Problems of Population

Pollution is today, probably, the greatest problem that faces mankind. There is a saying in English that it is an ill bird that fouls its own nest. And human beings are, collectively speaking, fouling their nest at the increasing rate. It is an alarming picture. We are polluting the air, which we need to breathe, the earth from which we derive all our food, and the waters from which wÐ µ, of course, derive the water we need to live, to drink, and for other purposes.So all the elements that surround us — there are three elements: earth, air and water — are being increasingly polluted by the activities of man, by industrial activities, for the most part. And pollution, of course, knows no frontiers. One country that pollutes will export its pollution to others. The radioactivity, for example, that was generated by the catastrophe at Chernobyl four years ago, four or five years ago, was carried in clouds across Europe, and some of these radioactive clouds, something which have been polluted in this way, actually produced rains which fell on parts of England and Wales.And we had a problem with the cattle which, of course, are fed on the grass, which had been rained on, by this polluted water, so we had problems in a small way, even in England as a result of the Chernobyl disaster. But Chernobyl is only one instant among many. It's a dramatic and alarming symptom of what is going on in all societies, all advanced industrial societies, not just in one. It's common to all, it's a common problem for all of us and in fact, in a sense, it takes us beyond all our ideologies. We must, in fact, look at this in a totally neutral, totally objective way, not for the blaming of one, one economic system or another.It's something which is, as I say, frowning both ends. It's really a function of advanced economic systems, of advanced, technologically advanced countries. It's basically the result of the industrialization which began in England, of course, in the 18th century and in above all, it's a result of the increasing use of and dependence upon fossil fuels: primarily, of course, coal and oil for both industry and transport. Modern industry, modern civilizations are run almost entirely on oil, mainly upon oil, to a less extent also, of course, on coke coal.But oil, it is which quite literally†¦ oil's the wheels of industry throughout the world. And the desire for oil, the need for oil, the craving for oil, the guzzling of oil is one of the great phenomena of the 20th century. We know, of course, that oil is decreasing, the stocks of it are decreasing inevitably and that someday, since they are finite, they will run out altogether, but that day hasn't, of course, arrived yet and new reserves are found from time to time, to enable us to go on in the same old ways.But the problems are increasing. Some day we shall have to find means of transportation by means of inventing some new kind of transport which isn't dependent on oil, perhaps, upon electri ­city or something, like that, because electricity itself is oil-dependent to some extent. We shall have to become less and less oil dependent, whereas, in fact, in the last hundred years or so, we've become more and more oil-dependent.First, dependence on oil is something, this rapidly dwindling resource is one of our major problems, and we hope all of us, I think, that the scientists will be able to find alternative sources of energy, solar energy, water, water power from the seas, so to say, and it is derived from sea power and the building of dams. This will take the place, we hope, on the oil-fired and coal-fired power stations which we depend on so much today and, of course, nearly all our transport is fuelled by oil or by its derivative — petrol. We shall have to do something about that.There'll be required a technological revolution. We hope we'll acquire one, at any rate, which will bring about a new way of life which is less pollutant, less polluting , less dangerous to our environment. Let us look a little bit at the picture today. We are all aware of this; in order to satisfy our almost boundless need for oil today we send huge tankers to trans ­port it from one country, where it is to be found, where it is drawn up from the ground, to many countries, of course, that have no oil of their own at all, they have to import it.And then, of course, the issued tankers sometimes sink and sometimes have collisions, and vast spillages occur, causing oil slicks which are sometimes miles and miles long. As a result of these oil slicks, which gradually come towards the coast, we have a poisoning of fish life and sea birds, and this makes the beaches unfit for either the local residents or for holiday-makers to use. And it's a dirty sight, a tragic sight.The sea birds, for example, are covered in thick black oil, and they have no chance of survival unless people can get to them early and clean their bodies, clean their wings of this oil. So the great cost to natural life – we've been endangering the other creatures of Earth in our greed for more and more oil. And the cost, the economic cost of cleaning up these oil slicks is enormous and, of course, fines that are imposed upon owners of tankers from which oil is spilt, but the fines themselves are derisory, they are not nearly heavy enough.Many tanker captains deliberately flush out the holes of their vessels in foreign ports leaving the foul mess for other people to clean up, and the fines they pay, if they are caught, which is not always the case, are literally peanuts. And then there are carbon emissions from our factories and from the traffic. The emissions from the exhausts of cars and other vehicles on the roads are largely responsible for the atmospheric pollution from which we are suffering these days.These emissions cause acid rain which, when it falls upon the ground, is harmful to plant life, and to some extent, to animal life too. We are told by t hose who are supposed to know about these things that the atmospheric temperature throughout the world, the average temperature is rising very slightly, and the result of this is so-called global warming, which is only by 1 or 2 degrees, perhaps not even as much as 2 degrees will be the so-called greenhouse effect.This can be described in the following way: the ice at the Pole caps, the North Pole and the South Pole, will begin to melt as a result of this global warming and causing the level of the oceans to rise, and this, in turn, will flood low-lying coastal areas in various parts of the world, thereby, of course, not only causing disaster to people who live there, but also depriving man of some of the soil — the earth which he needs to grow his food on.And as further results will be, this happens, that the climate in many parts of the globe will change, maybe, of course, some parts will become warmer and may be better from that point of view, but others undoubtedly will s uffer. We can't know in total whether this will be a good or a bad thing, but we shouldn't just assume blindly that all will be well. We must try and plan and look on the gloomy side in a sense. We must assume the worst; we must take the worst case analysis, as it's called in England.It’s really quite a moot point today whether mankind will perish by flood or by frying, whether it'll be flooded out of existence or fried out of existence. For many decades after the Second World War, once the atom bomb had been invented, people were afraid above all of a nuclear war. Nu ­clear war was what it was feared, would wipe out mankind because, if there were Ð ° wÐ °r and nuclear weapons were used, and rockets with the nuclear warheads — â€Å"nukes† as the Americans call them – were used, then there's little hope for mankind, there'll be no victor in such a war.Everybody would be vanquished and, of course, the pollution would occur as a result, as well as the d evastation would probably, or could easily wipe out mankind, or if not wipe out mankind, then make lives, all life that was left unbearable, as to be almost not worth thinking about, not worth contemplating. There is, of course, an ever horrifying doomsday scenario, from which it's to be really gloomy about this sort of thing. This is the possibility of the Sun baking us all, frying us all.I haven't spoken about the possibility of flood from melting of the ice caps at the two Poles, but there's this other possibility which is opening up now as a result of man's activity in space and on earth, of course. Some of the hydrocarbons that we release into the atmosphere, es ­pecially those from the aerosol cans together, it is believed, with the rockets that we launch into space cause holes to appear, large holes to appear in the ozone layer above the Poles, above the Pole caps.And it is this layer, and this alone, incidentally, this ozone layer which prevents us all and which protects u s, in fact, from the harmful effects of the ultraviolet rays given off by the sun. Were there no ozone layer, Ð ¾f course, life would not be life as we know it. It would not be sustainable, and for our type of life it would be too hopeless, the rays, ultraviolet rays would harm us. We know this is so, when we go sunbathing we give caution not to expose our bodies too much to