Theses and Dissertations at Montana State University (MSU)
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Item The evolution of prospective elementary teachers' competencies : procedural knowledge, mathematical knowledge for teaching, attitudes, and enactment of mathematical practices(Montana State University - Bozeman, College of Letters & Science, 2015) Samuels, Shari Lynne; Chairperson, Graduate Committee: Elizabeth BurroughsThe purpose of this research was to explore the evolution of prospective elementary teachers' competencies (in practices, knowledge, and attitudes); examine the relationships that occur between knowledge, attitudes and practices; and develop an idea of how certain prospective elementary teachers grow and progress in their enactment of two of the Common Core Standards for Mathematical Practice, persevering in problem solving and constructing viable arguments. This was conducted as a case study of the first two of three inquiry-based mathematical content courses for elementary teachers. Both qualitative and quantitative data was collected from a cohort of students moving through the curriculum over the course of a year. Results showed there was an increase in prospective elementary teachers' mathematical knowledge for teaching scores over time, but no change in their procedural knowledge or attitude scores. Positive, linear relationships existed between all of the pair-wise comparisons between mathematical knowledge for teaching, procedural knowledge, and attitudes toward mathematics. Overall, students grew in their ability to problem solve and construct viable arguments in mathematics while moving through the curriculum, with a few exceptions. Three factors contributed to students' learning in the curriculum: the amount of effort made by the student, the atmosphere and attitudes of students in the class, and the nature of the content and questions asked in the curriculum. Another important consideration which arose from the data analysis was the opportunities the curriculum allowed for the practice of written versus verbal explanations, and what was formally assessed. Designers of teacher education programs using a similar curriculum should evaluate the importance of written versus verbal explanations in the goals of the course, and appropriately assess the students.Item Rational numbers and the common core state standards : a descriptive case study(Montana State University - Bozeman, College of Letters & Science, 2014) Fischer, Roger Mark; Co-chairpersons, Graduate Committee: Elizabeth Burroughs and Brian LindamanResearch suggests that some students and teachers have a problematic sense of number. While previous standards documents have emphasized facility with representations of rational numbers, the Common Core State Standards for Mathematics (CCSS) places comparatively more emphasis on rational numbers in general and the repeating decimal representation of rational numbers in particular. The limited body of literature on repeating decimal concepts suggests that teachers are ill-equipped to teach rational numbers in the way described in CCSS. The purpose of this study was to describe the ways that inservice middle grades teachers, selected from a sample, understand rational numbers, how they interpreted a statement that rational numbers can be written as a repeating decimal, and how these understandings manifested during instruction. The setting is a mid-size town in the Rocky Mountain region. The study was conducted in two phases. The interview phase consisted of 40-60 minute interviews with ten different inservice middle school teachers regarding their rational number sense. The observation phase involved observing four of the ten interview participants delivering a lesson on repeating decimals. Data were analyzed both between and within cases using a standardized open-ended interview protocol designed by the researcher as a framework for comparisons. Results indicated that participants primarily understood rational numbers as a collection of mutually exclusive sets. Rational numbers were defined primarily in terms of their decimal representation, with a majority of participants describing rational numbers as "not irrational." The statement "A number is rational if it can be written as a repeating decimal" was interpreted as incomplete by many participants due to a sharp distinction drawn between terminating and repeating decimals. Each observed teacher displayed at least one understanding of rational number in classroom instruction that manifested during the interview, with many participants repeating their understandings verbatim. Recommendations for inservice teacher professional development are offered, as well as suggestions for future research.Item Preservice elementary teachers' mathematical content knowledge of prerequisite algebra concepts(Montana State University - Bozeman, College of Letters & Science, 2007) Welder, Rachael Mae; Chairperson, Graduate Committee: Linda SimonsenResearch illustrating that student achievement is affected by teachers' knowledge advocates for K-8 teachers to be knowledgeable regarding prerequisite algebra concepts: (1) numbers (numerical operations), (2) ratios/proportions, (3) the order of operations, (4) equality, (5) patterning, (6) algebraic symbolism (including letter usage), (7) algebraic equations, (8) functions, and (9) graphing. The theoretical framework for the knowledge for teaching mathematics built for this study suggests that the mathematical content knowledge needed for teaching consists of specialized content knowledge in addition to common content knowledge. Specialized mathematical content knowledge extends beyond solving mathematical problems to encompass how and why mathematical procedures work and an awareness of structuring and representing mathematical content for learners. The effects of an undergraduate mathematics content course for elementary education students on preservice teachers' common and specialized content knowledge of prerequisite algebra concepts was investigated, using a pre-experimental one-group pretest-posttest design.Item Cognitive presence among mathematics teachers : an analysis of tasks and discussions in an asynchronous online graduate course(Montana State University - Bozeman, College of Letters & Science, 2008) Colt, Diana Lynn; Chairperson, Graduate Committee: Jennifer LuebeckHigher order learning, in terms of both process and outcome, is