Preserve Toothbrushes Essay Example for Free

Preserve Toothbrushes Essay Strategic planning is an important aspect for any business or corporation. Without it, an organization has no way to achieve a stated goal. They have no way to march towards their vision. In the case of Eric Hudson, an entrepreneur with a vision, he successfully created a business with a goal in mind and implemented a strategic plan to make it happen. Every small business startup has to have some edge in order to compete; otherwise they will be forced out by larger, pre-existing competitors. For Preserve, this meant identifying and filling a niche in an industry largely consisting of multinational conglomerates. For Eric Hudson, this meant thinking critically, having a plan, and implementing it in order to carry out his vision. Preserve found its niche in environmentally friendly toothbrushes. Preserve created an environmentally friendly toothbrush manufactured from recycled plastic. Further enhancing the product was a unique 45 degree angled head designed by Hudson’s father, an industrial engineer. At the time, the â€Å"green† movement was largely a niche in itself. While not a first-mover, others were already in the â€Å"green† products industry; Preserve had a modified approach of the second-mover strategy. It did not create the â€Å"green† industry, but it did create a new product. It was often a small market for â€Å"green† products, and the products that existed were often less effective and of poor quality. However, for Eric Hudson, he made it his vision to provide an environmentally friendly product without sacrificing quality or performance. Initially, Preserve’s toothbrush was marketed through grassroots movements and sampling. However, it was at one of the sampling events, that an employee from Stonyfield Yogurt, an organic yogurt company, approached Preserve and relayed the information that they consistently had leftover plastic from producing yogurt containers. With this bit of information, Preserve had a new strategic alliance that provided them with the raw materials to produce its product, and Stonyfield had the ability to promote tself as an environmentally friendly corporation as a result of its efforts. Preserve is a small, niche brand, and like most other small niche brands, their marketing budget pales in comparison to its larger competitors. In Preserve’s case, they focus on generating publicity to promote and advertise their brand. The best advertisement is free advertisement, and Preserve tends to generate a lot of magazine articles and television appearances. This in turn generates word of m outh testimonials, the most effective form of advertising. Businesses in general should continually strive to enhance their business model and update their strategic plan. A SWOT analysis is a useful tool for examining a business and how it is meeting its basic mission. By comparing the firm’s strengths and weaknesses with external opportunities and threats, a SWOT analysis provides managers with a view of the organization’s internal and external environments. In turn, this analysis can be used to determine if the firm is meeting expectations or if it is on track to meet expectations. Preserve’s director of marketing, C. A. Webb, admits that she doesn’t actually perform SWOT analysis’s, but certainly uses that form of thinking consistently when implementing new products, distribution channels, designs, or other decisions. In fact, it was this style of thinking that led to Preserve to focus its effort on packaging its unknown product in order to appeal to a more sophisticated market and consumer. It was crucial to target a specific audience because retailers could not devote large amounts of shelf space to a small brand such as Preserve. So Preserve made it their mission to go up against their large competitors in the only way possible—out marketing them. Business strategy and strategic plans are crucial for any business, small, medium, or large. Preserve certainly provides an excellent example of the necessity of a strategic plan and how following one can lead to success, provided the organization has the necessary strengths and environment as can be determined by a SWOT analysis. Preserve’s success can certainly be attributed to a successful business plan, as well as careful planning. Both are important lessons to be taken away from Preserve’s market presence.

John Brown: Violent Abolitionist :: essays research papers fc

John Brown was born on May 9, 1800 in Torrington, Connecticut. When he was about five years old, his father moved the family to Hudson, Ohio. There, John was filled with the heavy anti-slavery sentiment that was present in that area. This, combined with personal observations of the maltreatment of blacks and the influence of Calvinism, started John Brown on his crusade to abolish slavery. In 1855, Brown and several of his sons moved to Kansas, a territory deeply divided over the slavery issue. Brown was captured after the raid, sentenced to death, and was hung on December 2, 1859 in a field near town. On October 16, 1859, abolitionist John Brown and several followers seized the United States Armory and Arsenal at Harpers Ferry. The actions of Brown's men brought national attention to the emotional divisions relating to slavery On Pottawotamie Creek, on the night of May 24, 1856, Brown and his sons murdered three men who supported slavery, although none actually owned slaves. Brown and his sons escaped. Brown spent the next three years collecting money from wealthy abolitionists in order to establish a colony for runaway slaves. To accomplish this, Brown needed weapons and so decided to capture the arsenal at Harpers Ferry. In 1794, President George Washington had selected Harpers Ferry, Virginia, and Springfield, Massachusetts, as the sites of the new national armories. In choosing Harpers Ferry, he noted the benefit of great waterpower provided by both the Potomac and Shenandoah rivers. Producing of weapons began in 1796. In 1817, the federal government contracted with John H. Hall to manufacture his patented rifles at Harpers Ferry. The armory and arsenal continued producing weapons until its destruction at the outbreak of the Civil War. In the summer of 1859, John Brown, using the alias Isaac Smith, too k up residence near Harpers Ferry on a farm in Maryland. He trained a group of twenty-two men, including his sons Oliver, Owen, and Watson, in military training. On the night of Sunday, October 16, Brown and all but three of the men marched into Harpers Ferry, capturing several watchmen. The first victim of the raid was an African-American railroad baggage handler named Hayward Shepherd, who was shot and killed after confronting the raiders. During the night, Brown captured several other prisoners, including Lewis Washington, the great-grand-nephew of George Washington. There were two keys to the success of the raid. John Brown: Violent Abolitionist :: essays research papers fc John Brown was born on May 9, 1800 in Torrington, Connecticut. When he was about five years old, his father moved the family to Hudson, Ohio. There, John was filled with the heavy anti-slavery sentiment that was present in that area. This, combined with personal observations of the maltreatment of blacks and the influence of Calvinism, started John Brown on his crusade to abolish slavery. In 1855, Brown and several of his sons moved to Kansas, a territory deeply divided over the slavery issue. Brown was captured after the raid, sentenced to death, and was hung on December 2, 1859 in a field near town. On October 16, 1859, abolitionist John Brown and several followers seized the United States Armory and Arsenal at Harpers Ferry. The actions of Brown's men brought national attention to the emotional divisions relating to slavery On Pottawotamie Creek, on the night of May 24, 1856, Brown and his sons murdered three men who supported slavery, although none actually owned slaves. Brown and his sons escaped. Brown spent the next three years collecting money from wealthy abolitionists in order to establish a colony for runaway slaves. To accomplish this, Brown needed weapons and so decided to capture the arsenal at Harpers Ferry. In 1794, President George Washington had selected Harpers Ferry, Virginia, and Springfield, Massachusetts, as the sites of the new national armories. In choosing Harpers Ferry, he noted the benefit of great waterpower provided by both the Potomac and Shenandoah rivers. Producing of weapons began in 1796. In 1817, the federal government contracted with John H. Hall to manufacture his patented rifles at Harpers Ferry. The armory and arsenal continued producing weapons until its destruction at the outbreak of the Civil War. In the summer of 1859, John Brown, using the alias Isaac Smith, too k up residence near Harpers Ferry on a farm in Maryland. He trained a group of twenty-two men, including his sons Oliver, Owen, and Watson, in military training. On the night of Sunday, October 16, Brown and all but three of the men marched into Harpers Ferry, capturing several watchmen. The first victim of the raid was an African-American railroad baggage handler named Hayward Shepherd, who was shot and killed after confronting the raiders. During the night, Brown captured several other prisoners, including Lewis Washington, the great-grand-nephew of George Washington. There were two keys to the success of the raid.