the effect of the sunrays. As if all this were not enough, we pollute our water in various other ways.We pollute it not only, that is to say, with oil slicks, spillages of one source or another. In many places, and certainly this is true in England, and probably true in other countries, I'm sure, it's true in many continental countries on the Mediterranean coast, for example, in many places untreated sewage is discharged directly into the sea, instead of being treated and used on the land, as would seem to be possible, of course. Side by side with this, we use huge quantities of chemical ferti ­lizers in our agriculture.Some of these fertilizers seep down into the underground water shelves and aquifers and finds its way into the river system together with chemicals discharged by factories, which are often sited near rivers and lakes, of course, straight into the river or the sea. We are polluting our waters with chemicals, with oil and with untreated sewage. And, of course, the oceans are huge, of course, they cover more of the earth surface than land, as we all know, but they can't endlessly prove a kind of flushing system, purification system for modern civilizations.The harmful chemicals which are deposited in one way or another into our rivers, our seas, our lakes and our oceans, get into parti ­cularly harmful metals, such as lead and cadmium; get into the food chain and the water supply. And, of course, we are absolutely depen ­dent on food, on safe food and water which is fit to drink. However, we mustn't be too gloomy about this, the mankind is capable not only of dirtying, of messing up the planet, it is also capable of cleaning it up, if he applies himself rationally to this problem.Many rivers have in recent years been cleaned up, that is to say, made a lot cleaner and the matter is clean as long as they would wish and they certainly have been improved immensely, the Thames in London is an example of this. Fish which have not been seen in the river Thames for decades are now reappearing there now. Of course, many of them are put in deliberately as the river is restocked. But the fish which would not have stood a chance of surviving in the Thames a few years ago are now able to survive in that environment, which is very encouraging, of course.There is another form of pollution I'd like to speak of briefly. It is not quite so harmful to the human race as a whole, but it is certainly deleterious, has a deleterious effect on the environment. It is one another environmental problem we have to deal with. This is the, what I call, noise pollution. This can come from various sources, for example, aircraft with the loud engines. These engines can be made quieter, and there is much effort going into making them less noisy all the time, but they are still enormously disruptive in their effects.Anybody who lives near an airport knows what a terrible noise these planes can make after they take off or when they are land ­ing. Some people can adapt to this, but not everybody can. It is not true that if you live near a noisy place, you'll adapt to it. I know from personal experience that a noisy traffic can have a continuously bad effect on one's health, because it disturbs one's sleep, keeps one awake and keeps one in a nervous state. Some people adapt to it, but many people, quite a large minority of people never adapt to noisy conditions.And the traffic noise and the bubble aircraft noise are the worst offenders in this respect. It's not only, of course, aircraft or road vehicles which cause a noise, but other things, such as the well-kn own ghetto blasters, as they call these, they are very loud hi-fi systems, or loudspeaker systems that they have in pubs and other places of entertainment. These are played at full blast, hence the name â€Å"blasters†, and they are very harmful to people's health. In the long run they can affect hearing.There is no doubt about it, scientists have shown that young people who are habitually exposed to very loud noises, to this deafening loud music, will in a course of years suffer an impairment in their hearing and can, in some cases, become deaf, which is a heavy price to pay for listening to loud rock music or something else of the kind. They can enjoy just as well at a lower volume surely, but the fashion today is to play these things as loud as possible, without regard for those around who, perhaps, don't want to hear these things.It's not uncommon for people in our country to play their transistors in their cars and then to open the car window and the sound comes out, and everybody hears whether they want to or not. I usually don't want to hear it. This is an offensive thing to do to one's fellow creatures to impose a sudden noise on them if, so to say, they don't want to hear. It's antisocial, to say the very least of it. It annoys me in ­tensely.And some shops where they sell hi-fi equipment will have that equipment, will have some music playing usually, usually junk music, I call it, playing very loud and such you can hear from the street. Why should I be, why should I be punished, as if my ears be afflicted with the sounds of music I don't wish to hear, which I don't regard in any way artistic or aesthetic, just because some other people are thoughtless, too thoughtless to turn the sound down. So we have created a problem, perhaps, of a nuisance. The same thing happens, of course, to transistors.Young people sometimes take a transistor with them, say, to a beach in the country and even in the town, and play it loud. And so all people are force d to listen to, are forced to hear it. This shouldn't be possible in fact, on our railways, I'm not sure about the buses, and other forms of public transport, certainly on our railways, it's illegal, it's technically illegal to play a transistor. It's perfectly acceptable to play one of these players, one of these personalized hi-fis which you wear, you just have earphones, walkmans, but not to play a transistor.We've created smokeless zones in our cities to rid ourselves of some of the pollution. Now, of course, we are not allowed to burn coal on our fires in most areas in England. In the country you still can have this kind of smoke, but in the towns one is not allowed to have an open fire which burns coal. One has to have a special smokeless fuel, and this, of course, has reduced the pollution and the fog and the smoke in the towns enormously. And even in my life-time I've noticed a huge difference in this respect.When I was a young man, it was common in November, particularly at the end of the year, to expe ­rience terrible fogs — â€Å"pea-soupers† we used to call them. And that was a mixture, of course, of industrial smoke, of smoke from all the chimneys in the houses and fog, and it really did look green, and the description of â€Å"pea-soupers† is very appropriate. It did look greenish, a horrible colour. It choked you, of course, and you got black deposits in your nose and so on. It was foul. People used to walk around with handkerchiefs and scarves wrapped around their noses and their eyes.They got into your eyes, as well, and it's really quite dreadful. We've managed to overcome that problem by the introduction of smokeless zones, I think some time in the late 1950s or early 60s. We could and should do something about the noise that we're creating, that is creat ­ing a nuisance for us. In a similar way, we could, in fact, impose re ­strictions on people. We have started in a small way but we need to go a lot further in th is respect. So all these problems, these problems of pollution are man-made problems.It is we, the human race who have caused these problems by failure to appreciate the extent of the damage we're doing to our environment by mismanagement, even when we do know the effects, we don't always take measures to secure a clean and safe environment. But we know now more and more of the matter, how very delicately balanced the ecology of our world is and that if we go on as we are doing, we might, well, disrupt it and alter it irreversibly, and to the detriment of all, all the human race. We've got to be extremely careful how we're moving in this respect.There is, as we say, only one spaceship — earth; we have only one planet, we are all in the same spaceship, all in the same boat, as we say. And we could perish unless we alter our attitudes, alter our industrial methods and ruthlessly punish those who are responsible for polluting parts of our Earth. And if we don't, I can quite trut hfully say that the sands of time are running out for mankind. *Sort some of the underlined vocabulary under the following headings (to have at least 5 items for each): a.words and phrases that show the attitude of people to environmental issues; b. the hottest environmental problems we are facing; c. the dramatic consequences of man’s activities; d. reasons for / causes of /sources of pollution and other ecological problems; e. ways of dealing with these problems; words and phrases that show the attitude of people to environmental issues; the hottest environmental problems we are facing; the dramatic consequences of man’s activities; reasons for / causes of /sources of pollution and other ecological problems; ways of dealing with these problems;

