Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
Browse
6 results
Search Results
Item The mathematics in mathematical modeling(Montana State University - Bozeman, College of Letters & Science, 2017) Fulton, Elizabeth Anne White; Chairperson, Graduate Committee: Elizabeth BurroughsThe purpose of this research was to investigate how teachers interact with mathematics while teaching mathematical modeling to elementary students. To conduct this study, I used a case study approach with four elementary teachers. Each teacher participated in professional development on mathematical modeling prior to the study and incorporated mathematical modeling into their classroom. Modeling task lessons were observed and teachers participated in interviews before and after each lesson. I qualitatively explored what mathematical decisions teachers made while teaching mathematical modeling and how students' mathematical contributions influenced the modeling cycle. This analysis took place through three analytical lenses: the mathematics used, the teacher's interactions with their student's mathematical ideas, and as compared to components of the mathematical modeling cycle. Findings indicate that students engaged in meaningful mathematics to explore real-world problems. Across all cases, teachers prepared students to use mathematics by creating tasks with mathematical opportunities and by orienting the students towards using mathematics to investigate the problems. Each teacher allowed their students to introduce most of the mathematical ideas used to investigate the modeling questions. Each task became a mathematical modeling task by the way it was implemented, through teachers' and students' contributions to the activity.Item From immunology to MRI data anlysis : problems in mathematical biology(Montana State University - Bozeman, College of Letters & Science, 2015) Waters, Ryan Samuel; Chairperson, Graduate Committee: Tomas GedeonThis thesis represents a collection of four distinct biological projects rising from immunology and metabolomics that required unique and creative mathematical approaches. One project focuses on understanding the role IL-2 plays in immune response regulation and exploring how these effects can be altered. We developed several dynamic models of the receptor signaling network which we analyze analytically and numerically. In a second project focused also on MS, we sought to create a system for grading magnetic resonance images (MRI) with good correlation with disability. The goal is for these MRI scores to provide a better standard for large-scale clinical drug trials, which limits the bias associated with differences in available MRI technology and general grader/participant variability. The third project involves the study of the CRISPR adaptive immune system in bacteria. Bacterial cells recognize and acquire snippets of exogenous genetic material, which they incorporate into their DNA. In this project we explore the optimal design for the CRISPR system given a viral distribution to maximize its probability of survival. The final project involves the study of the benefits for colocalization of coupled enzymes in metabolic pathways. The hypothesized kinetic advantage, known as 'channeling', of putting coupled enzymes closer together has been used as justification for the colocalization of coupled enzymes in biological systems. We developed and analyzed a simple partial differential equation of the diffusion of the intermediate substrate between coupled enzymes to explore the phenomena of channeling. The four projects of my thesis represent very distinct biological problems that required a variety of techniques from diverse areas of mathematics ranging from dynamical modeling to statistics, Fourier series and calculus of variations. In each case, quantitative techniques were used to address biological questions from a mathematical perspective ultimately providing insight back to the biological problems which motivated them.Item Selected topics in theoretical mathematics(Montana State University - Bozeman, College of Letters & Science, 1988) McAtee, Deborah GibsonItem Rotation and dynamics for simple solenoidal maps of Tori(Montana State University - Bozeman, College of Letters & Science, 2012) Mathison, Mark Tyler; Chairperson, Graduate Committee: Jaroslaw KwapiszThe rotation number for a circle map has provided a complete and useful classification for that class of maps. In higher dimensions there is still progress to be made towards obtaining a more complete understanding of the relationship between the map and its average rotation. In this dissertation, we explore a class of homeomorphisms on the d dimensional torus T d that preserve each leaf of a foliation of the torus into parallel lines densely winding on T d. First the rotation sets of such maps are explored, with particular emphasis on those maps that have a single fixed point; zero is necessarily an element of those rotation sets. Conditions are found that show when these maps have a non-trivial rotation set. When such maps, with non trivial rotation sets, are created as the time-one map of a flow it is shown that the existence of merely two, or infinitely many ergodic measures is connected