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Universal Truths in Math, 2005.
This paper examines some theories in order to determine if there are any universal truths in mathematics.
675 words (approx. 2.7 pages), 3 sources, $ 26.95
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Abstract
The paper looks at the theories of George Lakoff and Rafael Nunez, as well as those of Keith Devlin in order to explore if there are really any universal truths in maths. Set off by an excerpt from Robert Sawyer's novel "Computing God," the paper theorizes that there really are not any universal truths, at least none that can be defined until all forms of life are themselves defined. The paper points out that this is neither the quantification nor the metaphor and symbolism that math requires and uses.
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Biological Evolution and Mathematical Development, 2005.
This paper discusses if biology, evolution and the development of mathematics have a connection.
675 words (approx. 2.7 pages), 3 sources, $ 26.95
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Abstract
The paper examines the possibility that biology, evolution and the development of mathematics are linked more closely than mathematicians would necessarily have us believe. The paper challenges the basic Platonist assumption that abstract mathematical concepts possess concrete being and are consequently fundamental parts of the universe. Instead, the paper discusses the possibility that mathematics is a construction of the human mind and an evolutionary development.
From the Paper
"Most often we take mathematical truth for granted. Rather than understand it as an historical construction - not so different from any other human production, such as language - most people fully believe that mathematics is natural and etched into the very fabric of the cosmos. This is a classic Platonist view of the universe in which even abstract concepts have physical reality. Twentieth century theorists, especially in linguistics, have repeatedly challenged the efficacy of abstract concepts. But mathematics is still, in some part, understood to be the realm of the gods with right-brains their unerring prophets."
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Mathematics and Universal Truths, 2005.
This paper discusses whether mathematical thought can lead to fundamental truths and highlights the use of metaphor in mathematical thought.
675 words (approx. 2.7 pages), 3 sources, $ 26.95
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Abstract
The paper argues that fundamental truths cannot be arrived at by math. The paper is of the opinion that this is insofar as the questions we ask, the processes we use and the assumptions we make are shaped by environmental, biological and contextual factors that have little - if anything - to do with "rational" and purely objective thought. The paper places great emphasis upon the place of metaphor in the construction of mathematical thought.
From the Paper
"The question of whether there are unquestionable truths in mathematics is indeed a puzzling one. This paper will examine the matter by looking a few readings from our class notes. As will soon become apparent, there is much doubt that mathematics leads irrevocably to universal truth; indeed, in the limited space available, this paper will suggest that, because so much of mathematics is metaphorical in nature, Euclidean mathematics and other "relational" branches of math may lead us into the realm of creative metaphor and no further. In fact, as Sawyer seems to suggest, mathematical "truth" - all truth - is essentially the product of cultural epistemology and ontology."
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High School Math, 2005.
This paper provides a lesson plan for the teaching of mathematics in high school.
3,825 words (approx. 15.3 pages), 10 sources, $ 151.95
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Abstract
In this article, the writer develops a lesson plan for teaching quadrilaterals in high school math and considers some of the underlying pedagogical theory and how it applies. The writer notes that quadrilaterals are defined as polygons with four sides, and while this encompasses any such figure, the more important of these are parallelograms, squares, and rectangles. Further the writer shows how the student can discover certain relationships by looking to the real world.
From the Paper
"Below is a lesson plan for the instruction of high school students in the mathematics, specifically on the subject of quadrilaterals. This lesson is found in the larger subject area of Geometry. Quadrilaterals are defined as polygons with four sides, and while this encompasses any such figure, the more important of these are parallelograms, squares, and rectangles. The lessons in this subject area define these figures and address different mathematical concepts applying to them, including ways of determining area, angles, and other ratios. This lesson should introduce the students to the area of quadrilaterals defining this area of Geometry by describing the elements that make up a quadrilateral and the mathematical relationships that define this type of figure, as well as the formulae that are used to calculate different characteristics."
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Biology, Evolution and Mathematics, 2005.
This paper studies the connections between biology, evolution and mathematics.
675 words (approx. 2.7 pages), 6 sources, $ 26.95
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Abstract
This paper examines the question of what mathematical premises would be dependent on the biological and physical evolution of a given species, assuming of course that we knew other intelligent species had evolved. The writer discusses that some critics suppose that language and mathematics by extension are dependent upon the physical parameters set out by the body. The writer explains: ten fingers and hence a decimated numerical system. This essay probes that assumption.
From the Paper
"There is almost certainly a connection between biology and the ability to conceptualize. The basic logical processes that we, as humans, often take for granted are in reality quite dependent upon our own physical evolution. How likely is it that we would have developed a base ten numerical system if we didn't just happen to have ten fingers? It would be perfectly plausible to have a base six system or base twelve, for example. But the question becomes how much of mathematics is a product of biological evolution and how much of it exists unto itself."
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Astronomy from Ptolemy to Galileo, 2005.
This paper studies science, in particular astronomy, making use of the book "Science without Limits" by James Perlman.
675 words (approx. 2.7 pages), 1 source, $ 26.95
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Abstract
This paper examines the history of science in terms of changes in astronomy from the time of Ptolemy to Galileo, based on the book "Science without Limits" by James Perlman. The writer notes how the ancients saw science as a form of philosophy, while by the time of Galileo, observation was being joined with experimentation to examine concepts and find the truth.