Breakfast Eating Habit: A Statisical Research Project Essay

Every day millions of people around the globe turn off their alarm clocks and start their day. Some will jump out of bed and hit the door running full tilt because they are already 10 minutes late. Others are up hours before they are required to be anywhere, soak in the sunrise and settle into a predefined routine, of which breakfast may or may not be a part of. But who eats breakfast? Are there any significant factors that predispose certain people to eat breakfast or not eat breakfast? One approach to determining dependency of environmental variables and eating habits is to class individuals according to one of many possible variables. The topic for discussion in this paper is breakfast eating habits among women and the variability of children, specifically, whether women with pre teen children living at home eat breakfast more frequently than women without pre teen children. In theory, a person might assume that that women with pre teen children will tend to eat breakfast more so than women without pre teen children due in part to the responsibility they have to wake up early and prepare their children for the days events including school and sports. Different factors of each home can vary results in either direction but our research attempts to determine the dependency that pre teen may have on their mothers own breakfast eating habits. There is a tremendous amount of research done on the benefits of eating a health breakfast not only for adults, but children as well. Researchers have found that when healthy, lean women skipped their morning meal, it raised their cholesterol levels and diminished their bodies’ sensitivity to insulin, a hormone that helps regulate blood sugar levels. In addition, the women tended to eat more calories on breakfast-free days, suggesting that over time, skipping breakfast could spur weight gain. Past studies have also suggested that women who eat breakfast, particularly whole-grain cereals, have lower cholesterol and insulin levels. Along with past evidence, new findings suggest that making time for breakfast is likely to have long-term health benefits. Whether one of those benefits results in a smaller waistline remains unclear. Some research has found a direct correlation between eating breakfast, particularly whole-grain foods, and lower body weight. Conversely, other studies have found no such relationship exists. What is clear is that researchers have found evidence that Mom was right: breakfast may really be the most important meal of the day, especially for growing children. It is essential for children to regularly consume a balanced breakfast. Data supports that the more often adolescents eat breakfast, the less likely they tend to be overweight. Additional studies have shown children are more focused in school and do better academically than those who skip breakfast. Children show the tendency to have more energy throughout the day and are more likely to participate in school activities and sports. Psychologists have determined children mostly observe their parents and follow their actions. A healthy habit of eating breakfast would tend to be a learned trait by following generations and influence the children to continue eating breakfast into adulthood. Experts also surmise that if breakfast is regularly consumed as a family meal, it may reduce the likelihood of drug use as children mature into adults. The feeling of family  closeness and nurturing support is a significant factor in reduced propensity of crime related activities with children. In order to help determine whether or not any dependency exists between breakfast eating habits and having pre teen children, our team created a survey that asked very basic questions relating to age, pregnancy, children, and how many times those women ate a breakfast that consisted of more than water or coffee. Our research chose to exclude women that where pregnant with their first child in order to rule out any difference in opinion of whether or not those women had a child at all. Our research also specifically excluded women under the age of 18 due to a litany of factors that we will not discuss in this paper. The breakfast quantities were lumped into three groups: 0-2 times per week, 3-5 times per week, and 6-7 times per week. We then randomly distributed our survey via the internet in order to capture a diverse group of woman and collated the responses. Since our research focused on determining dependency between pre teen children and eating breakfast, our null and alternative statements were simple: Null Statement – There is no difference in breakfast eating habits of women in relation to having pre teen children or having no pre teen children living in the house. Alternative Statement – Women with pre teen children living in the house eat breakfast more frequently then women who do not have pre teen children living in the house. Failing to reject the null statement would be an indication that no dependency lies between having children and eating breakfast. Conversely and what we expected to see was that a dependency did exist between the two. Our survey yielded 39 respondents, of which the category breakdown is as follows: Breakfast 0-2 Breakfast 3-5 Breakfast 6-7 With Preteen Children 1 7 6 Without Preteen Children 4 4 17 While the data may indicate a correlation between having children and breakfast eating habits exist, performing a Chi Squared Test for Independence revealed that we are 95% confident that no such relationship exists as shown by the graph on the following page: Our research dictates that we need to fail to reject the null hypothesis thus determining that with 95% certainty, no relationship exists. In conclusion, while having pre teen children in the home has no bearing on whether or not women age 18 and higher eat breakfast more often, there may be other factors that would lend itself to a dependency. Actively working, reporting time, time of year, and perhaps a survey on men might be other explored variable in an effort to determine any dependency relating to breakfast eating habits. It was interesting to note that respondents that did not have children ate breakfast in the 6-7 times a week range disproportionately higher than the other two categories. Further research is needed to ferret out any identifiable variable that would support this evidence. It is important to note for further discussion that the test statistic calculated high enough to the critical value that based on a larger sample size, a possibility may still exist that a dependent relationship is real, but not within our sample. Eating breakfast has been determined beneficial for adults and children alike, if research could identify correlations, education could be tailored to specific at risk groups and encourage those within the preferred band of dependency.

