Comparison of the Early Civilizations Essay Example for Free

Comparison of the Early Civilizations Essay Paleolithic and Neolithic communities differed from each other in many ways, including their use of tools, their hunting of animals and gathering of food. First of all, the Paleolithic people were the most rudimentary in their use of tools. They did not manipulate them in any way. A rock was just used as it was for pounding or cutting a surface. In addition, the Paleolithic people were nomadic; they traveled around hunting and gathering food. When the supply was exhausted in one place, they moved on to the next in small groups of about twenty or thirty people (Spielvogel, 2003). For this reason, the Paleolithic people were not able to form stable communities. Their family and extended family made up their group, which could eventually die out if the conditions grew too harsh. Especially problematic was the Ice Age. Consequently, the Paleolithic people were very adaptable. They learned how to make fire, shelter and crude tools to hunt. They survived in this way until the end of the Ice Age in 8000 B. C. where cave drawings have survived them to reveal their existence to modern man (Spielvogel, 2003). Unfortunately, population issues forced the Paleolithic man into a new way of life. Hunting and gathering societies required huge areas of land, so as the population grew, so did the demand for land. In fact, Krieger (1994) estimates that for every 25 people the community needed at least 250 acres of land to sustain them. At the time, the population was about 10,000,000 people. Clearly a change was needed. However, the Neolithic people were named so because of the changes made in the style of living. The Neolithic people moved from crude tools to more sophisticated ones. They were able to take that rock from the Paleolithic era and manipulate it so that it formed a sharp edge, making that more of a tool for cutting. Later, toward the end of the era, the people discovered ways to melt certain rocks to form metals. In this way the Neolithic society developed more sophisticated knives, axes and hammers (Spielvogel, 2003). The Neolithic people changed from hunting animals to raising animals for food and from gathering food to growing food. They became agricultural and thus were able to stay in the same place for a long time. This permanence allowed the Neolithic people to form communities and villages. As a result, they could develop products of their own and trade with neighboring communities. Art existed in the Neolithic as well, with some artisans even trading jewelry to other communities. As society progressed, the Neolithic age gave way to the Bronze age about 3000 B. C (Spielvogel, 2003). The civilizations that grew in the river valleys of the Nile and of the Tigris and Euphrates bore some resemblance to the earlier periods of man, but they also are characterized by some marked differences. For example, geography played a role in these developments. First of all, the people were less nomadic than the Paleolithic people. They wanted to stay in the fertile areas mentioned above and to take advantage of its soil. However, because the rivers would sometimes flood, they were forced to develop methods of water control and irrigation in order to remain there (Spielvogel, 2003). The first known society was that of the Sumerians in the Tigris and Euphrates river valley, otherwise known as Mesopotamia. They, like the earlier peoples, learned to build shelters, not with wood or stone, which the Paleolithic and Neolithic people used, but with the limited resources they had, namely mud and clay ((Krieger et al, 1994). Using these bricks, they erected buildings and even walls around their cities. They became the leaders of the countryside, forming political city-states. The dominant form of rule here was a theocracy. The Sumerians built temples and believed that gods ruled the cities and divinely chose the kings to do their biddings. Some people were thus noble, and other people were made slaves. However, the economy was similar to that of the Neolithic people in that it was based on agriculture and trade (Spielvogel, 2003). The Sumerians developed a form of writing called cuneiform, which is a form of picture writing. However, few people learned to write (Krieger et al, 1994). Unfortunately, other communities in the area noticed the power of the Sumerians and wanted what they had. The areas was not very well protected by any natural land forms or waterways, so attacking each other was easy and constant (Krieger, et al, 1994). Thus, war was perpetuated in the world of early man. The Mesopotamian area was in a period of empire building and warring states for over thousands of years. The Nile river valley also provided the same geographic necessities as the Tigris and Euphrates did for the Sumerians. Where the Nile divides (called the Delta) is a very similar geographically as the people’s to the north. The flooding of the river provided very fertile lands, thus encouraging agricultural communities. The Nile was the main mode of transportation, so trade was relatively easy for these people. Unlike the people of the Mesopotamian communities, the Egyptians were secluded by deserts, the Nile, and the Red and Mediterranean Seas. Their lifestyle was not marred or changed by continual warfare. Like the people of the Mesopotamia, religion was also very important. Again, the rulers were determined by divine prophecy, creating dynasties of ruling families. One of the most memorable and interesting aspects of Egyptian religions is the use of pyramids and grave goods for burial. The architecture of these pyramids was so sound