From the Paper
"The history of astronomy shows the development of science as a discipline from the ancient world to the Renaissance, from the time of Ptolemy to the time of Galileo. Over that period, astronomy began to shift from a philosophy to a science. Science in the ancient world was not created out of whole cloth and was based on observations and the application of reason. Mathematics were also used to develop ideas about the universe. Mathematics is itself an application of reason, though aspects of mathematics have also been developed through observation and testing. By the time of Galileo, however, science was gaining a more experimental structure, and Galileo himself tested many ideas directly. His astronomy was also based on observations, but he was able to observe more directly and closely with the telescope. Perlman notes that "science in large part . . . is a matter of testing assumptions"."
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Matrices, 2005.
This paper present a study of the theory of matrices that includes its history, development and uses.
900 words (approx. 3.6 pages), 2 sources, $ 35.95
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Abstract
This paper discusses the theory of matrices, how it was developed, how it changed, some of the applications for which it has been used, and other aspects of the issue. The writer notes how the underlying ideas are ancient and began with the Babylonians and Chinese and then resurfaced in the seventeenth century with the world of Cayley and others. Further the writer points out that the theory of matrices has led to uses in physics, chemistry, and economics as well as mathematics.
From the Paper
"Matrices are a means of visualizing mathematical concepts and relationships in graphic form. A matrix is a rectangular set of elements viewed as a single entity, identified by the number of rows and columns of which it is made. Matrices can be added or multiplied on the basis of an algebra of matrices, and one application of this sort of operation is seen in vector analysis and in the solving of systems of linear equations. The basis for the matrix is found in the Cartesian system of Rene Descartes, whose contribution to mathematics was in the development of analytical geometry, closely tied with the development of the Cartesian system of mapping on a grid or graph, for Descartes saw that a function or polynomial can be represented graphically by points."
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Mathematics Education Dissertations, 2005.
This paper describes two distinct mathematics education dissertations.
2,250 words (approx. 9.0 pages), 2 sources, $ 89.95
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Abstract
This paper explains that the field of mathematics education provides considerable support for a variety of perspectives, which include new and innovative ideas and concepts. The author points out that graduate-level mathematics students are typically required to develop and submit a comprehensive dissertation to demonstrate their knowledge and skills. The paper presents two distinct mathematics education dissertations in greater detail, emphasizing the key strengths and weaknesses of each argument and the supporting literature reviews.
From the Paper
"The field of mathematics education provides considerable support for a variety of perspectives, which include new and innovative ideas and concepts that provide valuable contributions to the subject. It is evident that today's mathematics educators provide valuable knowledge, information and skills to mathematics students of all ages, and that there is a wide body of research that exists regarding mathematics education that is critical to the field. Graduate-level mathematics students are typically required to develop and submit a comprehensive dissertation to their respective schools in order to demonstrate their knowledge and skills in order to earn a graduate degree. The following discussion evaluates two dissertations written in the field of mathematics education, promoting different concepts in unique ways. A comparison and contrast is introduced, along with an evaluation of the key strengths and weaknesses of each dissertation."
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Mathematics Pedagogy, 2005.
This paper discusses of teaching mathematics.
3,825 words (approx. 15.3 pages), 13 sources, $ 151.95
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Abstract
This paper examines issues of mathematics pedagogy and the degree of the contextualization of the subject matter in teaching mathematics. The author points out that mathematics is often presented more as a more abstract examination of numbers and measurements that appear, when mathematics really is always relevant and should be seen in the context of the real world. The paper states that mathematics pedagogy needs to develop a way present mathematics within this real world context.
From the Paper
"The issue of relevance in education is often a question of the contextualization of subject matter, meaning that the subject relates to the lives of the students because it can be seen in the context of their lives, with issues understandable because they are applicable to the real world. Mathematics is often presented more as a pure Mathematics has the dual character of being both a language (a symbol system) and an underlying model of relationships among actions with objects. As such, it fits closely with the Vygotskian description of sign-sign relationships and de-contextualized knowledge. At the same time, its development in relation to human actions on objects gives it a prominent place in Piagetian analysis. Furthermore, mathematics teaching requires the recognition of mathematics as a sociocultural achievement worthy of reproduction in new generations."
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Mathematics Education, 2005.
This paper analyzes if it is possible to test the understanding of mathematics.
3,600 words (approx. 14.4 pages), 10 sources, $ 142.95
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Abstract
This paper is a report on a questionnaire given out to students in college to test their understanding of mathematics. The author points out that this research investigates the difference between knowledge and understanding and seeks the way to assess understanding. The paper concludes that the questionnaire derived from the GED in mathematics is a way to test understanding of high school mathematics for students who have graduated from high school.
From the Paper
"The purpose of this analysis is to see if it is possible to test understanding, specifically the understanding of mathematics. Such an analysis tests both mathematics teaching and mathematics learning, though at this preliminary stage it is not clear whether the teaching method is what is most important or the learning style of the student. Testing understanding is different from testing knowledge, for the latter shows that the student has assimilated ideas and even processes, while the former shows that the student has learned the underlying theory and can apply it in different situations. In mathematics, testing understanding is perhaps more common in normal testing than would be the case in certain other disciplines where simple facts are more common. In mathematics, of necessity the student must show an understanding of theory in order to apply mathematical concepts to written problems and arrive at the correct answer."
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