The Broken Column and Olympia Essay Example | Topics and Well Written Essays - 750 words

The Broken Column and Olympia - Essay Example The essay "The Broken Column and Olympia" focuses on the comparison of two paintings, "The Broken Column" and "Olympia". Common artistic practice and subjects within cultures suggests Ancient Egyptian art was created as a means of commemorating important people and the ancient Greeks made art to help them worship their gods and goddesses and to preserve their cultural myths. The Romans seem to have adopted elements of the Greek style and fused it with elements of the Egyptian style to develop an artistic approach intended to inspire and celebrate their cultural achievements. After the fall of Rome, the art of the Middle Ages became dominated by themes of Christian religious myths in order to reject the Paganism of the fallen Empire, incorporating religious, political and educational purpose. From here, the art of the Renaissance reflects the re-discovery of more ancient art forms fused to new scientific advancements giving it a historical and experimental purpose. As this skip throug h art history demonstrates, it is easily understood that art can be used for several purposes and it is often inspired or developed in some way so as to build on the past while reflecting the attitudes of the present. As an aspiring animator, the art that appeals most to me is that which takes risks and adopts a more modern or simplified representation of subject such as Edward Manet's groundbreaking "Olympia" entering into the modern period or Frida Kahlo's "The Broken Column" as an expression of surrealism. ... More than that, she is clearly successful in that she has a servant and lives within a very rich environment. Her servant is so dark that she almost becomes lost in the background color of the painting even though she stands at the edge of the far side of the bed. Because of the way he created the painting, Manet leaves it widely open to interpretation. â€Å"The public nakedness of a beautiful woman sometimes becomes a question of politics †¦ which actions are permitted under which unspoken and frequently changing rules† (Friedrich 1). Her appearance within the rigidly proper Victorian culture was shocking because she was seen as a very successful, beautiful and independent woman as compared to the mostly powerless women of the age. Therefore, her representation within polite society was a direct challenge to the rigid ideals of propriety and purity because the painting suggested that her behavior and character were accepted and even condoned within her society. In †Å"The Broken Column† (1944), Kahlo uses dreamlike images to illustrate her personal pain and suffering and line to hold everything together while also demonstrating an independent woman. The broken column of the title refers to the internal structure of Frida’s spinal column. This spinal column is seen through the torn skin and loosely twined bandages that are the only things holding her together. The broken column symbolizes her lack of a support structure and represents her sense of imbalance at the same time that it indicates one of her major sources of pain. Her skin is pricked all over with tiny pins, demonstrating that she is never free of the smaller pains of daily life. The pins continue down the right side of her body, which is the side that had been crippled by both

Week 5 individual Essay Example | Topics and Well Written Essays - 250 words

Week 5 individual - Essay Example Quality management International, Inc is a full service consulting, management, and professional service firm that aims at helping organizations successfully attain their objectives. As registered auditors, certified trainers, and accomplished advisors, the company helps organizations achieve health, safety, quality, integrity, sustainability, and security (Quality Management International Inc., 2014). This company helps leaders and their teams in organizations to apply process-based management systems in order to add value and to also prevent loss. The Texas Department of State Health Services is a local institution that has a quality management branch which aims at assuring that contractors who are funded by the Texas Department of State Health Services (DSHS) comply with the requirements and standards of the department (Texas Department of State Health Services, 2012). This branch also assists contractors to identify avenues and opportunities that will help improve services. The participation of this branch helps in maintaining high quality in services while ensuring that services are delivered in a cost effective manner. The participation of these organizations is crucial in the quality performance of organizations because the trainings offered by these organizations results in continual capacity development and quality improvement. In the long term, these organizations help their clients to