to the solvability of a cohomological equation. An example, of the infinitely many ergodic measure case, is provided. Finally, we explore on T ² maps without a fixed point that happen to also have a point whose orbit has bounded deviation from the mean rotation. Such maps are seen to be akin to circle maps with irrational rotation number; the irrationally sloped foliation leads to the map being semi-conjugate to an irrational translation of T ².Item Teachers' beliefs regarding effective teaching strategies for American Indian students in mathematics(Montana State University - Bozeman, College of Letters & Science, 2008) Vallines Mira, Raquel; Chairperson, Graduate Committee: Maurice J. BurkeExtensive research has been conducted on teaching strategies that are effective for American Indians in mathematics. Despite the variety of cultural, linguistic, socioeconomic, and geographic factors influencing student learning within and among American Indian communities, common characteristics of learning styles and effective teaching practices have been identified. Though the wording in each definition varies, research based on a variety of theoretical frameworks and using a variety of methodologies and instruments suggests that among American Indian students, there is a tendency to learn better when the following three strategies are used: contextualization, modeling and demonstration, and joint productive activity. Despite the general agreement in education research that the beliefs that teachers hold about mathematics teaching and learning greatly impact their instructional decisions in the classroom, few, if any, of those studies have examined teachers' beliefs regarding effective strategies for American Indians in mathematics. The main purpose of this study was to add the voices of four teachers to the research community conversation about effective teaching strategies for American Indians in mathematics. Two elementary and two high school teachers from two schools in Montana were selected for this study for their experience with and commitment to the mathematics education of American Indian students. Two are American Indians and two are White. Using a combination of classroom observations and a modification of videoclip interviews, the beliefs of the four teachers were identified with particular focus on the three teaching strategies mentioned above. The study shows that teachers' definitions of research-based strategies often differ from those intended by the research. Teachers' views about these strategies seemed to be idiosyncratic to individual teachers and appeared to be shaped by multiple lenses. In this study, some of those lenses emerged including, among others, school structures and teachers' cultural backgrounds. In light of the results of the study, future efforts for constructive bi-directional communication between the research community and practitioners are recommended.Item An examination of the integration of graphing calculators in formal assessments that accompany high school mathematics textbooks(Montana State University - Bozeman, College of Letters & Science, 2005) Graham, Kimberly Joy; Chairperson, Graduate Committee: Maurice J. BurkeTo support teachers in their efforts to integrate graphing calculator technology in the assessment of student learning, mathematics educators need to know the extent and the quality of graphing calculator usage in assessment materials that accompany published mathematics textbooks. If improved student understanding through the use of graphing calculators is a goal of the curriculum and if the use of graphing calculators as recommended by the Principles and Standards (NCTM, 2000) is truly valued, but graphing calculators are not integrated into assessments, then this fact demonstrates a lack of alignment of the curriculum. The researcher analyzed and compared the extent and quality of graphing calculator use in formal assessments that accompany three thirdyear textbooks used in NSF-funded curricula and that accompany seven Algebra 2 textbooks used in non-NSF-funded curricula. Quantitative data were collected using a rubric constructed by the researcher. The rubric was constructed based on Senk, Beckmann, and Thompsonαs (1997) coding scheme and the recommendations of the Principles and Standards (NCTM, 2000). In addition, the researcher examined the use of graphing calculator technology in the textbooksα instructional materials that support the formal assessments. The study demonstrated that the issue of analyzing and comparing curricula on the extent and quality of graphing calculator use in formal assessments is very complex with many factors involved. Results of the study raised many questions, including βWhat is meant by a Standards-based use of graphing calculators?γ The researcher found differences in the extent and quality of graphing calculator use between the ten curricula examined. Regarding the use of graphing calculators, the goals and assessments were in found to be in general alignment for the ten curricula. The extent and quality of graphing calculator use was also generally aligned between the textbooks and their formal assessments, with some inconsistencies associated with the quality of use.