My thesis will consist of extensions to results that I proved at the 2004 East Tennessee State REU. Most of these results have to do with graph pebbling and various probabilistic extensions. Specifically, in Chapter 2 we compute the cover pebbling number for complete multipartite graphs and prove upper bounds for cover pebbling numbers for graphs of a specified diameter and order. We also prove that the cover pebbling decision problem is NP complete. In Chapters 3 and 4 we examine domination cover pebbling. In Chapter 5, we obtain structural and probabilistic results for deep graphs, and in Chapter 6 we compute cover pebbling probability thresholds for the complete graph.

abstract: Researchers have documented the importance of seeing a graph as an emergent trace of how two quantities’ values vary simultaneously in order to reason about the graph in terms of quantitative relationships. If a student does not see a graph as a representation of how quantities change together then the student is limited to reasoning about perceptual features of the shape of the graph. This dissertation reports results of an investigation into the ways of thinking that support and inhibit students from constructing and reasoning about graphs in terms of covarying quantities. I collected data by engaging three university precalculus students in asynchronous teaching experiments. I designed the instructional sequence to support students in making three constructions: first imagine representing quantities’ magnitudes along the axes, then simultaneously represent these magnitudes with a correspondence point in the plane, and finally anticipate tracking the correspondence point to track how the two quantities’ attributes change simultaneously. Findings from this investigation provide insights into how students come to engage in covariational reasoning and re-present their imagery in their graphing actions. The data presented here suggests that it is nontrivial for students to coordinate their images of two varying quantities. This is significant because without a way to coordinate two quantities’ variation the student is limited to engaging in static shape thinking. I describe three types of imagery: a correspondence point, Tinker Bell and her pixie dust, and an actor taking baby steps, that supported students in developing ways to coordinate quantities’ variation. I discuss the figurative aspects of the students’ coordination in order to account for the difficulties students had (1) constructing a multiplicative object that persisted under variation, (2) reconstructing their acts of covariation in other graphing tasks, and (3) generalizing these acts of covariation to reason about formulas in terms of covarying quantities. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2017

This study examined the effects of using graphing calculators with a numerical approach designed by the researcher on students learning of limits and derivatives in an Applied Calculus course at a community college. The purposes of this study were to investigate the following: (1) students achievement in solving limit problems (Skills, Concepts, and Applications) with a numerical approach compared to that of students who solved limit problems with a traditional approach (primarily an algebraic approach); and (2) students achievement in solving derivative problems (Skills, Concepts, and Applications) with a numerical approach compared to that of students who solved derivative problems with a traditional approach (primarily an algebraic approach). Students (n = 93) in all four daytime sections of an Applied Calculus course in a community college participated in the study during the spring 2005 semester. One of two MWF sections and one of two TR sections served as the treatment groups; the other two sections served as the control groups. Two instructors other than the researcher participated in the study. Instructor A taught one treatment group (a TR section) and one control group (a MWF section); instructor B taught one treatment group (a MWF section) and one control group (a TR section). Dependent variables were achievement to solve skill, concept, and application limit problems and skill, concept, and application derivative problems, measured by two teacher-made tests. A pretest administered on the first day of class determined that no significant difference existed between the groups on prerequisite algebra skills. Separate ANCOVA tests were conducted on the skill, concept, and application portions of each of the limit and derivative exams. Data analyses revealed the following: (1) there was no significant difference found on the skill portion of the limit topic (unit 1 exam) due to instruction or to instructor; (2) there was a significant difference found on the concept portion of the limit topic due to instruction and to instructor; (3) there was a significant difference found on the application portion of the limit topic due to instruction but not due to instructor; (4) the interaction effects between instructor and instruction were not significant on the skill, concept, and application portions of the limit topic; (5) there was a significant difference found on the skill portion of the derivative topic (unit 2 exam) due to instruction but not due to instructor; (6) there was a significant difference found on the concept portion of the derivative topic due to instruction and to instructor; (7) there was a significant difference found on the application portion of the derivative topic due to instruction but not due to instructor; and (8) the interaction effects between instructor and instruction were not significant on the skill, concept, and application portions of the derivative topic. All significant differences were in favor of the treatment group.

