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Philosophy of Mathematics, 2007.
An analysis of the universal nature of mathematics and developments in the philosophy of mathematics.
1,899 words (approx. 7.6 pages), 6 sources, APA, $ 60.95
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Abstract
This paper considers some of the major developments in the philosophy of mathematics regarding the capacity of mathematics to be universally valid and applicable. It presents some of the basic arguments and schools of thought of the philosophy of mathematics. The paper then analyzes whether, at its foundation, mathematics can have a legitimate claim to be universal.
Table of Contents:
The Problem Of The Ideal And The Real
Math As Logic
Math As Structure
Application And Universality
From the Paper
"This problem, Russell's paradox, proved to be an intractable problem for Frege which, after it was pointed out to him, he could not overcome. The impact upon the philosophy of math was major. An important attempt to boil math down to logical principles had proven unsuccessfully, and eventual efforts to rescue the project by Russell and others were unable to develop a logicism that showed math as both consistent and complete. Therefore math cannot be said to be universal by appeal to logic alone."
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Stephan Korner's "The Philosophy of Mathematics", 2004.
Summary and review of Stephan Korner's "The Philosophy of Mathematics: An Introductory Essay".
1,091 words (approx. 4.4 pages), 2 sources, MLA, $ 38.95
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Abstract
The first part of this paper expounds on Stephan Korner's discussion, in "The Philosophy of Mathematics, of the nature of mathematics, and the three main schools of thought relating mathematics to philosophy. The paper continues with a discussion on logicism and why it provides the clearest way to look at mathematical concepts and the best way to explain mathematical philosophy.
From the Paper
"Mathematics is an indispensable science that justifies and confirms many aspects of other scientific subject matter. Mathematics relies on conclusions not assumptions and evidence is required to confirm theoretical entities as true. Of course the debates exist as to which school of thought holds the most validity. Mathematical realism will always be different to each of these philosophical schools and arguments can be found to both support and reject each school of thought."
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"The Problems of Philosophy", 2004.
This paper discusses "The Problems of Philosophy" by Bertrand Russell (1872-1970), widely regarded as one of the great philosophers of the 20th century.
1,415 words (approx. 5.7 pages), 5 sources, APA, $ 47.95
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Abstract
This paper explains that Bertram Russell is considered the founder of analytic philosophy, the tradition dominating 20th century Anglo-American philosophy. He is know for his writings in the fields of epistemology, logic, the foundations of mathematics, ethics, political and social philosophy, the philosophy of science and antiwar. The author points out that Russell in Chapter V of "The Problems of Philosophy" emphasizes that the knowledge of things is comprised of two components: (1) "Knowledge of Acquaintance" wherein the person is knows directly through his or her acquaintance with the object, without the intermediary of any process of inference or any knowledge of truths and (2) "Knowledge by Description" wherein, similar to Plato's "Forms" analysis, the person knows a description and knows that there is just one object to which this description applies. The paper relates that Russell states that people process information in different ways, but everyone must use the same fundamental steps to arrive at an accurate perception of the world and to understand it.
Table of Contents
Introduction
Review and Discussion
Background and Overview
Knowledge of Acquaintance
Knowledge by Description
Evaluation and Critique of Russell's Position and Arguments
Conclusion
From the Paper
"Russell was not trying to develop a comprehensive definition of his ideas about how and why people think about the world in the ways they do; rather, he was attempting - to borrow a phrase from the academicians - operationalize the terms involved in understanding. Certainly, in order to understand the subtle nuances of how people think about "things," "ideas," and "truths," there must be some solid basis for describing the components involved. For instance, Russell sums up Chapter V by pointing out that "We shall not at this stage attempt to answer all the objections which may be urged against this fundamental principle. For the present, we shall merely point out that, in some way or other, it must be possible to meet these objections, for it is scarcely conceivable that we can make a judgment or entertain a supposition without knowing what it is that we are judging or supposing about.""
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A History of Mathematics, 2002.
This paper discusses some aspects of the history of mathematics from the earliest mathematical records to the modern era.