My Teaching Philosophy Essay Example | Topics and Well Written Essays - 500 words

My Teaching Philosophy - Essay Example As such, there are a number of things needed in order to facilitate effective teaching and learning. Among the areas of concern are flexibility, knowledge ability and the global orientation. To begin with, flexibility is of critical importance in order to be successful in teaching and learning. Flexibility is the ability of changing from an ordinary course to a more effective course in order to meet the demands of a system. Flexibility is important since it allows the trainer or the teacher to deal with a number of problems that might deter the learners from achieving the best if things were to be done the way they have been done before. This is achieved through cognitive flexibility. (Chieu, 2007, p. 33). Students also get the most out of a flexible learning system since issues that are a hindrance to their better understanding are dealt with when the teacher is flexible. Secondly, flexibility allows for learning of new skills and strategic deviation from the old ways of doing things. This eventually leads to a well skilled team of learners that are able to effectively compete within the educational system, since cognitive issues are individually based. (Palincsar, 199 8, p. 346). Thirdly, flexibility acts as a motivation to learners since it facilitates the introduction of new things that act as novel stimuli. (Casey & Wilson, 2012, p. 82). For instance, new techniques in dealing with problems Mathematical problems can be adopted. Being knowledgeable is also important in effective teaching. The teacher ought to be knowledgeable about the subject he or she is teaching in order to effectively impart the same skills on the learner. Being knowledgeable about what one is doing is of great importance in a number of ways. Firstly, being knowledgeable creates a leeway of better teaching methodologies. (Metzler & Woessmann, 2010, p. 2). A Knowledgeable teacher will know

Networking Assignment Reflection Paper Essay Example | Topics and Well Written Essays - 500 words

Networking Assignment Reflection Paper - Essay Example e is really passionate about market research, has a strong sense of responsibility, and very much knowledgeable about the situation of the different industries in the market. I believe that her professionalism enables her to generate interviewees from industry players and trade associations. Her innate passion for market research coupled with her strong sense of responsibility allows her to finish each project to the best of her ability. These values also encourage her to work amidst the difficulties in data gathering. Her interest in the market and knowledge on how it operates makes her as a credible source and enables her to giver accurate reports. Written skill is a very important requirement for her job because it enables her to communicate her knowledge about the market. The ability to make rational forecast on the market will perform is also important as it is a very crucial part of the industry reports. The capability of gathering data and analyzing them are keys to having accurate results. Compared to the interviewee, I am more comfortable in verbal communication than writing out my ideas. Being inexperienced, I am not adept in making forecasts about industries. However, I am also confident of my skills in gathering data and analyzing them. I love to do researches even though I know that I still need to learn more about judging the quality of the data and how they can be used in order to come up with rational results. I have always been interested in market research yet I am not really motivated to pursue this profession when given a chance. What I really want when I choose this interviewee is how various think-tanks like AC Nielsen come up with the market reports which they present to business organizations. Also, interviewing a market research analyst enabled me to understand her work and the important characteristics needed to excel in her career. The primary challenge when approaching an unknown professional is the fear of being rejected. Noting

What is good marketing Essay Example | Topics and Well Written Essays - 500 words

What is good marketing - Essay Example disturbing situation leads to a viable business opportunity, how we can follow up without appearing exploitative?’ (Lears, 1995, p.78) It is only through a satisfactory and conscientious introspection that the often ethically dodgy marketing industry can make amends. According to a leading British management scholar, a good marketing approach would translate to the company â€Å"pursuing the new opportunity carefully and raising awareness of the issue without tying it directly to a sensitive incident. They need to set the stage by building awareness of their overall positioning among a horizontal audience†. (Krebsbach, 2006, p.30) Unless such ethical considerations are catered for, marketers will not be able to achieve good marketing standards. Ethical marketing is a phrase much bandied about, and at one level, it appears a genuine concept. However, skeptics suggest that â€Å"brands seizing on the fad for an ethical bent are merely displaying a selfish reaction to consumer pressure, which, while dressed up in the guise of saving the earth, is simply intended to keep profits flowing† (Campbell, 1999, p.106). But in spite of all the criticism one can attach to the marketing industry as a whole, some corners of the industry is trying in earnest to move towards acceptable marketing standards. However, unless the basic motive of marketers is modified, there won’t be any perceptible change in the prevailing situation. And the term â€Å"good marketing† will continue to remain an oxymoron. What gives hope is the invention of the concept ‘social marketing’. It is defined as â€Å"the application of commercial marketing technologies to the analysis, planning, execution and evaluation of programs designed to influence the voluntary behavior of target audiences in order to improve their personal welfare and that of society†. (Bloom & Novelli, 1981, p.83) Commercial marketing, on the other hand is defined as the understanding, targeting and advertisement of

Civil Rights Movement Around 1960s Essay Example | Topics and Well Written Essays - 500 words

Civil Rights Movement Around 1960s - Essay Example What appears to be beneficial to one segment of society is sentimental to another. It was depicted in the U.S. Supreme Court decision of Brown v. Board of Education of Topeka, Kansas (1954) ending racial segregation in public schools. The doctrine of ‘separate but equal’ was no longer adopted sweeping aside 88 years of sound judicial precedent. No doubt there was resistant expressed by politicians and one expressed his case by using his power of office to uphold the internal affairs of the Georgian state.Moreover, there were other resistance groups such as the Ku Klux Klan in Mississippi which tried to maneuver the cause of white college students volunteering in what was known the Mississippi Freedom Summer. These students traveled to Mississippi to stand the cause of blacks in casting their votes and teaching them their history. It turned out that despite the intensity faced and resistant encountered the cause contributed to the success of the Voting Rights Act of 1965. Other philosophers that have stood the test of time were Mahatma Gandhi who proved non-violence against war as the key to gaining peace and freedom for Indians. His conviction enabled different sects in India to march through cities. Likewise Martin Lurther King did not use violence in his fight for the black people in America. In August 28 of 1963 he led the largest civil rights demonstration in history with nearly 250,000 people in attendance delivering his famous speech of I have a Dream.

