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.

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.

A total of 84 students from a lower economic area, aged 8 to 14, were interviewed about their understanding of decimal fractions. Results showed that most students could give a context in which they saw decimal fractions outside of school. The vast majority could draw a diagram of how a cake or field could be divided equally among 10 or 100 people. However, few students under 14 could give either decimal fraction symbols or common fraction symbols to represent these divisions. Less than half of the students at ages 10, 11 and 12 could visualize what might come between 0 and 1. About half of the students aged 11 and 12 could indicate what 0.1 or 0.01 meant. It was inferred that difficulty in relating these symbols to referents might be an important source of difficulty in understanding decimal fractions. Therefore, these interviews were followed by an intervention study that examined if working with contextualized decimal fractions aided understanding of these numbers when they were presented without context. Half of a group of 16 similar students, aged 11 and 12, were asked to solve problems in which numbers that incorporated decimal fractions were contextualized, and the other half were asked to solve similar problems given in purely numerical form. Students worked in pairs, on problems which incorporated common misconceptions. The group who worked on contextualized problems gained significantly more understanding than did the group that worked on purely numerical problems, as measured by the difference between pretest and posttest scores. Transcripts of the students' discussions were analysed for the effect of prior learning, aspects of peer collaboration that appeared to be beneficial to learning, and the effect of cognitive conflict. The students who gained most from collaboration were not too distant in initial expertise, showed a degree of social equity, and worked on contextualized problems. Much of students' learning appeared to result from needing to reconsider their views following a conflict between their expectations and the results of operating on a calculator or in writing, or hearing an alternative view. / Subscription resource available via Digital Dissertations only.

A total of 84 students from a lower economic area, aged 8 to 14, were interviewed about their understanding of decimal fractions. Results showed that most students could give a context in which they saw decimal fractions outside of school. The vast majority could draw a diagram of how a cake or field could be divided equally among 10 or 100 people. However, few students under 14 could give either decimal fraction symbols or common fraction symbols to represent these divisions. Less than half of the students at ages 10, 11 and 12 could visualize what might come between 0 and 1. About half of the students aged 11 and 12 could indicate what 0.1 or 0.01 meant. It was inferred that difficulty in relating these symbols to referents might be an important source of difficulty in understanding decimal fractions. Therefore, these interviews were followed by an intervention study that examined if working with contextualized decimal fractions aided understanding of these numbers when they were presented without context. Half of a group of 16 similar students, aged 11 and 12, were asked to solve problems in which numbers that incorporated decimal fractions were contextualized, and the other half were asked to solve similar problems given in purely numerical form. Students worked in pairs, on problems which incorporated common misconceptions. The group who worked on contextualized problems gained significantly more understanding than did the group that worked on purely numerical problems, as measured by the difference between pretest and posttest scores. Transcripts of the students' discussions were analysed for the effect of prior learning, aspects of peer collaboration that appeared to be beneficial to learning, and the effect of cognitive conflict. The students who gained most from collaboration were not too distant in initial expertise, showed a degree of social equity, and worked on contextualized problems. Much of students' learning appeared to result from needing to reconsider their views following a conflict between their expectations and the results of operating on a calculator or in writing, or hearing an alternative view. / Subscription resource available via Digital Dissertations only.

A total of 84 students from a lower economic area, aged 8 to 14, were interviewed about their understanding of decimal fractions. Results showed that most students could give a context in which they saw decimal fractions outside of school. The vast majority could draw a diagram of how a cake or field could be divided equally among 10 or 100 people. However, few students under 14 could give either decimal fraction symbols or common fraction symbols to represent these divisions. Less than half of the students at ages 10, 11 and 12 could visualize what might come between 0 and 1. About half of the students aged 11 and 12 could indicate what 0.1 or 0.01 meant. It was inferred that difficulty in relating these symbols to referents might be an important source of difficulty in understanding decimal fractions. Therefore, these interviews were followed by an intervention study that examined if working with contextualized decimal fractions aided understanding of these numbers when they were presented without context. Half of a group of 16 similar students, aged 11 and 12, were asked to solve problems in which numbers that incorporated decimal fractions were contextualized, and the other half were asked to solve similar problems given in purely numerical form. Students worked in pairs, on problems which incorporated common misconceptions. The group who worked on contextualized problems gained significantly more understanding than did the group that worked on purely numerical problems, as measured by the difference between pretest and posttest scores. Transcripts of the students' discussions were analysed for the effect of prior learning, aspects of peer collaboration that appeared to be beneficial to learning, and the effect of cognitive conflict. The students who gained most from collaboration were not too distant in initial expertise, showed a degree of social equity, and worked on contextualized problems. Much of students' learning appeared to result from needing to reconsider their views following a conflict between their expectations and the results of operating on a calculator or in writing, or hearing an alternative view. / Subscription resource available via Digital Dissertations only.

