Abstract This paper examines the origins and significance of hyperbolic geometry. Specifically,it briefly discusses the men who conceived of it, as well as how hyperbolic geometry differs from Euclidean geometry. Finally, and most importantly, the paper looks at the significance of hyperbolic geometry when it comes to exploring the universe around us.
From the Paper "Delving deeper, the contemporary significance of non-Euclidean geometry grows more and more unavoidable - even to those disinclined to give it its "due". For one thing, it is well-known that hyperbolic geometry has shed some light on the immersion and curvature of spaces. More importantly, Einstein's theory of relativity is, at least in part, indebted to non-Euclidean geometry - though it is admittedly not clear from the available literature the precise extent to which hyperbolic geometry made his revolutionary findings vis-a-vis relativity possible. In any event, this writer - drawing upon course work completed in previous introductory classes that dealt with geometry and its relationship to modern cosmology - would be remiss if he did not also point out the fact that the "empty" regions in outer space where no matter exists can really only be described adequately using a hyperbolic model. In effect, understanding the Hubble Constant involves understanding and appreciating non-Euclidean, hyperbolic geometry."
Abstract The paper is an in depth explanatory narrative on the subject of linear transformation. After an analytical definition of the term, the paper gives examples of the many applications of linear transformation and explains that linear transformation is a method of altering geometric figures into another form. The paper also explains the basic requirements for this and quotes examples. The paper also provides descriptive explanations of linear transformation (also referred to as the algebra of matrices )and interprets how the process occurs. The paper further relates an extensive explanation on conic sections and how they are determined. Throughout the paper the various terms are fully explained together with examples and methods of application.
From the Paper "For then transformation (of the plane), let S be the set of points in the plane. A transformation of the plane is then a one-to-one mapping from S to S. The most important transformations of the plane are the linear transformations, meaning those that can be represented by linear equations. For a linear transformation T, there are constants a, b, c, d, h and k such that T maps the point P with coordinates (x, y) to the point P' with coordinates (x', y') where h = k = 0. The origin O is a fixed point, since T maps O to itself, at which point the transformation can be written x' = Ax. Such transformations include rotations about O, reflections in lines through O, and dilatations from O. Translations are examples of linear transformations in which O is not a fixed point."
Abstract The paper outlines the paradigms within the programming arena that have a close link to mathematical logic and provides an explanation for that link. The paper points out that it is difficult to separate the mathematical logic from the programming paradigms, which highlights how connected each programming logic is to the mathematical concepts, functions, methods and logic.
From the Paper "The procedural paradigm refers to the programming language that specifies steps it takes to reach a desired state. Within the language the operations contain a series of steps that are completed to finalize a desired action. The object-oriented paradigm is the programming language where each object is considered a separate entity that translates processes, receives and sends data throughout the process, (Hudak 360). The objects are collectively responsible for operations, but each object has its clearly defined role within the system. Functional programming paradigm uses mathematical functions for processing and evaluating data, and focuses on the application of functions as the avenue for programming languages. The logic paradigm is the mathematical concepts for computer programming, with the programming language utilizing logic for the problem solving and model development process."
Abstract The paper looks at an article "Enhancing Curriculum and Instruction Through Technology" by S. Rigeman and N. McIntire that outlines some ways in which computer technology can help students in middle and high school classrooms bolster their math skills and give classroom instructors a tool with which to be more responsive to the varied needs of their pupils. The paper discusses some of the limitations inherent in using the Rigeman and McIntire math program and presents an alternative Instructional Technology Plan. The paper supports interactive computer technology which allows students to move at their own pace and in an individualized context.
From the Paper "To begin with, Sally Rigeman and Nancy McIntire (2005) state that Iowa's Area Education Agencies (AEA) district superintendents met recently to discuss how technology could be applied to the augmentation of classroom instruction. 17 of Iowa's 21 AEA districts chose to participate in the implementation of a "technology-rich, research-based, National Science Foundation (NSF)-designated 'exemplary' mathematics program - Cognitive Tutor Algebra I" (Rigeman & McIntire, 2005, p.31). The other four districts stayed with their existing math programs (all of which were NSF-approved) and acted as controls (Rigeman & McIntire, 2005). The Cognitive Tutor Algebra I curriculum used 6 research -based strategies in shaping student learning: "real-world situations; mastery learning; cooperative learning; direct instruction; group and individual presentations; and student use of technology" (Rigeman & McIntire, 2005, p.31). Within the Cognitive Tutor Algebra I classrooms of the participating districts, teachers actually guided classroom instruction about 60 percent of the time while students used the other 40 percent of the time to progress sequentially through sections of the Computer Tutor program at their own pace; the program, apparently, is also geared to accommodate the individual needs of students, as well."
