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."

Tags:logistic, bertrand, russell, arithmetical, geometrical, reasoning, theorems, formalism, intuitionists, kant, logical

Abstract
This paper considers some of the major developments in the philosophy of mathematics regarding the capacity of mathematics to be universally valid and applicable. It presents some of the basic arguments and schools of thought of the philosophy of mathematics. The paper then analyzes whether, at its foundation, mathematics can have a legitimate claim to be universal.

Table of Contents:
The Problem Of The Ideal And The Real
Math As Logic
Math As Structure
Application And Universality

From the Paper
"This problem, Russell's paradox, proved to be an intractable problem for Frege which, after it was pointed out to him, he could not overcome. The impact upon the philosophy of math was major. An important attempt to boil math down to logical principles had proven unsuccessfully, and eventual efforts to rescue the project by Russell and others were unable to develop a logicism that showed math as both consistent and complete. Therefore math cannot be said to be universal by appeal to logic alone."

Tags:Euclidean, geometrics, paradox

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 Ones 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)."

Tags:Mathematics, reason, thought, universe, true, truth, doubt

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."

Tags:mathematics, teaching, learning

Abstract
This paper was about the effect Nazism had on mathematics and mathematicians. It covers the state of these subjects pre Nazi, goes to mathematics and mathematicians under the Nazi regime (incompatible), gives specific examples of what happened to some of the eminent mathematicians and how the study of mathematics was changed by Hitler. Lastly, the benefits and losses the rest of the world and Germany suffered as a result of these years.

From the Paper
Mathematics and Mathematicians in the PreNazi Era Prior to the Nazi era, Germany was the center of the mathematical world. Two institutes in particular attracted not just Germans but mathematicians from all over the world, as both lecturers and students. Gottingen was probably the most eminent with the University of Berlin following very closely. According to an American student at the time "the Mathematical Institute at Gottingen in 1931-1932 was a dynamic and successful model of a top mathematical center" (Maclane, 1135). There were many world famous mathematicians teaching, lecturing and researching there during those years: Gauss, Riemann, Dirichlet, Klein, Minowski, Hilbert, Weyl, Landau,

Tags:mathematics, nazism, incompatibility

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."

Tags:mathematics, language, syntax

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."

Tags:mathematics, education, dissertations

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."

Tags:assessment, mathematics, education

Abstract
The paper reveals that Africa's influence on mathematics dates back thousands of years to before the beginning of recorded history, for archeological evidence indicates that ancient African societies made significant contributions to the study of mathematics beginning as far back as prehistoric times. The paper offers the example that the Ishango bone found by archeologists in Zaire has been dated circa 18,000 BCE and a much older bone found in Namibia dated circa 35,000 BCE both indicate a knowledge of basic arithmetic, for they were inscribed with equally spaced tally marks used for counting.

From the Paper
"Africa's influence on mathematics dates back thousands of years to before the beginning of recorded history, for archeological evidence indicates that ancient African societies made significant contributions to the study of mathematics beginning as far back as prehistoric times. For example, the Ishango bone found by archeologists in Zaire has been dated circa 18,000 BCE and a much older bone found in Namibia dated circa 35,000 BCE both indicate a knowledge of basic arithmetic, for they were inscribed..."

Tags:african, mathematics, influence

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."

Tags:mathematics, teaching, practices, standardized, science, education