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 examines issues of mathematics pedagogy and the degree of the contextualization of the subject matter in teaching mathematics. The author points out that mathematics is often presented more as a more abstract examination of numbers and measurements that appear, when mathematics really is always relevant and should be seen in the context of the real world. The paper states that mathematics pedagogy needs to develop a way present mathematics within this real world context.
From the Paper "The issue of relevance in education is often a question of the contextualization of subject matter, meaning that the subject relates to the lives of the students because it can be seen in the context of their lives, with issues understandable because they are applicable to the real world. Mathematics is often presented more as a pure Mathematics has the dual character of being both a language (a symbol system) and an underlying model of relationships among actions with objects. As such, it fits closely with the Vygotskian description of sign-sign relationships and de-contextualized knowledge. At the same time, its development in relation to human actions on objects gives it a prominent place in Piagetian analysis. Furthermore, mathematics teaching requires the recognition of mathematics as a sociocultural achievement worthy of reproduction in new generations."
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."
Abstract The paper discusses the two statistical analysis forecast methods. The paper explains how they can both be used to trend market areas, one on a broad basis, while the other can be extremely detailed and therefore, more accurate.
From the Paper "Although there are many approaches to determining accurate forecasting there is one approach, which can be used when little data is available on a local level. This approach is known as the "build up" method and when applied is used to gain basic market information (i.e. market population, product market share and product demand percentage). This data is used to determine market size potential within a given area and is based on the entire market, not segments (Barnett, 1988, p. 28). In addition to this the "build up" method does not take into consideration the initial goals of the company but takes into consideration market conditions only. An example of build up forecasting would show total consumer sales on automobiles across the nation."
Abstract This paper discusses Godel's theory on mathematical truths as being that they cannot be found in any set of axioms or rules and ultimate truth cannot be achieved. The paper suggests that Kurt Godel's incompleteness theorem encompasses the fact that all formal systems turn out to be incomplete by their very nature and it discusses the implications of this theory.
From the Paper "Godel stated that there can be no proof of any statement (P). If P is true, there is no proof of it. If P is false, there is a proof that P is true. This is a contradiction. It cannot be decided whether P is true in a symbolic system."
Abstract This six page paper looks closely at the ancient and historical figure of Eratosthenes, who died around 195 BC. He made many significant contributions to the fields of science, mathematics, astronomy, geography, and many others. His discovery of the diameter of the earth has been proven to be highly accurate today. As thus, his work is of lasting importance.
From the Paper "Eratosthenes, a Greek scholar from about 276-195 B.C, is remembered chiefly for his scientific measurements of the earth's circumference. His work, albeit somewhat unacknowledged by his contemporaries, resulted in fantastic scientific experiments which are comparably accurate even today. By looking briefly at his biography, and then the results of his experiments, Eratosthenes will be shown to be both a highly important as well as a highly innovative thinker of his age, regardless of how he was considered at the time of his life work. Born in North Africa, Cyrene, Eratosthenes spent much of his educational time in Athens. In Athens he received the education..."
