Abstract In this article, the writer discusses the notion of the 24-hour period used in the day/night cycle. The writer explains that this cycle began in ancient Egypt, while the sixty divisions of degrees, minutes and seconds is derived from the number system based on sixty (sexagesimal) of the Mesopotamians. The writer examines this approach to dividing the day and night into like segments. Further, the writer looks at ancient peoples' observations about the motion of the sun and discusses how this ultimately results in the system that is used to measure angles today.
From the Paper "Given its ancient origins, the reason angles are measured in degrees, minutes and seconds today has likewise been forgotten by many modern observers. In fact, the basis for this method was developed almost five thousand years ago in Sumeria based on their use of sundials to track time. In her book, Time's Pendulum: The Quest to Capture Time -- from Sundials to Atomic Clocks, Jo Ellen Barnett reports that the convention of the 24-hour period used in the day/night cycle began in ancient Egypt, while the sixty divisions of degrees, minutes and seconds is derived from the number system based on sixty (sexagesimal) of the Mesopotamians; because the Mesopotamians had not yet invented fractional numbers, they preferred whole numbers which could be divided in several different ways, and the number 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30."
Abstract This brief review of literature explores the integral connections between mathematics and board games so that the subject matter at hand will be fully understood. Extrapolations are then made as to why these connections may not necessarily mean that there is a connection between enjoyment of board games and math class success.
Table of Contents:
Introduction
Literature Review
Methodology
Questionnaire
From the Paper "According to Heather Jenkins (2004), the mathematical omnipresence in the world is undeniable. However, many people completely overlook the fact that "math forms the basis of many forms of entertainment (and perhaps addictions)." (Jenkins 2004) The very field of probability was born of games of chance which have been played for a very long time, when a dice player became curious about betting outcomes and consulted with the mathematicians Pascal and Fermat. Mathematics is not just an "academic" occurrence. Since the game of dice was played before the field of probability was born, even though probability is the mathematical drive behind the game, it can be extrapolated that an understanding or enjoyment of the study of mathematics itself is not necessary in order to enjoy and excel at activities which are based on math."
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."
Abstract This paper explains that obvious similarities conclude that human language may be reducible to mathematical formulation. The author points out that that mathematics consists of sets of axioms in which statements can be either true or not. The paper relates, while this does not necessarily seem very much like language, Godel's Incompleteness Theorem relates that meaning can exist outside of axiomatic sets, providing a new basis for similarity.
From the Paper "It should not be surprising that mathematicians and linguists have drawn parallels between these two disciplines. There are obvious similarities that have made many believe that human language may be reducible to mathematical formulation. Some have even attempted to use the assumption to teach machines how to speak, constructing complex utterances based on a limited number of syntactical rules. However, these efforts and others to fully connect mathematics and language have proved largely unsuccessful. The following paper will briefly examine some of the similarities between language and mathematics. By its nature, language has a combinational structure, known as syntax or grammar, that permits the communication of complex ideas (Devlin "Born")."
Abstract This paper relates that one often hears people say, "I am good with languages but useless at math" and vice versa as if the two were entirely opposite ways of thinking. The author points out that closer examination of human language and mathematics reveals a surprising number of similarities. The paper states that the most obvious similarity between the two is that both natural human languages and mathematics have a formal syntax i.e. a set of rules that governs them.
From the Paper "Human languages and mathematics seem on the face of it to be very different things. One often hears people say "I am good with languages, but useless at math", and vice versa, as if the two were entirely opposite ways of thinking. However, closer examination reveals a surprising number of similarities. The most obvious similarity between the two is that both natural human languages and mathematics have a formal syntax, i.e. a set of rules that governs them. In the case of language, this is a set of rules that governs how the words may be put together. "
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."
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."
Abstract This paper discusses the theory of matrices, how it was developed, how it changed, some of the applications for which it has been used, and other aspects of the issue. The writer notes how the underlying ideas are ancient and began with the Babylonians and Chinese and then resurfaced in the seventeenth century with the world of Cayley and others. Further the writer points out that the theory of matrices has led to uses in physics, chemistry, and economics as well as mathematics.
From the Paper "Matrices are a means of visualizing mathematical concepts and relationships in graphic form. A matrix is a rectangular set of elements viewed as a single entity, identified by the number of rows and columns of which it is made. Matrices can be added or multiplied on the basis of an algebra of matrices, and one application of this sort of operation is seen in vector analysis and in the solving of systems of linear equations. The basis for the matrix is found in the Cartesian system of Rene Descartes, whose contribution to mathematics was in the development of analytical geometry, closely tied with the development of the Cartesian system of mapping on a grid or graph, for Descartes saw that a function or polynomial can be represented graphically by points."
Abstract This paper examines the question of what mathematical premises would be dependent on the biological and physical evolution of a given species, assuming of course that we knew other intelligent species had evolved. The writer discusses that some critics suppose that language and mathematics by extension are dependent upon the physical parameters set out by the body. The writer explains: ten fingers and hence a decimated numerical system. This essay probes that assumption.
