A case study approach was used to explore second-chance mathematics through two larger courses and one individual study program. A different teaching approach, by committed experienced teachers, was used in each course. In evaluating their effectiveness, I focused on affective change in the students, relating this to their achievement. This study contributes to research on understanding good teaching of mathematics to adults. Qualitative and quantitative data were collected over several years. Methods included: a questionnaire (including mathematics attitude and belief scales as well as demographic and open questions); interviews with students to gather more affective data and explore their reactions to the course approach; and the individual supervised study course was audio-taped for six months. Teachers of the larger courses were also interviewed about their goals for, and experiences with, the students. These multiple strands of evidence provide a complex overall picture of three, largely successful, teaching approaches. Each measure had its own contribution to make, and taken together they illuminated the ways in which affective change was related to ackevement in the three contexts. The higher achieving groups in each of the two larger courses entered the courses with more positive attitudes and beliefs than the lower achieving groups and subsequent affective changes reinforced these differences. The lower achieving groups completed the courses affectively worse off than when they started, Students' reactions to these approaches were compared and found to reflect the nature of the approach. In addition to this finding, successhl students' beliefs about mathematics changed in two of the courses. In the one-to-one course the teacher focused initially on understanding the students' fear of mathematics and early mathematical experiences. The student-focused teaching approach trusted and encouraged the growth of ths student's mathematical thinking. Six months later the student felt empowered and had come to believe that mathematics as a creative and enjoyable process of discovering patterns. The second course focused on the mathematization of realistic situations. Successful students came to regard mathematics as useful, interesting, relating to real life. Successful students in the third course appreciated the carefully structured reintroduction to mathematics and were pleased they could finally do the mathematics they hadn't been able to understand at high school.

A case study approach was used to explore second-chance mathematics through two larger courses and one individual study program. A different teaching approach, by committed experienced teachers, was used in each course. In evaluating their effectiveness, I focused on affective change in the students, relating this to their achievement. This study contributes to research on understanding good teaching of mathematics to adults. Qualitative and quantitative data were collected over several years. Methods included: a questionnaire (including mathematics attitude and belief scales as well as demographic and open questions); interviews with students to gather more affective data and explore their reactions to the course approach; and the individual supervised study course was audio-taped for six months. Teachers of the larger courses were also interviewed about their goals for, and experiences with, the students. These multiple strands of evidence provide a complex overall picture of three, largely successful, teaching approaches. Each measure had its own contribution to make, and taken together they illuminated the ways in which affective change was related to ackevement in the three contexts. The higher achieving groups in each of the two larger courses entered the courses with more positive attitudes and beliefs than the lower achieving groups and subsequent affective changes reinforced these differences. The lower achieving groups completed the courses affectively worse off than when they started, Students' reactions to these approaches were compared and found to reflect the nature of the approach. In addition to this finding, successhl students' beliefs about mathematics changed in two of the courses. In the one-to-one course the teacher focused initially on understanding the students' fear of mathematics and early mathematical experiences. The student-focused teaching approach trusted and encouraged the growth of ths student's mathematical thinking. Six months later the student felt empowered and had come to believe that mathematics as a creative and enjoyable process of discovering patterns. The second course focused on the mathematization of realistic situations. Successful students came to regard mathematics as useful, interesting, relating to real life. Successful students in the third course appreciated the carefully structured reintroduction to mathematics and were pleased they could finally do the mathematics they hadn't been able to understand at high school.

We study a type of calculus for proving inequalities between subgraph densities which is based on Jensen's inequality for the logarithmic function. As a demonstration of the method we verify the conjecture of Erdos-Simonovits and Sidorenko for new families of graphs. In particular we give a short analytic proof for a result by Conlon, Fox and Sudakov. Using this, we prove the forcing conjecture for bipartite graphs in which one vertex is complete to the other side.

We study a type of calculus for proving inequalities between subgraph densities which is based on Jensen's inequality for the logarithmic function. As a demonstration of the method we verify the conjecture of Erdos-Simonovits and Sidorenko for new families of graphs. In particular we give a short analytic proof for a result by Conlon, Fox and Sudakov. Using this, we prove the forcing conjecture for bipartite graphs in which one vertex is complete to the other side.