that many of them still stand today (Spielvogel, 2003). Art was also popular among the Egyptians as it was among the Sumerians and even earlier people. From the Egyptians, come many modern standards of living. Craftsmen used wood and metals to create sculptures, furniture and drawings. The Egyptians also developed a system of writing and began the basis of an educational system, especially writing and mathematics. These people are credited with inventing the calendar and the process of embalming the dead (Krieger, et al, 1994) The lives of the early civilizations were undoubtedly difficult. Our modern society owes a debt to these people for beginning the seeds of society that has become now a convenient and fulfilling way of life. References Krieger, et al. (1994). World History: Perspectives on the Past. Massachusetts: Heath. Ohio: McGraw Hill. pp. 16-37 Spielvogel, Ph. D. , Jackson. (2003). World History. Ohio: McGraw Hill. pp. 19-53

Affirmative Action :: essays research papers

Affirmative Action in the words of Louis Fairykhan (humor) Affirmative Action is Necessary for Minorities to be Competitive in the Workplace The audience that I intend to attract are those that are minorities and agree with the idea of affirmative action. The persona that I chose is that of Louis Farrakhan, the leader of The Nation of Islam. I imagine that this is a speech to thousands of African-Americans that agree with Mr. Farrakhan. I appeal to tha emotions of my intended audience because I feel it is a strong tool in writing a good paper. I am notan authority on affirmative action but I feel that I know enough about it to present a good case on the subject. Good day my beautiful brothers and sisters. I'm sure you all know me by now because of my extensive involvement with the African-American movement. I, Louis Farrakhan, your delicious leader in this march towards equal rights and equality, feel that among our many obstacles in "White America" the most important is Affirmative Action. Affirmative Action is definitely a necessity in our long drive towards freedom. It certainly seems that we as a minority have come a long way since the days of slavery. Unfortunately, we are not yet considered the equal by our Vanilla brothers and sisters. It has been a long and treacherous journey, my savory brothers and sisters, but I feel that we can do more as a community if we can work side by side with the "crackers." It is not the white man's or the Jew's problem that we are not treated equally, it is ours. I was reading a well-known African-American newspaper the other day and I came across something that I felt was interesting. Our strong black brother, Ron Brown of the NAACP stated, "We're not ready to be color-blind because racial discrimination continues to afflict our society. Affirmative action continues to be needed not to redress grievances of the past, but the current discrimination that continues to exist"(Khalenberg). Can I hear an Amen? Was that not pinning the tail right on the hide of the milky-white donkey? We definitely have a strong black brother that knows his stuff and does his thing for our lovely, delicious community. Mr. Brown is obviously an intellectual and brilliant man because he has the last name Brown, which is a good reference to our beautiful chocolate colored skin. Affirmative Action :: essays research papers Affirmative Action in the words of Louis Fairykhan (humor) Affirmative Action is Necessary for Minorities to be Competitive in the Workplace The audience that I intend to attract are those that are minorities and agree with the idea of affirmative action. The persona that I chose is that of Louis Farrakhan, the leader of The Nation of Islam. I imagine that this is a speech to thousands of African-Americans that agree with Mr. Farrakhan. I appeal to tha emotions of my intended audience because I feel it is a strong tool in writing a good paper. I am notan authority on affirmative action but I feel that I know enough about it to present a good case on the subject. Good day my beautiful brothers and sisters. I'm sure you all know me by now because of my extensive involvement with the African-American movement. I, Louis Farrakhan, your delicious leader in this march towards equal rights and equality, feel that among our many obstacles in "White America" the most important is Affirmative Action. Affirmative Action is definitely a necessity in our long drive towards freedom. It certainly seems that we as a minority have come a long way since the days of slavery. Unfortunately, we are not yet considered the equal by our Vanilla brothers and sisters. It has been a long and treacherous journey, my savory brothers and sisters, but I feel that we can do more as a community if we can work side by side with the "crackers." It is not the white man's or the Jew's problem that we are not treated equally, it is ours. I was reading a well-known African-American newspaper the other day and I came across something that I felt was interesting. Our strong black brother, Ron Brown of the NAACP stated, "We're not ready to be color-blind because racial discrimination continues to afflict our society. Affirmative action continues to be needed not to redress grievances of the past, but the current discrimination that continues to exist"(Khalenberg). Can I hear an Amen? Was that not pinning the tail right on the hide of the milky-white donkey? We definitely have a strong black brother that knows his stuff and does his thing for our lovely, delicious community. Mr. Brown is obviously an intellectual and brilliant man because he has the last name Brown, which is a good reference to our beautiful chocolate colored skin.

