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.
Abstract This paper discusses how few ancient civilizations have given so much as have the ancient Egyptians. Like their Greek counterparts, the Egyptians' innovations in the areas of mathematics, architectural design, mythology, literature (albeit in the Egyptian case, hieroglyphics) and government were seized upon as exemplars by later empires in both the Western and Middle Eastern worlds. The paper examines a certain aspect of ancient (and still fairly inscrutable) Egyptian society known as the Sed Festival.
Abstract The paper explains that few mathematical figures have achieved the status that the Golden Ratio has throughout the historical past and well into the modern and post-modern era. The paper discusses how the Golden Ratio (GR) has also been termed the Golden Mean or the Divine Proportion because of its seemingly endless recurrence in nature as well as its perpetual application not only by mathematicians but by artists and architects alike, as well as others (Clawson b. 33). The paper explains that artists and architects seem to enjoy the predictability of the GR as well as its symmetry.
From the Paper "The GR has been attributed to the Greeks whose quest for knowledge, 0x01 graphic as employed by the Greeks as being representative of the GR in many respects where, "Golden Means. F = AB/BC = CH/BC = IC/HI = 2DE/EF = EG/2DE" (Clawson a. 121). In this respect the GR has also been related to other unique and fascinating mathematical principles."
Abstract This paper explains that Pythagoras left no written works; what is known about him and about his school is from a book written by the Pythagorean, Philolaus of Tarentum, which influenced Plato's philosophy. The author points out that Pythagoras considered himself a philosopher, which is clearly mentioned in Diogenes Laertios' letters, and the dialogues of Cicero, inspired by the Greek Iore. The paper states that the Pythagorean philosophy is related with the theory of numbers, which are not only the symbols of reality but the very substance of real things.
From the Paper "The theory of numbers is strongly connected with the harmony of spheres, this way, number ten being the perfect one has a body under the earth moving parallel with it and which is invisible to us. The five planets, the sun, the moon, and the earth with its counter-earth, moving from west to east at rates of speed proportionate to the distance of each from the central fire, produce eight tones which give an octave, and, therefore, a harmony. In their psychology and their ethics the Pythagoreans used the idea of harmony and the notion of number as the explanation of the mind and its states."
Abstract This paper reviews and discusses the literature 'Understanding the Concepts of Limit and Continuity in Calculus Courses at the High School and/or Early College Level'. According to the paper, the ten fastest growing career fields include five that are computer-related which rely heavily on the knowledge of mathematics.
Outline:
Background and Overview
Studies in the Concept of Limit and Continuity in High School and Early College-Level Calculus Courses
Interactions of Methods and Questions for Further Study
From the Paper "Clearly, then, just going through the motions and mechanics of calculus may serve to help familiarize these young learners with the basic steps involved. There are a number of studies to date, though, that have confirmed that these students typically experience a number of problems in understanding key calculus concepts that adversely affect their ability to master them (Bezuidenhout, 2001). To help facilitate the process, just as handheld calculators have been introduced into some early mathematics classes to facilitate routine calculations to allow more time for instruction, Heid (1988) suggested early on that computers could likewise be used to help beginning calculus students learn more by automating the algorithmic functions required. Although there is clearly a trade-off involved in such an approach, this author maintains that this approach would allow more class time for development of higher mathematics concepts."
Abstract The paper explores the Study Island computer-based mathematics program that was developed and designed to help children learn mathematical skills and to increase standardized math test scores. The paper looks at research that measured the effects of the utilization of this web-based program on the students' reference competency test mathematics scores. The paper relates that the research showed that students who were enrolled in a class where Study Island was used did better on a standardized test than other students. The paper concludes that computer-aided instruction has more advantages than disadvantages and suggests students use the Study Island mathematics program for the next state standardized test.
Outline:
Introduction
What is Study Island?
Research Study
Background and Significance of the Problem
To What Extent Did Computer Assisted Instruction Increase Standardized Mathematics Achievement Scores?
What Benefits and Advantages Emerged When Using Computer Aided Instruction With Sixth Grade?
How Can the Web-Based Mathematical Instruction Study Island Program Influence Student's Perception of Leaning?
Conclusion
From the Paper "Mathematics is one area that is being affected by the use of such technology. There have been several computer-based mathematics programs developed and designed to help children learn mathematic skills and to increase standardized math test scores. One such program is called Study Island. Study Island was developed in 2000 and offers practical and web-based instruction for students in all grade levels. The program is based around state standards so that students can use this program to prepare for state standardized testing and increase overall testing scores. Study Island is currently being used in over 6000 schools around the nation and is helping over two million students."
Abstract This extensive paper describes the use of Computer Algebra Systems (CAS) in helping students develop their mathematical skills. The research contained in this report addresses the use of CAS in the mathematics classroom. It also addresses the attitudes shared by teachers and students alike as it relates to the use of this technology. In particular, the Maple CAS system is evaluated. The author states that the purpose of this research is to take a systematic approach to the design and evaluation of the teaching, learning and assessing mathematics courses using the CAS Maple. The focus of the evaluation are first year service mathematics courses at a university. The effectiveness of different ways of incorporating Maple activities into such courses is also examined.
