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.

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.

We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations.

We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations.

We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations. / Graduate Studies, College of (Okanagan) / Graduate