Abstract This paper explains that H.E. Huntley, in his book "The Divine Proportion" claims that beauty exists as a principle, which is external and transcendent to any individual human being's ability to create either equations or art. The paper also discusses Huntley's arguement that the 'Golden Ratio', also known as phi, is the supreme proof that God is a mathematician and that the mathematician and creator God appreciates nature. The paper concludes that Huntley's book is clearly not aimed at mathematicians, given that he is trying to defend his profession and the beauty of math; however, most people lacking fairly solid math skills would find this book a very difficult read, except for its first and last chapters.
From the Paper "Huntley's last chapters shift somewhat from the defense of the 'Golden Ratio' as proof of the existence of universal ideals of beauty and proportionality, and moves on into a more general defense of mathematics as a discipline that is in pursuit of beauty no less than sculpture or art. But why does mathematics need to defend itself as beautiful, to hold its place beside art, poetry, and philosophy? The divisions between the disciplines that did not exist for the ancient Greeks say more about the development of our culture into a split between the sciences and the arts than a failure to recognize the capabilities of mathematics' contribution to the world in general."
Tags: phi, golden rule, fibonacci sequences, defense proportionality
Abstract This paper explains that Mario Livio's book "The Golden Ratio: The Story of Phi, the World's Most Astonishing Number", chronicles the history of, not of a person, thing, or concept, but a number. The paper then relates that this number phi, or notion of proportionality or the 'Golden Ratio', however, has been invested with so much cultural, emotional, and religious importance that it has taken on a character of its own. Next, the paper points out that the reason that phi is astonishing is because, for centuries, our fascination with proportion and beauty has made its properties an object of wonder. The paper concludes that, although Livio ultimately deflates the mystery of phi, his book is a helpful explanation not just of the number but also of why balance and symmetry dominates so many modern discussions of art and architecture.
From the Paper "But ultimately, astrophysicist Mario Livio says that creating this mysterious proportion is no different than a person cutting a piece of string into pieces. While the 'Golden Ratio' appears in many natural phenomena, some supposed appearances are really not true 'Golden Ratios' (such as the Pyramids and Parthenon) and all appearance of perfection is based in human notions of proportionality. It is evidence of humans looking at nature, not that nature or God through nature looking back at us. We see perfection and proportionality because we are looking for it in nature."
Abstract This paper explains the importance of a summer mathematic program is because of new requirements in Michigan, which will immediately endanger the graduation track of students who struggle early in their ninth grade Algebra course. The author presents the rational for a summer support algebra program and reviews the literature upon which to develop the project. The paper summarizes this literature by stating the need for new innovative methods of teaching specifically relevant to the instruction of Algebra. In addition, the author states that the traditional algebra instruction methods have left a generation of students who not only see no practical need for algebra but also view it as a frivolous waste of academic time and resources.
Table of Contents:
Introduction
Problem Statement
Importance and Rationale of the Project
Background of the Project
Statement of Purpose
Research Objectives
Limitations of the Project
Literature Review
Mathematics Curricula
Computer-Assisted Instruction (CAI) Programs
Instructional Process Programs
Summary
From the Paper "Another program used in addressing student achievement in Algebra is 'The Algebra Online Program' as reported by the Louisiana Department of Education - Center for Educational Technology. This program involved a team of planners all of whom are certified in teaching mathematics who met to discuss, design, format, supplementary textbook selection and implementation of the course. This is a distance-learning curriculum."
Tags: alternative, personal curriculum, college-level tutorial collaborative
Abstract The paper explores whether tracking groups for students according to non-biased indicators will allow the students to increase their state testing scores in math. The paper defines the relevant terms, provides a literature review, outlines the methodology and research designs and explains the anticipated outcomes.
