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The Quaternion Algebra And Its Connections To Medical Imaging, Eli Q. Brottman
The Quaternion Algebra And Its Connections To Medical Imaging, Eli Q. Brottman
Honors Capstones
Pure mathematics topics have widely been regarded as having few practical applications; however, over time, many applications have arisen. One such application is using the quaternions, an abstract algebraic structure and extension of the complex number system, to enhance image quality. Quaternion numbers take the form z = a + bi + cj + dk, where a, b, c, d are real numbers, and i, j, k are distinct square roots of −1. By having three distinct square roots of −1, rather than just one (as in the standard complex number system), unique mathematical properties and practical uses arise. Quaternions …
Mathematical Models Of Biochemical Switch Networks, Cassandra A. Mohr
Mathematical Models Of Biochemical Switch Networks, Cassandra A. Mohr
Honors Capstones
This project centers on mathematical applications to biochemistry. Specifically, the use of a dynamical system to model a special type of biochemical network and determine the effect of initial concentrations on the existence of several constant solutions. Many biochemical networks act as biological switches that are responsible for important biological functions such as cell differentiation and cell death; consequentially, the ability to better predict and manipulate their outcome is of great importance. One particularly insightful and relatively simple form of biochemical mechanism is that of the reversible substrate inhibition reaction. Utilizing basic principles of mathematics and chemistry, it is possible …
Motivation In Mathematics, Andre J. Richmond
Motivation In Mathematics, Andre J. Richmond
Honors Capstones
I conducted a review of existing mathematics education research on students’ motivation related to mathematics learning. My initial research questions were: “What are the constructs of student motivation as related to mathematics learning?” and “How do they these constructs inform mathematical teaching practice?” The research method that I decided to used was secondary research. I found that motivation can be understood in terms of self-efficacy, cognitive engagement, skill development, achievement motivation, self-regulation, and mindset about mathematics. Students’ unproductive beliefs can undermine their progress in mathematics learning. Motivation is what drives a person to do something. As a future teacher, I …
Students With High Incidence Disabilities: What Teaching Strategies And Interventions Work?, Tori A. Letizia
Students With High Incidence Disabilities: What Teaching Strategies And Interventions Work?, Tori A. Letizia
Honors Capstones
No abstract provided.
Equitable Mathematics Education Through Pursuing Quantitative Reasoning, Kyle Kimball
Equitable Mathematics Education Through Pursuing Quantitative Reasoning, Kyle Kimball
Honors Capstones
The purpose of this project is to develop means of addressing discrepancies between the mathematics education of students from different socioeconomic backgrounds. The central question of the project is: How can I as a teacher foster quantitative reasoning so that underrepresented students gain access to meaningful understanding of mathematics? Access to mathematics is important in order to gain access to economic opportunities, meaning it is essentially a social justice issue. The project entailed secondary research of existing paradigms and research projects. The major finding of the project is that some of the classroom causes of inequity in mathematics education can …
To Track Or Not To Track : Refining Middle School Mathematics, Jennifer E. Patton
To Track Or Not To Track : Refining Middle School Mathematics, Jennifer E. Patton
Honors Capstones
This research project discusses the issue of tracking, or ability grouping, in the education system. Using this type of system, students are grouped into low, medium,and high ability groups in all or at least several of their subjects in school. This type of grouping is the most commonly used instructional method to facilitate for students' differences. However, educational literature and research shows that although students have differences in abilities and learning styles, tracking is not the most effective, efficient, or equitable way of accommodating for these differences. Hence, this research project not only discusses the evidence for and against tracking, …
The Problem Of Adequately Defining Numbers, D. Stephan Delong
The Problem Of Adequately Defining Numbers, D. Stephan Delong
Honors Capstones
Natural numbers, although they pervade much of mathematics, are among the most difficult entities for which to provide definitions. Although it is often overlooked, as the efforts of pure mathematics are directed toward the maximization of rigor, the development of sound definitions for numbers can be viewed as one of the most critical objectives of the discipline. This paper is an examination and a support for the efforts in this area of the German logician Gottlob Frege, and in particular of his landmark treatise die Grundlagen der Arithmetik. This work marked the first successful attempt to define numbers through appeal …
Design And Analysis For Bayesian Approximation Of Solutions To Linear Differential Equations, Karole Herzog
Design And Analysis For Bayesian Approximation Of Solutions To Linear Differential Equations, Karole Herzog
Honors Capstones