How to raise a child Essay

Amy Chua’s article â€Å"Why Chinese Mothers are Superior† appeared in the Wall Street Journal on January 8, 2011. When this article was published the controversy began. One article in argument to Amy Chua’s was James Bernard Murphy’s article â€Å"In Defense of Being a Kid† which also appeared in the Wall Street Journal on February 9, 2011. Murphy contends by stating Amy Chua’s method of how to raise a child will do nothing but turn children into neurotic, self-absorbed and unhappy adults. James Murphy, author of â€Å"In Defense of Being a Kid† and professor of government at Dartmouth College argues that children should live childhood and enjoy childhood innocence, not be pressured or forced to prepare for their adulthood and the pressure that comes with it. â€Å"Part of the point of childhood is childhood itself. ‘ (Summers 279) Childhood takes up a quarter of one’s life and it would be nice if children enjoyed it. Murphy continues to explain what the unique blessings of childhood are. First, children have a gift of moral innocence, children are unaware of what is to come in their future and the burdens, and therefore they put their trust in us fully. Children are open to new adventures and unaware of time thus cannot be wasted. We as adults forget that most of us produced our best art, asked our deepest philosophical questions, and most readily mastered new gadgets when were children. We as parents need to take a step back from teaching our children and realize how much we can learn from them. Murphy uses emotion when he states â€Å"children are people with distinctive powers and joy. † He realizes what children are capable of if they are given space to imagine and explore ideas of the world that we have forgotten. Murphy thinks like a child and is defending their youth. It is important to know when to give a child space to allow them to become an individual. In defense Murphy argues, â€Å"most of us would like Tom’s childhood followed by Mill’s adulthood. But as parents we are stuck with trying to balance the paradoxical demands of both preparing our children for adulthood and protecting them from it. † The article seems to indicate you can’t have that childhood and young adulthood. I disagree because that is exactly how I grew up. Yes we did have responsibilities on the farm, but when chores were done we did what we wanted to do. We were taught what was right and what was wrong. I believe if you are raised with good morals, respect for yourself and others you can be very successful. I feel the drive to succeed comes from a supportive family and the want to be successful at what you love to do, not what you are forced to do. Murphy uses logic with the comparison of the ancient Greek philosopher Aristotle and Jesus. Two of which did not have the same beliefs of children. I disagree with Aristotle when he said â€Å"no child is happy†, the only time a child is happy is when they have thoughts of the achievements as an adult. When a child is given space it gives them the opportunity to imagine, to think outside of the box and capable for intellectual activity. We have to encourage and embrace their uniqueness and imagination or as Jesus did praise our children. I strongly agree with Murphy’s fourth thought, â€Å"We forget that most of us produced our best art, asked our deepest philosophical questions, and most readily mastered new gadgets when we were mere children. † (Murphy 279) As children we are more carefree and have less sense of our surroundings and what people think of us. We are eager to learn and curious about adulthood but should not be rushed to become one. I believe in realizing the capacity of a child, you need to know their capabilities and their limits. Work Citied Behrens, Laurence, and Leonard J, Rosen. Writing and Reading Across the Curriculum. 12th ed. Boston: Pearson, 2013. Print

Little Ball of Fur The Black Cat Essay - 781 Words

Edgar Allan Poe was born January 19, 1809. Many consider him to be the master of mystery and horror. He only lived until he was forty, and during that time he endured a lot of hardships. For one, every woman he ever cared for ended up dead, usually by the dreaded tuberculosis. That included his wife Virginia. The men in his life who were supposed to be there to guide him, well, they were not there. I think that helped him write a lot of his dark and macabre stories. His most famous work is The Raven, a narrative poem that was first published in January 1845. He also wrote other highly regarded short stories such as, The Murders in the Rue Morgue, The Tell- Tale Heart, and The Black Cat. I will be doing my review on the latter†¦show more content†¦Poe creates a narrator that loses his humanness. The narrator kills his cat and wife due to mere fits of rage. A sane person would not do that. Poe basically has his narrator starts telling the story at his mad point . Poe had his narrator realize his deterioration but also an inability to stop it from happening. The narrators madness was exasperated by an addiction to alcohol. From the story, But my disease grew upon me - for what disease is like alcohol? (Poe 199). I think Mr. Poe is saying that alcohol can push people over the edge. The narrator already had his demons that he had been able to keep in check until he went on a drinking binge. From there on out, things unraveled. It is said that Mr. Poe himself struggled with alcoholism (cite). It is quite possible that he drew up that aspect of the narrator makeup from his own life. And on a minor note, Edgar Allan Poe also uses this story to present the age old no crime goes unpunished bit. He has a narrator that believes that he has gotten away with his crime. Poe presents a person who is so ho-hum after a heinous crime, the murder of his wife, but that does not last very long. Poe uses the cat as an antagonizing figure that foil s his narrators perfect crime. I have to agree with Edgar Allan Poe in regards to the alcoholism. I think we have all seen to some capacity the negative affect alcohol can have on an already strained situation. It really is like throwing a gallon of gasolineShow MoreRelatedA Crazy Cat Lady1322 Words   |  6 Pagesam the epitome of a crazy cat lady. Don’t take me to a humane society, it will end with me attempting to smuggle out a cat, and getting caught because...well, stealing cats is apparently frowned upon in society. Yes, I am unhealthily obsessed; I love my cat more than Iove my best friend, more than my sister, and on some days, I love him more than I love my parents. The sickness started in August of 2015. Well, no, that’s actually a lie. That was when I got my first cat. I’d wanted one since I wasRead MoreEssay On High School1552 Words   |  7 Pagesthe common room. Their Headmistress stood in front of them. 