Enron Corporation and Anderson case study Essay

Analyzing the fall of two Giants This case results in the publishing of Sarbanes-Oxley Act of 2002 and relevant to the Securities and Exchange Commission. Also, it is related to SAS 103: Auditing, Quality Control, and Independence Standards and Rules. [1] What were the business risk Enron faced, and how did those risks increase the likelihood if material misstatements in Enron’s financial statements? The business risks Enron faced are as following: †¢Using complex business model †¢extensive using special purpose entities †¢using untraditional ventures to expand business rapidly †¢limitations in GAAP The complex business model used in Enron lead overstate its revenue while not disclose the exact value of debt. Numbers of special purpose entities are used to keep debt off the books. The untraditional ventures incense the business expansion rapidly and risky. Also, the limitation of GAAP makes it possible that management took advantages of complex standards to hide the actual economic substance. All of these above increase the likelihood of material misstatements in Enron’s financial statements. [2] (a) What are the responsibilities of a company’s board of directors? (b) Could the board of directors at Enron—especially the audit committee—have prevented the fall of Enron? (c) Should they have known about the risks and apparent lack of independence with Enron’s SPEs? What should they have done about it? The responsibilities of a company’s board of directors include: †¢Protect the shareholders’ assets and provide a return on investment †¢Make important decisions that affect shareholders (dividends) †¢Decide on which executives to hire / fire The fall of Enron could have been prevented by the board of directors. The board should responsible for the company’s financial reports. However, they are failed to disclose the off books liabilities to the public, which led  the Enron fall. What is more, the board and the audit committee do not question any of the high risk transactions. They should have known about the risks and apparent lack of independence with Enron’s SPEs. They should recognize that the high risk transactions with SPE will have huge effects on Enron. Meanwhile, they should ask SPE to disclosure financials properly. [4] What are the auditor independence issues surrounding the provision of external auditing services, internal auditing services, and management consulting services for the same client? Develop arguments for why auditors should be allowed to perform these services for the same client. Develop separate arguments for why auditors should not be allowed to perform non-audit services for their audit clients. What is your view, and why? Auditors should not be allowed to perform non-audit services for their audit clients, because auditors need to be independence. If an auditor provide management consulting services for his audit client, he is just audit what he have done, which ,I think, is meaningless. On the contrary, some people may agree that auditors should be allowed to perform their services for the same client. First, choosing one firm to do all of these services can save a great deal of money. Second, the auditors will much more familiar with the client’s business and its industry, which make their work efficient. [6] Enron and Andersen suffered severe consequences because of their perceived lack of integrity and damaged reputations. In fact, some people believe the fall of Enron occurred because of a form of â€Å"run on the bank†. Some argue that Andersen experienced a similar â€Å"run on the bank† as many top clients quickly dropped the firm in the wake of Enron’s collapse. Is the â€Å"run on the bank† analogy valid for both firms? Why or why not? Yes, I think the â€Å"run on the bank† analogy valid for both firms. The fraud of Enron’s financials leads a collapse of investor, customer, and trading partner confidence. Its stocks experience a sharp slump. Meanwhile, Standard & Poor’s re-classify Enron’s stocks as junk bonds, making almost every stockholder feel unsafe. The price drops to $0.26 per share in couple of days. Even worse, debts holders begin to call the loans because of the diminished stock price, which lead the collapse of Enron directly. Andersen experiences a similar situation. The damaged reputation of Andersen results  in losing many top clients and partnerships oversea. [9] What has been done, and what more do you believe should be done to restore the public trust in the auditing profession and in the nation’s financial reporting system? The Sarbanes-Oxley Act of 2002 is a good way to restore the public trust in the auditing profession and financial report. The Act required top management to certify the accuracy of financial information individually, and increase the independence of outside auditors. As the most severe act in history ever, I believe SOX can help to restore the public trust.