1,400 words (approx. 5.6 pages), 9 sources, $ 53.95
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Abstract
This paper only touches on some selected aspects of a broad and encompassing subject. The author begins by outlining some of the key developments as a whole before further subdividing into three sections: Greek mathematical developments; Chinese and Middle Eastern developments; and Western developments. The paper concludes by drawing attention to the enormous scope of the history of mathematics.
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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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Career Possibilities for Women In Philosophy, 2001.
This paper examines which career paths a women with a degree in philosophy may be able to follow.
1,100 words (approx. 4.4 pages), 8 sources, MLA, $ 38.95
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Abstract
The writer explains that because a degree in philosophy is so broad and all-encompassing, the career possibilities are endless. The paper examines options in the following fields: teaching, law, ministry, mathematics and journalism.
From the Paper
"Philosophy is not a hard science requiring that problems be worked out with formulas and centuries-old givens. The American Philosophical Association says that, ?Philosophy is a basic field of inquiry. Its range encompasses ideas and issues in every domain of human existence; and its methods apply to problems of an unlimited variety. The major in philosophy can develop not only philosophical skills and sophistications but also intellectual abilities that are readily applicable to pursuits in everyday social and personal life.? (APA)"
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Language and Mathematics, 2006.
Discusses the similarities between natural human languages and mathematics.
1,350 words (approx. 5.4 pages), 3 sources, $ 53.95
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Abstract
Normally, natural human languages and mathematics are regarded as being diametrically opposed to one another. Mathematics is formal and is marked by precision; the objects of theory must be carefully defined so that the informal can be formalized. Natural human language on the other hand is flexible, and one term can denote not just multiple meanings but opposing ones as well. This paper explains that, in spite of these differences, human language and mathematics actually share common ground such as the fact that both human language and the language of mathematics actually have a precise formal structure.
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Mathematics Instruction in English in Bilingual Classrooms, 2005.
Research proposal for examining the effects of mathematics instruction in English in bilingual classrooms.
2,211 words (approx. 8.8 pages), 14 sources, APA, $ 68.95
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Abstract
This paper proposes a research project that would examine the effectiveness of English instruction of mathematics on Second Grade ELL (English language learners) students as compared to the effectiveness of instruction in their native language. The proposal is in response to the controversy surrounding the issue of how best to teach mathematics to children from non-English-speaking backgrounds, since it has been found that the best way for children to learn to use mathematics to organize, understand, compare, and interpret their experiences is by making a connection between mathematics and their everyday lives. The paper examines whether ELL students should be taught how to make this connection in their native language with gradual exposure to English in language classes, or whether they should be immersed in English as early as possible. The paper includes an annotated bibliography and an observation checklist of lessons taught in class.
Introduction
Setting
Problem/Issue
Research Question
Hypothesis
Methodology
Subjects
Instrumentation
Significance of the Study
From the Paper
"Mathematics is a powerful tool for interpreting the world. Research has shown that for children to learn how to use mathematics to organize, understand, compare, and interpret their experiences, mathematics must be connected to their lives. Such connections help students to make sense of mathematics and view it as relevant. There has, however, been controversy with regard to children from non-English backgrounds and the best ways to get them to make those connections. Questions are raised regarding how to instruct these children who are referred to as English language learners (ELL?s). Should they initially be taught in their native language with gradual exposure to English in language classes, or should they be immersed in English as early as possible."
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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 Curriculum Review, 2002.
A comprehensive analysis of the problems in the elementary school's mathematics curriculum.
3,545 words (approx. 14.2 pages), 6 sources, APA, $ 99.95
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Abstract
This paper examines the question of how to reverse the trend of lack of educational progress, specifically in the world of mathematics. This is considered through an evaluation of three elementary schools' stated mathematics curriculum, and how they compare to the standards of the National Council of Teachers of Mathematics published standards. The process of this evaluation is a point by point comparison between the NCTM standards and the printed curriculum guidelines for these schools. Specific points which are supportive, and which may fail to reach the guidelines are identified and discussed for each school. The purpose of this evaluation is not to approve or reject these curricula, but rather to identify specific applications which can be either improved through change, or strengthened by building upon existing positive initiatives.