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Its closest pre-domesticated ancestor is believedRead MoreDescriptive Essay - Original Writing1226 Words   |  5 Pagesnight laid my doggie and I relaxing while watching the dramatic series of â€Å"90210†. I began to sink deeper into my comfy bed beginning to doze off, until I heard a creaking sound that seemed to be coming from below my bedroom floor carpet. Mimi’s little coal colored ears perked up in curiosity. I watched her every move, as she hopped down from the bed and just sort of wandered around my bedroom floor. She scoped out the area for a few seconds and seemed satisfied enough to resume her spot at theRead MoreA Short Story Of A Book : Finding Out 2080 Words   |  9 Pages | Prologue: Finding Out | Creekroot fluffed up his already short fur as the chilly wind brushed through it. He was sitting with his tail over his front paws and head tilted slightly backwards to look at the night sky. He enjoyed watching the stars. The stars of his fallen friends and family. Creekroot begged in his mind that they would give some sort of sign of hope for his Clan. . . . He clenched his teeth as the dreadful reality of his Clan’s doom overwhelmed him. The silver, mackerel tabbyRead MoreMr. Snuffles: A Short Story About a Monstrous Creation1195 Words   |  5 PagesMr. Snuffles Large black clouds covered the sky above a lovely Victorian Manor; thunder crashed and shook the sturdy foundation as a young man ran through the Manor’s dark halls and down its many winding stairs, the only light guiding him came from the flash of constant lightning. He ran down and down until he finally reached his destination, a large wine cellar refurbished into a crude laboratory with a large pool to the side, many different wires, and pipes leading in and out adding mixtures andRead MoreEssay about Creative Writing - Belonging1278 Words   |  6 PagesCreative Writing I jump back as a black feline shoots across the path in-front of me. I catch a glimpse of its matted fur; a screeching hiss pierces the night air as it encounters a mammal in the undergrowth. They say a black cat at night is bad luck, but I honestly doubt my luck could get any worse in light of recent events. â€Å"Get out, you don’t belong here anymore†, â€Å"Just looking at you makes me sick. Leave now and get out of our lives†. It’s funny how the ones who are meant to love you the mostRead MorePesky Goat1604 Words   |  7 PagesThe Pesky Goat    Doà ±a Josefina lived on the outskirts of Cià ¡les, a small town in the highlands of Puerto Rico. Her tiny house was as spotless as a porcelain teacup. Although she lived alone, except for her dog, Ladrà ³n, and her cat, Misifà º, Doà ±a Josefina was too busy to ever be lonely.   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   She spent all her waking moments tending her beloved garden. Doà ±a Josefina was known throughout the island for her  mano santa--the blessed hand that turned all plants she touched into breathtaking specimensRead MoreMy Sisters And I Are Completely Different1220 Words   |  5 Pagesmom calls her a Pomapooski because Zoey has three different dog breeds mixed in her: Pomeranian, Poodle, and Shitzu. Zoey was a Mother’s Day gift for my mom. Zoey was such an adorable little puppy! When my parents got her at Sadie’s Flea Market in Dothan, Alabama, my mom sent me a picture of a little ball of fur sticking its head outside of the pocket of my dad’s khakis. When they came home that afternoon with her she was very much-loved on. My sisters and I couldn’t put her down that night becauseRead MorePugs6576 Words   |  27 Pagesdisagreement exists regarding the manner in which pug came to be applied to these endearing, impish, sprite like, solid-as-a-barge, sometimes demonic little monkeys that were great favorites at court if not with courtesans. Many observers believe that pug first was applied to monkeys and, after certain facial resemblances between monkeys and the little dogs with the curly tails had been noted, the word was applied to the dogs, too. (This application was noted as early as 1731 in England.) Persons subscribing

For Adult Learners

Factors Influencing Unemployment - 700 Words

According to economists, employment and unemployment results when the supply and demand for human resources or labour is out of balance. Supply and demand are influenced by a range of forces that are the result of the interaction of economic, structural and policy factors. Economic Factors Economic factors affect both the supply and demand sides of labour. Demand for goods and services stimulate production which, in turn, generates employment. The resulting demand for workers affects the supply side as more workers are attracted to a vibrant labour market. The market never reaches this ideal state of balance due to a number of factors. Business cycles  ¾ Agreement among economists is rare, but they do agree that market-driven†¦show more content†¦The matching of skills in demand with those available is a common and persistent cause of unemployment. Education and training  ¾ Companies continually complain that the literacy levels of the work force do not meet the skill needs of the economy. As Canada shifts to a more knowledge-based economy, the availability of jobs for those without high levels of education will shrink.Show MoreRelatedMacroeconomic Variables Of The Stock Market974 Words   |  4 PagesExecutive Summary The belief of macroeconomic variables influencing the stock market has been a highly debated discussion for the past decades. There has been no clear conclusion whether or not macroeconomic variables impact the stock market or inversely. 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A business idea to help move in the right direction and hopefully start to lower the unemployment rate in New Zealand is to start a market where people who are unemployed and willing to work can sell food and gener ate income for themselves