Geometric Thinking - Free Essay Example

Sample details Pages: 26 Words: 7715 Downloads: 1 Date added: 2017/06/26 Category Statistics Essay Did you like this example? Students Geometric Thinking 8 CHAPTER 1 Introduction In the last 20 years, the perception of learning as internalization of knowledge is criticized and problemized in mathematics education society (Lave Wenger, 1991; Sfard, 2000; Forman Ansell, 2001). Lave and Wenger (1991) describe learning as a process of increasing participation in communities of practices (p.49). Sfard (2000) also emphasized the new understanding of learning as Today, rather than speaking about acquisition of knowledge, many people prefer to view learning as becoming a participant in a certain discourse (p.160). This new perspective in the understanding of learning brings different views to mathematics teaching practice. While the structure of mathematics lessons is organized in the sequence of Initiation- Response-Evaluation (IRE) in the traditional mathematics classrooms, with the reform movement, participation of the students become the centre of the mathematics classrooms (O Connor, 1993; Steele, 2001). Initiating topic or problems, starting or enhancing discussions, providing explanations are the role of the teacher in the traditional classrooms but these roles become a part of students responsibilities in the reform mathematics classrooms (Forman, 1996). Don’t waste time! Our writers will create an original "Geometric Thinking" essay for you Create order Turkey also tries to organize their mathematics curriculum according to these reform movements. With the new elementary mathematics curriculum, in addition to developing mathematical concepts, the goal of mathematics education is defined as enhancing students problem solving, communication and reasoning abilities. Doing mathematics is no more defined only as remembering basic mathematical facts and rules and following procedures, it also described as solving problems, discussing the ideas and solution strategies, explaining and defending own views, and relating mathematical concepts with other mathematical concepts and disciplines (MEB, 2006). Parallel to new understanding of learning, reform movements in mathematics education, and new Turkish elementary mathematics curriculum, students roles such as developing alternative solution strategies and sharing and discussing these strategies gain great importance in mathematics education. Mathematics teachers are advised to create classroom discourse in which students will be encouraged to use different approaches for solving problems and to justify their thinking. This means that some researches and new mathematics curriculum give so much importance to encourage students to develop alternative problem solving strategies and share them with others. (MEB, 2006; Carpenter, Fennema, Franke, Levi Empson, 1999; Reid, 1995). One of the aims of the new mathematics curriculum is that the students stated their mathematical thinking and their implications during the mathematical problem solving process (MEB, 2006). According to new curriculum, the students should have opportunity to solve the problems using different strategies and to explain their thinking related to problem solving to their friends and teacher. Moreover, the students should state their own mathematical thinking and implications during the problem solving process and they should develop problem solving strategies in mathematics classrooms (MEB, 2006). Fraivillig, Murphy and Fuson (1999) reported that creating this kind of classrooms requires that teacher has knowledge about students mathematical thinking. One of the most important studies related to childrens mathematical thinking is Cognitively Guided Instruction (CGI). The aim of this study is to help the teachers organize and expand their understanding of childrens thinking and to explore how to use this knowledge to make instructional decisions such as choice of problems, questions to ask children to acquire their understanding. The study was conducted from kindergarten through 3rd grade students. At the beginning of the study, researchers tried to explore students problem solving strategies related to content domains addition, subtraction, multiplication and division. The findings from this investigation is that students solve the problems by using direct modeling strategies, counting strategies derived facts strategy and invented algorithms. In order to share their findings with teachers, they conducted workshops. With these workshops, the teachers realized that the students are able to solve the problems using a variety of stra tegies. After this realization, they started to listen to their students mathematical explanations, tried to elicit those strategies by asking questions, tried to understand childrens thinking and encouraged the use of multiple strategies to solve the problems in their classrooms (Franke, Kazemi, 2001, Fennema, Carpenter, Franke, 1992). At the end of the study, the students whose teachers encourage them to solve the questions with different strategies and spend more time for discussing these solutions showed higher performance (Fennema, Carpenter, Franke, Levi, Jacobs, Empson, 1996). Similar finding is also observed the study of Hiebert and Wearne (1993). They concluded that when the students solve few problems, spend more time for each problem and explain their alternative solution strategies, they get higher performance. As indicated the new curriculum in Turkey (MEB,2006), the teacher should create a classroom in which students develop different problem solving strategies, share these with their classmates and their teacher and set a high value on different problem solving strategies during the problem solving process. Encouraging the students to solve the problems is important since while they are solving the problems, they have a chance to overview their own understanding and they take notice of their lack of understandings or misunderstandings (Chi Bassock, 1989, as cited in Webb, Nemer Ing, 2006). Moreover, Forman and Ansell (2001) stated that if the students develop their own problem solving strategies, their self confidence will be increase and they ca n build their mathematical informal knowledge. Not only mathematical thinking, but also geometrical thinking has very crucial role for developing mathematical thinking since National Council of Teachers of Mathematics in USA (2000) stated that geometry offers an aspect of mathematical thinking that is different from, but connected to, the world of numbers (p.97). While students are engaging in shapes, structures and transformations, they understand geometry and also mathematics since these concepts also help them improve their number skills. There are some studies which dealt with childrens thinking but a few of them examine childrens geometrical thinking especially two dimensional and three dimensional geometry. One of the most important studies related to geometrical thinking is van Hiele Theory. The theory categorizes childrens geometrical thinking in a hierarchical structure and there are five hierarchical levels (van Hiele, 1986). According to these levels, initially students recognize the shapes as a whole (Level 0), then they discover the properties of figures and recognize the relationship between the figures and their properties (level 1 and 2). Lastly the students differentiate axioms, definitions and theorems and they prove the theorems (level 3 and 4) (Fuys, Geddes, Tischler, 1988). Besides, there are some other studies which examined geometrical thinking in different point of view. For example, the study of Ng (1998) is related to students understanding in area and volume at grade 4 and 5. But, Battista and Clements (1996) and Ben-Chaim (1985) investigated students geometric thinking by describing students solution strategies and errors in 3-D cube arrays at grades 3, 4 and 5. On the other hand, Chang (1992) carried out a study to understand spatial and geometric reasoning abilities of college students. Besides of these studies, Seil (2000), Olkun (2001), Olkun, Toluk (2004), zbellek (2003) and Okur (2006) have been conducted studies in Turkey. Generally, the studies are about students geometric problem solving strategies (Seil, 2000), the reason of failure in geometry and ways of solution (Okur, 2006), the misconceptions and missing understandings of the students related to the subject angles at grade 6 and 7 (zbellek, 2003). In addition to these, studies has been done to investigate the difficulties of students related to calculating the volume of solids which are formed by the unit cubes (Olkun, 2001), number and geometry concepts and the effects of using materials on students geometric thinking (Olkun Toluk, 2004). When the studies are examined which has been done in Turkey, the number of studies related to spatial ability is limited. Spatial ability is described as the ability to perceive the essential relationships among the elements of a given visual situation and the