This research study is an investigation that describes how intermediate mathematics teachers use laptop technology in their classrooms and the influence of this technology on mathematics teaching and learning using the framework of the Ten Dimensions of Mathematics Education (McDougall, 2004). This study was a qualitative analysis of the experiences of six teachers, as well as the classroom environments they created with the use of technology. Data were gathered using a variety of methods including observation, field notes, interviews, and surveys. Previous research suggests that mathematics teachers have not used laptops frequently, and when they are used, they are used in traditional ways. This study shows that there is potential for mathematics teachers to be effective implementers of laptop technology. The role of the teacher in the studied one-to-one laptop classrooms became one of a facilitator. These teachers were able to be more student-centred in their delivery of the mathematics curriculum. These teachers were also more creative and they were able to use multiple resources to demonstrate mathematical concepts. Because of the wide variety of resources available, these laptop classrooms were more exploratory in nature. These teachers faced barriers such as students being distracted and the extra time it took to plan lessons. The participants indicated that these barriers could be overcome by being patient with their students and by collaborating with their colleagues. Using the framework of the Ten Dimensions of Mathematics Education, it was found that the presence of a one-to-one laptop environment in these classrooms influenced mathematics teaching and learning in a few key areas. The teachers in this study perceived that meeting individual needs, increased use of manipulatives and technology, and appropriate use of assessment techniques were the aspects that changed the most when laptops were present in the classroom. One-to-one laptop technology can change the teaching and learning that takes place in schools. The researched classrooms became more student-centred, exploratory, and engaging for students. Thus, this study shows that the presence of laptop technology has the potential to move mathematics classrooms towards a more reform vision of teaching and learning.

This research study is an investigation that describes how intermediate mathematics teachers use laptop technology in their classrooms and the influence of this technology on mathematics teaching and learning using the framework of the Ten Dimensions of Mathematics Education (McDougall, 2004). This study was a qualitative analysis of the experiences of six teachers, as well as the classroom environments they created with the use of technology. Data were gathered using a variety of methods including observation, field notes, interviews, and surveys. Previous research suggests that mathematics teachers have not used laptops frequently, and when they are used, they are used in traditional ways. This study shows that there is potential for mathematics teachers to be effective implementers of laptop technology. The role of the teacher in the studied one-to-one laptop classrooms became one of a facilitator. These teachers were able to be more student-centred in their delivery of the mathematics curriculum. These teachers were also more creative and they were able to use multiple resources to demonstrate mathematical concepts. Because of the wide variety of resources available, these laptop classrooms were more exploratory in nature. These teachers faced barriers such as students being distracted and the extra time it took to plan lessons. The participants indicated that these barriers could be overcome by being patient with their students and by collaborating with their colleagues. Using the framework of the Ten Dimensions of Mathematics Education, it was found that the presence of a one-to-one laptop environment in these classrooms influenced mathematics teaching and learning in a few key areas. The teachers in this study perceived that meeting individual needs, increased use of manipulatives and technology, and appropriate use of assessment techniques were the aspects that changed the most when laptops were present in the classroom. One-to-one laptop technology can change the teaching and learning that takes place in schools. The researched classrooms became more student-centred, exploratory, and engaging for students. Thus, this study shows that the presence of laptop technology has the potential to move mathematics classrooms towards a more reform vision of teaching and learning.

The present work is divided in two parts. The first is concerned with the dynamics on the Grassmann manifold of k-dimensional subvector spaces of an n dimensional real or complex vector space induced by a linear invertible transformation A of the vector space into itself. The Grassmann map GA sends p to Ap, and one asks, what are the dynamics of GA? In the second part, I consider dynamics induced by a linear cocycle covering a diffeomorphism of a compact manifold, acting on the Grassmann bundle of k-dimensional linear subspaces of TN. I prove a Kupka-Smale theorem for the space of cocycles covering diffeomorphisms of a compact manifold. The proof of this theorem implies the same type of results for derived cocycles parametrized in the space of diffeomorphisms. The results of the second part can be generalized without effort to cocycles covering endomorphisms of N.