Critical literature review Research Paper Example | Topics and Well Written Essays - 1500 words

Critical literature review - Research Paper Example Globalization involves the process of globalizing which implies particularly the growth of a tremendous integrated global economy marked particularly by the free flow of capital, free trade and the tapping of low-cost foreign labor markets. The world has witnessed various advances in the field of communication and enhanced technology that have both made globalization possible. Globalization has become a common phenomenon in the world wherein the most production of goods and services has become international. A key element of the current wave of increased globalization is foreign production chains that presently allow workers to pair easily up across various borders. Although globalization has been seen to have benefits, there are various arguments put forward by lobby societies and protestors who oppose the conditions under which globalization operates. Likewise, the incidence of inequality has been on the rise with many nations not getting a fair share of this global trend. The foll owing literature addresses these concerns and provides more insight into the issue of globalization and inequality. Joyce (2008) in his research addresses the gap that exist between the richest and poorest nations in terms the average GDP per capita levels. The author argues that despite the tremendous increase in globalization, developing nations have continued to drag behind economic wise. In his research, the author provides a review of the evidence on the contributing factors of the disparities in GDP per capita, which has concentrated on the role institutions play in fostering development. These institutions mainly reflect prevailing domestic conditions; however, globalization, as indicated by the Joseph, has an impact on development of these nations. The work adds to the knowledge that globalization has direct effects on economic activity, which impacts the occurrence of poverty,

An evaluation of contemporary leadership and governance challenges Research Paper - 1

An evaluation of contemporary leadership and governance challenges among universities in Africa - Research Paper Example By identifying and evaluating the major challenges that face leaders in African universities, it is possible to provide research based solutions to these problems, which in turn will enhance effective leadership strategies as the path for development in these countries. Objectives of the study Main Objectives To evaluate the challenges that are facing the leaders in African Universities in their administrative role. To investigate poor leadership qualities within the administrative bodies of the university. Specific Objectives To identify, through data collection, the major challenges that leaders in African Universities have to confront in their day to day duties. To identify weaknesses that exist within the university leadership and governance that has contributed to the problem of poor leadership. To analyse the results of the survey in to establish the common leadership problems among the African universities. To recommend possible solutions to the problems that are threatening l eadership and governance in African Universities. ... world countries and the path to rise to international heights has become rough, making these countries remain stagnated in terms of growth and development. This has generated a lot of attention among many researchers to investigate the barriers that have kept the pace of growth in these countries slow and unyielding (Task Force on Higher Education and Society, 2000). A research by Petlane (2009) indicated that one of the major challenges in these countries is the poor leadership and governance in this country that has failed to drive the country to economic success. The findings of this research have triggered significant research to investigate the challenges that University leaders have faced in implementing development goals in the country. A recent research conducted by Kuada (2010) was meant to investigate on the knowledge gaps that exist within African leaders that have undermined the development of third world countries. The findings of this research indicate that the weakness of African leaders emanates from the shortage of development skills and knowledge that exist within them. Other researchers have identified the need shortage of technological knowledge among the leaders which is a necessity in a technologically growing environment. Previous research by Hall and Symes (2005) provided that the only way to enhance development in African countries is by maintaining effective leadership in tertiary institutions and impacting leadership mentorship in upcoming professionals. Although researchers have reached a concession that the reason why African countries have failed to shine in the global scope is because of the many challenges that confront leaders in these countries, researchers have failed to identify the specific challenges that face leaders especially