ability to mentally manipulate one or two elements and is logically related to learning geometry (as cited in Moses, 1977, p.18). Some researchers claimed that it has an important role for mathematics education since spatial skills contribute an important way to the learning of mathematics (Fennema Sherman,1978; Smith, 1964) and Anderson (2000) claimed that mathematical thinking or mathematical ability is strongly related with spatial ability. On the other hand, Moses (1977) and Battista (1990) found that geometric problem solving and achievement are positively correlated with spatial ability. So, developing students spatial ability will have benefit to improve students geometrical and also mathematical thinking and it may fost er students interest in mathematics. Problem Statement Since spatial ability and geometric thinking are basis of mathematics achievement, then one of the problems for researchers may be to investigate students geometric thinking (NCTM, 2000; Anderson, 2000; Fennema Sherman, 1978; Smith, 1964). For this reason, generally this study will focus on students geometrical thinking. Particularly, it deals with how students think in three-dimensional and two-dimensional geometry, their solution strategies in order to solve three-dimensional and two-dimensional geometry problems, the difficulties which they confront with while they are solving them and the misconceptions related to geometry. Also, whether or not the students use their mathematics knowledge or daily life experiences while solving geometry questions are the main questions for this study. Purpose Statement The purpose of this study is to assess and describe students geometric thinking. Particularly, its purpose is to explain how the students approach to three-dimensional geometry, how they solve the questions related to three-dimensional geometry, what kind of solution strategies they develop, and what kind of difficulties they are confronted with when they are solving three-dimensional geometry problems. Also, the other purpose is to analyze how students associate their mathematics knowledge and daily life experience with geometry. The study attempt to answer the following questions: 1. How do 4th, 5th, 6th, 7th and 8th grade elementary students solve the questions related to three-dimensional geometry problems? 2. What kind of solution strategies do 4th, 5th, 6th, 7th and 8th elementary students develop in order to solve three-dimensional geometry problems? 3. What kind of difficulties do 4th, 5th, 6th, 7th and 8th elementary students face with while they are solving three-dimensional geometry problems? 4. How do 4th, 5th, 6th, 7th and 8th elementary students associate their mathematics knowledge and daily life experience with geometry problems? Rationale Most of the countries have changed their educational program in order to make learning be more meaningful (NCTM, 2000; MEB, 2006). The development of Turkish curriculum from 2003 to up till now can be assessed the part of the international educational reform. Particularly, the aim of the changes in elementary mathematics education is to make the students give meaning to learning by concretizing in their mind and to make the learning be more meaningful (MEB, 2006). In order to make learning more meaningful, knowing how the students think is critically important. For this reason, this study will investigate students mathematical thinking especially geometrical thinking since geometry provides opportunity to encourage students mathematical thinking (NCTM,2006). The result of the international exams such as Trends in International Mathematics and Science Study (TIMSS) and the Programme for International Student Assessment (PISA) and national exams Secondary School Entrance Exam Ortaretim Kurumlar renci Seme Snav (OKS) show that the success of Turkish students in mathematics and especially in geometry is too low. Ministry of National Education in Turkey stated that although international average is 487 at TIMSS-1999, Turkish students mathematics average is 429. Moreover, they are 31st country among 38 countries. When the sub topics are analyzed, geometry has least average (EARGED, 2003). The similar result can be seen the Programme for International Student Assessment (PISA). According to result of PISA-2003, Turkish students are 28th county among 40 countries and Turkish students mathematics average is 423 but the international average is 489. When geometry average is considered, it is not different from the result of TIMSS-1999 since intern ational geometry average is 486 but the average of Turkey is 417 ((EARGED, 2005). As it can be realized from result of both TIMSS-1999 and PISA-2003, Turkish students average is significantly lower than the international average. Since in order to get higher mathematical performance, being aware of childrens mathematical thinking has crucial role (Fennema, Carpenter, Franke, Levi, Jacobs, Empson, 1996). For this reason, knowing students geometric thinking, their solution strategies and their difficulties related to geometry problems will help to explore some of the reasons of Turkish students low geometry performance in international assessment, Trends in International Mathematics and Science Study (TIMSS) and Program for International Student Assessment (PISA), and in national assessment, Secondary School Entrance Exam Ortaretim Kurumlar renci Seme Snav (OKS). As a result, when geometry and being aware of students problems solving strategies and their difficulties when they are solving geometry problems has important roles on mathematics achievement are taken into consideration, studies related to geometry and students geometric thinking are needed. Besides, Turkish students performance in international assessments is considered; it is not difficult to realize that there should be more studies related to geometry. For these reasons, the study will assist in Turkish education literature. Significance of the Study Teachers knowledge about childrens mathematical thinking effect their instructional method. They teach the subjects in the way of childrens thinking and they encourage students to think over the problems and to develop solution strategies. With such instructional method, classes are more successful (Fennema, Carpenter, Franke, 1992). Geometry is one of the sub topic of mathematics (MEB,2006) and it has crucial role in representing and solving problems in other sub topics of mathematics. Besides, geometry has important contribution to develop childrens mathematical thinking. On the other hand, in order to understand geometry, spatial ability is useful tool (NCTM, 2000). Battista et al.(1998), Fennema and Tartre (1985) and Moses (1977) emphasized that there is a relationship between spatial ability and achievement in geometry. Moreover, mathematical thinking and mathematical ability is positively correlated with spatial thinking (Anderson 2000). Since geometry, spatial ability and mathematical thinking are positively correlated, being successful in geometry will get higher mathematics achievement. To increase geometry achievement, the teachers should know students geometric thinking. Particularly, how students solve problems, what kind of strategies they develop, and what kind of difficulties they face with while t hey are solving the problems are important concepts in order to get idea about students thinking (Fennema, Carpenter, Franke, 1992). With this study, the teachers will be informed how children think while they are solving geometry problems especially three-dimensional geometry problems, what kind of strategies they develop to solve them, what kind of difficulties they face with related to geometry problems. Furthermore, university instructors will benefit from this study to have knowledge about childrens geometric thinking and this knowledge may be valuable for them. Since they may inform pre-service teachers about childrens thinking and the importance of knowing childrens thinking while making instructional decisions. As a result, knowing students geometric thinking will benefit to increase their geometry achievement and also mathematical achievement, and consequently, this will help to raise the Turkish students success of the international exams CHAPTER 2 Literature Review Geometry can be considered as the part of mathematics and it provides opportunities to encourage students mathematical thinking. Also, geometry offers students an aspect of mathematical thinking since when students engage in geometry, they become familiar with shape, location and transformation, and they also understand other mathematics topics (NCTM, 2000). Therefore, understanding of students geometrical thinking, their geometry problem solving strategies and their difficulties in geometry become the base for their mathematical thinking. Also, since geometry is a science of space as well as logical structure, to understand students geometrical thinking requires knowledge of spatial ability and cognitive ability (NCTM, 1989, p.48). This chapter deals with some of