Master of Science / Department of Mathematics / Andrew G. Bennett / The purpose of my research is to identify what features of a graph are important for college teachers with the intention of eventually developing a system by which a machine can recognize those features. In particular, I identify the features that college algebra teachers look at when grading graphs of lines and how much disagreement there is in the relative importance graders assign to each feature. In the process, eleven students from college algebra classes were interviewed and asked to graph six linear functions of varying difficulty. Eleven experienced college algebra graders were asked to grade the selected graphs, and interviewed to clarify what features of the graphs were important to them in grading. Altogether, a general grading rule appears to be: slope is worth 4 points, y-intercept is worth 4 points, labeling of intercepts, points and graph is worth 1 point. After that, add 1 point if everything is correct. All graders considered slope and y-intercept to be very important. Only some of them considered labeling to be important. Anything else was a matter of a single point adjustment. Furthermore, the graders judged slope and intercept from two points(the y-intercept and the first point to the right). Returning to the students’ work, I saw that the students also placed extra importance on points to the right of the y-axis. I conclude that this grading style may have a role in students’ learning to think only about two points in a line (but nothing else), and that replicating human grading may not be the best use of machine grading.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / "Just In Time Teaching" (JiTT) is a teaching practice that utilizes web based technology to collect information about students' background knowledge prior to attending lecture. Traditionally, students answer either multiple choice, short answer, or brief essay questions outside of class; based on student responses, instructors adjust their lectures "just-in-time." In this study, modified JiTT techniques in the form of online review modules were applied to a first semester calculus course at a large midwestern state university during the spring 2012 term. The review modules covered algebra concepts and skills relevant to the new material presented in calculus lecture (the "just-in-time" adjustment of the calculus lectures was not implemented in this teaching experiment). The reviews were part of the course grade. Instead of being administered purely "just-in-time," the reviews were assigned ahead of time as part of the online homework component of Calculus-I. While previous studies have investigated the use of traditional JiTT techniques in math courses and reported student satisfaction with such teaching tools, these studies have not addressed gains in student achievement with respect to specific calculus topics. The goal of this study was to investigate the latter, and to determine whether timing of the reviews plays a role in bettering student performance. Student progress on weekly Calculus-I online assignments was tracked in spring of 2012 and compared to student scores from weekly Calculus-I online assignments from spring 2011, when modified JiTT instruction was not available. For select Calculus-I online assignments during the spring 2012 term, we discovered that the review modules significantly increased the number of students receiving perfect scores, even when the reviews were not purely administered ``just-in-time." Analysis of performance, success of review assignments, and future implications are also discussed.

Doctor of Philosophy / Department of Physics / N. Sanjay Rebello / Calculus is used across many physics topics from introductory to upper-division level college courses. The concepts of differentiation and integration are important tools for solving real world problems. Using calculus or any mathematical tool in physics is much more complex than the straightforward application of the equations and algorithms that students often encounter in math classes. Research in physics education has reported students’ lack of ability to transfer their calculus knowledge to physics problem solving. In the past, studies often focused on what students fail to do with less focus on their underlying cognition. However, when solving physics problems requiring the use of integration, their reasoning about mathematics and physics concepts has not yet been carefully and systematically studied. Hence the main purpose of this qualitative study is to investigate student thinking in-depth and provide deeper insights into student reasoning in physics problem solving from multiple perspectives. I propose a conceptual framework by integrating aspects of several theoretical constructs from the literature to help us understand our observations of student work as they solve physics problems that require the use of integration. I combined elements of three important theoretical constructs: mathematical resources or symbolic forms, which are the small pieces of knowledge elements associated with students’ use of mathematical ideas; conceptual metaphors, which describe the systematic mapping of knowledge across multiple conceptual domains – typically from concrete source domain to abstract target domain; and conceptual blending, which describes the construction of new learning by integrating knowledge in different mental spaces. I collected data from group teaching/learning interviews as students solved physics problems requiring setting up integrals. Participants were recruited from a second-semester calculus-based physics course. I conducted qualitative analysis of the videotaped student conversations and their written work. The main contributions of this research include (1) providing evidence for the existence of symbolic forms in students’ reasoning about differentials and integrals, (2) identifying conceptual metaphors involved in student reasoning about differentials and integrals, (3) categorizing the different ways in which students integrate their mathematics and physics knowledge in the context of solving physics integration problems, (4)exploring the use of hypothetical debate problems in shifting students’ framing of physics problem solving requiring mathematics.