The main objective of this thesis is to implement stochastic correlation into the existing structural credit risk models. There are two stochastic models suggested for the covariance matrix of the assets' prices. In our first model, to induce the stochasticity into the structure of the correlation, we assume that the eigenvectors of the covariance matrix are constant but the eigenvalues are driven by independent Cox-Ingersoll-Ross processes. To price equity options on this framework we first transform the calculations from the pricing domain to the frequency domain. Then we derive a closed formula for the Fourier transform of the Green's function of the pricing PDE. Finally we use the method of images to find the price of the equity options. The same method is used to find closed formulas for marginal probabilities of defaults and CDS prices. In our second model, the covariance of the assets follows a Wishart process, which is an extension of the CIR model to dimensions greater than one. The popularity of the Heston model, which uses the CIR process to model the stochastic volatility, could be a promising point for using Wishart process to model stochastic correlation. We give closed form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives. For the calculation of our pricing formulas we make a bridge between two recent trends in pricing theory; from one side, pricing of barrier options by Lipton (2001) and Sepp (2006) and from other side the development of Wishart processes by Bru (1991), Gourieroux (2005) and Fonseca et al. (2006, 2007a, 2007b). After obtaining the mathematical results above, we then estimate the parameters of the two models we have developed by an evolutionary algorithm. We prove a theorem which guarantees the convergence of the evolutionary algorithm to the set of optimizing parameters. After estimating the parameters of the two stochastic correlation models, we conduct a comparative analysis of our stochastic correlation models. We give an approximation formula for the joint and marginal probabilities of default for General Motors and Ford. For the marginal probabilities of default, a closed formula is given and for the joint probabilities of default an approximation formula is suggested. To show the convergence properties of this approximation method, we perform the Monte Carlo simulation in two forms: a full and a partial Monte Carlo simulation. At the end, we compare the marginal and joint probabilities with full and partial Monte Carlo simulations.

The purpose of the study is to determine how coaching can be used effectively to improve instruction and student achievement while exploring teachers’ specific emotions during mathematics education reform initiatives that challenge the teacher’s beliefs about teaching and learning in mathematics. It also examines how teachers incorporate the reform changes into their practice in order for the new instructional practices to have the expected effect. I explored teacher learning which refers to the correct use of reform strategies by mathematics teachers so that they have the intended effects on student achievement with the support of a coach during reform initiatives. Through questionnaires, interviews, observations and archival material, the study determines the relationship between teachers’ specific emotions, teacher learning and teacher coaching in secondary school mathematics classrooms. As a result, the study highlights the issues associated with the implementation of mathematics education reform initiatives and implications. The findings show that mathematics education reforms produce emotional responses that can be described as both negative and positive. For example, some emotions include pride, joy, fear, feeling drained and ineffective. The four teachers in the study experienced these emotions because of factors such as a lack of knowledge of how to implement mathematics reform, beliefs about teaching and learning in mathematics that were inconsistent with the reform initiatives, the nature of coaching, and gains in student achievement and engagement. They also experienced negative emotions because of favorable in-school factors such as an administration that supported teacher efforts to implement mathematics reforms. The study shows that: a) coaching may not help teachers reconstruct their professional self-understanding when it fails to address their self-image issues; b) teacher learning may occur even when the teacher’s beliefs are inconsistent with reform initiatives; and c) even when teacher learning results from coaching, reforms do not present themselves as expected in the classroom. Coaches experienced positive and negative emotions as a result of how well the reforms were being implemented by teachers. The experiences of the two coaches during mathematics reforms indicate a need to support coaches as they help teachers use the reform strategies. The directions for future research are described.

The main objective of this thesis is to implement stochastic correlation into the existing structural credit risk models. There are two stochastic models suggested for the covariance matrix of the assets' prices. In our first model, to induce the stochasticity into the structure of the correlation, we assume that the eigenvectors of the covariance matrix are constant but the eigenvalues are driven by independent Cox-Ingersoll-Ross processes. To price equity options on this framework we first transform the calculations from the pricing domain to the frequency domain. Then we derive a closed formula for the Fourier transform of the Green's function of the pricing PDE. Finally we use the method of images to find the price of the equity options. The same method is used to find closed formulas for marginal probabilities of defaults and CDS prices. In our second model, the covariance of the assets follows a Wishart process, which is an extension of the CIR model to dimensions greater than one. The popularity of the Heston model, which uses the CIR process to model the stochastic volatility, could be a promising point for using Wishart process to model stochastic correlation. We give closed form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives. For the calculation of our pricing formulas we make a bridge between two recent trends in pricing theory; from one side, pricing of barrier options by Lipton (2001) and Sepp (2006) and from other side the development of Wishart processes by Bru (1991), Gourieroux (2005) and Fonseca et al. (2006, 2007a, 2007b). After obtaining the mathematical results above, we then estimate the parameters of the two models we have developed by an evolutionary algorithm. We prove a theorem which guarantees the convergence of the evolutionary algorithm to the set of optimizing parameters. After estimating the parameters of the two stochastic correlation models, we conduct a comparative analysis of our stochastic correlation models. We give an approximation formula for the joint and marginal probabilities of default for General Motors and Ford. For the marginal probabilities of default, a closed formula is given and for the joint probabilities of default an approximation formula is suggested. To show the convergence properties of this approximation method, we perform the Monte Carlo simulation in two forms: a full and a partial Monte Carlo simulation. At the end, we compare the marginal and joint probabilities with full and partial Monte Carlo simulations.

