# Mathematics

Permanent URI for this collectionhttps://hdl.handle.net/11274/15813

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Item TAKS Scores: an evaluation of standardized test performance at a low performing school(2006-12) Davis, Jessica; Mallam, Winifred; Edwards, Don; Thompson, FrancesThe purpose of this study was to examine whether students’ scores on the Mathematics Texas Assessment of Knowledge and Skills (TAKS) are affected by ethnicity, gender, or socioeconomic status. Data for students in Texas, a large urban district within the state, and a low-performing high school within the district taking the Exit Level (Grade 11) TAKS mathematics section in Spring 2005 were obtained from the Texas Education Agency. The research question asked which categories demonstrated the highest level of achievement. Three hypotheses were tested using analysis of variance (ANOVA) for effects due to ethnicity, socioeconomic status, and gender. An important finding of the study was that at the low performing high school, the categories of interest had no significant bearing on the test performance. Based on the findings, recommendations were made to guide educators in the preparation of students for standardized tests with respect to ethnicity, gender, and socioeconomic status.Item Fourier transform: Constructing and deconstructing overtones(May-23) Wiersema, Lukas; Navarra-Madsen, Junalyn; Smith, Shawnda; Falley, BrandiThe purpose of this thesis is to examine why different instruments sound different even when playing the same note. This is done through the use of the Fourier transform on sample recordings of different instruments. The thesis will include a bases for understanding waves and how the Fourier transform is used, as well as some basic music theory, before examining the differences in the instruments.Item TAKS scores: An evaluation of standardized test performance at a low performing school(2006-12) Davis, JessicaItem Relating plane transformations with stereographic projection(2022-12-01T06:00:00.000Z) McClintock, Patrick; Navarra-Madsen, Junalyn; Falley, Brandi; Smith, ShawndaStereographic projection is a type of transformation mapping points on a sphere of dimension n+1 onto a plane of dimension n. It has properties such as continuity and preserving of certain angles which lend to exploring properties of the plane in relation to the sphere. Here stereographic projection is used as a method of drawing relationships between different two-dimensional spaces. Each of the complex plane, the split-complex plane, and the hyperbolic plane are examined in their relation to the Euclidean plane. First, Mobius transformations are considered on the complex plane. It is shown that every Mobius transformation can be represented via a movement of the sphere between two stereographic projections. Second, similar transformations are represented on the split-complex plane. Laguerre Transformations, analogous to Mobius transformations, explored with respect to split-complex numbers. Finally, a model of the hyperbolic plane is constructed using a pair of lateral projection and central projection.Item Analysis of factors contributing to student’s academic performance in college’s STEM program(1/18/2022) Ayeni, Abimbola A; Falley, BrandiThe purpose of this study is to analyze different variables in understanding how they affect the academic performance of students taking STEM courses at a mid-sized college in the southern United States. These factors are classified into pre-college variables and college variables. The pre-college variables are ACT/SAT scores, gender, ethnicity, and first-generation student or not. The college variables are college GPA, Pell grant recipient, the student’s continuance with the degree major in sophomore year. Implementing regression analysis and using backward regression to remove variables that are not correlated to the response variable (CGPA), we were able to arrive at the best model using the three variables: ACT mathematics, ACT English, and gender. The resulting variables selected show us how the value of R^2 has helped us in making sense of the model selected by 14.2%.Item Leveraging commonly used admission criteria to identify traditionally overlooked applicants(5/27/2021) Gengo, Gregory B.; Hamner, Mark S.Universities have long had criteria that must be met in order to be admitted to the institution. The purpose of admissions criteria is to determine students who are well suited to begin their academic journey at the university level. The threshold level that these criteria must meet varies across the university landscape, but most, if not all, universities have some commonality with regard to the actual criteria that they require for consideration. In the course of this study, we were unable to find any significant literature that addressed holistically how the threshold levels, against which applicants are measured, were determined. Therefore, this study’s goal is to determine a statistical framework for which any university can determine these levels – to better identify applicants that are likely to be successful as students. A logistic regression model will be built using an innovative dependent variable, to predict the probability of an applicant accruing a specific number of semester credit hours in a given time period, the specifics of which we will discuss through this study. Because of the predictive accuracy, this model can serve as a framework for predicting the likely success of an applicant as a student.Item Dealing with missing data in the work environment at King Khalid University(9/8/2020) Asiri, Zahra; Falley, Brandi; Hamner, Mark S.; Edwards, Don E.; Demuynck, MarieThis study is about missing data mechanisms developed by Rubin (1976), including missing data completely at random, missing data at random, and missing data not at random. This study utilizes a scenario at King Khaled University where potential employees complete a Post-Graduate General Aptitude Test (PGGA) to represent techniques for handling missing data. There are both traditional