Table of Contents
Introduction and Statement of the Research Questions
Literature Review
How People Learn Mathematics and the Role of Technology
Review of Studies Related to the Use of Technology in the Classroom
Utilizing Computer Algebra Systems
CAS in the Classroom
Survey Papers
The Research Methodology of the Study
Conclusion
From the Paper "The purpose of this research is to take a systematic approach to the design and evaluation of the teaching, learning and assessing mathematics courses using the CAS Maple. Of particular interest are first year service mathematics courses at RMIT University. The effectiveness of different ways of incorporating Maple activities into such courses will also be examined.
The investigation will be conducted as a research and development activity through which Maple activities are designed and evaluated in a feedback cycle and we follow an Action Research methodology. Initially, examples from the literature and relevant theories concerning mathematical understanding were sought in order to inform the development of new resources. Student's responses to the first cycle of activities in 2003 were obtained. The conclusions drawn are informing the development of resources for the next cycle. This process will continue over the course of six semesters. The research methods utilized are observations of classes, analysis of student's work, responses to specially designed test instruments, use of feedback questionnaires and structured interviews. Some use will be made of video will also be utilized to record and analyse methodology to evaluate the teaching and learning of mathematics using Maple in a computer lab."
Abstract This study analyzes results of the Virginia Standards of Learning tests. The author contrasts one group of high school students who used online computerized methods of testing versus the second group, who were tested with the traditional "paper-pencil" method. The author argues that computer-based testing is in its preliminary phases. This study, therefore, is intended to let scholars research the dependability of these tests. This research begins to fill this gap and offer future direction for additional research to be carried out.
Outline:
Abstract
Statement of Problem
Rationale of Study
Research Objectives
Literature Review
Hypothesis
Methodology
Participants
Measures
Procedure
Implications and Conclusion
References
From the Paper "The control of computers is the seeming dominant feature at the closing stages of the twentieth century. In the 1970s computers operated behind closed doors to tally books, record paychecks, organize weather newscasts, and perform whichever errands whose qualities frequently incorporated repellent recurring processes. The 1980s saw a transformation. Computers took a permanent position in the forefront of firms and businesses. Soon after, the computers replaced the human employees in the department of customer services. Personal computers with telecommunication were more common. Nowadays, the computers handle far more complicated procedures than what they did before. "Machine intelligence", "Inference engines", and "Expert Systems" are expressions that are gradually growing in trend. "
Abstract In this article, the writer conducts a mixed methods research in order to measure the effect of the utilization of a computer-assisted instruction (CAI) program. The writer explores the web-based Study Island program on the student's reference competency test mathematical scores. In addition, the research focuses on how the students involved in the study related to the interactive program. The writer concludes that CAI, when combined with traditional instruction, has been shown to significantly increase standardized test scores for students in mathematics testing. Further the writer points out that research has indicated that the use of computers in schools both supports learning and is useful in the development of higher order skills such as critical thinking and scientific inquiry, because the students are engaged in complex tasks in a collaborative learning environment.
Outline:
Objective
Introduction
Background and Significance of Problem
Purpose of the Study
Importance of the Study
Research Questions
Literature Review
Summary of Literature Review
Methodology
Research Design
Data Collection/Analysis
Bibliography
From the Paper "The Study Island program was designed for the purpose of assisting students in their mastery of the content of the New Jersey Core Curriculum Content Standards for Mathematics. The interface of this program is user-friendly allowing students to go through the program by steps. This program can be used by students at any location with Internet access. Further, this program is inclusive of questions that are styled just as the New Jersey GEPA providing an accurate assessment of student progress as the school year progresses. Students are able to learn at their own pace using this program and the student is able to choose from several different formats of learning. The beauty of this program is that statistics are kept in the program which can be viewed by educators throughout the year which allows assessment of individual student's learning and the program itself as the year progresses."
Abstract This paper reviews dyscalculia, an academic developmental disorder that is believed to affect about 3 to 6 percent of the population. According to the paper, an individual with dyscalculia may struggle with a number of different issues such as, linguistic skills, perceptual skills, attention skills and mathematical skills.
From the Paper "Dyscalculic individuals can usually learn the sequence of counting words, but can have trouble moving back and forth, especially in twos, threes or more. Estimating numbers is also more of an issue in comparison to others their age. With dyscalculia, the lack of understanding number magnitudes, which is typical of children in the age group of 7 to 11, is absent. Math vocabulary may also to lead to problems for students, especially when they have to use several different words at a time, such as "add," "plus," and "combine" that can be used interchangeably. Other terms, such as "hypotenuse" and "to factor" do not occur in normal discussions and must be learned specifically for mathematics. At times, an individual may understand the underlying concept but not how to use a specific term correctly."
Abstract In this article, the writer studies the character of Josiah Willard Gibbs, a mathematician and physicist. The writer discusses that he managed to achieve great things during his lifetime and lead the world on to greater and better scientific discoveries. The writer points out that Josiah Willard Gibbs has been recognized as one of the greatest American scientists of the nineteenth century. Further the writer notes that it is Gibbs who managed to provide a sound thermodynamic foundation to physical chemistry, to America and to the entire world.