Cloud Free Essays

string(63) " people respond differently to the two sides of this equation\." One of the most important findings of mathematics education research carried out In Britain over the last twenty years has been that all pupils constantly ‘Invent’ rules to explain the patterns they see around them. (Askew and William 1995) While many of these invented rules are correct, they may only apply in a limited domain. When pupils systematically use incorrect rules, or use correct rules beyond the their proper domain of application, we have a misconception. We will write a custom essay sample on Cloud or any similar topic only for you Order Now For example, many pupils learn early on that a short way to multiply by ten is to ‘add a zero’. But what appends to this rule, and to a child’s understanding, when s/he is required multiply fractions and decimals by ten? Askew and William note that It seems that to teach in a way that avoid pupils creating any misconceptions is not possible, and that we have to accept that pupils will make some generalizations that are not correct and many of these misconceptions will remain hidden unless the teacher makes specific efforts to uncover them. 1995: 13) According to Malcolm Swan Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalization that the pupil has made. It may in fact be a natural stage of development. (2001: 154) Although we can and should steer clear of activities and examples that might encourage them, misconceptions cannot simply be avoided (Swan 2001: 150). Therefore It Is Important to have strate gies for remedying as well as for avoiding misconceptions. This paper examines a range of significant and common mathematical mistakes made by secondary school children. Descriptions of these mistakes are followed by discussions of the nature and origin of the misconceptions that may explain them. Some strategies for avoiding and for remedying these misconceptions are then suggested. The paper ends by relating some general features of the recommended strategies to the educational theories of Jean Pigged and Level Boycotts. Misconceptions 1 and 2: Algebra mm + 6 = mm 11 The type of error highlighted in the heading for this section arose in each of the three classes to which I have taught algebraic topics. There are at least two misconceptions that could explain this kind of error: (1) that letters represent objects rather than numbers, (2) that an answer should not contain an operator symbol such encountered the error of simplifying, say, mm + 6 to mm in two different mathematical intents. These contexts were: (I) expanding brackets containing an unknown and (it) simplifying expressions by collecting like terms. Although pupils in one class were only asked to remove the brackets from expressions such as 3(m + 2), and not to simplify, some pupils attempted such simplification. The result was that many who correctly expanded 3(m + 2) to give mm + 6, went on to incorrectly simplify this to mm. Being aware that such mistakes often originate in the misconception that numbers represent objects (discussed below), I did everything I could to avoid encouraging such a belief when teaching the collection of like terms. I strongly emphasized the fact that letters represent numbers. In keeping with this, I read mm as three-lots-of- m; making it clear that mm involves multiplying 3 by m (or m by 3) and therein implying that the letter represents a number. The group coped reasonably well until they met examples such as mm + m + 6. Many pupils would have responded to this with mm rather than the correct mm + 6. Although several pupils were helped by my repetition of such mantras as â€Å"adding six is different to adding six-lots-of-m,† many pupils persisted in this kind of error. Misconception 1: Letters as Objects After this lesson the normal class teacher and a second observer suggested that perhaps I would have had more success with a different ‘more concrete’ approach. Both recommended that in future I use what others have called the fruit salad’ approach (Thirst et al. 1998). This approach involves reading AAA + b + b + AAA as 3 apples plus 2 bananas plus a banana plus 4 apples, which naturally becomes 7 apples plus 3 bananas or AAA + b, which is correct. Unfortunately this would have helped least where I needed help most, and it would also have encouraged the same mistakes! 2 As Doug French points out, if k is interpreted as kangaroos, when faced with the expression k + k + 4, the students’ â€Å"instinctive thought is likely to be ‘4 what?. And the obvious answer is 4 kangaroos, giving k altogether† (2002: 11). This is not the only problem that can arise from thinking of letters as representing objects. If a and b represent apples and bananas, then what does ABA mean? Indeed, if letters are used in this manner and we are asked to convert the equivalence a week is seven days into a mathematical formula relating days and weeks, we could easily be led to = ad rather than d = owe (where w is the number of weeks, and d the number of days, in a given period). Given the popularity of the fruit salad approach among teachers at the school in question, I would not be surprised if such a misconception was the underlying cause of much of the confusion on the topic of expanding brackets. However, my experience with the collection of like terms led me to search for other possible explanations for such errors. Misconception 2: Operations vs.. Answers My conclusion, confirmed by the literature (Thirst et al. 1998), was that a large part of The presence of the operator symbol, +, makes the ‘answer’ appear unfinished. In short, pupils see such symbols as +, x, and * as invitations to do something, and if something is still to be done, then they ought to do it. If we have to remove the symbol by doing what it tells us to do, in this case adding, it is only natural that mm + 6 should become mm. Readers who find it difficult to understand this tendency may like to consider their own response to the following statement 3+40- 3 140 Most people respond