Abstract This paper examines the life and the work of the 19th century German mathematician Friedrich Bernhard Riemann, whose ideas concerning geometry of space had a profound effect on the development of modern theoretical physics, including providing the foundation for the concepts and methods used later in relativity theory.
From the paper:
"An examination of the facts of Riemann's family background would not have led one to suspect that he would have become the great mathematician that he would develoo into. He was the second of six children of a Lutheran pastor and it was this pastor/father who gave him his first formal education. Indeed, much of his early education was centered in his family, which was by all accounts both happy and deeply devout. He later attended the local high school, where he made quick and substantial progress in mathematics, soon moving beyond the ability of his teachers to educate him further (Laugwitz 38-41). He quickly mastered calculus and theory of numbers of Adrien-Marie Legendre. After graduating from the high school (or gymnasium), he studied at the universities of G?ttingen and Berlin from 1846-51. It was at this point in his education that he became interested in problems concerning the theory of prime numbers, elliptic functions, and geometry, theoretical interests that would guide much of his later work."
Abstract Fermat's Last Theorem (FLT) has been one of the most fascinating theorems in mathematics. This paper looks at the conformities and the disparities of this statement. It assesses the theorem and the problem it comes to solve as well as the theorem's proof. It gives a detailed mathematical exercise which he solves using the theorem.
From the Paper "Pierre de Fermat was born near Montauban in 1601. He was born in a family reared by a leather-merchant who was his father and was educated at home. He was essentially a lawyer and was an amateur mathematician. Throughout his life, Fermat published only one mathematical paper, which was written anonymously and appeared as an appendix to a book. He died in 1655."
Abstract This paper helps to explain and justify the creation of calculus. Isaac Newton helped to solve some of the most perplexing problems the world has imagined, and the method he used in doing so is still used for the same purpose today. Newton's creation of calculus and ideas of using it to prove the universal laws of nature made human reason the most powerful method of thought and most definite route to seeking the truth.
From the Paper ?In mathematics, a certain surprising thing happens again and again. Someone poses a simple question, a question so simple that it seems no useful result can come from answering it. And yet it turns out that the answer opens the door to all kinds of interesting developments, and gives great power to the person who understands it.? (Saywer 3) This quote from a prestigious professor of mathematics parallels the story of Isaac Newton and his development of calculus. Isaac Newton helped to solve some of the most perplexing problems the world has imagined, and the method he used in doing so is still used for the same purpose today. There is a popular myth that Newton was sitting under a tree when an apple fell from it, and he asked himself what force could pull the apple to the Earth. Whether this story is true or not is uncertain, but the image is clear of Newton getting hit on the head with the apple of epiphany. He used calculus to prove that gravity pulled objects to Earth and held the planets together, and also to prove his world-renowned three laws of motion. By forming this revolutionary method of problem solving, Newton not only paved the way for new roads in mathematics but also changed the way that people thought and sought out answers. Newton's creation of calculus and ideas of using it to prove the universal laws of nature made human reason the most powerful method of thought and most definite route to seeking the truth.
Abstract A look at the different discoveries by Archimedes focusing on what he is most greatly known for - discovering the solution of pi. His approximation of pi between 3-1/2 and 3-10/71 was the most accurate of his time, and with this discovery he devised a new way to approximate square roots.
From the Paper "Little known details remain about the life of Archimedes who was one of antiquity's greatest mathematician, Archimedes. Most of the facts about Archimedes? life come from a biography written by the Roman biographer Plutarch. What is known, is that he was born in Syracuse, Sicily in the year 287 B.C., and died in 212 B.C. at the age of 75 in Syracuse. I was able to come up with an astonishing amount of information on Archimedes for this paper. It seems that there is no end to his accomplishments, and I tried not to leave out any of them. "
Tags: archimedes, burning, cicero, greek, marcellus, mathmetician, mirrors, pi
Abstract This paper presents a detailed examination of the history of calculus. The writer takes the reader on an exploratory path through the origins of mathematics and then on to the history of calculus. The people who are credited with its invention as well as the forms that it took are all included in the discussion.
From the Paper "The history of mathematics is one in which the topic follows the actual subject. Mathematics are taught by building on foundational blocks. Each block is taught and mastered and when that is completed the next block is introduced. The origin and history of mathematics follows the same path. The history of calculus is perhaps the most interesting of the mathematical techniques. The history and origin of calculus is founded in philosophy as well as science and it is one of the most fascinating of the mathematical theories and practices."
Abstract This paper looks at the life and work of Karl Gauss. It examines his theory on Plate Tectonics, the theory of Motion of Heavenly Bodies and several other theories that were developed during his lifetime. The writer first briefly gives a bio of Gauss and then attempts to explain the theories in laymen's terms.
From the Paper "There are many well known mathematicians from history whose work is well known and position widely recognised. However, there are also many lesser known mathematicians that have also made equally valuable contributions. Karl Friedrich Gauss is one of these, and as such is a worthwhile individual to study. Gauss developed many ideas and theories which are still in use today. He is best known for his theory of plate tectonics and his work entitled "Theoria Motus Corporum Coelestium" ; Theory of the Motion of Heavenly Bodies in 1809. With Wilhelm E. Weber; a physicist he also developed a theory concerning geomagnetism. Much of his work is still used today, including work in the fields of physics, astronomy, and his statistical theories are even used in software algorithms. In this we see man who has made large contributions to the world of mathematics and related disciplines (Schaaf, 1964)."
Examines causes & effects of gender gap in learning math, social sterotypes & teacher bias, anxiety & self-image, examples, cultural support and solutions.
3,150 words (approx. 12.6 pages), 20 sources, 1997, $ 111.95
From the Paper "The purpose of this review of literature is to examine factors that contribute to the under-achievement of girls in mathematics and projects and programs which have been used to remediate the situation. The review begins with a brief description of the problem, and ends with the formulation of conclusions regarding factors producing the problem and interventions that will reduce and or eradicate the problem.
Description of the Problem
In general, at both elementary and high school levels, boys tend to have higher achievement levels in mathematics than do girls; moreover, girls tend to have significantly more negative attitudes toward mathematics than boys (Froebe, 1996). In ..."
This paper discusses that mathematics is at the core of understanding business and social sciences: Financial statements, supply and demand, forecasts, linear regression, equilibrium and elasticity.
1,350 words (approx. 5.4 pages), 5 sources, 1994, $ 47.95
From the Paper "Mathematics is at the core of understanding business and social sciences. Both disciplines make use of arithmetic, quantitative methods, statistics, linear regression and calculus as they seek to describe, predict and analyze the vast array of numerical data available in the fields. This research examines the application of math in these areas with a particular emphasis on math with regard to the supply and demand function.
Anyone selling a product or providing a service uses basic arithmetic to determine how much money they take in and how much money they pay out. When the expenses are less than the revenues, they make a profit. This simple accounting principle becomes more complex as the items associated with the various components increases in complexity. Revenues can be based on cash received, or they may be placed on accounts receivable."
From the Paper "Statisticians work with large masses of data. Before any conclusions can be drawn from such data, it must be condensed and arranged in a usable form. One of the most common ways to summarize and describe a mass of data is to arrange a frequency distribution table. These tables can then be graphed with the frequency scale on the y-axis and the interval being graphed on the x-axis. Above each interval a horizontal line is drawn which corresponds to the frequency of the interval, resulting in a stair-step histogram pattern. Connecting the midpoints of these class intervals produces a frequency polygon and an interval curve. Distribution curves which can be "folded" vertically so that the two halves of the curve are essentially the same are said to be bilaterally symmetrical. Perfectly symmetrical curves which have a bell shape are said to be normal curves, or Gaussian curve ... "
This paper examines an application of the statistical procedure of population sampling: Describes theory and techniques and assesses validity of application in population sampling.
1,350 words (approx. 5.4 pages), 5 sources, 1995, $ 47.95
From the Paper "This research examines an application of the statistical procedure of population sampling. The initial part of the examination describes sampling procedures, and illustrates the use of the procedures in an application. Following the description and illustration, the accuracy and appropriateness of the application is discussed.
Description of the Procedure, and An Illustration of the Use of the Procedure in An Application
Population sampling procedures are described in this section. This description is followed by an illustration of the use of the procedures in an application.
Abstract Kids decoder rings in cereal boxes, the puzzles in the comic pages of the daily newspapers and high-tech encryption all have something in common, they are all variations of cryptography. The paper shows how, ever since the early days of civilization, people have been trying to encode massages to keep secrets from falling into the hands of the wrong person. Today the science and math of cryptography go way beyond switching letters around according to a certain pattern, but if a person remembers that the basic idea is the same, cryptography can be a fascinating endeavor into math, science, and even into language itself. This paper reviews the history of cryptography and the many things encryption has been used for in the past. It then looks at how encryption is used in modern times and for what purposes. The paper explains cryptography from a mathematical point of view, following the development of encryption and cryptography mathematically. Finally, it looks at the future of this science.
From the Paper "One of the most important developments came in the form of the Wheel Cipher. The Wheel Cipher was created by Thomas Jefferson, possibly with the help of Dr. Robert Patterson, a mathematician at the University of Pennsylvania. In 1913, Captain Parket Hitt reinvented the Wheel Cipher in strip form. This lead to the creation M-138 -A, used in World War II. Just a few years later in 1916, Major Joseph O. Mauborgne ut Hitt's strip cipher back into the wheel form, strengthened the alphabet construction, and produced the device that would lead to the M-94 cipher device. These devices, along with encryption courtesy of the Navajo people, helped the allies defeat Germany, Japan, and Italy in World War II."
Abstract Euclid gave the world much of the information it has on planar geometry in his five postulates. The paper shows that while the first four are relatively easy to understand, the fifth one is very difficult in relation to the others. It is this fifth postulate that many people feel can never be proven. The paper discusses how there are those that say it is simply incorrect, those that say it's both true and false and others that say there is no possible way to prove it, and Euclid himself may have realized that the task was impossible. The author of the paper surmizes that if someday the fifth postulate is proven to be either true or false, and the decision is agreed upon, then it could change the way mathematics are done and the way geometry is looked at.
From the Paper "Theoretically it would be possible for the lines to move toward one another so slowly, because of the low degree of angle, that they take a huge amount of space to come together at the end. But is it possible to have such a slight angle that the lines are almost parallel? They would be so close to parallel at that point that the impression that they are drawing closer together wouldn't be noticed unless they were looked at over miles at one time. That must be possible, but they still must meet somewhere in infinity.
Perhaps Euclid was right and the lines do meet somewhere, but the angles can be so minute that the lines go on almost to infinity, and we don't have the capabilities to calculate just how far that is yet. Perhaps Euclid is wrong and lines will go on into infinity still never touching, but only being a hair's width apart. Mathematicians may never know, since they haven't discovered any way to prove Euclid's fifth postulate by now."
Abstract Diophantus is referred to more often than not as the "Father of Algebra", although algebra predated Diophantus. His contributions to the study of algebra, however, have led to this attribution. This paper reviews his life, his mathematics, his place in the history of mathematics and the relevance of his work in the 21st century. The review is presented in discussions of his life, his work, his place in mathematics history and the contemporary relevance of his contributions.
From the Paper "Diophantus lived in the third century A.D. The best estimates of his birth and death years are 200 A.D. and 284 A.D. Other conjectures of these data range from 150 B.C. to 350 A.D. Exactly when he lived, however, is not nearly as relevant to contemporary society as is what he accomplished while he lived. What is generally agreed upon about Diophantus is that he was a normal man who married, had children, and lived a normal but scholarly life. Not all of his work has survived, at least not in a recorded form that may be attributed directly to him. That work which has survived and which can be directly attributed to him, however, has established him as mathematics theoretician of worthy note (Heath (Vol. I) 15-16)."
Tags: mathematics, numbers, equation, Arithmetica, formula
Abstract Reviews the article "Morning, Noon, Night and Math" and its discussion of Diane McCarty's approach to teaching the relevance of math in everyday life. As an educator, McCarty sought to dispel the myth that mathematics is not needed to perform daily tasks. McCarty designed a math unit with the following goals in mind: 1) experience the role of math in everyday life, 2) recognize relationships among different aspects of mathematical processes, 3) become more familiar with the use of mathematical precepts in various careers, 4) relate the use of math to common human activities, and 5) enhance students understanding of mathematics.
From the Paper "The math unit created by McCarty was very effective in showing the students the importance of mathematics in everyday life. The children found that math was an instrumental part of all three environments"this was especially true in the work environment. The interviewees encouraged children to learn as much as they could about math even if math wasn"t their favorite subject. The interviewees were very effective in demonstrating to the students the relevance of math in the work environment."
Tags: math, in, the, bath, bulletin, board, importance, of, mathematics, mathematics, unit
The paper analyzes the phenomenon of Mathematics Disorder, and the ways teachers, administrators and parents can contribute to eradicating this phobia.
Abstract The paper discusses the nature of the phobia as being an irrational fear of being unable to acquire good mathematics skills. The paper explores ways to get rid of this phenomenon using curriculum material designed for weak students, multimedia and integrating parent cooperation, planning learning models, interacting with pedagogic experts and using workshops.
From the Paper "Teachers, administrators as well as parents of children are becoming more aware of the deficiency in not meeting the challenge of change and adopting Math as part of the curricula [Reys et al 1999]. The focus of the standardized test is to achieve the purpose of dissemination of math knowledge to children and testing them based on the skills acquired. This model in the olden days was called the NCTM Standards in which all recommendations of implementation activities are provided while in the modern day the NSF Standards have replaced the NCTM and has been able to make significant improvement on the old."
Tags: change, failure, uncertainty, strategy, school