Outline:
Introduction
Statement of the Problem and Purpose of the Study
Background and Significance of the Problem
Definitions and Terms
Literature Review
Research Questions
Brief Description of the Methodology and Research Design
Anticipated Outcomes
From the Paper "The work entitled: "Equitable Practices" states that "...despite prevailing practices, research over the last two decades has demonstrated the negative results of sorting students according to perceived motivation or ability." (NWREL, 2001) Furthermore, research had indicated that lower tracks tend to be disproportionately composed of lower-income and ethnic minority students, thus compounding the disadvantage many students already face." (NWREL, 2001) The data also has indicated that "in some cases students of color with the necessary scores for high-track placements are less likely to be placed in those classes than their European American peers." (NWREL, 2001)"
Abstract The paper discusses community, classroom and student characteristics that include geographic factors, community and school population, socio-economics, race/ethnicity, community stability and classroom rules and routines, grouping patterns, scheduling and arrangement and how they affect the teaching/learning process. The paper explains that contextual factors also acknowledge the impact of aspects like attitudes, perceptions, expectations, abilities, gender, socio-cultural background and maturity on every learning experience. The paper also looks at how community, classroom and student characteristics influence instructional planning and assessment.
From the Paper "It generally goes without saying that contextual factors play an important role in mathematical classroom via the way they affect the teaching/learning process. Among these factors are environmental (geographic location), community and school population, socio-economics, race/ethnicity, community stability, political climate and community support for education as well as classroom factors represented by rules and routines, grouping patterns, scheduling and classroom arrangement. Student characteristics should also be examined when designing instruction and assessing learning, such as age, gender, race/ethnicity, special needs, achievement/developmental levels, culture, language, interests, learning styles/modalities and skill levels."
Abstract The paper explains that strategy/implicit instruction is a student-centered approach, which focuses on the general skills, rules and processes required for learning a particular concept. The paper highlights the advantages of using this method and refers to several literary sources on the strategy/implicit instruction. The paper presents the conclusion that the combination of strategy/implicit instruction and direct instruction is the ideal method for teaching mathematics in the classroom.
From the Paper "In order to make the right decision concerning the choice of instructional strategy for middle school mathematics class it appears necessary to take into consideration the general school mathematics standards and the peculiar needs, behaviors and interests of middle school students. Besides complying with the standards, an efficient strategy should promote successful and productive learning. When it comes to middle school, the instructional elements, which could be extremely useful, are the following: clear routines, integrated curriculum, cooperative groups, combination of challenge and support, resorting to real-life connections. All of the above can provide valuable assistance to the teacher. The environment in the class should promote inquiry- and project-based, cooperative instruction. Engaging activities and connections with real life are sure to increase students' motivation and involvement. Thus, the challenging material will be easily tackled by them."
Abstract This paper discusses how mathematics is often treated as a distant and very different discipline from the arts even though the arts make use of mathematics in a number of ways. In particular, the paper looks at how paintings, drawings, and designs can be analyzed according to mathematical principles to see ways in which the artist balances different shapes and forms according to mathematical principles or draws on mathematical theory for inspiration. The paper also examines how the art of different periods may reflect different mathematical ideas.
From the Paper "The classical era was one in which mathematics was used quite consciously in developing artistic styles, and some of these styles have even been named with mathematical references. The artworks of a given era reflect the formalist, social, and economic realities of the period, exemplifying the prevailing artistic styles and the social and economic structures which influence the arts. In Greek art, the Geometric period was an era which produced a good deal of pottery and other geometrically regular works. The Geometric krater from the Dipylon cemetery from the eighth century B.C. (De La Croix, Tansey, and Kirkpatrick 130) exemplifies the style of the period. The Geometric period is the name given to the era between the end of the Mycenaean age and the beginning of the Classic age. "
Abstract In this article, the writer discusses how the reform movements impacted the teaching of math and science. In addition, the writer looks at the differences between traditional teaching and current practices in mathematics and science. The writer notes that the absence of a national curriculum means that how children learn varies greatly, yet the increased demand for accountability through frequent national standardized assessment limits curricular innovation on the part of teachers, as more conceptual learning may be more time-consuming and take longer to show immediate results. Additionally, the writer points out that current educators may not be familiar in the ways to teach such subjects. The writer concludes that when contemplating educational reform in math and science, America seems to be caught in a paradox--America demands quick, demonstrable improvement but is unwilling to relinquish local control, current testing standards, or different ways to fund and teach scientific and mathematical concepts.