differently to the two sides of this equation. You read "Cloud" in category "Papers" The left-hand-side looks like a question, namely â€Å"What is three divided by forty? The right-hand-side, however, is not a question but simply a fraction: three fortieths. In a sense these are just two different ways of writing the same thing, but this may also be seen as revealing that in both arithmetic and algebra some expressions lead a dual existence as both process and product. M + 6 can be seen as a set of instructions for 13 calculating a numerical value, but also as mathematical object in its own right (French 2003). The resistance to accepting mm + 6 as an answer is easily understood. In ordinary arithmetic it is always possible to remove the operator signs (unless there are infinite n number), and the final answer has not been reac hed until they are all gone.. Strategies and Remedies For reasons that should already be clear, despite its obvious appeal the fruit-salad’ approach seems best avoided. It may also be possible to over use ‘realistic’ contexts in the early stages of teaching algebra, as such contexts often lead us (and textbook authors) to use letters in ways that might invite a letters-as-objects interpretation (m means miles, h means height and the like). In the place of such ‘realistic’ examples, I would suggest using number puzzles, tricks and games. Pupils find these very nagging and algebra can be a very powerful tool for solving and explaining in these contexts. In my more recent teaching of collecting like terms, I began the topic with a think of a number’ trick, requiring a sequence of mental operations. When the results were collected in, nearly everyone got the same answer. Those who did not were quickly helped to realism their mistakes. Algebra was then introduced as a tool that could explain why this happened, why it is that we could all start with different numbers, but all reach the same final answer after performing the operations in question. The topic of collecting like terms fits neatly with such tricks, as the simplifications work Whatever numbers the letters represent’, Just as in the trick everyone reaches the After working through some examples on the board, including some of the problematic form mm + m + 6, I stopped to ask the class what is funny about the answer mm + 6. The class immediately volunteered that it is funny because it contains a plus symbol. After commending this response, I briefly commented that â€Å"this happens sometimes in algebra†, and set the class to work through some questions for themselves. When one student became stuck on a question of this form, all I said was â€Å"It’s another of those ones with a funny answer† and he immediately knew how to continue. 14 Another useful approach with students who have already made such errors is to substitute a particular value (or values) into the two expressions believed to be equivalent (French 2002: 12). This approach has the benefit of bringing about a â€Å"cognitive conflict† (Swan 2001). On the one hand the pupil believes that the two expressions are equivalent, on the other they can see that they give different results when values are substituted into them. In this situation the pupil can see that something has to be wrong, and can even be invited to explain and resolve this conflict for him- or herself. Even if the pupil is unable to resolve this conflict, the awareness of it is likely to make him or her more receptive to resolutions offered by the teacher or by other students. Substitution seems especially powerful in communicating equivalence (and the lack thereof) when each of a set of values is substituted into the expressions in question. Two expressions are equivalent if, and only if, they give the same result for all values that we choose to substitute. Misconceptions 3 and 4: The Equals Sign Equals as Makes / Equals as a Logical Connective The first misconception to be discussed in this section has been present in almost every class I have taught or observed, including high and low attaining groups in all years. One instance occurred in a Y 10 lesson on finding fractions of a quantity. Students were taught a standard method for tackling such problems as â€Å"Find % of 16† The method is to first find h of 16 (by dividing 16 by 4) and then multiply the result by 3. In solving this problem, several pupils in the class would have written the allowing: 16 + 4 = 4 x 3 = 12. While the answer reached is correct, the written statement is not. That statement (unlike the pupil) asserts, among other things, that 16 + 4 = 4 x 3, which is equivalent to asserting that 4 = 12, which is clearly false. The same problem occurs with any two-step calculation. A similar but importantly different error is often seen in algebraic contexts, Doug French points out that â€Å"it is common to see erroneous statements like: xx – 5 = 7 = xx= 12 = x = 4† (2002: 14). Again, this statement asserts both that 12 = x and that x = 4, and therefore entails the leasehold that 12 = 4. Both of these mistakes can be explained in terms of alternative interpretations of the equals sign. 15 The misconception behind the first kind of mistake is well summarized by Candida Morgan, who observes that there is a persistent use of an ‘operator’ concept of the equals sign by students at all levels which suggests that it is fulfilling a material rather than a relational role. (1998: 81) In writing 16 * 4 = 4 x 3 = 12 pupils seem to show that they use the equals sign not to state an equivalence between two expressions but as an instruction to calculate the alee of the previous expression. So, in our example it is as though the two equals signs connect only the emboldened portions in which they are contained 16 * 4 = 4 x 3=12. In each of these cases the term following the equals sign does indeed give the value of the previous expression. Doug French explains the misconception and its likely origin as follows: Students interpret equals as an instruction to do something to determine a result rather than as a symbol that indicates the equivalence of two expressions. This arises in a natural way through the use of equals in numerical calculations. It is also encouraged by the presence of a key labeled with an equals sign on many