Sunday, March 31, 2019

The importance of geometry

The importance of geometryThis chapter includes the importance of geometry and the importance of learn how to solve traditionalistic book of account tasks by learners in school mathsematics. The concerns of maths education stakeholders nigh article chore resolve based on national and external assessments and the suggestions provided by researchers and educators to improve students murder when solution sound out conundrums atomic number 18 in whatever case follow-uped. The theories and empirical studies that focus on comprehension, representation, and solution of intelligence of honor riddles atomic number 18 summarized.Although utilize math, and in particular geometry, to model situations from work places has been part of education for centuries, the review of the literature starts with the beginning of the late nineteenth century, with the exception of Ren Descartes (1596-1650) doctrine of hassle solving (Encyclopedia Britannica, 1983). The review includes recommendations from important publications that inform mathematics education. Research-based theoretical and conceptual frameworks that support the solution process of mathematics word troubles ar utilise to develop a research hypothesis for examination in this study. line of work result and resolving Word Problems Some mathematics educators and researchers believe that a trouble lies as an obstruction between two ends, the problem and the solution, without any clearly defined ways to traverse (Brownell, 1942 Mayer, 1985 Polya, 1980). This interpretation may as well be applied to word problems beca give umpteen researchers include math word problems in problem solving research (Kilpatrick, 1985). The logic behind this definition can be traced back to Ren Descartes (1596-1650) philosophy which suggests that method is necessary to put out the truth of nature. The following excerpt from Encyclopedia Britannica (1983) on Descartes Dis raceway on Method is worth mentioning as part of his doctrine of problem solving1The Discourse is a philosophical classic. It hides the implicit in(p) assertion that the human mind is essentially sound and the only means of attaining truth never to accept anything as true which I you did not clearly and distinctly see to be so. Descartes thus implies the rejection of all accepted ideas and opinions, the determination to doubt until convinced of the inappropriate by self-evident facts. The second rule is an instruction to analyze the problem to be figure out. Once cleared of its prejudices, the mind, use the deterrent example set by mathematicians, must divide each of the difficulties under examination into as many separate as possible that is, disc over what is relevant to the problem and shave it as far as possible to its simplest data. The third rule is to transport my thoughts in order, beginning with objects that are the simplest and easiest to know and so proceed, gradually, to knowledge of the more complex . The stern rule is a warning to recapitulate the chains of reason out to be certain that there are no omissions. These simple rules are not to be considered a mere automatic formula they are to be regarded as a mental discipline, based on the example of numeral practice. (p. 600) Schoenfeld (1987) summarized the four forms of Descartes problem solving plan. The idea in build I is to reduce an algebra problem to a single variable equality for solving. Phase II suggests reducing a mathematics problem to an algebra problem and solving it according to phase I. In phase III, any problem situation is converted to a mathematics problem by mathematizing. In phase IV, the problem is then(prenominal) solved victimization the ideas in phase I and II. In two of his many rules (rules XIV and XV), Descartes suggested the drawing of diagrams as an aid to solving problems (pp. 29-36). It is noted from the above excerpt of Descartes problem solving process that a problem should be broken dow n to its parts before attempting to solve it. Each part should likewise be understand separately. For example, a word problem can usually be solved if one can understand the words (vocabulary), their meaning, their interconnection, the objects they represent, and the relevance of those objects in the problem. Solving a word problem is also sometimes referred to as problem solving. jibe to Branca (1987), problem solving is an alternative meaning of applying mathematics to polar circumstances (p. 72). That means if a situation is explained in words, or in a word problem, then applying mathematics as a shit to solve that problem situation may be treated as problem solving. Also, Brown, Cronin, and McEntire (1994) deferd that assessment on word problems has different names, including math reasoning, problem solving, word problems, as well as story problems (p. 32). Although word problems have been extensively employ in problem solving research, the coincidence and differences b etween word problems and problem solving should be clarified. A word problem is also a problem to solve, according to the definitions previously mentioned. many an(prenominal) educators think solving word problems require the problem solving skills. For this dissertation, word problems will refer to problems of the type that appear in standardized assessments and tests much(prenominal) as the NAEP, the sassy Jersey HSPA, the SAT, and the ACT. They are not problems related to public human life without unstated facts where students have to wander, collect facts for mathematizing the situation before solving them. The problems in this study can be attempted exploitation general heuristics (Polya, 1945 Schoenfeld, 1985), as well as through the application of Descartes problem solving principle and early(a) methods based on