From the Paper "Ever since Horace Mann began his innovative educational reforms in the public schools programs of the 19th century, American education has tended to stress practical skills in its curricular approach and local control of schools. These two impulses have often existed in tension, as Americans have strived to remain competitive in math and science education and wish to see gains in the performance on standardized tests by its nation's youth. However, there is often great resistance to changes in the ways that such subjects are taught and standards are set by government agencies.
"Math and science education is seen as vital for the nation, economically, and also in terms of its national security. The resolve to put a man on the moon was accompanied by a new emphasis in technical education. "
Abstract This paper first explains that measures of central tendency are those descriptive statistics that describe the point or points about which a distribution centers. The paper then provides a description of the three measures which are used to describe central tendency and identify the advantages and disadvantages of each, as well as describing a situation in which each of these measures might be used. A summary of the research and salient findings are presented in the conclusion.
Table of Contents:
Review and Discussion
Introduction
Mean
Median
Mode
Summary and Recapitulation
Table:Summary of the Three Measures of Central Tendency
Conclusion
From the Paper "This measure of central tendency is sometimes referred to as the arithmetic mean or "average". According to Cai, Lo and Watanabe (2002), seven properties of the arithmetic average are as follows: (a) the average is located between the extreme values; (b) the sum of the deviations from the average is zero; (c) the average is influenced by values other than the average; (d) the average does not necessarily equal one of the values that was summed; (e) the average can be a fraction that has no counterpart in physical reality; (f) a value of zero."
Presents an extensive discussion on the teaching of elementary grade mathematics including a plan for teaching fifth graders the concepts of elementary geometric measurements.
Abstract This paper explains that, because of increased demands for teacher and student accountability, identifying better ways of delivering educational methods for teaching young learners about mathematics concepts is important. The author reviews extensively the Texas Education Agency report on the teaching of mathematics to the state's 5th grade students. The paper uses the materials from this Texas report to develop a guide for teaching the concepts of area, perimeter and volume. The instructional strategy is based on a popular taxonomy used in educational design, Gagne's nine events of instruction. The author concludes that significant learning will take place among the fifth grade pupils according to the constructivist learning theory.
Table of Contents:
Problem Statement and Needs Analysis
Background of the Problem
Definition of the Problem
Needs Analysis
Rationale for the Need for Instruction
Available Resources
Goal Statement
Learner Analysis
Demographic Information
Relevant Group Characteristics
Prior Knowledge of Topic
Entry Level Knowledge and Skills
Attitudes and/or Motivation toward the Subject
Task Analysis
Area
Area: Questions for Reflection
Perimeter
Volume
Performance Objectives
Instructional Strategies and Supporting Learning Theories
Learning Theory Discussion
From the Paper "Absent hands-on exercises, though, many young learners will not have an opportunity to construct an understanding of the process of measurement or a concept of measurement unit which can frequently result in mechanical and inappropriate applications of measurement knowledge and tools. For instance, Baroody and Coslick point out that many elementary-level children tend to confuse area with perimeter and vice versa; some common types of errors that are made by these young learners when using a ruler."
Abstract The paper states that cooperative learning is an effective way to develop the ability to communicate with others. The paper notes that teaching is said to be the epitome of efficient and effective communication, and since mathematics is also considered as an area that requires communication between teacher and student, it would seem that teachers of mathematics would embrace such teaching methods in order to teach more effectively. This paper discusses the reasons behind these ideas and discusses how cooperative learning, and other pedagogical techniques can be employed in educational mathematics environments in order to facilitate learning. The paper notes that cooperative learning encompasses many areas of pedagogy with discussions and small group activities being paramount in usage.
From the Paper "Implementing techniques in the mathematics classroom can be relatively simple in nature. One study suggests that cooperative learning is best enhanced when, "students are assigned to work in teams of four. Introductory in-class, team-building activities in which teams discuss rules and expectations can foster a positive learning experience" (Doyle, Beatty, Shaw, 1999, p. 73). Fostering a positive learning environment in a mathematics classroom (that is likely perceived as not the most exciting of courses) is likely a key factor in learning in that classroom. In many regards mathematics as it is taught today may not have that positive learning environment. By allowing the students to interact, working together in small groups to discover answers and the step by step process of doing so could be very positive in nature, and would surely add to the positive classroom environment being sought. "