the literature in four areas related to the problem of this study. The first section of this chapter is related to the van Hiele theory since van Hiele theory explains the level of childrens geometrical thinking (van Hiele, 1986). The second section of this chapter deals with the research studies related to students mathematical and geometrical thinking. The third section is devoted to research studies related to spatial ability. And the last section of this chapter reviews the research related to relationship between spatial ability and mathematics achievement. Section 1: The van Hiele Theory The van Hiele theory is related to childrens thinking especially their geometrical thinking since the theory categorizes childrens geometrical thinking in a hierarchical structure (van Hiele, 1986). According to theory of Pierre and Diana van Hiele, students learn the geometry subjects through levels of thought and they stated that the van Hiele Theory provided instructional direction to the learning and teaching of geometry. The van Hiele model has five hierarchical sequences. Van Hiele stated that each level has its own language because in each level, the connection of the terms, definitions, logic and symbol are different. The first level is visual level (level 0) (van Hiele, 1986). In this level, children recognize the figures according to their appearance. They might distinguish one figure to another but they do not consider the geometric properties of the figures. For instance, they do not consider the rectangle as a type of a parallelogram. The second level is descriptive leve l (level 1). In this level, students recognize the shapes by their properties. For instance, a student might think of a square which has four equal sides, four equal angles and equal diagonals. But they can not make relationships between these properties. For example, they can not grasp that equal diagonal can be deduced from equal sides and equal angles. The third level is theoretical level (level 3). The students can recognize the relationship between the figures and the properties. They discover properties of various shapes. For instance, some of the properties of the square satisfy the definition of the rectangle and they conclude that every square is a rectangle. The fourth level is formal logic level (level 4). The students realize the differences between axioms, definitions and theorems. Also, they prove the theorems and make relationships between the theorems. The fifth level is rigor level (level 4). In this level, students establish the theorems in different postulation sy stems (Fuys, Geddes, Tischler, 1988). As a result, the levels give information about students geometric thinking to the researchers and mathematics teachers. Mathematics teachers may guess whether the geometry problem will be solved by students or not and at which grade they will solve them. Section 2: Children thinking The van Hiele theory explains the students thinking level in geometry. The levels are important but how students think is as important as their thinking level. To ascertain how students think related to mathematics and especially geometry, a number of studies have been conducted (Carpenter, Fennema, Franke, 1996; Chang, 1992; Battista, Clements, 1995; zbellek, 2003; Olkun, 2005; Ng, 1998; Okur, 2006). Some of these studies are related to mathematical thinking and some of them geometrical thinking. Carpenter et al. (1999) and Olkun (2005) studied childrens mathematical thinking and Chang (1992), Battista and Clements (1995), Ben-Chaim (1985), Olkun (2001), zbellek (2003), Okur (2006) and Ng, (1998) carried out research studies related to childrens geometrical thinking. An important study related to mathematical thinking has been conduct by Carpenter, Fennema and Franke initiated over 15 years ago in USA and the name of this study is Cognitively Guided Instruction (CGI) which is described as the teacher development program. Cognitively Guided Instruction sought to bring together research on the development of childrens mathematical thinking and research on teaching (Franke, Kazemi, 2001). Carpenter, Fennema and Franke (1996) stated that Cognitively Guided Instruction (CGI) focuses on childrens understanding of specific mathematical concepts which provide a basis for teachers to develop their knowledge more broadly. The Cognitively Guided Instruction (CGI) Professional Development Program engages teachers in learning about the development of childrens mathematical thinking within particular content domains. (Carpenter, Fennema, Franke, Levi, Empson, 1999). These content domains include investigation of childrens thinking at different problem situat ions that characterize addition, subtraction, multiplication and division (Fennema, Carpenter, Franke, 1992). In order to understand how the children categorize the problems, Carpenter et al. (1992) conducted a study. According to this study, Fennema, Carpenter, and Franke (1996) portrayed how basic concepts of addition, subtraction, multiplication, and division develop in children and how they can construct concepts of place value and multidigit computational procedures based on their intuitive mathematical knowledge. At the end of this study, with the help of childrens actions and relations in the problem, for addition and subtraction, four basic classes of problems can be identified: Join Separate, Part-Part-Whole, and Compare and Carpenter et all. (1999) reported that according to these problem types, children develop different strategies to solve them. The similar study has been carried out by Olkun et al (2005) in Turkey. The purpose of these two studies is the same but the s ubjects and the grade level are different. Olkun et al (2005) studied with the students from kindergarten to 5th grade but the students who participated in Carpenters study is from kindergarten through 3rd grade (Fennema, Carpenter, Franke, 1992). Furthermore, CGI is related to concepts addition, subtraction, multiplication and division but the content of the study done in Turkey is addition, multiplication, number and geometrical concepts (Olkun et al, 2005). Although the grade level and the subjects were different, for the same subjects, addition and multiplication, the solution strategies of the students in Olkuns study are almost the same as the students in CGI. But the students in the study of Carpenter used wider variety of strategies than the students in Turkey even if they are smaller than the students who participated in Olkuns study. This means that grade level or age is not important for developing problem solving strategies. On the other hand, there are some studies related to childrens geometrical thinking which are interested in different side of geometrical thinking. Ng (1998) had conducted a study related to students understanding in area and volume. There were seven participants at grade 4 and 5. For the study, she interviewed with all participants one by one and she presented her dialogues with students while they are solving the questions. She reported that students who participated in the study voluntarily have different understanding level for the concepts of area, and volume. She explained that when students pass from one level to another, 4th grade to 5th grade, their thinking becomes more integrated. With regard to its methodology and its geometry questions, it is valuable for my study. On the contrary to Ng, Chang (1992) chose his participants at different levels of thinking in three-dimensional geometry. These levels were determined by the Spatial Geometry test. According to this study, students at lower levels of thinking use more manipulative and less definitions and theorems to solve the problems than high level of thinking. On the other hand, the levels of two-dimensional geometry identified by the van Hiele theory. The results were the same as the three-dimensional geometry. In this case, Chang (1992) stated that the students at the lower levels of thinking request more apparatus and less definitions and theorems to solve the problems. Moreover, for both cases, the students at the higher levels of thinking want manipulative at the later times in the problem-solving process than the students at the lower level of students. The result of this study indicated that using manipulative require higher level of thinking. By providing necessary manipulative, I hope th e students use higher level of thinking and solve the problems with different strategy. Besides of these studies, Ben-Chaim et all. (1985) carried out the study to investigate errors in the three-dimensional geometry. They reported four types of errors on the problem related to determining the volume of the three-dimensional objects which are composed of the cubes. Particularly, they categorize these errors two major types which students made. These major types of errors defined as dealing with two dimensional rather than three and not counting hidden cubes (Ben-Chaim, 1985). The similar study was conducted by Olkun (2001). The aim of this study is to