From the Paper "The second work that Gibbons published in the same year was "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces". From the years from 1876 to 1878, Gibbs published two memoirs, which were later to be combined into one work, entitled, "On the Equilibrium of Heterogeneous Substances". Added to this, Josiah Willard Gibbs has contributed to various other spheres, like for example, crystallography, the determination of planetary and comet orbits, and also to electromagnetic theory. The most interesting phenomenon that Gibbs managed to achieve was that he made the practical side of science appealing and fascinating. Gibbs was also recognized as a 'theoretical physicist' of international stature, and he received a patent in the year 1866 for an improved type of railroad brake."
Tags: thermodynamics, science, accomplishments, mathematical, field
Abstract This paper discusses John von Neumann's contributions to the fields of quantum physics, functional analysis, set theory, economics, computer science, numerical analysis, hydrodynamics, statistics and other mathematical fields. The paper also discusses his contribution to the creation of the hydrogen bomb. It goes on to describe some of his most influential achievements.
From the Paper "John Von Neumann inarguably contributed a wealth of knowledge to the development of computers, and without his contributions the face of technology today would be primitively underdeveloped. However, Neumann may have canceled out the "good" he did in an act of self-fulfilling equivalent exchange with his work in the realms of math and science with the contributions he made to warfare and massive weaponry. The name Von Neumann is associated as much with the Atomic Bomb as it is with computer programs (Wilson), and Neumann may have had even more devastating projects on the horizon at the time of his unexpected death from cancer. During the Second World War, von Neumann worked as a consulted to both armed forces and civilian agencies that were involved in wartime projects. Neumann's genius was in high demand, and he was able to design an implosion method for bringing nuclear fuel to explosion, as well as playing an integral part in the development of the hydrogen bomb. (Cabrera) According to one of Neumann's biographers, "It has been stated that von Neumann's electronic computer hastened the hydrogen bomb explosion on November 1, 1952." (Bochner)"
Abstract This paper addresses both Zipf's Law and Benford's Law. It describes the theories of each law in detail and discusses how they can be applied to various situations. It describes Zipf's Law as not a theoretical law, but rather an experimental law. The paper then discusses the significance of these two laws to the field of mathematics.
From the Paper "However, there is some lack of precision where this is concerned. Most items have to occur a number of times that is actually an integer (Li, n.d.). In other words, a word that is seen within a document cannot appear in that document 2.5 number of times. It either must appear, in this example, 2 times or three times, since there cannot be an area of the document where only 1/2 of the word appears. Despite the fact that there is some variation and lack of precision, however, when wide ranges are examined and one only desires to have a relatively close approximation, many of the natural phenomena that are seen in this world do obey Zipf's law (Li, n.d.). This is seen to hold true as long as the individual examining the issue is not looking for scientific precision and will accept the slight variation that is seen."
Tags: integer, distributions, phenomenon, frequency
Abstract This paper explains that Pafnuty Chebyshev's lifelong work, which left a lasting legacy that influenced the study of mathematics and statistics worldwide, included many subjects such as probability theory, quadratic forms, orthogonal functions, the theory of integrals, the construction of maps and the calculation of geometric volumes. The author points out that, during his pursuit of a doctorate degree, Chebyshev wrote an important prize-winning book "Teoria Sravneny" from which his profound knowledge of probabilities greatly aided the Russian insurance industry.The paper relates that his most notable students were Aleksandr Lyapunov, and Andrew Markov.
From the Paper "Chebyshev's family moved to Moscow in 1832 mainly for their eldest son's educations. Chebyshev was taught by one of the best teachers in Moscow, P.N. Pogorelski. Pogorelski taught Chebyshev math and physics. Pogorelski was regarded as one of the best elementary math teachers; he had written numerous books about elementary mathematics. Chebyshev was taught by highly known professionals for other subjects as well. With his knowledge of the French language, it helped him speak internationally about math."
Abstract The paper examines how, though not always apparent, there are a number of significant connections between mathematics and art. The purpose of this paper is to demonstrate how the fundamental similarity between math and art can be exploited as a means to teach difficult mathematical concepts to students. To show how this could happen, a particularly complex--if intellectually intriguing--mathematical concept is explored: the concept of distance in hyperbolic geometry, specifically in a Poincare disk.
Outline:
Introduction
Context: What Is Hyperbolic Geometry?
Context: Who Is M.C. Escher?
Developing an Appropriate Class Project
Conclusions
Works Cited
From the Paper "Since mathematics education produces singular anxiety for many students, this confluence with art presents significant possibilities for the imaginative educator (Granger 10). It is possible that we could, as educators, use art as a physical and visual means of explaining complex mathematical concepts in other than abstract terms. Over reliance on complex equations and difficult language can and will stymie many students. By endeavoring to ground mathematical theory in artistic reality, students can leans mathematical lessons in the process of seeing how math and art aren't really all that dissimilar."