calculators. (2002: 13-4) Misconception 4: Equals as a Logical Connective The mistake made in the algebraic context is different, and involves the equals sign playing a dual role. In some contexts it functions as we would expect, but in others it plays the role of such words as therefore, so, leads to, or entails. Using Fresh’s example, we can easily restructure the incorrect statement into a well formed argument by replacing every other equals sign with a genuine connective. In this way, xx – 5 = 7 = 12 = x = 4, becomes X-5=7 xx=12 so, Hence, The above quoted passage from Morgan continues: Another common role [for the equals sign] is as a logical connective between statements, for example xx+ 3 = xx- 15 = xx- 18. [This is] likely to be considered to be mathematically incorrect by a secondary school teacher-assessor; a recent handbook for mathematics teachers picks out use of the equals sign as a connective as its single â€Å"example of bad practice† to illustrate writing that does not ‘make sense when read aloud’ (1998: 81) 16 such mistakes is the practice of reading the offending statement aloud. I would suggest that when using this strategy, it might sometimes be valuable to replace the word equals with a synonym such as is-the-same-as. This strategy is intended to help the pupil see the logical implications of their claim. In the case of 16 x 12, it might help the pupil to see that their written statement implies that 16 + 4 = 12, which they know is false. Again, this has the benefit of creating a cognitive conflict, a conflict between their mental methods and the implications of the accompanying written methods. This conflict can then be used to encourage the accepted interpretation and use of the equals sign. It is also important to model the correct use of the equals sign in your board-work. If the teacher uses the sign incorrectly, this can only increase the likelihood of pupils doing the same. Other useful strategies include emphasizing the symmetry of the equals sign, for example by not reversing the final line in the following piece of working out: xx=XX-3 XX+3=XX It may also be useful to limit pupils to one equals sign per line. Misconception 5: The Addition of Fractions Add the tops, add the bottoms In this section I consider an error that I encountered in teaching the addition (and abduction) of fractions to both Y and Y. The error in question is to ‘add’ two or more fractions by simply adding the numerators and adding the denominators as follows. 2 3+2 5 4 4+3 7 With both classes, I built towards teaching the correct method by first considering examples where the fractions were already written on a common denominator, and after a lesson on equivalent fractions, moved on to consider cases such as the 17 above. Despite their success in the first two lessons, many pupils found the third lesson extremely difficult, in most cases because they struggled to find appropriate moon denominators. During the lesson, then, nearly all pupils attempted to follow the correct method, although they found it difficult to use. Later homework and tests My overall Judgments about these classes were very much in line with the conclusions of K. Hart. A very common error in the addition of fractions was to use a rule ‘add tops add bottoms’. This was more prevalent in examples where the two denominators were different. It was also interesting to note that this particular error occurred more when the question was posed in computation form than in [word] problem form. (Hart AAA: 75) Misconception 5: Add the tops, add the bottoms In older children, who have met the multiplication of fractions, this error might be explained as a ‘local generalization’ that involves assuming that we apply any given operation to a pair of fractions simply by applying that operation to the numerators and denominators taken separately. Unfortunately, the pupils in question had not yet met the multiplication of fractions, and so the explanation is false in this case. It may, however, be true in others. Indeed, this might explain one of Hart’s other observations: â€Å"The ability to solve addition and subtraction computations [with reactions] declines as the child gets older† (1981 a: 79). A more likely explanation for pupils adopting this procedure is that they conceive fractions as test-scores, at least when written in their standard notation. Indeed, a test score of 32 out of 50 is usually written in the form 32 150. AS Malcolm swan (2001 : 149) points out, when we add two test scores together we rightly adopt the rule of adding the tops and bottoms; a score of 10 120 on one test and 120 on another does give an overall score of 25/40. As with misconceptions 3 and 4, a useful strategy in this case is to encourage pupils o read the question aloud, and to read it in a way that naturally suggests the correct approach. The addition 18 should not be read as one-over-five plus three-over-five, but rather as one-fifth plus three-fifths. Fifths are objects that most pupils are comfortable adding; one of them plus three of them is obviously four of them in total. A cognitive conflict strategy is also available for this misconception. The vast majority of pupils can tell you that a half plus a half is one (or one whole). But if you get the pupils to write down the question and apply their general strategy, something interesting happens: 1+1 2+2 2. I used this strategy with some success with Y. This example might not be sufficient to shake the misconception, as it involves adding two fractions with the same denominator. In that case we might also try adding a half to three-quarters. Pupils will be able to see that the answer should be more than one, and that their method gives an answer of 5/6, which is less than one. Again, the experience of cognitive conflict will not only lead the pupil to seek a resolution of their own but will make them more receptive the resolutions proposed by others. Misconception 6: Reflection The same, but on the other side of the line In an end of term test, I recall watching one pupil as she attempted to answer the following question in Figure 1 (all figures are at the end of this document). The girl’s initial answer is shown in figure 2. I was pleased to see her rub this answer out, believing that