methods of handling missing data and modern methods that are more sophisticated for subsequent analyses and offer specific advantages. This study will go through the process of imputing data to understand how to deal with missing data depending on the missing data mechanism. This study concludes by providing recommendations for handling missing data primarily through regression imputation and multiple imputation, which are exemplified through the researcher’s simulated data related to the PGGA and job performance. Strengths and limitations of different techniques are discussed.Item Dealing with missing data in the work environment at King Khalid University(9/8/2020) Asiri, Zahra; Falley, Brandi; Hamner, Mark S.; E. Edwards, Don; Demuynck, MarieThis study is about missing data mechanisms developed by Rubin (1976), including missing data completely at random, missing data at random, and missing data not at random. This study utilizes a scenario at King Khaled University where potential employees complete a Post-Graduate General Aptitude Test (PGGA) to represent techniques for handling missing data. There are both traditional methods of handling missing data and modern methods that are more sophisticated for subsequent analyses and offer specific advantages. This study will go through the process of imputing data to understand how to deal with missing data depending on the missing data mechanism. This study concludes by providing recommendations for handling missing data primarily through regression imputation and multiple imputation, which are exemplified through the researcher’s simulated data related to the PGGA and job performance. Strengths and limitations of different techniques are discussed.Item Affect and achievement: creating an optimal learning experience in mathematics(7/14/2020) Skousen, Elizabeth; Navarra-Madsen, JunalynThis paper is an analysis of cross-curricular studies about motivation, affect, and engagement in a classroom setting. We further determine appropriate tools for measuring student engagement and affect. Student self-efficacy is a determining factor in motivation and engagement in the classroom. Three instructional methods are of particular interest in increasing student engagement and motivation: problem-based learning, mastery learning, and student self-assessment. We consider each of these instructional methods in turn as ways to enhance student self-efficacy and positive affect and conduct a statistical analysis on the correlations between these measures. Problem-based learning, mastery learning, and student self-assessment are positively correlated with student affect, motivation, and engagement, which contribute to student achievement and future learning of mathematics.Item Modeling nurse-entry success by integrating student characteristics with gateway course performance(2/5/2020) Appicelli, Emily; Hamner, Mark S.Decades of undergraduate retention research has uncovered common predictors of student success and methods, such as early-alert intervention, that promote persistence among academically at-risk students. This study addresses low retention rates among nurse-entry majors enrolled at Texas Woman's University (TWU) by integrating a gateway course performance variable with other student characteristic variables and using predictive analytics to identify at-risk nurse-entry students after their first semester of coursework. A logistic regression model was built to predict the probability of being admitted into the upper division nursing program after persisting for two to three years. Significant interaction between gateway course performance and first-semester GPA was detected, conveying valuable insight into the odds of success for these students based on first semester behavior. Because this model exhibits exceptional predictive accuracy, it may realistically serve as a basis for early-alert intervention programs among TWU nurse-entry students in the future.Item Ray tracing in 3D graphics(2000-12) Bishop, Robert B.; Edwards, Don E.; Marshall, David.; Zimmerman, WayneThe purpose of the study is to analyze the properties and functions of ray tracing, and to describe realistic 3D graphics applications. Ray tracing can be described as a global illumination based rendering method. Rays of light trace from the eye back through the image plane into the scene. Then the rays are tested against all objects in the scene. The basic algorithm goes back to 1968, when Appel was attempting to solve the hidden surface problem when rendering 3D objects. The algorithm works in object space, where we visualize a point in the image plane and trace into the scene with color and intensity from that point. In 1637, René Descartes wrote a treatise, describing his experiments and observations, to explain the shape of the rainbow. Recursive ray tracing, the difficulties of geometrical calculation and the long run-times of early work are discussed. Ray tracing geometry can create a situation where rays spawn more rays. Direct illumination and global illumination are presented with examples. Shadows are created when objects interrupt the light source vector with an additional ray created, called the “shadow feeler.” There are difficulties in the reflection-illumination model, in aliasing, sharp shadows and signature problems. This is investigated as distributed ray tracing. Ray tracing and anti-aliasing are discussed, along with potential improvements. Recent references provide new techniques with promise for future systems, such as interpolation, video textures and a psychophysically-based light reflection model.Item Modeling autoimmune disease with differential equations(10/3/2018) Anderson, Jennifer Ann; Grigorieva, EllinaIn this project, I will build a mathematical model of a developed autoimmune process considering cell autoimmunity that plays the main role in any autoimmune disorder using a system of three non-linear differential equations. As model variables, I will use the concentration of target cells not bearing damage, concentration of cytotoxic T- lymphocytes against given cells, and the concentration of the tissue-specific antigen formed because of the destruction of the target cells. All concentrations will be expressed in the moles per liter. We will investigate the model over the time interval [0, T] given either by months or days analytically as well as numerically.Item The inverse problem to the Voronoi diagram(2007-05) Winter, Lisa G.; Zimmerman, Wayne J.; Hamner, Mark S.; Hogan, TurnerThe primary purpose of this thesis is to address the problem of solving the Inverse Problem for the Voronoi Diagram where the Inverse Problem is: Given a diagram that is in fact a Voronoi Diagram find the set of points X = {x1, x2, x3,…,x n} in R2 that will generate the diagram. In formulating a solution to the Inverse Problem it was necessary that we consider the problem of characterizing Voronoi Diagrams. In developing an algorithm for determining the generating set we considered questions of the form: Is the solution to the Inverse Problem unique? If the solution to the Inverse Problem is not unique, then what properties characterize the Voronoi Diagram? In addition we will develop solutions to a set of problems related to Voronoi Diagrams. These problems are related to: Given a finite set of points X = {x1, x2, x3,…,x n} in R2 find the domain of points Nk such that for every x &egr; N k, xk is the nearest point to x, that is,Item Making the grade: A study of academic success(2013-12) Acosta, Christina; Edwards, Don E.; Marshall, David, Ph. D.; Carlsen-Landy, BeverlyFreshmen account for more than fifty percent of students who drop out before completing their degree, which means that increasing the success rates for freshmen is a critical challenge for many colleges and universities. In order for these academic institutions to increase student retention, programs or policies, geared towards identifying and assisting at-risk students must be successfully developed and implemented. One such study was the Comprehensive Student Success Program, which was implemented and developed for high failure and withdrawal rate mathematics courses to provide additional support such as supplemental lessons, tutoring, and peer mentoring administered by course assistants outside of the classroom at Texas Woman's University. The results of the study showed no significance in success rates among courses that provided the additional assistance when compared to traditionally taught courses. However, students who completed the additional supplemental lessons and tutoring requirements had a significant increase in success rates over students who only attended tutoring or weren't required to attend supplemental lessons or tutoring. In addition, students with additional support showed increases in academic skills such as time management, use of study aids, and most notably motivation.MasterItem Euler's Rotation Theorem: Rotating objects in 3-space(2014-08) Taylor, Kasie; Navarra-Madsen, Junalyn; Marshall, David; Edwards, Don E.The purpose of this thesis is to explore Euler's Rotation Theorem as it applies to the rotation of objects along various paths. Matrices can be used to represent these rotations along with the equation for a specific sphere. After these matrices are selected Maple programming will be used to calculate and further animate the rotation of a sphere (the earth) along an elliptical path, while another sphere (the moon) is rotating in a circular path around the first sphere. The computations in this paper were performed by using Maple TM. Maple is a trademark of Waterloo Maple Inc. These rotations and the matrices that are yielded are known as orthogonal matrices. Even more specifically they can be thought of as special orthogonal matrices. This thesis investigates the various properties of these rotational matrices along with the relationship between orthogonal matrices and special orthogonal matrices.Item Challenges the airline industry faces at present(1/31/2019) Tran, Y Thi Thuy; Falley, BrandiThe competition among businesses operating in the airline industry has become highly intense. This has resulted in creating different types of obstacles for those organizations. For instance, companies are now required to invest large amounts of resources in marketing and promotion activities to attract more and more customers. The main aim of conducting the present study is to identify various issues faced by the airline industry. For the present study, the research has adopted an inductive approach focusing on general assumptions then comes to the specific and applies to the industry, the research has emphasis on selecting exploratory research design technique. The sample size for the present study is 101 respondents from more than 10 different airline companies at Dallas-Fort Worth airport in Dallas, Texas includes American Airlines, JetBlue Airways, Spirit Airlines, Delta Airlines, Frontier Airlines, United Airlines, Alaska Airlines, Sun Country Airlines, Korean Airlines, and Qatar Airlines. From the study carried out, it can be concluded that there are large number of issues and challenges that are being faced by the companies operating in the airline industry. At present, conducting smooth flow of all operations and activities is no longer an easy task for companies. Companies are not able to determine suitable strategies that can support in attracting new customers and retaining the old ones in every possible manner. It is recommended to the airline companies to carry out external analysis on a continuous basis through PESTLE (Political environment, Economic factors, Sociocultural factors, Technological changes, Legal issues, Environmental aspects) as through this it is possible for them to know the external challenges that affect business operations.Item A polychotomous, multivariate regression model predicting performance level in a core mathematics course(2008-12) Ingram, Paul Burton; Hamner, Mark S.; Marshall, David; Thompson, FrancesEnrolling students in a mathematics course commensurate with their performance ability is of great importance to mathematics departments at all