International management and change Essay Example for Free

International management and change Essay When Pascal made this statement, globalization had yet to take place. People lived in different societies remaining cut-off from each other. Each society had its own perception of truth and reality. Then came along the advancement in technology bringing along inventions such as computer, internet and telecommunications making the world a global village. Perceptions started changing as information flow across the globe happened at the speed of light. People from all across the globe started sharing their experiences with each other thus reducing the differences in various cultures. However, the act of globalization has yet to reach its peak. Cultural differences still exist within different societies and the level of information and technology is still heterogeneous around the globe. I would now highlight some distinguishing features of the management styles of various regions. We observe that the beliefs and values of people vary across various cultures. The Japanese work as a group and organizational system is based on community work. The reward system is based on the level of seniority and is also group based. The organizational structure is cooperation based whereas American organizations often are based on competitive style where individuals are rewarded based on their performance levels. Similarly the management style of French is also very different from that of US. French follow a more creative thinking pattern and do not like to adhere to strict rules and regulations. The European management style has some key points that distinguish it from the American management style. The European management style even differs within the European countries and two countries deserve a special focus France and Germany because, among other reasons, the bureaucrats in those countries have long been regarded as mandarins by the field of public administration (Dogan 1976). European management style can be classified into different clusters based upon Hofstede’s cultural dimensions and Trompenaars and Hampden-Turner’s cultural variables (Goliath, 2004) according to which European management style can be sub-divided into: Anglo-Saxon culture (Ireland, UK, and USA), Nordic culture (Denmark, Norway, Netherlands, Sweden, and Finland), Germanic culture (Austria, Belgium, Germany, and Switzerland), Frankophile culture (France, Greece, Portugal, and Spain) and Italian culture. These cultures also differ greatly from the US and Japanese styles of management thus requiring different management practices. The pay for performance system can work with the European management style as employees in Europe are mostly achievement oriented and perform good work for better rewards. The cultural beliefs and values of Europeans are pretty much different from the people of US and hence the marketing techniques used in Europe should be different from those applied in US. For example, Europeans are very much reluctant to providing personal information as compared to US customers; also the credit card usage in Europe is considerably lower than in US (Heilbrunn, 1998). The introduction of new logo and implementation of matrix structure allowing a flatter organizational structure can work relatively better in the European business environment The company operations in South America demand a completely different approach as Latin American culture significantly differs from US culture in terms of communication process. The US culture emphasizes on completion of task and sentences are interpreted in their direct meaning. However Latin American society focuses on relationships and context of the communication may involve more than just the spoken words (Wederspahn, 2001). So the marketing and sales efforts in South American cannot be standardized with the US practices since the Latin American customers tend to perceive communication messages in a very different way from their US counterparts. The new logo can be introduced, however introduction of a flatter organization with matrix structure may not be feasible in South America as the type of culture prevalent in this society is not conducive to the working of non-hierarchical organizations. If we study the culture of Middle Eastern countries, it is found that the business practices again differ significantly from those of US. The business is mostly chaotic and based upon word of mouth rather than written agreements, the cultural values are mostly Islamic and respect for relationships is held supreme; also the marketing regulations in Middle Eastern countries are very strict as compared to US or rest of the world (Kwintessential Ltd). The middle eastern management style is mostly authoritarian and based upon Douglas McGregor’s Theory X motivational leadership style (Daniel Workman , 2008). Thus employees have little or no motivation to work on their own and will perform only when directed by the superiors. Thus bright managers from US and other Western states often try to avoid working in Middle East and so they must first be provided proper cultural training in authoritative styles thus enabling them to adapt to the management practices of the Middle East. The Australian business is more similar to that of US as compared to the rest of regions as described above. The language and dress code is pretty much similar. However, Australian society is strictly based upon egalitarian principle where nobody like being bossed around. People don’t like to consider others as superiors and there is almost no hierarchical system (Slideshare, 2009). So authoritarian style of management will not be welcomed in the Australian business. The meeting and negotiation styles as well as the general beliefs of the Australian society are pretty close to the American beliefs. However according to a study on behavior of 35,000 managers from Australia and New Zealand, the Australian managers consistently try to avoid responsibilities and do not take initiatives (Gettler, 2002). Thus FES should provide cultural and leadership training to Australian manager before implementing any kind of central strategy. Discrimination issues also prevail in Australia that can make life for foreign managers very difficult. However, recently steps have been taken to strongly implement anti-discrimination laws. The roles of front line managers are also changing in Australia as they are provided more and more freedom and responsibility (James Saville, Mark Higgins, 1994). The sales and marketing strategy in Australia can be aligned with the US strategies relatively easily because of the similarity in both societies. So the dilemma of FES is choosing between centralized and decentralized approach. Centralization can be defined as, the degree to which decision-making authority is kept at top levels of management. while Decentralization can be defined as, the degree to which decision making authority is pushed down to lower levels of the firm (Schilling 2008). Both approaches have their own pros and cons. FES has seen tremendous growth over the years and has been expanding into new regions at a very quick rate. Thus the organization requires a large amount of flexibility in its decision making that can be provided by a decentralized approach. However, due to rapid expansion, the organization requires tighter control of finances and close monitoring of all its operations to reduce costs and maintain quality. These measures can be achieved through a more centralized approach. So what approach suits FES bests can be determined by a cost benefit analysis where each advantage and disadvantage of delegation authority has to be carefully analyzed in light of the company objectives. Thus aligning the decision making process with the vision of the organization. A centralized approach can benefit the organization by significantly reducing the administrative cost related to employee management. Infrastructure handling will also be more efficient and it will be easier for the headquarter to align organizational goals with the functional and departmental goals ( Dezaree Seeds, Alan S. Khade , 2008). Centralized approach will also make reporting procedure simple resulting in standardized organizational policies. Thus in this way it will further help the organizational objective of flattening the organizational structure and in creation of a matrix organizational setup. A decentralized approach on the other hand can benefit the organization by providing flexibility and empowering employees at the divisional and functional level. As explained above, different regions in which FES is operating have different working environments and the condition of energy industry in Europe, Australia, Middle East, US and South East Asian regions are very different from each other. Thus decentralization of HR and marketing policies will allow the organization to cater to the customer needs in the ways most suited to specific regions. It will also allow greater local control and the ability to act quickly to gain local resources ( Dezaree Seeds, Alan S. Khade , 2008). Decentralization will also allow the organization to better deal with the cultural diversity and provide training to the employees as and when the need arises. One thing that business, institutions, governments and key individuals will have to realize is spiders and starfish may look alike, but starfish have a miraculous quality to them. Cut off the leg of a spider, and you have a seven-legged creature on your hands; cut off its head and you have a dead spider. But cut off the arm of a starfish and it will grow a new one. Not only that, but the severed arm can grow an entirely new body. Starfish can achieve this feat because, unlike spiders, they are decentralized; every major organ is replicated across each arm (Beckstrom and Brafman 2008). The type of approach taken by FES depends upon the nature and present condition of the global energy industry. The energy industry around the world is in a continuous state of flux. As more and more energy companies around the world become private and become free from the control of state, the nature of competition is changing (ExxonMobil, 2004). Demand for power is increasing at an exponential rate and the competition from private as well as government owned companies is fierce. In the power generation sector, each country has established different laws and tariff rates for multinational organizations. Thus on a whole it seems wiser to maintain a decentralized organizational setup which is flexible enough to quickly respond to the changing conditions of the industry. FES should maintain its present functional structure by keeping the finance, production and HR departments centralized. However using the same marketing and sales techniques all across the globe has a lot of negative implications. Thus allowing the regional marketing departments to work according to their specific environment is the best possible option. So in order to maintain decentralization in the marketing and sales department, the top management has to establish strong ties with all of its regional departments. A strong organizational culture based upon a clear vision and mission of the company should be communicated to all of its employees. Perhaps FES can create a hybrid model which allows centralization of decision making with respect to financial and important managerial decisions and at the same time, the marketing campaign is allowed flexibility according to the specific needs of the region. A key concern for FES is to harmonize its rapid global growth. Power and gas generation has to be taken in new countries and new plants need to be setup. The international operations have expanded a lot and efforts must be made to consolidate the existing business while growing in new regions of the world. As part of the consolidation effort FES must conduct extensive employee training program instilling in each employee the vision and objective of the organization. The company should consolidate its major decision making power in the Headquarter thereby allowing Headquarter to maintain a close monitory system on all of its subsidiaries. By centralizing its finance and management activities, the company can also approve marketing budgets for different regions from its Headquarter, thus building coherence in the finance and marketing goals. However the marketing and sales strategies for each region should be decentralized thus instilling greater flexibility and freedom among marketing departments to cater to their local markets. The pay for performance system can work smoothly in few of the regions but as explained earlier, some regions such as Middle East and some parts of South East Asian cultures demonstrate a more hierarchical structure and it will be extremely difficult to introduce pay for performance system in these regions. Also the parameters should be clearly defined before introducing the pay for performance system in different regions. Because same results usually do not show same amount of performance in different regions. What kind of performance and what kind of pay should be decided and communicated to all regional offices. The parameters for monitoring performance should be based upon different standards for different regions. Similarly some of the major HR policies should be centralized but mostly HR of each region should be independent in choosing the who, what, when and where about its employees. For example, Middle Eastern region is very warm in summers as compared to the European region and the work habits and optimum level of work during different seasons is different for both these regions. Thus separate working policies should be established for each of the region. The RD department should remain centralized at the American Head Office. As a centralized RD approach is more beneficial for a company like FES which has not expanded business into a variety of categories (AB, 2001). However taking technological inputs from all across the globe may help the RD department in improving its efficiency and effectiveness. Thus technological integration should be built where all global units of FES are directly integrated with its Head Office and the information flow between Head Office and different regions should be quick and efficient. The production department should carry out its role from the center, managing the manufacturing process across the globe from the Head Office. So it can be seen that different aspects of FES operations require different level of centralization and decentralization. Also different regions require different level of authority delegation based upon their internal environment and national cultures. It is relatively easier to introduce standard policies in Europe and Australia as both of these cultures are very similar to that of US. However, standardizing procedures across Middle East and South America is not a feasible option. The regional business units should remain decentralized as there should be some form of flexibility to make the decision making process quicker. The regional heads should be sent directly from the Head Office, however if some regions in Middle East are not being managed effectively by American managers, then leadership services of local managers should be availed after providing them extensive cultural training programs at the Head Office. The regional units should be linked to the Head Office through the fastest technology available so that there is no or little information gap between the center and regional units. Thus in light of the cultural differences prevailing in various regions of the world, the best suited approach should be to take a hybrid approach while showing some restrain in application of a centralized approach and making some variations in the standards for implementation in different regions. The integration of overall business can be achieved by proper use of technology and defining proper hierarchical system for flow of information between Head Office and various regional units.