she was about to replace this answer with a correct one. Unfortunately she only rubbed it out to increase the length of the rectangle she had drawn. In fact, she did this several times before she was happy that she had done it correctly. Many there pupils in the class made the same error. Pupils performed much better on the other two reflection questions, in which the mirror-lines were vertically and horizontally orientated. Informs that mistakes of this kind are common: â€Å"Children may have particular difficulties when the mirror-line is not horizontal or vertical but the slope of the object is horizontal/vertical† (KÃ'ËœAchaean 1981: 143). The APP (1980)) 19 studies revealed a similar tendency when the mirror-line is horizontal or vertical but the slope of the object is not. For example, many respondents answered the question n Figure 3 as shown in Figure 4. The examples given here suggest that pupils may understand reflections as ‘completing a shape, by making it the same on both sides of the mirror-line. However, examples discussed by KÃ'ËœAchaean reveal that the given shape does not need to meet the mirror line for this kind of mistake to occur. Reflecting shapes in vertical and horizontal lines is certainly easier than reflecting them in lines of any other orientation. Reflections are also easier when the slope of the object is oriented either parallel or perpendicular to the mirror-line. Understandably, teaching on reflection often begins with examples of this sort. In a context such as this, many of the resulting reflections can be explained by a rule that does not extend to more complex cases. The rule is this: flip the object over in such a way as to leave it’s slope unchanged, now translate the object in the direction of its slope until it lies on the other side of the mirror-line, and at the same distance from that line. This is much easier to do than to explain. This rule fits the case of Figures 3 and 4 exactly, and makes the answer in fugue 2 appear quite natural the slight lack of fit combined tit the obvious fact that the object (and so the image) meets the mirror-line in two places, also seems to explain the girl’s dithering over exactly what length to make the resulting rectangle. KÃ'ËœAchaean suggests that these misconceptions may be best avoided by tying the idea of reflection to that of folding, and the essential properties of reflection teased out through investigative work. The actions and the representations are both highly indubitable so that it should be possible to develop such an approach in ways that are meaningful to most children. The transformations can be internalized in gradual steps, by focusing first on the actions themselves and then on the representation of imagined actions. In addition, the resulting drawings can be checked at each step by a return to the actions. The approach advocated is one that directs children towards discoveries from which the rules and properties of the transformations can be surmised and against which they can be tested. (KÃ'ËœAchaean 1981: 157) A similarly investigate approach could be used in conjunction with a dynamic geometry package. This would allow students to investigate what properties of a â€Å"reflection† remain unchanged as the image, object and mirror-line are manipulated 0 discussed in the next section. Misconceptions and Theories of Learning Many of the above strategies and remedies fit well with the theories of learning due to Jean Pigged and Level Whisky. Due to his emphasis on personal interactions, particularly in the use of language, Level Hoosegows theory of learning is often thought of as a calculators theory. Whisky believed that the ideal learning relationship was that of a novice learning as an apprentice under the tutelage of an expert. Nevertheless, like Pigged, he would have seen much value in peer discussions that, ender the guidance of a teacher, overcome differences of opinion and in doing so achieve a shared understanding. Hoosegows theories also provide a theoretical backing for strategies involving reading aloud. What is at first read aloud is, of course, soon expected to become self-speech (also referred to as private-speech or inner-speech). According to Whisky, as the principle form of self-direction and self-regulation, such self-speech is the foundation of all higher cognitive powers (Beer 1997: 248). According to Pigged, all cognitive change can be classified as one of two types: adaptation and organization. Organization is a largely internal process involving rearranging and linking up items of previous learning to form a â€Å"strongly interconnected cognitive system† (Beer 1997: 213). More important for our purposes is adaptation, which itself comes in two varieties: assimilation and accommodation. In assimilation the learner simply fits new concepts, skills and information into his or her existing cognitive framework. However, on some occasions new items of learning cannot be fitted into the existing cognitive framework, and that framework must be changed in order to make room for them. This is accommodation. The awareness of a need for a change in one’s cognitive framework is brought about by a realization that something important ‘doesn’t fit in’. For this reason, Malcolm Swan and others in the Diagnostic Teaching Project have seen Piglet’s views as providing theoretical Justification for their view that the best way to overcome a misconception is by engineering a cognitive conflict (Swan 2001). Many of the strategies suggested above fit into this category. When those strategies were described, they were described as strategies for use with individual 21 students. However, Swan and others suggest that the strategy may be as, if not more, effective when the conflict in question is between members of the same peer group, who must then come to a resolution through discussion. A [pupil] might superficially accept a [teacher’s] perspective without critically examining it, out of an unquestioning belief in the [teacher’s] authority. Pigged also asserted that clashing viewpoints – arguments Jarring the [learner] into noticing a peer’s point of view – were necessary for peer interaction to stimulate movement towards logical thought. (Beer 1997: 253) How to cite Cloud, Papers