Descartes philosophy. According to Kilpatrick (1987), in recent age, some researchers in mathematics education have used problems with increasing level of diff iculty and learning opportunity that require the sweet combination of rules and reasoning. A some similar problems were used in this research. (See vermiform appendix K for sample problems) However, these problems are infrequently found outside of tests or class assignments. Solving Word Problems A Goal of maths pedagogy Learning to solve problems is the principal reason of studying mathematics (National Council of Supervisors of Mathematics, 1977, p. 2). The NCTM (Krulik Reys, 1980) also suggested that problem solving be regarded as the major goal of learning school mathematics from 1980 to 1989 and repeated that recommendation more recently (NCTM, 2000). Mathematics transaction of students, which includes problem solving, became a major concern in the U. S. with the dismissal of A Nation at Risk (U. S. Department of study, 1983). This publication recommended snap on the teaching of geometric and algebraic concepts and real-life importance of mathematics in solving probl ems. The low word problem solving ability of U.S. students of 9, 13, and 17 years of suppurate was verified by the first data from the NAEP conducted in 1973. While analyzing the results of that assessment, Carpenter, Coburn, Reys, and Wilson (1976) concluded It is most disturbing to ascertain the suggestion that many students receive very little opportunity to learn to solve valet problems. The assessment results are so poor, however, that we wonder whether this is not the case. A consignment to working and thinking about word problems is needed for teachers and their students. (p. 392) Table 2.1 shows the outgo scores of NAEP on mathematics obtained by U.S. students in grades 4, 8, and 12, on a 0 to 500 outdo, from 1990 to 2007. Table 2.2 shows the percent of different types of word problems decent answered by the students in grades 8 and 12. According to Braswell et al. (2001), the work levels of 249, 299, and 336 are considered deft levels for fourth-, 8th-, and 12th-g rade students, respectively. Table 2.1 indicates very small improvements in the NAEP test scores for fourth-grade and eighth-grade students over the span of 17 years (1990 to 2007). However, these scores are below the suggested proficiency levels. It may be noted from Tables 2.1 and 2.2 that improvement, either in overall capital punishment or in word problem solving skills for all dynamic U.S. students, is trivial. Also the scores that hover around 230 for grade 4, 275 for grade 8, and ternion hundred for grade 12 on a 0 to 500 crustal plate are too low. Of particular concern is an middling of only 4% correctly answered questions for the years 1990 to 2000 (Table 2.2) by U.S. grade 12 students on multitude and surface area related problems. International assessments such as the FIMS in 1965, the SIMS in 1982, the PISA in 2003 and 2007, and the TIMSS in 1995 and 2003 further attested U.S. students poor problem solving skills and highlighted their low mathematical achievement in comparison to students from new(prenominal) active countries. The FIMS and SIMS conducted mathematics assessment of 13year-old students and high school seniors (National Council of Educational Statistics, 1992). According to the NCTM (2004), the PISA measures the numerical skills and problem solving aptitude of 15-year-old students on a scale of 0 to 500whereas the TIMSS measures fourth and eighth grade students ability on concepts on a scale of 0 to atomic number 19. The NCTM also give notice (of)ed that the NAEP, TIMSS, and PISA, which are low-stakes tests, relent group performance results of students. High-stakes tests, like New Jerseys HSPA or other state mandated tests, as well as the SAT and ACT, focus on the performance of individual students. Of the three assessments, NAEP, TIMSS, and PISA, TIMSS and NAEP have the most in common in terms of mathematical concepts and cognitive necessity (NCTM). The findings from the mathematics results of the PISA of 2000 and 2003 rep orted by Lemke et al. (2004) indicated that U. S. performance in algebra and geometry was lower than two-third of the participating OECD countries. Even the top 10% of the participants in the U.S. were outperformed by more than half of their OECD counterparts in solving problems. The then U.S. Education Secretary emphasized the need to reform high schools on top priority basis (U.S. Department of Education, 2005). The latest PISA (2007) results indicated that the mathematical accomplishment of U.S students is lower than the international average. According to TIMSS (2003), U.S. students of fourth and eighth grades scored on average 518 and 504, respectively in mathematics. These scores were higher than the average score of 495 of the fourth-grade students in the 25 participating countries and the average score of 466 of the eighth-grade students in the 45 participating countries. However, these scores were lower than the 4 Asian countries and 7 European countries for fourth grade an d lower than the 5 Asian countries and 4 European countries for eighth grade. Although the average score of U.S. eighth-grade students improved by only 12 points from 492 in 1995 to 504 in 2003, there was no change reported by TIMSS in their score from 1999 to 2003. Overall, these scores on a scale from 0 to 1000 indicate that students in grades four and eight in the U.S. only achieved about 50% mastery of the concepts tested. National (NAEP, 2007) and international (FIMS, 1965 SIMS, 1982 TIMSS, 1995, 1999, 2003) assessments indicate that student achievement in mathematics remains a major educational concern. Those assessments use multiple choice, short-response, and open-ended word