Types or Remedies Essay Example for Free

Types or Remedies Essay Compensatory A money award equivalent to the actual value of injuries or damages sustained by the aggrieved party. When a contract has been breached, the court orders the party that breached to pay the amount of direct losses done to innocent party. Consequential Special damges that compensate for a loss that is not direct or immediate (for example, lost profits0. The special damages must have been reasonably foreseeable at the time the breach or injury occurred in order for the plaintiff to collect them. The party that breached the contract must reimburse the innocent party for the indirect losses even if they were foreseeable damages. Punitive Money damages that may be awarded to aplaintiff to punish the defendant and deter future similar conduct. A monetary award the court orders on the guilty party to punish them for a breach of contract. Nominal A small monetary award (often one dollar) granted to a plaintiff when no actual damage was suffered or when the plaintiff is unable to tshow such loss with sufficient certainty. When the plaintiff cannot prove damages or no damages were done, the court may provide a small amount of money, â€Å"often one dollar†. Liquidated An amount, stipulated in the contract, that the parties to a contract believe to be a reasonable estimation of the damages that will occur in the event of a beach. For the court to order this remedy, the contract must be in writing. Liquidation takes place of compensatory and consequential remedies. In the contract, it must say â€Å"if contract is breached† breaching party must pay this much to the other party. Rescission A remedy whereby a contract is canceled and the parties are returned to the positions they occupied before the contract was made; may be effected through the mutual consent of the parties, by their conduct, or by court decree. This remedy requires both parties to â€Å"undue† the contract and put each party into the pre-contract position. Restitution An equitable remedy under which a person is restored to his or her original postion prior to loss or injury, or placed in the position he or she would have been in had the breach not occurred. Very similar to a rescission, but both parties must return each other into previous states but instead of canceling out the contract and returning each other into pre-contract state, they would be returned to pre-breached state. Specific performance An equitable remedy requing the breaching oarty to perform as promised under the contract; usally granted only when money damages would be an inadequate remedy and the subject matter of the contract is unique (for example, real property) The court orders breaching party to complete the specific action that was promised in the contract. Reformation A court ordered correction of a written contract so that it reflects the true intentions of the parties. Quantum meruit Literally, â€Å"as much as he deserves’-an expression describing the extent of liability on a contract implied in law. An equitable doctrine based on the concept that one who benefits from another’s labor and materials should not be unjustly enriched thereby but should required to pay a reasonable amount for the benefits reecived , even abesnt of a contract.

Vedic Mathematics Multiplication

Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure Vedic Mathematics Multiplication Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure