Abstract This ten-page undergraduate level paper that discusses the effects of technology on teaching geometry. It will reflect the current trends in mathematics education from grades six to twelve and will point out the concerns of teaching geometry with the help of technology.
Abstract This paper discusses how people, in general, like to get a visual picture of what they are hearing about and how, through the media and constant representation of statistical data as hard fact, numbers can control people's opinions on issues. It shows how one of the largest issues regarding statistics and their appealing nature is the fact that most of us are innumerate. It also shows how, in addition to innumeracy, the public's opinion of ideas often leads to skewed views on issues; statistics can become so alluring to activists that they can say something that will change a large group of people's minds on an issue, and then they will get what they want.
From the Paper "Although even though some statistics are wrong, people want to believe them so bad that they will ignore all logic just so that they will have a numerical view of the situation. Perhaps the biggest real life example of this is a social statistic that Joel Best-in his book Damned Lies and Statistics-describes as "The worst social statistic ever... Every year since 1950, the number American children gunned down has doubled" (Best 1). To anyone using this statistic to promote gun control, this statistic is gold, and it sounds believable too. But if you analyze it you'll find otherwise."
Abstract This paper summarizes, analyzes, and critiques John Corcoran's work, in which he puts the different definitions of the word "implications" into distinct, well-defined, recognizable contexts. The paper reviews Corcoran's work from the perspectives of history, logic, philosophy, and linguistics.
From the Paper "In his paper, Corcoran lists several "implication" phrases in common English usage which presuppose that the premise ?A? is true and that validates the conclusion B. Corcoran also critiques the philosopher/mathematician Frege in that Frege's thesis on logic is constrained by his instincts. Corcoran avers that Frege included pure logic in his thinking and did not make room for deduction as a major contributor to the concept of implication. ?Frege's strategy was to show that no appeal to intuition is required for the derivation of the theorems of number theory. This in turn required that he show that the latter are derivable using only rules of inference, axioms, and definitions that are purely analytic principles of logic(Ref)??which he did not."
A research study using co-integration analysis to study the relationship between the stock index cash and futures market in relation to price discovery.
Abstract Co-integration has come to represent an econometric data analysis method that has been utilized to determine the long-run equilibrium relationships among nonstationary economic variables. This paper uses co-integration analysis to determine the relationship between the stock index cash and futures market in relation to price discovery, market stability, and market efficiency. The data for the study was collected from the Athens Derivatives Exchange S.A. (ADEX), with the main data for the study being the returns of the FTSE/ASE-20 futures and spot index. In order to study the relationship between the ADEX stock index cash and futures market in this paper, daily closing price returns of the FTSE/ASE-20 Index are considered for the period from 3 January 2000 to 27 July 2003. The paper includes several graphs and tables.
Paper Outline:
Data
Methodology
OLS Results
Co-integration Results
References
From the Paper "GARCH modeling represents an important data analyses procedure as it provides a means of further understanding and modeling volatility, taking into account excess kurtosis (i.e., fat tail behavior) and volatility clustering, two important characteristics of financial time series. It provides accurate forecasts of variances and covariances of asset returns through its ability to model time-varying conditional variances. As a consequence, the application of GARCH models has been identified as useful in risk management, portfolio management and asset allocation, option pricing, foreign exchange, and the term structure of interest rates."
This paper explores the ways in which graph theory can be joined to computer simulation programs to make the planning stages of road redesign more efficient and more accurate.
Abstract This paper explains that graphs allow for a simplification of the real world, doing away with extraneous details without sacrificing any information necessary for the task. The author points out that graph theory is the best tool to use to solve the problem of road conversion because it is the most parsimonious. The assumptions made by graph theory more closely than any other model or theory match the real-world conditions. The paper states that, in converting roads, the effect in the real world will be a disruption of what had been the shortest paths between different points, and urban planners will include some consideration of shortest path issues. Tables and figures.
Table of Contents
Introduction
Graph Theory as the Basis for Conversion of Two-Way Roads
Undirected Graphs to Digraphs
Reachability Problem
Shortest Path Problem
Importance of Simulation
Comment on Validity of Data
Conclusion
From the Paper "The specific shortest path method that is most useful in this particular type of problem is Pallottino's graph growth algorithm with two queues. It must be emphasized that the determination of the shortest path algorithms is perhaps the most important component of any network analysis. It is also quite often the first step in any network analysis as the determination of the shortest path is often needed as a key datum in making later choices."
Abstract Discusses collecting, organization, analyzing, and presenting data. Presents two basic types of data: categorical and quantitative. Discusses various approaches to organizing data and approaches to other tasks, averages and variations, and the theory of probability.
From the Paper The term "statistics" to many (perhaps most) people implies a collection of numerical data about a topic. The extent to which most people have confidence in the validity of such a collection of data depends upon ..."
Abstract Examines elements of the scientific method, including concepts, definition, hypotheses, and theory. Describes statistical analysis as the process through which data becomes knowledge. Cites alternative models for statistical analysis.
From the Paper "The underlying basis of statistical analysis is the scientific method. The foundations of the scientific method are (1) concepts, (2) definition, (3) hypotheses, and (4) the..."
Abstract Discusses history from the Early Chinese remainder theorem to the recent AKS system. Discusses the concept behind the two postulates, the search for a method to efficiently test for prime numbers, and the classical definition of a prime number.
From the Paper "History of Primality Testing from the Chinese Remainder Theorem to the Agrawal-Kayal-Saxena (AKS) Algorithm
Introduction
This paper will present a non-linear analysis of the historical concept of primality testing from the earliest days of mathematical ...."
Contends that the relationship between gender, math, and science achievement remains unclear.
1,800 words (approx. 7.2 pages), 7 sources, 2003, $ 63.95
Abstract The paper shows that the differences between math and science levels for males and females is not consistent across age groups. It cites factors for males performing better than females in math and science.
From the Paper "The Relationship Between Gender and Math/Science Achievement
Introduction
Public schools have shifted to a racial and linguistic composition that includes Hispanics and other ethnic populations. Many students are English language learners and many have ..."
Abstract This paper examines two of the systematic mistakes that humans tend to make when they make decisions that they are likely to consider to be rational. These include mistakes or inclinations toward both pessimistic and optimistic biases.
From the Paper "Except, of course, that we?re not. But it is true that humans are relatively bad at purely rational thinking. This should not perhaps be surprising to us: We are not, after all, computers, which are far better than are humans at making rational decisions and providing rational calculations about situations. This is not entirely a bad thing: Humans have apparently (though the process of evolution) sacrificed the ability to make perfectly rational calculations for the ability to excel at what those who are trying to teach computers to think like humans call fuzzy thinking. We are good, for example, at being able to read another person's internal emotional state by the tilt of their eyebrows but we are relatively bad at calculating the odds of whether to take another card in blackjack ? to the unending enrichment of the Las Vegas casinos."
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."
Abstract This paper discusses the video, "The Proof," a NOVA episode aired on PBS, which presents a look at one man's obsession with proving or disproving a theory, Fermat's Last Theorem, written over two hundred years ago and never proved. Specifically, it summarizes and reviews the video, with a focus on what the video tells us about how people learn to do mathematics. It looks at how "The Proof" is more than just a video about solving a complex mathematical problem and how it is a story of determination, setting goals, and finding out that solutions come from many different places and ideas.
From the Paper "The program then delves into how Wiles began obsessing about the "proof" when he was ten years old, and began a lifelong process of proving Fermat's Theorem. While the story is clearly mathematical, it becomes more than that during the course of the story. It becomes a tale about a man who cannot let go of his obsession, and how to creatively find the solutions to complex problems, whether they are mathematical or not. One mathematician in the show talks about making "good mistakes," and how difficult it is. This is the key to learning about mathematics, and solving mathematical problems. You will make mistakes. Learning how to make "good" mistakes is quite difficult. However, if you can learn from your mistakes, or your mistakes lead you in another direction, they are valuable, and can keep you always learning about mathematics, and other complex problems."
Abstract This paper looks at the life of William Gosset, who worked as a chemist in the Guinness brewery in Dublin in 1899 ,and who also carried out crucial experiments on statistics. It explores how the conditions of brewing gave Gosset an insight to work as a statistician and how he took his data from the different examples of brewing to experiment, which was the best combination of factors. In particular, it examines how these experiments led to the invention of the t-test to calculate and manage small samples for quality control in brewing and how, under the name "Student", Gosset developed the form of the t distribution by a combination of mathematical and empirical work with random numbers on the basis of the early application of the Monte Carlo method.
From the Paper "In 1903, Gosset, came up with methods that could calculate standard errors. In 1904 he wrote on the brewing of beer. After reading this new report written by William Gosset, Karl Pearson consulted Gosset and also they met Pearson in July of 1905. They discussed the developments and reports for a long time. Pearson, helped Gosset understand the theory of standard errors in less than two hours. Gosset after understanding the procedure went back to the brewery and practiced those methods to develop something new for the next year. The meeting was successful because Pearson motivated Gosset to take up the study of the law of error."
Abstract This paper presents an overview of the life of Omar Khayyam, born on 18 May 1048 at Nishapur, the provincial capital of Khurasan. The writer explores all aspects of his amazing life, as painter, mathematician, musician, writer and philosopher. The paper begins with his early life in Persia through to his death in Nishapur on 4th December 1131. The writer believes that Omar Khayyam was an outstanding astronomer and astrologer and his contributions to this field are invaluable still today. The paper includes a number of drawings of the man and examples of his writing.
From the Paper "Omar Khayyam was well known as a poet, philosopher, mathematician, astronomer and physician. His full name was Ghiyath al-Din Abu?l-Fath Omar ibn Ibrahim Al-Nishapuri al-Khayyami. A literal translation of the name al-Khayyami means "tent maker" which maybe derived from his father's trade or he may have practiced this skill at one time. Khayyam played on the meaning of his own name when he wrote; ?Khayyam, who stitched the tents of science, Has fallen in grief's furnace and been suddenly burned, The shears of Fate have cut the tent ropes of his life, And the broker of Hope has sold him for nothing!?."
Abstract This paper examines some of the ways to teach statistics that will best overcome some of the main problems that students encounter while learning statistics and offers solutions to these problems.
From the Paper "Students do not normally encounter statistics until they are in college--at least not on any kind of practicable level--unless they are in extremely advanced mathematics classes at their high school. Even so, not every high school offers statistics as a course, while almost every college does. Teaching and learning statistics is problematic for most college students and teachers because to learn and understand statistics, it is necessary to first have a grasp of some of the properties and features of higher mathematics. Many college students do not have these skills upon entering college, and many professors assume that they do have these skills when beginning to teach a statistics course."