institutions of higher learning. This research will utilize pre-existing historical data from Texas Woman's University containing readily available and easily measured factors, which most institutions of higher learning will have available, to develop a predictive model that can be used to place a student in an appropriate mathematics course. Multiple logistic regression and ordinal logistic regression methods are used to construct dichotomous and polychotomous models predicting performance level in Elementary Statistics and Elementary Analysis. External validation will be conducted on different data sets than those used in model construction to evaluate how accurately the models predict performance. Both dichotomous and polychotomous models were found to have performed well, predicting within four percent for Elementary Statistics and within ten percent for Elementary Analysis.Item Mathematics behind Google's PageRank algorithm(8/28/2018) Moor, Brian; Navarra-Madsen, Junalyn; Edwards, Don E.Google has become a name synonymous with web searching, to the point where Merriam Webster defines google as “verb: to use the Google search engine to obtain information about (someone or something) on the World Wide Web.” Google, Inc. was founded in 1998, and quickly became the leader of Internet search engines. With its use of the algorithm it called PageRank, it retrieved search results that were much more relevant to the interests of its users. The PageRank algorithm works on the basic theory that the more important and useful a page is, the more other pages will link to it. Therefore a page that has many other pages linking to it is more important, and will appear higher in the search results. This proved much more effective than competing search engines at the time, many of which were still searching primarily by keywords, something that was very easily abused. Since the introduction of the PageRank algorithm, there has been many efforts towards the improvement and optimization of the mathematics behind it. There has also been much work to apply the PageRank algorithm to other fields and areas of research. These include determining the relative importance of authors in published in scientific journals, finding potential interactions in proteins, and determining the relative importance of various species in a food web. In this thesis, we will explore the history and mathematics behind the PageRank algorithm, and the optimizations and expanded uses it has found over the years.Item Statistical package utilizing the general linear model for R(6/28/2018) McNally, Alexander; Marshall, David, Ph. D.The purpose of this thesis is to formulate a statistical package for the program R. This package will utilize matrices for ease of calculation to make use of the general linear model to provide meaningful data. With this statistical package, I hope to create a refined, simple, and powerful tool to aid mathematicians, statisticians, and researchers alike. In order to create this statistical package, I will be using both R and R-Studio. Both of these programs utilize the same programing language and will be able to make full use of this package. The package will be able to read-in both Excel and CSV spreadsheets. The output of this package will be presented in a clear and concise manner. The general linear model that will be used in this thesis can be represented with the following equation, as it is found in, “The General Linear Model: A "New" Trend in Analysis of Variance,” by Maurice Tatsuoka. The matrices of the equation are as follows: Y matrix is the observations matrix, X is the design matrix, Θ is parameter matrix, and ε is the error matrix. (Tatsuoka, 1975) With these matrices, meaningful statistical data will be calculated including but not necessarily limited to: parameter estimation, statistical significance, and hypothesis testing. To calculate these values, the author will start by invoking the principle of least squares, which is given by the following equation. Q=ε'ε=(Y-XΘ)'(Y-XΘ) To re-parameterize the general linear model, and invoke the principle of least squares, the author will utilize the following matrices to estimate the parameters of the general linear model. With the parameters and error estimated for a given dataset, the author will then proceed with a few intermediate calculations to arrive at the variance-covariance matrix, V(Θ*), in the following manner. Using these equations, the author will work through an example in the text to compare two groups. The aforementioned equations are the same equations that will be programmed into the R package. The following two matrices will be partially used to calculate the t-statistic, or an analogous value.Item Using a statistical model to predict student success in Texas Woman's University mathematics program(7/23/2018) Ibrahim, Esther A; Marshall, David, Ph. D.Admission applications from high school students are received yearly by colleges but majority of the students that are being admitted eventually do not graduate. Colleges need to base their admission requirements on some specific factors to determine the students who will succeed. The purpose of this research was to utilize the data from first time incoming students who were enrolled full time and graduated from the mathematics department within five years to predict the success of future students seeking admission into the department. Data from Fall 2003 to Fall 2012 was used to build the predictive model. Success in this research is defined as the students that graduated from the mathematics program within five years of admission. Two models were developed, one from doing a forward stepwise logistic regression on all the datasets and the second was using cross validation to build a model with the training datasets and checking the effectiveness on the testing datasets. The findings were that the SAT Mathematics score is the best predictor of success. A student with an SAT Math score ≥ 590 has a probability of 0.5 of graduating within five years while those <590 are at risk of not graduating.