explain students difficulties which they faced with calculating the volume of the solids. He concluded that while students were finding the volume of the rectangular solids with the help of the unit cubes, most of the students were forced open to find the number of the unit cubes in the rectangular solids. Also, the students found the big prism complicated and they were forced open to give life to the organization of the p rism which was formed by the unit cubes based on the column, line and layers in their mind, i.e. they got stuck on to imagine the prism readily. (Olkun, 2001). The categorization of students difficulties will be base for me to analyze difficulties related to geometry problems of the students who are participant of my study. Besides of these studies, Battista and Clements (1996) conducted a study to understand students solution strategies and errors in the three-dimensional problems. The study of Battista and Clements (1996) was different from the study of Ben-Chaim (1985) and Olkun (2001) in some respect such as Battista and Clements categorized problem solving strategies but Ben-Chaim and Olkun defined students difficulties while reaching correct answer. Categorization of the students problem solving strategies in the study of Battista and Clements (1996) is like the following: Category A: The students conceptualized the set of cubes as a 3-D rectangular array organized into layers. Category B: The students conceptualized the set of cubes as space filling, attempting to count all cubes in the interior and exterior. Category C: The students conceptualized the set of cubes in terms of its faces; he or she counted all or a subset of the visible faces of cubes. Category D: The students explicitly used the formula L x W x H, but with no indication that he or she understood the formula in terms of layers. Category E: Other. This category includes strategies such as multiplying the number of squares on one face times the number on other face. (Battista Clements ,1996). At another study of Battista and Clements (1998), their categorization was nearly the same but their names were different than the study which has done in 1996. In this study, they categorized the strategies as seeing buildings as unstructured sets of cubes, seeing buildings as unstructured sets of cubes, seeing buildings as space filling, seeing buildings in terms of layer and use of formula. Battista and Clements (1996, 1998) concluded that spatial structuring is basic concept to understand students strategies for calculating the volume of the objects which are formed by the cubes. Students should establish the units, establish relationships between units and comprehend the relationship as a subset of the objects. Actually, these studies are important for my study since they gave some ideas about different solutions for solving these problems. Also, different categorization of students geometry problems strategies will help me about how I can categorize students strategies. Also, In addition to these studies, Seil (2000), Olkun (2001), Olkun, Toluk (2004), zbellek (2003) and Okur (2006) have been conducted studies in Turkey. Seil (2000) has investigated students problem solving strategies in geometry and Okur (2006) have studied the reason of failure in geometry and ways of solution. In the study of zbellek, the misconceptions and missing understandings of the students related to the subject angles at grade 6 and 7. Also, studies has been done to investigate the difficulties of students related to calculating the volume of solids which are formed by the unit cubes (Olkun, 2001) and the effects of using materials on students geometric thinking (Olkun Toluk, 2004). As a result, in order to understand children thinking, several studies has been conducted. Some of them were related to children mathematical thinking and some of them were interested in childrens geometrical thinking. These studies dealt with childrens thinking in different aspects and so their findings are not related to each other. But the common idea is that spatial ability and geometrical thinking are correlated positively. Since spatial reasoning is intellectual operation to construct an organization or form for objects and it has important role to for constructing students geometric knowledge (Battista, 1998). Section 3: Spatial Ability The USA National Council of Teachers of Mathematics (2000)explained that the spatial ability is useful tool to interpret, understand and appreciate our geometric world and it is logically related to mathematics (FennemaTartre, 1985). On the other hand, McGee (1979) describes spatial ability as the ability to mentally manipulate, rotate, twist or invert a pictorially presented stimulus object. Since spatial ability is important for childrens geometric thinking, the development of it has been investigated by several studies. First and foremost study has been carried by Piaget and Inhelder (1967). Piaget et al. (1967) defined the development of spatial ability in young children and the properties of the task they accomplish as they grow up. Piaget divides the development of children into four stages (Malerstein Ahern, 1979). According to the Childs Conception of Space (1967), in the first stage, sensorimotor stage, the children recognize only the shapes, not recognizes differences between the shapes. In the second stage, preoperational stage, children recognize figures different shapes, differentiate lines from curve. In the third stage, operational stage, children understand the X-Y axis. This means that they coordinate the point according to the reference point. And finally, in the fourth stage, formal operational stage, children reach the concept of proportionality for all dimensional relations. In the Childs Conception of Geometry (1960), Piaget and Inhelder connect the spatial ability and geometric understanding. They describe the childrens understanding of conservation, measurement of length, area, and volume. When the children reach stage three, they comprehend the measurement, conversation, area and volume. According to Piaget, understanding of children changes when they grow up, from stage to stage. On the other hand, some researchers claimed that spatial ability does not depend on the age. It depends on learning.) have demonstrated that ability to represent three-dimensional objects with two-dimensional drawings can be learned at any age. As a result, spatial ability is positively correlated with geometry learning. Piaget and Inhelder (1967) claimed that spatial ability develops with increasing age but other researchers do not agree the claim of Piaget (Bishop, 1979; and Presmag, 1989). They demonstrated that spatial ability is not related to age, it is related to function of learning. Section 4: The Relationship between Spatial Ability and Mathematics Achievement Large number of researchers thought that spatial ability has an important role in mathematics learning. Battista (1980), Fennema and Sherman (1977), Fennema and Tartre (1985), Ferrini-Mundy (1987), and Moses (1977) have been carried out studies to explore the relationship between spatial ability, mathematical problem-solving and mathematics achievement. The results of these studies are inconsistent and unclear. The relationship between spatial ability and mathematical achievement are different from one study to another. In the field of mathematics, some mathematicians have claimed that all mathematical tasks require spatial thinking (as cited in Lean Clements, 1981, p.267). Moses (1977) reported that in order to improve students mathematical performance, they should be trained on spatial tasks. The results of several studies supported this claim and they showed that spatial ability and mathematical achievement are positively correlated (Aiken, 1971; Battista, 1980; Fennema Sherman ,1977) and Battista (1990) explained this correlation in the range of .30 to .60. The study of Fennema and Sherman (1977) verified this result and they specified that spatial ability and mathematical achievement are positively related. This means that there is direct proportion between spatial ability and mathematical problem-solving and Smith (1964) confirmed that if a person solve high-level mathematical problem, s/he generally have greater spatial ability than person who cannot solve high-level mathematical problem. N evertheless, Battista (1990) investigated the role of spatial thinking and logical reasoning in high school geometry. According to the results of the study showed that spatial thinking and logical reasoning are significantly related to geometry achievement and problem-solving. Particularly, geometric problem-solving correlated higher with spatial thinking than logical reasoning. But on the other hand, Fennema and Tartre (1985) reported that spatial ability does not guarantee success in problem-solving in their later study. Battista et al. (1982) agreed their findings since after investigating the relationship between spatial ability and mathematics performance, the role of spatial