frequently cited as the goal of higher education (Garrison, Anderson, & Archer, 2000). However, the adoption of computer mediated communication in higher education has far outpaced our understanding of how this medium can best be used to promote higher order learning (Garrison, Anderson, & Archer, 2004). Researchers have examined quantitative components of computer mediated communication, but little work has been done to examine the cognitive aspects of online discussion. Those studies that do exist demonstrate inconsistent evidence of higher order learning in online discussions (Kanuka & Anderson, 1998; Littleton & Whitelock, 2005; McLoughlin & Luca, 2000; Meyer, 2003). Researchers conjecture that this could be due to the nature of the tasks that instructors implement for discussion purposes (Arnold & Ducate, 2006; Meyer, 2004; Murphy, 2004; Vonderwell, 2003). This study explored whether one component of instruction, the tasks assigned to students, had an effect on the level of cognitive presence that existed in the mathematical discussions of practicing mathematics teachers enrolled in an online graduate course. Through the method of content analysis, discussion transcripts were analyzed to look for evidence of higher-order learning based on the cognitive presence coding protocol developed by Garrison, Anderson, and Archer (2001). Seventeen students in a History of Mathematics course form the primary sample for this study. The results of the content analysis were triangulated with qualitative data from a questionnaire on student backgrounds and demographics and a post-course survey assessing student perceptions of their learning experiences. The researcher concluded that the MATH 500 course discussions did provide evidence of higher order learning in terms of cognitive presence. Task type, as defined in this study, was not directly related to the levels of cognitive presence achieved in the course. This finding does not negate the possibility of such a relationship, but in this study the effects of task type could not be isolated from other features of the course structure and assignments.Item A study of the relationship between introductory calculus students' understanding of function and their understanding of limit(Montana State University - Bozeman, College of Letters & Science, 2009) Jensen, Taylor Austin; Chairperson, Graduate Committee: Maurice J. BurkeIntroductory calculus students are often successful in doing procedural tasks in calculus even when their understanding of the underlying concepts is lacking, and these conceptual difficulties extend to the limit concept. Since the concept of limit in introductory calculus usually concerns a process applied to a single function, it seems reasonable to believe that a robust understanding of function is beneficial to and perhaps necessary for a meaningful understanding of limit. Therefore, the main goal of this dissertation is to quantitatively correlate students' understanding of function and their understanding of limit. In particular, the correlation between the two is examined in the context of an introductory calculus course for future scientists and engineers at a public, land grant research university in the west. In order to measure the strength of the correlation between understanding of function and understanding of limit, two tests-the Precalculus Concept Assessment (PCA) to measure function understanding and the Limit Understanding Assessment (LUA) to measure limit understanding-were administered to students in all sections of the aforementioned introductory calculus course in the fall of 2008. A linear regression which included appropriate covariates was utilized in which students' scores on the PCA were correlated with their scores on the LUA. Nonparametric bivariate correlations between students' PCA scores and students' scores on particular subcategories of limit understanding measured by the LUA were also calculated. Moreover, a descriptive profile of students' understanding of limit was created which included possible explanations as to why students responded to LUA items the way they did. There was a strong positive linear correlation between PCA and LUA scores, and this correlation was highly significant (p<0.001). Furthermore, the nonparametric correlations between PCA scores and LUA subcategory scores were all statistically significant (p<0.001). The descriptive profile of what the typical student understands about limit in each LUA subcategory supplied valuable context in which to interpret the quantitative results. Based on these results, it is concluded that understanding of function is a significant predictor of future understanding of limit. Recommendations for practicing mathematics educators and indications for future research are provided.Item Environments that encourage mathematics graduate teaching assistants : the effects of institution type and availability of training(Montana State University - Bozeman, College of Letters & Science, 2007) Latulippe, Christine Lynn; Chairperson, Graduate Committee: Linda Simonsen; Dana Longcope (co-chair)This dissertation examined factors which are related to mathematics graduate teaching assistants' (GTAs') teaching attitudes and perceptions of the support for good teaching in their respective math departments. The research questions addressed differences between math GTAs at four groups of universities, categorized using an institution's Carnegie Classification and the availability of teacher-training for GTAs, in regard to math GTA perceptions of the support for good teaching in their department and in regard to math GTA attitudes toward teaching. Additionally, correlations between math GTA perceptions for support and math GTA attitudes toward teaching were examined. Finally, through use of multiple data sources, a qualitative analysis of the primary teaching support structures available to math GTAs at the participating universities in the four groups was conducted.Item The effects of a framework for procedural understanding on college algebra students' procedural skill and understanding(Montana State University - Bozeman, College of Letters & Science, 2006) Hasenbank, John Fredrick; Chairperson, Graduate Committee: Ted HodgsonThis dissertation examined the effectiveness of an instructional treatment consisting of lecture content, homework tasks, and quiz assessments built around a common Framework for Procedural Understanding. The