granted until we become ill. It is so that we recognize the worth of being without complaints. It is so that we appreciate experiencing strong, robust and healthy. Bing healthy and, being physically and mentally sound, is associated with one # 8217 ; s satisfaction with life. Developmental wellness psychologists # 8211 ; specializers who study the interaction of age, behaviour, and wellness and geriatricians # 8211 ; specializers in the scientific discipline of aging # 8212 ; are patching together the inside informations of diet, exercising, personality and behaviour that make it practical to hit for 80, or even 120. Peoples nearing in-between age can anticipate a fillip of several old ages of excess populating thanks to go oning medical advancement against malignant neoplastic disease, bosom disease and shot. Specialists in the field of aging, developmental wellness psychologists, and geriatrician, concentrate their country of survey on finding wellness position over the class of maturity, and finding the nature and beginning of age-related diseases. They are besides concerned with depicting the effects of wellness on behaviour and depicting the effects of behaviour on wellness. The ends of these specializers are: bar of diseases, saving of wellness, and improved quality of wellness for those enduring from disablement and disease. What does it intend to be healthy? Health is a province of complete physical , mental, and societal wellbeing and non simply the absence of disease, unwellness or frailty. It is of import to separate between disease and wellness. Disease is the forecast of a peculiar upset with a specific cause and characteristic symptoms. On the other manus, unwellness is the being of disease and, the person # 8217 ; s perceptual experience of and response to the disease. Whether in illness or in wellness age and the patterned advance through life play a big portion in our wellness and our developmental position. The function of age in respect to wellness is listed below: Most immature grownups are in good wellness and experience few restrictions or disablements. About 71 % of grownups older than 65 life in a community study their wellness as excellent, good, or really good. # 8211 ; Health and mobility diminution with age particularly after age 80. Disease is more common among older grownups. Most of the diseases of subsequently life have their beginning old ages earlier. # 8211 ; Income is related to perceptual experience of wellness. # 8211 ; The older the person, the more hard it is to retrieve from emphasis. # 8211 ; As an single age, acute conditions lessening in frequence while chronic conditions addition in frequence. 4/5 of grownups over 65 have at least one chronic status. # 8211 ; Older grownups may hold multiple upsets and centripetal shortages that may interact. Treatments may besides interact. In contrast to younger age groups, the aged are likely to endure from physical wellness jobs that are multiple, chronic, and treatable but non curable. Acute unwellness may be superimposed on these conditions. Although there are factors that affect our wellness and the ripening procedure that are non in our control, In a World Health article, K. Warner Schaie ( 1989 ) , research manager of the Andrus Gerontology Center at the University of Southern California, cites three grounds for optimism about future old age: The control of childhood disease, better instruction, and the fittingness revolution. ( p.2 ) The control of childhood disease frequently eliminates jobs that occur subsequently in life as a consequence of these diseases. Alternatively of traveling off, the child assaults suffered by the organic structure from disease, maltreatment and disregard can hold # 8220 ; sleeper # 8221 ; effects. For illustration Chicken syphilis in a kid can take much later in life to the rubing diseases known as herpes zosters. Vaccines and other medical specialty have eliminated many of the childhood disea Ses. Schaie predicts that people who will go old 30 or 40 old ages from now will non hold childhood diseases. Most people who are now old have had them all. Better instruction is besides a ground for drawn-out and fitter lives. Where a grade school instruction was typical for the older coevals, more than half of all Americans now 30 or 40 old ages old have completed at least high school, and surveies show that people with more instruction live longer. They get better occupations, suffer less economical emphasis, and be given to be more engaged with life and more receptive to new thoughts. Finally, the fittingness revolution has changed our wonts with regard to diet and exercising and self-care. An article in Generations, Joyce Carrol Oates ( 1993 ) provinces, # 8221 ; per capita ingestion of baccy has dropped 26 per centum over the past 15 old ages, and the bead is speed uping, assuring a lessening in lung malignant neoplastic disease. # 8221 ; Life-style alterations and improved intervention of high blood pressure have already produced a dramatic national lessening in shots. ( p.13 ) In add-on being cognizant of and accepting ripening is an of import procedure in aging. It is of import to acknowledge that life non merely has a beginning and a center but besides an terminal. It is of import to acknowledge the unrecorded rhythm and all that goes with it. The big life rhythm is divided into three chief parts: Young grownups, middle-aged, and older grownups. There are features of each division of maturity. First, we will look at immature grownups. They are in by and large good wellness. Respiratory nutriments are their primary wellness jobs ( cold, concern, and fatigue ) . Allergies are their most common chronic unwellness. Fatal diseases are rare. The prima causes of deceases are for males, accidents and for females malignant neoplastic disease. Aids is besides a menace for this age group. The following country of concern is the middle-aged grownups. There is a drastic alteration in wellness position from immature grownup to middle maturity. Daily unwellnesss Begin to happen such as: respiratory complaints, and musculoskeletal complaints. Chronic conditions begin emerging that bounds day-to-day activities such as arthritis, high blood pressure, chronic sinusitis, bosom conditions, and hearing damages. Fatal diseases besides begin to look for illustration: bosom disease, high blood pressure, diabetes, arterial sclerosis, emphysema, and malignant neoplastic disease. The prima causes of decease among the middle-age are bosom disease and malignant neoplastic disease. Again there is a drastic alteration in wellness position from middle-age to older grownups. Musculoskeletal jobs are more common and more terrible. Acute jobs become more terrible. Chronic conditions are dominant and more terrible. Death rates rise quickly among the aged. The taking cause of decease is bosom disease, shot, and malignant neoplastic disease. In drumhead, people # 8217 ; s rating of their physical wellness diminution with age. Daily symptoms alteration with age. Acute conditions lessening with age. Chronic conditions addition with age. Nonfatal and life threatening diseases increase steadily with age. Of equal concern is the consequence of gender and race on wellness and ripening. However, unequal attending has been given to the scope of fluctuations in societal, cultural, and wellness features within and between minorities and adult females. The clip has come for more deliberate, purposeful, and thoughtful accounts of the effects of race and gender on wellness. Understanding grownup development requires an apprehension of the relationship between wellness, disease, disablement and aging. Understanding and non being afraid of the aging procedure may decelerate the procedure. Mention Oates, Joyce Carrol. ( 1993, Spring/Summer ) . The ageless ego. Coevalss, ( Vol. 17 Issue 2, p13 ) . Schaie, Warner K. ( 1989, Nov ) . Looking in front. World Health, pp.2-4. Health, Disease, and Disability