A total of 84 students from a lower economic area, aged 8 to 14, were interviewed about their understanding of decimal fractions. Results showed that most students could give a context in which they saw decimal fractions outside of school. The vast majority could draw a diagram of how a cake or field could be divided equally among 10 or 100 people. However, few students under 14 could give either decimal fraction symbols or common fraction symbols to represent these divisions. Less than half of the students at ages 10, 11 and 12 could visualize what might come between 0 and 1. About half of the students aged 11 and 12 could indicate what 0.1 or 0.01 meant. It was inferred that difficulty in relating these symbols to referents might be an important source of difficulty in understanding decimal fractions. Therefore, these interviews were followed by an intervention study that examined if working with contextualized decimal fractions aided understanding of these numbers when they were presented without context. Half of a group of 16 similar students, aged 11 and 12, were asked to solve problems in which numbers that incorporated decimal fractions were contextualized, and the other half were asked to solve similar problems given in purely numerical form. Students worked in pairs, on problems which incorporated common misconceptions. The group who worked on contextualized problems gained significantly more understanding than did the group that worked on purely numerical problems, as measured by the difference between pretest and posttest scores. Transcripts of the students' discussions were analysed for the effect of prior learning, aspects of peer collaboration that appeared to be beneficial to learning, and the effect of cognitive conflict. The students who gained most from collaboration were not too distant in initial expertise, showed a degree of social equity, and worked on contextualized problems. Much of students' learning appeared to result from needing to reconsider their views following a conflict between their expectations and the results of operating on a calculator or in writing, or hearing an alternative view. / Subscription resource available via Digital Dissertations only.

A total of 84 students from a lower economic area, aged 8 to 14, were interviewed about their understanding of decimal fractions. Results showed that most students could give a context in which they saw decimal fractions outside of school. The vast majority could draw a diagram of how a cake or field could be divided equally among 10 or 100 people. However, few students under 14 could give either decimal fraction symbols or common fraction symbols to represent these divisions. Less than half of the students at ages 10, 11 and 12 could visualize what might come between 0 and 1. About half of the students aged 11 and 12 could indicate what 0.1 or 0.01 meant. It was inferred that difficulty in relating these symbols to referents might be an important source of difficulty in understanding decimal fractions. Therefore, these interviews were followed by an intervention study that examined if working with contextualized decimal fractions aided understanding of these numbers when they were presented without context. Half of a group of 16 similar students, aged 11 and 12, were asked to solve problems in which numbers that incorporated decimal fractions were contextualized, and the other half were asked to solve similar problems given in purely numerical form. Students worked in pairs, on problems which incorporated common misconceptions. The group who worked on contextualized problems gained significantly more understanding than did the group that worked on purely numerical problems, as measured by the difference between pretest and posttest scores. Transcripts of the students' discussions were analysed for the effect of prior learning, aspects of peer collaboration that appeared to be beneficial to learning, and the effect of cognitive conflict. The students who gained most from collaboration were not too distant in initial expertise, showed a degree of social equity, and worked on contextualized problems. Much of students' learning appeared to result from needing to reconsider their views following a conflict between their expectations and the results of operating on a calculator or in writing, or hearing an alternative view. / Subscription resource available via Digital Dissertations only.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / The main objective of the study is to identify the general characteristics of groups within a typical Studio College Algebra class and then adapt aspects of the course to best suit their needs. In a College Algebra class of 1,200 students, like those at most state funded universities, the greatest obstacle to providing personalized, effective education is the anonymity of the students. Data mining provides a method for describing students by making sense of the large amounts of information they generate. Instructors may then take advantage of this expedient analysis to adjust instruction to meet their students’ needs. Using exam problem grades, attendance points, and homework scores from the first four weeks of a Studio College Algebra class, the researchers were able to identify five distinct clusters of students. Interviews of prototypical students from each group revealed their motivations, level of conceptual understanding, and attitudes about mathematics. The student groups where then given the following descriptive names: Overachievers, Underachievers, Employees, Rote Memorizers, and Sisyphean Strivers. In order to improve placement of incoming students, new student services and student advisors across campus have been given profiles of the student clusters and placement suggestions. Preliminary evidence shows that advisors have been able to effectively identify members of these groups during their consultations and suggest the most appropriate math course for those students. In addition to placement suggestions, several targeted interventions are currently being developed to benefit underperforming groups of students. Each student group reacts differently to various elements of the course and assistance strategies. By identifying students who are likely to struggle within the first month of classes, and the recovery strategy that would be most effective, instructors can intercede in time to improve performance.

Master of Science / Department of Mathematics / Andrew G. Bennett / In recent years motivation research has emerged as an area of interest within educational research. Increasing student achievement is not the only aspect of education being studied. Improving the quality of the learning experience and investigating how this improvement affects student achievement is an area of growing interest. Additional investigations also consider what aspects of instruction and teaching affect the quality of the learning experience. Many mathematical organizations have voiced a concern that post-secondary mathematics courses should adapt curriculum and instruction based on results of this research. The current study is an investigation into the effectiveness of suggestions made by these organizations, as well as the effect instructional adaptations have on student attitude and achievement.

The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.