The problem we are addressing is that of designing and analyzing experiments for numerically solving linear differential equations of the form: Σ-(i=0)^n [g_i (t) y^((i) ) = h(t)] for all t in a specified domain. In general, for a linear differential equation of the order n, we are given n pieces of information to use as boundary conditions. This information may consist of a boundary condition for y and each derivative of y up to y(n-1); or, we may be given multiple boundary conditions for y and/or any of the (n-1) derivatives, as long as there are n pieces of …
A Workshop For Building Confidence In Learning Mathematics, Joseph David Rich
A Workshop For Building Confidence In Learning Mathematics, Joseph David Rich
Honors Capstones
This study was undertaken in an attempt to decrease the mathematics anxiety and thereby improve the performance of a group of twenty-four freshmen college students in a special admissions program at Northern Illinois University. The study was conducted in an intermediate algebra class and used a behavior modification approach for reducing anxiety. A book entitled Building Confidence in Mathematics was written by the experimenter and provided for the students’ use. The workshop met twice a week for three weeks and was run by the experimenter and a counselor. The students were given a pre-test and post-test of the Phobos anxiety …
Fractal Geometry : Mathematics And Applications, Eileen Buckingham
Fractal Geometry : Mathematics And Applications, Eileen Buckingham
Honors Capstones
The first section, entitled “Introduction and Applications,” is self-descriptive. The paper begins by introducing and defining the new terms “fractals” and “fractal geometry.” At the same time, Benoit B. Mandelbrot is accredited for having developed this new field. The introduction is followed by simple examples. This section is completed with a number of practical applications involving a variety of sciences. Section 2 consists of the background mathematics necessary to understand the Mandelbrot set and the computer program illustrating it. To fully understand the section, some prior knowledge of complex analysis is necessary. Sections 3 and 4 are the program description …
Gender Differences In Mathematics Achievement, Denise D'Antonio
Gender Differences In Mathematics Achievement, Denise D'Antonio
Honors Capstones
Current research on gender differences in mathematics achievement is summarized. A cross-cultural study of sex differences in attitudes towards mathematics (confidence, individuality and liking) and in attributional patterns in mathematics (ability, effort, task, environment/luck) included American sixth graders (n=155, m=70, f=85) and British 11-12 year olds (n=42, m=18, f=24). Comparisons based on responses to a two-part survey concerning sex of student, sex of teacher, nationality, and achievement level were made. Significant results were obtained in the following categories concerning attitudes about mathematics. American female students with a female teacher liked mathematics more than American female students with a male teacher …
Stacker : A Microworld For Algebra, Melissa Elli Hauser
Stacker : A Microworld For Algebra, Melissa Elli Hauser
Honors Capstones
Research has been done which shows students have difficulty bridging the gap between arithmetic and basic algebra. Often, students who are able to perform symbolic manipulations still have an inadequate understanding of the underlying concepts involved. Further research has shown that emphasizing the link between arithmetic and algebra aids students in acquiring these concepts. A microworld is a type of computer program which allows exploration within the confines of an environment which models a conceptual framework. Research suggests that this kind of exploration is especially conducive to internalization of the framework being modeled. “Stacker: A Microworld for Algebra,” pictorially models …
Variables : What Are They And Why Are They Important?, Kathleen A. Gavin
Variables : What Are They And Why Are They Important?, Kathleen A. Gavin
Honors Capstones
As a student in MATH 412 last semester, I encountered literature that suggested Algebra I students’ difficulty with that subject stemmed from their misconceptions concerning the concept of variable. Indeed, the transition from junior high school to high school mathematics is challenging enough and these things that we call variables truly form the base from which the concepts of Algebra develop. A variable can be identified as a special type of mapping, from a set of objects onto a number system. This mapping is based upon the measurement of some characteristic of the objects. Frequently, a variable is not named …
Fluid Flow Within Porous Media, Greg Fischer
Fluid Flow Within Porous Media, Greg Fischer
Honors Capstones
The objective of this project is to observe the flow of fluid within a saturated, rectangular, two-dimensional, porous media that is being heated from the side.
What Is The Square Root Of 2?, Kristine A. Beernink
What Is The Square Root Of 2?, Kristine A. Beernink
Honors Capstones
In mathematics, we are always told that when there are two methods for doing some problem, they will always yield the same results. In this paper, I will see if this is true when using two methods for constructing the real numbers.
Expanding Horizons In Intermediate Mathematics, Sheri Pindel
Expanding Horizons In Intermediate Mathematics, Sheri Pindel
Honors Capstones
This project is aimed at providing teachers with activities that could be used to increase students’ understanding of mathematics. The activities listed are aimed for students in fifth through eighth grade. Many of the concepts presented have strong upper level mathematical background, but a complete understanding of the background of the material is not necessary to effectively present the material to students. A teacher who understands the material that is presented to the students in the everyday textbooks should find these supplement activities interesting and enjoyable to teach their students.