Ben and Jerrys Management Style free essay sample

Ben and Jerry’s ice cream has been in business since 1978. It began with Ben Cohen Jerry Greenfield and a combined life savings of $8,000. These two young men had a vision and were able to develop into one of the most successful ice cream businesses in America. Most importantly, they have employee’s that genuinely enjoy working for the company and are motivated to go to work. How does this company do it? This essay will explore how Ben and Jerry’s motivate its employees by having values and a goal-oriented business. There are some defining terms and concepts to have a general understanding of before reviewing the success of Ben and Jerry’s. One of those is understanding what individuals perceive and why they believe in a company or a product. The actual definition of perception is the act of using the senses to become aware of the environment (Bowman). People use perception to create their own beliefs about themselves and their surroundings, and in turn, create their values (Bowman). If managers can understand an employee’s â€Å"mental map†, or perception, they can then truly lead that person and create solutions to problems for a win-win solution (Bowman). The problem with individual perceptions is that they are not always correct and are easily transformed into what we want to believe or value. This is a big challenge for many employers to address with their employees. It is impossible to motivate or get productivity out of someone if they perceive the company negatively. Ben and Jerry’s have been able to combine perceptions, values and beliefs of the company with their employees, and customers to create their positive environment. It starts with the company’s product mission statement that states, â€Å"To make, distribute sell the finest quality all natural ice cream euphoric concoctions with a continued commitment to incorporating wholesome, natural ingredients and promoting business practices that respect the Earth and the Environment†. This statement clearly states the company’s goals for all employees and customers to know. The company wants to make sure that everyone can have pride in knowing they are using the finest quality ingredients and implementing business practices employees can take pride in. The unique thing to notice is that Ben and Jerry’s not only has a product mission statement, but also has an economic and social mission statement. In these two statements, the company shows it is serious about expanding opportunities for development, career growth for their employees, and sustaining a good quality of life. Another concept that can be applied to Ben and Jerry’s success with satisfied employees is the Employee Satisfaction Model. The job facets that are visible in Ben and Jerry’s are clear in their mission statements. Some of these facets include the ability for promotions and increasing pay rates, company policies, and the clear mission of the company to name a few (Scholl). Because these job facets are in place, it creates a positive cognitive and affective response, which in turn, creates job satisfaction (Scholl). The cognitive component, or an individual’s perception regarding the organization, is evident from the reward system they have in place to the satisfaction employee’s have with their role and tasks they accomplish on a day-to-day basis. The affective component creates satisfaction as well. Employees genuinely enjoy social interactions at work, which evoke a positive image of the organization (Scholl). How does all this affect motivation? First, it is important to understand that motivation is the set of processes that moves a person toward a goal (Allen). Goal setting is perhaps the strongest of all forces for personal motivation (Abernathy). There are many organizations that do not succeed because organizational goals are not communicated or created. An important fact to remember is that goals serve as guidelines for action, directing and channeling employee efforts (Organizational Goals). Ben and Jerry’s organizational goals are clear in the product, economic and social mission statements. Some factors that Allen states that affect work motivation include: individual differences, job characteristics, and organizational practices. Individual differences are the personal needs, values, attitudes, interests, and abilities that people bring to their jobs. The individual difference inside Ben and Jerry’s is one source of their success. The organization uses reward inducement system that offers money to employees for their own ideas on how to do things better or how to create a better work environment. This actually has potential to capture what individual employee’s value. Most importantly, most employee go to work for Ben and Jerry’s because they truly believe in what the company is doing and how they are doing it. Ben and Jerry’s are extremely environmentally friendly, which employee’s can take pride in. If you look at the Ben and Jerry’s website, you can see that their focus is on children and families, the environment and sustainable agriculture on family farms. Once again, these things are what their employees value. The organization has a casual dress requirement as well. Their philosophy is that employee’s should be comfortable at work. Comfort equals happy and motivated people, which then increase productivity. The company also brings in massage therapist for their 12-hour shift laborers to help motivate them. It is apparent that Ben and Jerry’s goal is to create a pleasurable work place and reinforce individuals self worth (Scholl). Generally, if employees have a positive affect, they tend to evaluate the organization positively (Scholl). From the beginning in 1978 to the present day Ben and Jerry’s has been a unique organization. It is evident that they have strived to be a company that people really want to work for. Creating this image and making it a success is extremely challenging. What makes this organization so great to work for starts with its product, economic, and social mission statements. There is no doubt, what the company stands for and what it wants to do in the future. This makes it so easy for their employees to understand what the organizations goals are. When a company can create an environment that is casual, comfortable, and thrives to hear from their employees motivation is easy to achieve.