There is a crisis in mathematics education (National Research Council, 1989). This crisis has caused stakeholders to question the purpose of mathematics education. Teaching mathematics for social justice is a pedagogy that uses mathematics as a tool to expose students to issues concerning power, resource inequities, and disparate opportunities between different social groups to illicit social and political action (Gutstein, 2006). This study uses action research to explore the effects of incorporating social justice issues in mathematics with affluent, middle school students. Findings indicate that integrating social justice issues into mathematics affected some students’ cognitive and affective domains and in some cases led to empowerment and action. The study also found that students’ perception of responsibility, their age and personal connections along with the amount of teacher direction may have affected students’ development of social agency. These findings help to inform teachers’ practices and contribute to literature on critical mathematics.

There is a crisis in mathematics education (National Research Council, 1989). This crisis has caused stakeholders to question the purpose of mathematics education. Teaching mathematics for social justice is a pedagogy that uses mathematics as a tool to expose students to issues concerning power, resource inequities, and disparate opportunities between different social groups to illicit social and political action (Gutstein, 2006). This study uses action research to explore the effects of incorporating social justice issues in mathematics with affluent, middle school students. Findings indicate that integrating social justice issues into mathematics affected some students’ cognitive and affective domains and in some cases led to empowerment and action. The study also found that students’ perception of responsibility, their age and personal connections along with the amount of teacher direction may have affected students’ development of social agency. These findings help to inform teachers’ practices and contribute to literature on critical mathematics.

According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this thesis is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the answer to the previous question under the very mild hypothesis of dynamical coherence is no. The thesis is organized as follows. In the first chapter we give the necessary background and results in partially hyperbolic dynamics needed for the rest of the work, studying in particular the geometry of the center foliation. Chapter two is devoted to a general discussion of compact foliations. We give proof or sketches of all the relevant results used. Chapter three is the core of the thesis, where we establish the non existence of Sullivan's type of examples in the partially hyperbolic domain, and generalize to diffeomorphisms whose center foliation has arbitrary dimension. The last chapter is devoted to applications of the results of chapter three, where in particular it is proved that if the center foliation of a dynamically coherent partially hyperbolic diffeomorphism is compact and without holonomy, then it is plaque expansive.

The purpose of this study is to examine secondary mathematics teachers’ perceptions about technology integration in teaching the grade 9 Linear Relations Unit and to investigate the impact of these perceptions and teachers’ everyday practices on the development of student tasks, construction of content knowledge, and facilitation of students’ mathematical communication within the context of the Linear Relations Unit in grade 9 mathematics. Case studies were conducted with three mathematics teachers teaching in three urban secondary schools in Ontario. Qualitative data was collected through a series of ongoing classroom observations of the teachers. Additionally, interviews were conducted at the beginning and end of the data collection phase with each teacher. The results from this study suggest that the teachers perceived that the integration of technology in the Linear Relations Unit assisted them to: 1) create interactive and dynamic learning environments which helped make the content meaningful to students; 2) guide their instruction and to closely monitor students’ understanding and track their progress, by providing real time feedback; 3) help struggling students move forward in their learning when they did not master the prerequisite skills required to build upon a new math concept and to help them develop math interpretative and problem solving skills; 4) differentiate instruction and address different learning styles and skills making abstract content more tangible and helping students connect words to images and graphs; 5) teach students to verify and validate their answers and check for their correctness, as well as to avoid relying only on the visual aspect of mathematics; and 6) assist students build mathematical communication skills. Implications of the findings for future research and suggestions to secondary mathematics teachers integrating technology, in the context of the Linear Relations Unit, are also included.