problems which are similar to those on the New Jersey HSPA, SAT, and ACT. Since students mathematical skills are measured using one or more of the above assessments, learning to solve word problems must be considered a major goal of mathematics education and a major component of assessing student achieve ment in mathematics. Further, learning to solve word problems related to real-life situations using mathematical concepts also helps students to be successful at work and in their lives. Geometry as a Cornerstone of Mathematics-History of Problem Solving and Geometry In ancient India, the rudiments of Geometry, called Rekha-Ganita, were formulated and applied to solve architectural problems for building temple motifs (Srivathsa, Narasimhan, Sasat 2003, p. 218). The 4000 years old mathematics that emerged in India during The Indus Civilization (2500 BC-1700 BC) proposed for the first time, the ideas of zero, algebra, and finding square and cube roots in Indian Vedic literature (Birodhkar, 1997 OConnor Robertson, 2000 Singh, 2004). The entailment of studying geometry is evident from the bypast mathematical records. The book, A History of Mathematics (Suzuki, 2002) provides the mathematical innovations made by the most brilliant mathematicians from ancient times until the 20th cent ury. Some of the mathematical developments presented in this book that are related to problem solving and geometry are discussed next. According to Suzuki (2002), the ancient Egyptians (3000 B.C.) demonstrated their skills in solving word problems by an Egyptian scribe on the mathematical papyri using the concepts of linear and nonlinear equations without any mathematical notations. That is, every problem solved by an Egyptian scribe was a word problem (p. 13). In order to redraw prop lines after the yearly flooding of the Nile, the Egyptians developed realistic geometry related geometric figures, but not their abstract properties. Also, their geometry is filled with problems relating to pyramids (p. 16). The Babylonians (1700 B.C.) also routinely solved more complicated and complex problems entirely verbally (Suzuki, 2002, p. 28) without any administration of mathematical notations. Their ways of solving interest relate problems show their march on mathematical skills. Accordin g to Suzuki, the Babylonians also developed methods for calculating the area of triangles, trapezoids and other polygons. Before Pythagoras (580-500 B.C.), the Pythagorean Theorem was well known to the Babylonians (p. 31). The development of pre-Euclidean geometry goes back to the age of Plato (427-347 B.C.). It is said that the entrance plaque to Platos school in Athens read, let No whizz Unversed In Geometry Come Under My crown (Suzuki, 2002, p. 74). According to Suzuki, Plato had probably discovered the word mathematics from the mathema, meaning the three liberal arts, arithmetic, geometry, and astronomy (p. 74). Later, Euclid (300 B.C.), who lived in Alexandria, Egypt, wrote the Elements, a conglomeration of 300 years of Greek geometrical development. The Elements was so important for the next two super acid years of mathematics that Euclidean geometry became an essential part of learning mathematics until it faced the first serious mathematical challenges (p. 86) in the 19th century. The significance of understanding geometry for high school students has been a part of recommendations of the committees on mathematics education in the U.S. since 1894 (Commission on Mathematics, 1959 National Education Association, 1894, National delegacy on Mathematical Requirements, 1923 Progressive Education Association (PEA) Committee and the articulation Commission, 1940 The National Committee of Fifteen, 1912). An account of these committees reports may be found in the 1970 yearbook of the NCTM, A History of Mathematics Education in the fall in States and Canada. A brief of the recommendations of these committees are presented below. The first national group of experts that turn to mathematics education was the subcommittee on mathematics of the Committee of Ten (National Education Association, 1894). They considered the goals and curriculum for mathematics education and recommended preparatory work on algebra and geometry in the upper elementary school curricu lum. On demonstrative geometry, the committee punctuate on the importance of elegance and finish in geometrical materialisation (p. 25). About demonstrative geometry, the committee further stated, there is no student whom it will not brighten and strengthen intellectually as few other exercises can (p. 116). This suggests all mathematics teachers engage their students in using the geometric concepts to visualize their surroundings and to geometrically demonstrate what they visualize. The final report of The National Committee of Fifteen on the Geometry Syllabus (National Education Association, 1912) recommended using realistic approaches to exercises in mathematics instruction. Eleven years later, its final report, The shake-up of Mathematics in Secondary Education (The National Committee on Mathematical Requirements, 1923) also stressed the importance of the studying geometry. The commission advocated that the course of study in mathematics during the seventh, eighth, and ninth years contain the fundamental notions of arithmetic, of algebra, of intuitive geometry, of numerical trigonometry, and at least an introduction to demonstrative geometry (p. 1). One of the practical aims of this ecommendation was to encourage familiarity with geometric forms common in nature and life, as well as the elementary properties and relations of these forms, including their measurement, the development of space-perception, and the exercise of spatial imagination.

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