thinking in mathematical performance did not described adequately. It is not known that how important spatial ability is to learn several topics in mathematics. Lean and Clementss studies (1981) supported the findings of Battista et al. Lean and Clements (1981) claimed that there is not any correlation between them. Moreover, Chase (1960) agrees the findings of Lean and Clements and Battista et all. Since Chase found that spatial ability did not have any contribution to the problem-solving ability. As a result, there has been several research related to spatial ability and mathematics achievement. Although some researches claimed that there is positive correlation between spatial ability and mathematics achievement, some of them reported that the relationship between them have not described adequately yet. Summary The aim of this literature review to present the result of earlier studies related to van Hiele theory, childrens thinking, spatial ability and lastly the relationship between spatial ability and mathematics performance. According to van Hiele, there is a hierarchical structure of the levels of childrens thinking and the progress of thinking depends on instruction. On the other hand, Piaget claimed that the progress of thinking develops when the child grows up. Several studies have been conducted to understand children thinking and most of the studies found that geometrical thinking is positively correlated with spatial thinking and spatial thinking is related to mathematical achievement. Since the purpose of study is to explore and assess the students geometrical thinking, these studies are related to my study and I get related information from them in terms of level of geometric thinking of van Hiele Theory, difficulties of students while solving geometry problems and categorization of the strategies to solve geometry problems. CHAPTER 3 Methodology This study is designed to explore and assess elementary students geometrical thinking. Particularly, it is concerned with how students solve the questions related to three-dimensional and two-dimensional geometry, what kind of strategies they develop, and what kind of difficulties they are confronted with when they are solving these kinds of problems. Sampling This study will be conducted in a private elementary school in Istanbul during the fall semester of 2007 and approximately 25 students from this school will be selected. In order to select the school and the students, convenience sampling method will be used. Since I plan to find out students different solution strategies, the criterion of the selection of the students will be mathematics and geometry achievement of the students. To increase variety of students solution strategies, the grade level of the students will be different and the students will be 5th, 6th, 7th and 8th grade students. Instrumentation In order to collect data, approximately 10 questions related to three-dimensional geometry will be asked. These questions have been taken from the articles ( Haws, 2002; Ben-Haim, 1985; Battista, Clements, 1996) and the dissertation (Ng, 1998). The questions were translated from English to Turkish using back translation method. In line with this method, I translated the questions from English to Turkish and one of my friends who is an English teacher translated Turkish version to English. Then I compared the original version of the questions with the translation of my friend. Also, I asked for advice to my advisor and other instructors. In order to select the questions, the important factor is that the students can solve the questions by using more than one solution method. The following questions are examples. In both questions, the possible solution method can be the followings. 1. Counting the cubes by using materials 2. Counting from the figure 3. Counting the layers of the cubes 4. Using the formula of volume of cube Moreover, to realize how students solve the questions, there will be some materials such as base-ten blocks and unit cubes. I provide these materials to make the students solve the problems with different methods. Consequently, while I was selecting the questions, I take care of having more than one solution strategies. Data Collection The data will be collected from approximately 25 students in an elementary school in stanbul in October and November of 2007. . During the data collection period, firstly I will interview with the students teacher. I will want them predict whether their students solve the questions or not, what strategies the students will be use, what kind of difficulties they may confront while they are solving them, and so on. The aim of this interview is to learn how much teachers know students geometric knowledge and thinking. Secondly, I will interview with students one by one. At the beginning of the interview, I will not tell anything to them. I want them think and solve the questions. While they are solving the questions, I will use the think aloud method to clarify students thoughts. Particularly, with think aloud method, I will make the students tell what they think while they are solving a problem. During problem solving process, I will encourage them tell how they find the solution of th e problem and I will ask some prompting questions to get more information about what they think while they are solving the problems. During the interview, if possible, I will videotape or audiotape. If it is not possible, I will take detailed notes. After I analysis the data, if there is some missing part or unclear part, I will interview with the same students again. When I finish the interviews with the students and analyze the data, I will share this information with the teacher and I try to state how much they know their students thinking or geometry achievement expressly. Data Analysis The data analysis mainly based on the study of Battista and Clements (1996) for some questions. In this study, they categorized students strategies for finding the number of cubes in a rectangular prism. In this categorization, there are 5 basic groups and each group has some sub groups. After I get the data, I will match students strategies with these strategies and the problem solving strategies which is presented by Ministry of National Education in Turkey. These strategies are listed below (MEB, 2006): Trial and error Using shapes, tables, etc. Using materials Searching pattern Working backwards Guess Using assumptions Expressing the problem differently Implication Using equations Animation and imagine Also, I will categorize the solution strategies by taking care of whether they use materials or formulas to solve them or not. On the other hand, I will explain the difficulties which the students confronted with while they are solving the questions. At the end of the study, I hope to get different solution strategies of students to categorize them. Since this study is qualitative study, while I am collecting and analyzing the data, I should establish the trustworthiness of the procedures such as voluntary participation and guarantee of anonymity, purposeful sampling, triangulation, prolonged engagement, natural situation, peer debriefing and member checks. The students who will participate in the study will be volunteers. If they did not want to participate in the study, I will not force them even if their geometry achievement is higher. The students will be selected from the elementary school which is convenience for me and they will be selected according to their mathematics and geometry achievement. Most probably, the students will be my students so there will be no problem related to mutual comfort. Also, the identity of the students will be secret. On the other hand, I will get data from different sources by interviewing the teachers and I will share my results with them. So they may control my results. Furthermore, in ter ms of member checking, I will share my interpretations with the students whether I comprehend and interpret their problem strategies or not. The data collection and data analysis part will take approximately 4 months. Limitations The aim of the study is to define students solution strategies and their difficulties. I will plan to categorize students strategies. If students do not solve problems by using different strategies, than I may not get sufficient information. To overcome his limitation, I will choose the participants according to their mathematics achievement and I may make them think over the problems to solve the problems in different way. The other limitation is that if I could not audiotape or videotape the interviews, I may be forced to collect data and analyze the data. Also, in order to provide mutual comfort, I plan to choose participants from the school which I will work. In this way, there will be communication between me and participants before starting interview. References Aiken, L. R. (1971). Intellective variables and mathematics achievement: Directions for research. Journal of School Psychology, 9, 201-212 Anderson, J. N (2000). Cognitive psychology and its application. (5th edition). New York: Worth Publishers. Battista, M.T. (1980). The importance of spatial visualization and cognitive development for geometry learning in preservice elementary teachers. Journal for Research in Mathematics Education, 13(5),332-340 Battista, M.T. (1990). Spatial visualization and gender difference in high school geometry. Journal for Research in Mathematics Education, 21(10),47-60. Battista, M. T. Clements, D. H. (1995). Enumerating cubes in 3-D arrays: Students strategies and instructional progress. A research report. Battista, M. T. Clements, D. H. (1996). Students understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27(3), 258-292. Battista, M. T. Clements, D. H. (1998). Finding the number of cubes in rectangular cube building. Teaching Children Mathematics,4, 258-264 Ben-Chaim, D., Lappan, G., Houand, R.T. (1985). Visualizing rectangular solids made of small cubes: Analyzing and effecting students performance. Educational Studies in Mathematics, 16, 389-409. Bishop, A. J. (1979).Visualizing and mathematics in a pre-technological culture. Educational Studies in Mathematics, 10, 135-146 Carpenter, T. P., Fennema, E., Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal,97 (1), 3-20. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (1999). Childrens Mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Chang, K.Y. (1992). Spatial and geometric reasoning abilities of college students. Unpublished doctoral dissertation, Boston University (UMI No: 9221839). Chase, C. I. (1960). The position of certain variables in the prediction of problem-solving in arithmetic. Journal for Research in Mathematics Education, 54(1), 9-14 EARGED (2003). TIMSS 1999 nc Uluslararas Matematik ve Fen Bilgisi almas Ulusal Rapor. Ankara: MEB. EARGED (2005). PISA 2003 Projesi Ulusal Nihai Rapor. Ankara: MEB. Fennema, E., Carpenter, T.P., Franke, M. L. (1992). Cognitively guided instruction. The Teaching and Learning of Mathematics, 1 (2), 5-9. Fennema, E., Carpenter, T.C., Franke, M.L, Levi, L., Jacobs, V.R., Empson, S.B. (1996). A longitudinal study of learning to use childrens thinking in mathematics instruction. Journal for Research in Mathematics Education, 27 (4), 403-434. Fennema, E. Sherman, J. (1977). Sex related differences in mathematics achievement, spatial visualization and affective factors. American Educational Journal, 14(1),51-71. Fennema, E. Tartre, L. (1985). The use of spatial visualization in mathematics by girls and boys. Journal for Research in Mathematics Education, 16,184-206. Ferrini-Mundy, (1987). Spatial training for calculus students: Sex differences in achievement and in visualization ability. Journal for Research in Mathematics Education, 18 (2), 126-140 Fraivilling, J. L., Murphy, L. A., Fuson, K.C. (1999). Advancing childrens mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education, 30 (2), 148-170. Franke, M. L., Kazemi, E. (2001). Learning to teaching mathematics: Focus on student thinking. Theory into Practice, 40 (2), 102-109. Forman, E. Ansell, E. (2001). The multiple voices of a mathematics classroom community. Educational Studies in Mathematics, 46. 115-142. Forman, E. (1996) Learning mathematics as participation in classroom practice: Implications of sociocultural theory for educational reform. In Steffe, L., Nesher, P., Cobb, P., Goldin, G. Greer, B. (Eds.) Theories of mathematical learning, pp. 115-130. Mahwah, NJ: Lawrence Erlbaum Associates. Fuys,D., Geddes, D. Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education. Monograph, Vol. 3, 1-196 Haws, L. (2002). Three-dimensional geometry and crystallography. Mathematics Teaching in the Middle School, 8(4), 215-221 Hiebert, J. Wearne, D. (1993). Instructional task, classroom discourse, and students learning in second grade-arithmetic. American Educational Research Journal, 30 (2), 393-425. Lave, J. Wenger, E. (1991) Situated learning: Legitimate peripheral participation. New York, NY: Cambridge University Press. Lean, G. Clements, M. A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12 (3), 267-299. Malerstein, A. J., Ahern, M. M. (1979). Piagets stages of cognitive development and adult character structure. American Journal of Psychotherapy,23(1), 107-118 MEB (2006). lkretim matematik dersi retim program 6-8.. snflar. Ankara: MEB. Moses, B.E. (1977). The nature of spatial ability and its relationship to mathematical problem solving. Unpublished doctoral dissertation, Indiana University, Bloomington (UMI No: 7730309). National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics.Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.Reston, VA: Author. Ng, G. L. (1998). Exploring childrens geometrical thinking. Unpublished doctoral dissertation, University of Oklahoma, Norman (UMI No: 9828779). OConnor, M.C. Michael, S. (1993). Aligning academic task and participation status through revoicing: Analysis of a classroom discourse strategy. Anthropology Educational Quarterly, 24 (4), 318-335. Okur, T. (2006). Geometri Dersindeki Baarszlklarn Nedenleri ve zm Yollar. Unpublished doctoral dissertation, Sakarya University. Olkun, S. (2001). rencilerin hacim formln anlamlandrmalarna yardm edelim. Kuram ve Uygulamada Eitim Dergisi, 1 (1), 181-190. Olkun, S., Altun, A., Polat, Z.S., Kayhan, M., Yaman, H., Sinoplu, B., Glbahar, Y., Madran, R.O. (2005). Retrieved April 28, 2007 from https://yunus.hacettepe.edu.tr/~hyaman/ Olkun, S., Toluk, Z. (2004). Teacher questioning with an appropriate manipulative may make a big difference. IUMPST: The Journal, 2, www.k-12prep.math.ttu.edu. zbellek, G. (2003). lkretim 6. ve 7. snf dzeyindeki a konusunda karlalan kavram yanlglar, eksik alglamalarn tespiti. Unpublished doctoral dissertation, Dokuz Eyll University Piaget, J., Inhelder, B., (1967). The childs conception of space. (F. Langton J. Lunzer, Trans.) New York: W. W. Norton Piaget, J., Inhelder, B., Szeminska, A. (1960). The childs conception of geometry. (E. Lunzer, Trans.) New York: W. W. Norton Presmeg, N. C. (1986). Visualization and mathematical giftedness. Educational Studies in Mathematics, 17, 297-311 Reid, J.R. (1995). Mathematical problem solving strategies: A study of how children make choices. Unpublished master thesis, The University of Western Ontario (Canada). Seil, S. . (2000). Onuncu snf rencilerinin geometri problemleri zme stratejilerine ynelik bir alma. Unpublished Master Thesis, Middle East Technical University, Ankara. Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27 (2), 4-13. Sfard, A. (2000). On reform movement and the limits of mathematical discourse. Mathematical Thinking and Learning, 2 (3), 157-189. Smith, I.M. (1964). Spatial Ability. San Diego: Robert Knapp Steele, D. F. (2001). Using sociocultural theory to teach mathematics: A Vygorskian perspetive. School Science and Mathematics, 101(8), 404-416. Van Hiele, P. M. (1986). Structure and insight. New York: Academic Press. Webb, N. M., Nemer, K. M. Ing, M. (2006). Small-group reflections: Parallels between teacher discourse and student behavior in peer-directed groups. The journal of the learning sciences, 15 (1),63-119

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The ideals of the declaration which is most important free essay sample

The main purpose of the Declaration of Independence was to declare that the American colonies were separating from England (thus declaring independence). The document also stated what the principles were as the foundation for seeking independence. The document can be broken down into four parts.Introduction -States the purpose of the document; which was to explain why the American people were declaring independence from Great Britain. Theory The theory of certain immutable rights possessed by all individuals; and that these rights are not granted by the government but rather theyre inherent to human nature. That the main purpose of a benevolent government is to secure and protect these rights. Additionally that government is based on the consent of the governed (the people) who are the sole source of this governmental authority. Finally, that should the government persistently violate the principle of a proper and good government, then the people have the right to overthrow it. We will write a custom essay sample on The ideals of the declaration: which is most important or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page The third part of the document is a list of grievances against King George III, who was singled out to represent the actions of the British government. These grievances are examples of actions that violated the criteria for good government stated in the second part of the Declaration of Independence. These grievances, therefore, justify separation from the Kings bad government and establishment of a good government to replace it. Sovereignty The fourth part of the document is an unqualified assertion of sovereignty by the United States of America. It proclaims the determination of Americans to defend and maintain their independence and rights.