The purpose of the study is to determine how coaching can be used effectively to improve instruction and student achievement while exploring teachers’ specific emotions during mathematics education reform initiatives that challenge the teacher’s beliefs about teaching and learning in mathematics. It also examines how teachers incorporate the reform changes into their practice in order for the new instructional practices to have the expected effect. I explored teacher learning which refers to the correct use of reform strategies by mathematics teachers so that they have the intended effects on student achievement with the support of a coach during reform initiatives. Through questionnaires, interviews, observations and archival material, the study determines the relationship between teachers’ specific emotions, teacher learning and teacher coaching in secondary school mathematics classrooms. As a result, the study highlights the issues associated with the implementation of mathematics education reform initiatives and implications. The findings show that mathematics education reforms produce emotional responses that can be described as both negative and positive. For example, some emotions include pride, joy, fear, feeling drained and ineffective. The four teachers in the study experienced these emotions because of factors such as a lack of knowledge of how to implement mathematics reform, beliefs about teaching and learning in mathematics that were inconsistent with the reform initiatives, the nature of coaching, and gains in student achievement and engagement. They also experienced negative emotions because of favorable in-school factors such as an administration that supported teacher efforts to implement mathematics reforms. The study shows that: a) coaching may not help teachers reconstruct their professional self-understanding when it fails to address their self-image issues; b) teacher learning may occur even when the teacher’s beliefs are inconsistent with reform initiatives; and c) even when teacher learning results from coaching, reforms do not present themselves as expected in the classroom. Coaches experienced positive and negative emotions as a result of how well the reforms were being implemented by teachers. The experiences of the two coaches during mathematics reforms indicate a need to support coaches as they help teachers use the reform strategies. The directions for future research are described.

A case study approach was used to explore second-chance mathematics through two larger courses and one individual study program. A different teaching approach, by committed experienced teachers, was used in each course. In evaluating their effectiveness, I focused on affective change in the students, relating this to their achievement. This study contributes to research on understanding good teaching of mathematics to adults. Qualitative and quantitative data were collected over several years. Methods included: a questionnaire (including mathematics attitude and belief scales as well as demographic and open questions); interviews with students to gather more affective data and explore their reactions to the course approach; and the individual supervised study course was audio-taped for six months. Teachers of the larger courses were also interviewed about their goals for, and experiences with, the students. These multiple strands of evidence provide a complex overall picture of three, largely successful, teaching approaches. Each measure had its own contribution to make, and taken together they illuminated the ways in which affective change was related to ackevement in the three contexts. The higher achieving groups in each of the two larger courses entered the courses with more positive attitudes and beliefs than the lower achieving groups and subsequent affective changes reinforced these differences. The lower achieving groups completed the courses affectively worse off than when they started, Students' reactions to these approaches were compared and found to reflect the nature of the approach. In addition to this finding, successhl students' beliefs about mathematics changed in two of the courses. In the one-to-one course the teacher focused initially on understanding the students' fear of mathematics and early mathematical experiences. The student-focused teaching approach trusted and encouraged the growth of ths student's mathematical thinking. Six months later the student felt empowered and had come to believe that mathematics as a creative and enjoyable process of discovering patterns. The second course focused on the mathematization of realistic situations. Successful students came to regard mathematics as useful, interesting, relating to real life. Successful students in the third course appreciated the carefully structured reintroduction to mathematics and were pleased they could finally do the mathematics they hadn't been able to understand at high school.

A case study approach was used to explore second-chance mathematics through two larger courses and one individual study program. A different teaching approach, by committed experienced teachers, was used in each course. In evaluating their effectiveness, I focused on affective change in the students, relating this to their achievement. This study contributes to research on understanding good teaching of mathematics to adults. Qualitative and quantitative data were collected over several years. Methods included: a questionnaire (including mathematics attitude and belief scales as well as demographic and open questions); interviews with students to gather more affective data and explore their reactions to the course approach; and the individual supervised study course was audio-taped for six months. Teachers of the larger courses were also interviewed about their goals for, and experiences with, the students. These multiple strands of evidence provide a complex overall picture of three, largely successful, teaching approaches. Each measure had its own contribution to make, and taken together they illuminated the ways in which affective change was related to ackevement in the three contexts. The higher achieving groups in each of the two larger courses entered the courses with more positive attitudes and beliefs than the lower achieving groups and subsequent affective changes reinforced these differences. The lower achieving groups completed the courses affectively worse off than when they started, Students' reactions to these approaches were compared and found to reflect the nature of the approach. In addition to this finding, successhl students' beliefs about mathematics changed in two of the courses. In the one-to-one course the teacher focused initially on understanding the students' fear of mathematics and early mathematical experiences. The student-focused teaching approach trusted and encouraged the growth of ths student's mathematical thinking. Six months later the student felt empowered and had come to believe that mathematics as a creative and enjoyable process of discovering patterns. The second course focused on the mathematization of realistic situations. Successful students came to regard mathematics as useful, interesting, relating to real life. Successful students in the third course appreciated the carefully structured reintroduction to mathematics and were pleased they could finally do the mathematics they hadn't been able to understand at high school.