Introduction
Discussion of the NCTM Standards
West New York Public Schools, West MY
Bogota Public Schools, Bogota, NJ
North Bergen Public School System, North Bergen, NJ
Bibliography
From the Paper
"According to national statistics, the mathematical educational progress of American elementary students has failed to keep progress with the rest of the world. This stinging indictment of the educational system of the most technologically advanced culture in the world has caused a serious evaluation of the standards and goals of the elementary system. According to the National Council of Teachers of Mathematics, there are knowledgeable teachers in the system. The teaching staff has adequate support and resources. In a society which depends daily on mathematics, there is opportunity for students to learn and apply math principles and facts. There also is an abundance of access to technology to support the educational process. Finally, if students are considering careers, those in math related fields, such as engineering, financial planning, accounting and many others are some of the highest paying positions in our current job market."
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Language and Mathematics, 2005.
This paper discusses the similarities of language and mathematics.
675 words (approx. 2.7 pages), 3 sources, $ 26.95
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Abstract
This paper explains that language and mathematics are similar in that they both have rules. The author points out that people make assumptions when it comes to language and mathematics, which may not be proven and only are assumed to be correct. The paper relates that mathematics and language have many similarities such as syntax and semantics.
From the Paper
""Colorless green ideas sleep furiously," are words with specific meaning but put together in a sentence they clearly lack meaning (Devlin, Born). Does language and communication mean the same thing? Do the formulas for mathematics always have the same answers? Language and mathematics do not always make sense without the formal rules of syntax. People make assumptions when it comes to language and mathematics that may not be proven and only assumed to be correct. Mathematics and language have many similarities such as syntax and semantics."
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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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Teaching Mathematics, 2005.
This paper explores challenges in teaching advanced mathematics.
3,825 words (approx. 15.3 pages), 10 sources, $ 151.95
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Abstract
This paper discusses that studies have demonstrated that there are a minimum of seven specific forms of intelligence including mathematics. The author identifies the emergence of standards-based education within the United States and its significance for teaching, particularly in regards to high-school and pre-college mathematics education. The paper examines the effectiveness of selected teaching methods used by educators to help convey mathematics to their students.
From the Paper
"Advanced mathematics, specifically content at the algebraic level and beyond, creates challenges for many students. Learning strategies, modes of learning, and types of student intelligence all impact the method through which the student approaches mathematics and the extent to which they acquire and are able to apply learned information. Yet increases in national standards demand that students perform at specific levels of academic abilities before they are passed into the next grade, and an emphasis upon standards-based testing as a component of admission to college further demands specific performance levels from students. This combination of factors stresses a need to help students achieve certain goals within their academic careers in regards to advanced mathematics. In order to explore the challenges in teaching advanced mathematics to students, this paper will survey three specific components. First, this paper will explore types of intelligence."
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Music and Mathematics, 1995.
Examines the inter-relations between music and mathematics. Discusses the theory and philosophy of music and focuses on the mathematical foundations of such composers as Mozart, Schoenberg, and Cage.
1,575 words (approx. 6.3 pages), 9 sources, $ 55.95
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From the Paper
"Music and mathematics are closely linked, and musical rhythm serves as an example of the practical use of different mathematical principles. It has recently been noted in fact that the mathematical regularity of certain music, such as that of Mozart, can be a spur to clearer thinking, at least for a short period of time after listening to a piece of music. Music has a psychological effect that is partly explained by its mathematical regularity, seen in the way music is divided into regular bars, beats, and different note lengths. Psychologists have discovered the importance of patterns in music and in aspects of human behavior. Music satisfies certain human needs for order and rhythm, and mathematics both explains and empowers this process.
Edward Rothstein writes about the relationship between music ..."
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David Hilbert and Mathematics, 2002.
Discussion of David Hilbert and his impact on the study of mathematics in the 20th century.
1,400 words (approx. 5.6 pages), 3 sources, $ 53.95
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Abstract
This paper is on David Hilbert and mathematics. He became famous for developing his "axiomatic" and "existential" methods. His proposal in 1900 of twenty-three problems for the coming century set the course of much subsequent mathematics. It was in this context that Hilbert came to be seen as the person who set the foundation for many mathematical questions.
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