Abstract The first part of this paper expounds on StephanKorner'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 mathematicalphilosophy.
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 Descartes aims to define a method of objective thinking by attempting to apply the precision of mathematics to all fields of knowledge. The paper explains that in prescribing this method for reasoning, Descartes laid the foundation for modern day psychology. Having determined a method of arriving at "true" knowledge, Descartes strives to "seek answers" to questions pertaining to the very nature of existence. This paper analyzes the book and describes its influence on the world.
From the Paper "Descartes? Discourse on the "Method for Conducting One's Reason Well" is his attempt to apply the precision of mathematics to all fields of knowledge. Descartes? Method involved regarding the value of formal education in largely teaching the languages ?"necessary for the understanding of classical texts"? (Part One, p 3), while the pursuit of true knowledge required independent thinking to ?"distinguish the true from the false, in order to see my way clearly in my actions"? (Part One, p6)."
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 extensive paper describes the use of Computer Algebra Systems (CAS) in helping students develop their mathematical skills. The research contained in this report addresses the use of CAS in the mathematics classroom. It also addresses the attitudes shared by teachers and students alike as it relates to the use of this technology. In particular, the Maple CAS system is evaluated. The author states that the purpose of this research is to take a systematic approach to the design and evaluation of the teaching, learning and assessing mathematics courses using the CAS Maple. The focus of the evaluation are first year service mathematics courses at a university. The effectiveness of different ways of incorporating Maple activities into such courses is also examined.
Table of Contents
Introduction and Statement of the Research Questions
Literature Review
How People Learn Mathematics and the Role of Technology
Review of Studies Related to the Use of Technology in the Classroom
Utilizing Computer Algebra Systems
CAS in the Classroom
Survey Papers
The Research Methodology of the Study
Conclusion
From the Paper "The purpose of this research is to take a systematic approach to the design and evaluation of the teaching, learning and assessing mathematics courses using the CAS Maple. Of particular interest are first year service mathematics courses at RMIT University. The effectiveness of different ways of incorporating Maple activities into such courses will also be examined.
The investigation will be conducted as a research and development activity through which Maple activities are designed and evaluated in a feedback cycle and we follow an Action Research methodology. Initially, examples from the literature and relevant theories concerning mathematical understanding were sought in order to inform the development of new resources. Student's responses to the first cycle of activities in 2003 were obtained. The conclusions drawn are informing the development of resources for the next cycle. This process will continue over the course of six semesters. The research methods utilized are observations of classes, analysis of student's work, responses to specially designed test instruments, use of feedback questionnaires and structured interviews. Some use will be made of video will also be utilized to record and analyse methodology to evaluate the teaching and learning of mathematics using Maple in a computer lab."
Abstract This paper discusses the techniques and strategies of teaching basic mathematical concepts to preschoolers. It has been found that children can learn pre-mathematical and mathematical ideas easily if teachers use developmentally appropriate strategies. Most educators are pf the view that in preschool, mathematics classroom should employ the constructivist learning principles which allow children to think, interpret and reach their own answers.
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 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 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 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 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."
A research paper that examines educators' perceptions of changes in reform-related practices in mathematics instruction since the implementation of state wide testing.
Abstract The paper examines the effects of mathematics reform on teacher practices and determines the perceptions of educators regarding it's effects on student achievement since the implementation of high stakes testing. The paper identifies reform-related practices in mathematics instruction that have increased, decreased, or not changed since the implementation of high stakes testing, based on educators' perceptions and determines educators' perceptions of the effects of reform-related practices on improving student achievement since the implementation of high stakes testing. The paper also addresses a significant number of research questions regarding the perceptions of educators, both generally and demographically, regarding the changes that have occurred within the classroom for students since the implementation of outcomes based testing.
Outline:
Abstract
Acknowledgements
List of Tables
Chapter 1
Introduction
Statement of the Problem
Purpose of the Study
Research Question
Significance of the Study
Proposed Methods and Procedures
Definitions of Terms
Literature Review
Introduction
Components of MERA
Perspectives of Educators Regarding Standardized Education Reforms Standards and Assessments
Changes in Curriculum and Modes of Instruction
The Effects of Accountability Systems on Individual Teachers
The Effects of Accountability Systems on a School's Capacity
The Effects of Accountability Systems on Student Learning
Alignment of Curricula and Instruction
Conclusion and Final Thoughts
Theoretical and Conceptual Frameworks
Methodology
Research Design
Sample Description
Survey Permission and Procedures for Human Subject
Protection Survey
Distribution
Survey Returns
Instruments, Measures, and Validity
Data Analysis
Specific Data Analysis Plan for Each Research
Question
Limitations
Results
Research Question One
Research Question Two
Research Question Three
Research Question Four
Research Question Five
Research Question Six
Research Question Seven
Summary and Discussion
Connecting the Theoretical Framework
Discussion
Implications of the Outcome of the Data Conclusion
Implications for Future Research
From the Paper "Another informative aspect of reform and a clear guide for future research will be real test scores, beyond marginal improvements. To accept reform as positive teachers and other educators must be shown more than marginal improvements on test scores, and they must also see real improvement for remedial as well as advanced and "normal" students. Student participation in creative solutions can and likely will play a part in these improvements, regardless of early concerns regarding issues of teachers "teaching to the test." Real world mathematics applications, performance based assessment for daily, weekly and quarterly personal improvement needs as well as many other teacher based creative reforms will likely continue to play a significant role in change."
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 In this article, the writer first discusses that most researchers believed that during the 1950s the topics that were not raised consistently as influences on educational theories like racial discrimination, status of women, etc., had more drastic effect than the issues that were addressed. The writer then relates that educational theories were more specialized and respected in the 1960s than any other on the basis of the incorporation of assessment procedures that guaranteed impartiality, lucidity, and practical inflexibility. Further, the writer notes that Thomas Kuhn's Structure of Scientific Revolutions in 1970 basically brought forth the notion of competition that was singly the most influential concept in the coming decades. The writer points out that the decade of the 1980s saw feminism, neo-Marxist notion, and the first drafts of the postmodern theory being considered as legal areas of concern in the sphere educational philosophy. Finally, the writer notes that the 1990s brought around tragedy, feelings of loss and uncertainty among the societies. The writer concludes that in the future, one can hope that the reforms and incorporations that have been made over the past decades in the education theories yield fruitful results by enhancing educational standards and efficiency of the execution of education philosophies.
Outline:
Educational Theory and Philosophy during 1950s
Educational Theory and Philosophy during 1960s
Educational Theory and Philosophy during 1970s
Educational Theory and Philosophy during 1980s
Educational Theory and Philosophy during 1990s
Conclusion
From the Paper "There were also quite a few articles that chose to concentrate only the numerous problems or aspects of the education policies present like the level of educational liberty available to both the students and the teachers, the educational as well as peripheral responsibilities of the academic management. An example of an essay of this format is the article written by Willard Spalding who was the Dean of the College of Education at the University of Illinois at the time. He in this article tried to design a completely fresh notion and theory of training the teachers. He believed that this could be achieved by invalidating what he believed to be the conservative or traditional comprehension of the association amid the traits and various units of education. He followed the viewpoint first brought forth by Aristotle that regarded the numerous units as the primary factors whereas the traits as the secondary factors. Spalding supported this viewpoint because he believed that this led to a more lucid understanding of the society's needs and requirements for the educational standards, temperaments and policies."