This mixed-method study investigated the impact of graphing calculator use on high school calculus students' reasoning skills through calculus problems when applying to concepts of the definite integral and its applications. The study provides an investigation of the effects on reasoning when graphing calculators are used, since it is proposed that, through reasoning, conceptual understanding can be achieved. Three research questions were used to guide the study: (1) Does the use of the graphing calculator improve high school calculus students' reasoning ability in calculus problems applying the definite integral? (2) In what specific areas of reasoning does use of the graphing calculator seem to be most and least effective? and (3) To what extent can students who have used the graphing calculator demonstrate ability to solve problems using pencil and paper methods? The study included a quantitative, quasi-experimental component and a qualitative component. Results of the quantitative and qualitative analysis indicate that (1) graphing calculators had a positive impact upon students' reasoning skills (2) graphing calculators were most effective in the areas of initiating a strategy and monitoring progress (3) students' reasoning skills were most improved when graphing calculators were used together with the analytic approach during both instruction and testing and (4) students who used the graphing calculator performed equally as well in all elements of reasoning as those who used pencil and paper to solve problems.

The purpose of this study was to investigate changes in selected children's Graphing Response Patterns to elemental changes in compositions in theme and variation form. The research problems were (1) to determine points and degrees of elemental change in the compositional structure of the musical examples; (2) to determine number, degree, and nature of changes in subjects' graphing response pattern to aurally presented musical examples; (3) to determine percentages of agreement between changes in graphing response patterns and points of elemental change within the compositional structures; (4) to determine the relationship of changes in subjects' graphing response pattern to the quality and magnitude of elemental change within the compositional structure. Twenty second- and fourth-grade children were individually videotaped as they listened to and graphed a series of aurally-presented musical examples. Each musical example was analysed according to such parameters as timbre, range/interval size, texture, tempo/meter, attack/rhythmic density, key/mode, dynamic level, and melodic presentation. Change in each parameter was scored using an interval scale reflecting change/no change and degree of change. Changes in graphing response pattern were determined by an interval scale which reflected the presence of change/no change and amount of change, using as analytical units speed, size, shape, type, and pause. The following conclusions were made: findings showed an observable, quantifiable relationship between changes in children's graphing response patterns and elemental changes in music parameters. This relationship encompassed not only change/no change judgements but also magnitude of response. Overall, frequency and magnitude/degree of student response was proportionate to the frequency and magnitude of change in the music parameter/s. Results indicated the existence of high-ranking correlations between student response and certain parameters regardless of the degree-of-change/points-of-change ratio. Findings showed that one degree of change in a single music parameter was not sufficient to cause an observable change in the attention of the young listener.

Thesis advisor: Lillie R. Albert / To explore student mathematical self-efficacy and understanding of graphical data, this dissertation examines students solving real-world problems in their neighborhood, mediated by professional urban planning technologies. As states and schools are working on the alignment of the Common Core State Standards for Mathematics (CCSSM), traditional approaches to mathematics education that involves learning specific skills devoid of context will be challenged. For a student to be considered mathematically proficient according to the CCSSM, they must be able to understand mathematical models of real-world data, be proficient problem solvers and use appropriate technologies (tools) to be successful. This has proven to be difficult for all students--specifically for underrepresented students who have fallen behind in many of the Science, Technology, Engineering and Mathematics (STEM) fields. This mixed-method design involved survey and case-study research to collect and examine data over a two-year period. During the first year of this study, pre- and post-surveys using Likert-scale questions to all students in the urban planning project (n=62). During the two years, ten high school students' mathematical experiences while investigating urban planning projects in their own neighborhoods were explored through interviews, observations, and an examination of artifacts (eg. presentations and worksheets) in order to develop the case studies. Findings indicate that real-world mathematical tasks that are mediated by professional technologies influence both students' mathematical self-efficacy and understanding. Student self-efficacy was impacted by causing a shift in students beliefs about their own mathematical ability by having students interest increase through solving mathematical tasks that are rooted in meaningful, real-world contexts; students' belief that they can succeed in real-world mathematical tasks; and a shift in students' beliefs regarding the definition of `doing mathematics'. Results in light of mathematical understanding demonstrate that students' increased understanding was influenced by the ability to use multiple representations of data, making connections between the data and the physical site that was studied and the ability to communicate their findings to others. Implications for informal and formal learning, use of GIS in mathematics classrooms, and future research are discussed. / Thesis (PhD) — Boston College, 2013. / Submitted to: Boston College. Lynch School of Education. / Discipline: Teacher Education, Special Education, Curriculum and Instruction.