From the Paper "There is almost certainly a connection between biology and the ability to conceptualize. The basic logical processes that we, as humans, often take for granted are in reality quite dependent upon our own physical evolution. How likely is it that we would have developed a base ten numerical system if we didn't just happen to have ten fingers? It would be perfectly plausible to have a base six system or base twelve, for example. But the question becomes how much of mathematics is a product of biological evolution and how much of it exists unto itself."
Abstract In this article, the writer develops a lesson plan for teaching quadrilaterals in high school math and considers some of the underlying pedagogical theory and how it applies. The writer notes that quadrilaterals are defined as polygons with four sides, and while this encompasses any such figure, the more important of these are parallelograms, squares, and rectangles. Further the writer shows how the student can discover certain relationships by looking to the real world.
From the Paper "Below is a lesson plan for the instruction of high school students in the mathematics, specifically on the subject of quadrilaterals. This lesson is found in the larger subject area of Geometry. Quadrilaterals are defined as polygons with four sides, and while this encompasses any such figure, the more important of these are parallelograms, squares, and rectangles. The lessons in this subject area define these figures and address different mathematical concepts applying to them, including ways of determining area, angles, and other ratios. This lesson should introduce the students to the area of quadrilaterals defining this area of Geometry by describing the elements that make up a quadrilateral and the mathematical relationships that define this type of figure, as well as the formulae that are used to calculate different characteristics."
Abstract The paper argues that fundamental truths cannot be arrived at by math. The paper is of the opinion that this is insofar as the questions we ask, the processes we use and the assumptions we make are shaped by environmental, biological and contextual factors that have little - if anything - to do with "rational" and purely objective thought. The paper places great emphasis upon the place of metaphor in the construction of mathematical thought.
From the Paper "The question of whether there are unquestionable truths in mathematics is indeed a puzzling one. This paper will examine the matter by looking a few readings from our class notes. As will soon become apparent, there is much doubt that mathematics leads irrevocably to universal truth; indeed, in the limited space available, this paper will suggest that, because so much of mathematics is metaphorical in nature, Euclidean mathematics and other "relational" branches of math may lead us into the realm of creative metaphor and no further. In fact, as Sawyer seems to suggest, mathematical "truth" - all truth - is essentially the product of cultural epistemology and ontology."
Abstract The paper examines the possibility that biology, evolution and the development of mathematics are linked more closely than mathematicians would necessarily have us believe. The paper challenges the basic Platonist assumption that abstract mathematical concepts possess concrete being and are consequently fundamental parts of the universe. Instead, the paper discusses the possibility that mathematics is a construction of the human mind and an evolutionary development.
From the Paper "Most often we take mathematical truth for granted. Rather than understand it as an historical construction - not so different from any other human production, such as language - most people fully believe that mathematics is natural and etched into the very fabric of the cosmos. This is a classic Platonist view of the universe in which even abstract concepts have physical reality. Twentieth century theorists, especially in linguistics, have repeatedly challenged the efficacy of abstract concepts. But mathematics is still, in some part, understood to be the realm of the gods with right-brains their unerring prophets."
Abstract The paper looks at the theories of George Lakoff and Rafael Nunez, as well as those of Keith Devlin in order to explore if there are really any universal truths in maths. Set off by an excerpt from Robert Sawyer's novel "Computing God," the paper theorizes that there really are not any universal truths, at least none that can be defined until all forms of life are themselves defined. The paper points out that this is neither the quantification nor the metaphor and symbolism that math requires and uses.
Abstract This paper is a proposal for a study of mathematics education to determine the effectiveness of such teaching at the high school level and some of the methods used by the most effective teachers. It is assumed that their methods are variations on what they learned in teacher education altered by experience and not yet articulated as a different teaching method.
From the Paper "Mathematics education has been analyzed from many perspectives, but there is a need for ongoing studies of the process of teaching mathematics to assure that the educational system is working and that students are learning what they need, that the field is keeping up with the changing technological and scientific requirements involving mathematics, that the teaching is preparing students for the working world into which they will one day graduate, and so on. Mathematics testing is conducted at various times over the high school career of the average student and should provide an understanding of what students are learning and so whether the teaching methods followed by instructors are effective. Where there are lapses in terms of different mathematical concepts, teachers might adjust their methods to give added attention to these areas or to shift their method so something more effective."
Abstract The paper discusses how the central problem under investigation in The Cambridge Quintet is whether it is possible that a machine can think. The paper further discusses how the exploration of the issue is primarily between Wittgenstein and Turing, who each maintain opposing views. The paper explains that whereas Turing relied on mathematics and computing logic Wittgenstein progressed from the philosophy of mathematics to the belief that philosophical problems arise from misunderstandings of the logic of language.
From the Paper "The central problem under investigation in The Cambridge Quintet is whether it is possible that a machine can think. The exploration of the issue is primarily between Wittgenstein and Turing who each maintain opposing views. Whereas Turing relied on mathematics and computing logic, Wittgenstein progressed from the philosophy of mathematics to the belief that philosophical problems arise from misunderstandings of the logic of language. Philosophy of language poses questions about meaning and truth, as well as about the nature of concepts, signs, and symbols. The dinner itself comprises an analogy between gastronomic and intellectual consumption, and is also intended as a link between two mental representations."