The Contraction Mapping Principle, Elaine Rosenbloom
The Contraction Mapping Principle, Elaine Rosenbloom
Honors Capstones
In the world of mathematics, it is often necessary to approximate the solution to a problem by using an iterative method. One such method, not totally dependent on the initial guess, is the contraction mapping method. This paper will explore the contraction mapping principle for the real line, some extensions of it, and the principle in other contexts.
Qualitative Theory Of Differential Equations, Sheri Homeyer
Qualitative Theory Of Differential Equations, Sheri Homeyer
Honors Capstones
The problem of stability is of primary concern in the qualitative theory of differential equations and has occupied mathematicians for the past century. The problem appears when considering solutions to the differential equation x-f(t,x) where x=( x1(t),…,xn(t) )T and f(t,x) is a nonlinear function of x1,…,xn. While no known method of solving this equation explicitly exists even for the case n=2, it is possible to discuss the qualitative properties of x1(t) and x2(t) where x1(t) and x2(t) denote, for example, the populations, at time t, of two competing species. The qualitative attributes under consideration include points of equilibrium and the …
Selection In Genetics : A Mathematical Model, Sarah J. Blean
Selection In Genetics : A Mathematical Model, Sarah J. Blean
Honors Capstones
There are many areas of study in the field of genetics which continually affect our highly technological society. In particular, the idea of being able to predict the time when, given the current distribution of a population, a specific inherited trait might disappear from the population is one important question that is being raised. In order to accurately make such a prediction, a mathematical model will be built. Since there are several variables that enter into such a model, certain assumptions will initially be made. After studying the original model, a selected group of the variables and their corresponding assumptions …
Some Insights Into Partitioning And Repartioning Behaviors, Gina Marie Conner
Some Insights Into Partitioning And Repartioning Behaviors, Gina Marie Conner
Honors Capstones
Recently, in mathematical educational research, there has been quite a bit of work focusing on the acquisition of rational numbers concepts by school age children. It is believed that partitioning, the act of dividing a quantity into a given number of parts which are quantitatively equal, is one of the necessary behaviors for students to obtain in order to fully understand the rational numbers. The first part of this paper will discuss the research that has examined the partitioning behaviors of school age children. The latter part of this paper will contain an analysis of data collected from videotapes of …
The Monte Carlo Method : An Alternative Method Of Numerical Integration, Karen A. Haas
The Monte Carlo Method : An Alternative Method Of Numerical Integration, Karen A. Haas
Honors Capstones
In the world of mathematics, one of the more fundamental ideas that exists is the idea of a function, mathematically described as a correspondence that associates with each element x, of a set S, a unique element y, of a set Y. This concept is basic to many of the applications of higher mathematics, as are the concepts of integrating a function and differentiating a function. It is the former concept which shall be dealt with here.
Networks And Graphs, Brenda K. Tsao
Networks And Graphs, Brenda K. Tsao
Honors Capstones
Graph theory is a part of mathematics that has many practical applications. The study of graph theory began in 1736 with Leonard Euler’s work on the Konigsberg bridge problem. Konigsberg was a city built on a river with two islands and seven bridges. The question was posed as to whether you could start anywhere, cross each bridge exactly once, and end up at the starting point. Other problems that arise include scheduling routes for transportation systems that meet certain requirements such as distance, time, places stopped, and profits; and choosing the best times for replacement of equipment and the best …
The Assignment Problem, Diane Ferrara
The Assignment Problem, Diane Ferrara
Honors Capstones
In everyday life, and especially in business, situations arise which involve trying to gain the maximum profit for work, or trying to minimize loss. If each of us would investigate linear programming procedures, we could solve some of these everyday problems very quickly and efficiently. Linear programming procedures provide for solving realistic problems of everyday life. Specifically, the assignment problem can be used to delegate tasks to specific jobs in the best possible manner so as to obtain maximum feedback. There exist many methods and variations to the assignment problem and many applications of these methods.
Ab?Ba : An Investigation Into The Historical Roots Of Noncommutative Algebra, Keith Chavey
Ab?Ba : An Investigation Into The Historical Roots Of Noncommutative Algebra, Keith Chavey
Honors Capstones
Abstract Algebra is a branch of mathematics in which a large amount of research is currently taking place. This research includes the investigation into different types of algebraic structures such as fields, rings, groups, and their properties. The history of algebra is as rich as the science itself. It is my intention to investigate a crucial step in the development of algebra: the beginning of noncommutative algebra.