With more and more images being uploaded to social networks each day, the resources for identifying a large portion of the world are available. However the tools to harness and utilize this information are not sufficient. This thesis presents a system, called PhacePhinder, which can build a face database from a social network and have it accessible from mobile devices. Through combining existing technologies, this is made possible. It also makes use of a fusion probabilistic latent semantic analysis to determine strong connections between users and content. Using this information we can determine the most meaningful social connection to a recognized person, allowing us to inform the user of how they know the person being recognized. We conduct a series of offline and user tests to verify our results and compare them to existing algorithms. We show, that through combining a user’s friendship information as well as picture occurrence information, we can make stronger recommendations than based on friendship alone. We demonstrate a working prototype that can identify a face from a picture taken from a mobile phone, using a database derived from images gathered directly from a social network, and return a meaningful social connection to the recognized face.

The purpose of this study was to evaluate the effectiveness of a mathematics enrichment activity used to improve the mathematics performance of students relative to participation in the State Agricultural Mechanics Career Development Event (CDE) and in mandated assessments. The treatment group (13 schools, 43 students) participated in a mathematics enrichment activity situated in an agricultural mechanics context. The control group (16 schools, 56 students) did not participate in the enrichment activity. Both groups, as part of the CDE, were tested with a 100-question word problem examination, completed a individual skill and team activity, and completed a demographic instrument regarding participation in agricultural mechanics CDEs, scholastic performance, use of graphing calculators, enrollment in STEM, agricultural science, and fine arts courses, and other information. After the survey was conducted, schools were asked to provide TAKS exit scores on participating students. These scores were compared between schools and against statewide TAKS scores. Results of the study showed a significant improvement in scores on the individual written examination and teams scores for the agricultural mechanics CDE and on the TAKS exit level mathematics assessment. Mean written examination scores for the treatment group were 69.53; non-cooperators were 57.16. Mean total team scores for cooperating teams were 420.39; non-cooperators had a mean score of 368.13. Mean TAKS exit level mathematics scores for cooperators were 2336.78; non-cooperators had a mean TAKS exit level score of 2331.77. Participation in the enrichment activity improved both CDE and mathematics achievement scores.

With more and more images being uploaded to social networks each day, the resources for identifying a large portion of the world are available. However the tools to harness and utilize this information are not sufficient. This thesis presents a system, called PhacePhinder, which can build a face database from a social network and have it accessible from mobile devices. Through combining existing technologies, this is made possible. It also makes use of a fusion probabilistic latent semantic analysis to determine strong connections between users and content. Using this information we can determine the most meaningful social connection to a recognized person, allowing us to inform the user of how they know the person being recognized. We conduct a series of offline and user tests to verify our results and compare them to existing algorithms. We show, that through combining a user’s friendship information as well as picture occurrence information, we can make stronger recommendations than based on friendship alone. We demonstrate a working prototype that can identify a face from a picture taken from a mobile phone, using a database derived from images gathered directly from a social network, and return a meaningful social connection to the recognized face.

This study investigates learning difficulties involving volumes of solids of revolution (VSOR) at two FET colleges in Gauteng province, in South Africa. The research question for this study was: Why do students have difficulty when learning about volumes of solids of revolution? In order to answer the research question five skill factors were identified as the conceptual framework, subdivided into 11 elements. The five skill factors are: I. Graphing skills and translating between visual graphs and algebraic equations/expressions, II. Three-dimensional thinking, III. Moving between discrete and continuous representations, IV. General manipulation skills and V. Consolidation and general level of cognitive development. Before collecting the main data for this study, a preliminary study and a pilot study were conducted. The data for the main study were then collected in six different investigations. The investigations consisted of two runs of a questionnaire, classroom observations, examination analysis; detailed examination responses and an interview with one student. The results from the questionnaire runs as well as the pilot study reveal that students performed poorly in tasks involving three-dimensional thinking (Skill factor II), moving between discrete and continuous representations (Skill factor III), and consolidation and general level of cognitive development (Skill factor V). Students' performance was satisfactory in tasks involving graphing skills and translating between visual graphs and algebraic equations/expressions (Skill factor I) and general manipulation skills (Skill factor IV). Students were also more competent in solving problems that involved procedural skills than those that required conceptual skills. The challenges that students were faced with in class, evident from the classroom observations allude to the fact that the topic of VSOR is difficult to teach and to learn. It is recommended that VSOR be taught and assessed more conceptually in line with the five skill factors; that curriculum developers must communicate with other stakeholders like industries and other institutions of higher learning and that the Department of Education must provide adequate training for these teachers and liaise with industry in this regard. It is also recommended that the suitability of this topic for the particular cohort of students be reconsidered as it appears to be of too high cognitive demand. / Thesis (PhD)--University of Pretoria, 2012. / Mathematics and Applied Mathematics / unrestricted