study addressed concerns about increasing numbers of students enrolling in remedial mathematics courses because they did not develop sufficient understanding in previous courses. The Framework-oriented instruction was designed to help students develop deep and well-connected knowledge of procedures, which has been shown to facilitate recall and promote future learning. Data collection spanned the Fall 2005 semester at a western land-grant university. In the quasi-experimental design, instructors from six intact sections of college algebra were matched into pairs based on prior teaching experience, and the treatment condition was assigned to one member of each pair. Equivalence of treatment and control groups was established by comparing ACT / SAT scores for the 85% of students for whom those scores were available. Data collection consisted of classroom observations, homework samples, common hour exams scores, procedural understanding assessments, supplemental course evaluations, and a final interview with treatment instructors. Analysis of covariance was the primary statistical tool used to compare treatment and control group performances while controlling for attendance rates and pre-requisite mathematical knowledge. Treatment group students scored significantly higher than control group students on the procedural understanding assessments. Moreover, although treatment students were assigned 18% fewer drill questions than controls and 8% fewer problems overall, the gains in procedural understanding were realized without declines in procedural skill. The relationship between understanding and skill was also examined, and students with greater procedural understanding tended to score higher on the skills-oriented final exam regardless of which treatment condition was assigned to them. Finally, the interview with the treatment instructors provided insight into the implementation issues surrounding the treatment. They expressed concerns about time constraints and reported initial discomfort with, but eventual appreciation for, using the Framework for Procedural Understanding to guide instruction. The Framework-oriented treatment was found to be effective at helping students develop deeper procedural understanding without declines in procedural skill. Additional implications and recommendations for future research are also discussed.Item Action research in mathematics education : a study of a master's program for teachers(Montana State University - Bozeman, College of Letters & Science, 2009) Segal, Sarah Ultan; Chairperson, Graduate Committee: Maurice J. Burke; Jennifer Luebeck (co-chair)Action research is a methodology that has been found to be valuable as a problem-solving tool. It can provide opportunities for reflection, improvement, and transformation of teaching. The purpose of this study is to better understand these claims about the benefits of action research. Several research questions stand out: How is action research experienced by teachers? Is it beneficial and practical for teachers who use it? How are action research findings typically validated? What factors influence whether teachers are able to continue to practice action research? What kind of change has it initiated for teachers? And, how does action research focused on improving student achievement affect high need students? For the past five years, forty-five teachers completing master's degrees in mathematics education at a northern Rocky Mountain land-grant university have been required to conduct an action research project, referred to as their "capstone project." By studying this group of graduates, gathering both qualitative and quantitative data through surveys and interviews, I have examined the effectiveness of action research. This data, combined with graduates' capstone projects, has provided partial answers to the above questions, restricted to faculty-mediated action research within master's programs for mathematics teachers. The extent to which such action research projects impact teachers' practices has not been investigated before. While acknowledging that this research relied primarily upon self-reported data, the results strongly support what the research literature generally asserts about action research. (a) It is beneficial and often transformational for teachers as a professional development tool by allowing them to engage in a focused study of their own practice. (b) When done less formally it becomes more practical. (c) Communicating with others in the field builds confidence in teachers as professionals. (d) It makes teachers more actively reflective and more aware of their teaching and their students' learning. (e) It is effective in understanding and addressing the particular needs of high need students. Continued practice was highly dependent on time and support for action research within the school. Teachers often expressed the importance of having an action research community while conducting their capstones.Item The use of computer algebra systems in a procedural algebra course to facilitate a framework for procedural understanding(Montana State University - Bozeman, College of Letters & Science, 2007) Harper, Jonathan Lee; Chairperson, Graduate Committee: Maurice BurkeThis dissertation study evaluated the implementation and effectiveness of an introductory algebra curriculum designed around a Framework for Procedural Understanding. A Computer Algebra System (CAS) was used as a tool to focus lessons on the Framework and help students gain a deeper, well-connected understanding of algebraic procedures. This research was conducted in response to the prevalence of remedial mathematics and addresses the need for students in remedial mathematics to have a successful learning experience. The curriculum was implemented in the Spring 2007 semester at a western land-grant university. In this quasi-experimental study, one section of introductory algebra was taught using the CAS/Framework curriculum. This treatment section was determined based on a pretest used to judge equivalency of groups. Data sources included procedural understanding assessments with follow-up student interviews, procedural skill exams, classroom observations, and a debriefing interview with the treatment instructor. Qualitative analysis of student and instructor interview transcripts was done to supplement independent observation reports to evaluate the implementation of the curriculum.