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.

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.

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.

Master of Science / Department of Mathematics / Andrew G. Bennett / An online placement exam was administered to 2800 entering freshmen, 700 of whom enrolled in College Algebra during the succeeding Fall semester. Problems on the placement exam were clustered using several different techniques including both expert analysis and item response theory. Student scores on these groupings of problems were then compared to their scores on the first two hour exams in the course (representing the first half of the material in the course) and also on ACT data. Based on this comparison, certain problems were selected as more or less informative for purposes of placement. A model was created using previously available ACT data along with the new placement data to predict initial student success in the course. This model explains 50% more of the variance in student scores than the previously available ACT data alone. Suggestions for improvements to the test and the placement methodology are made based on our analysis.

The van der Waals forces, which are forces between neutral atoms and molecules, play an important role in physics (e.g. in phase transitions), chemistry (e.g. in chemical reactions) and biology (e.g. in determining properties of DNA). These forces are of quantum nature and it is long being conjectured and experimentally verified that they have universal behaviour at large separations: they are attractive and decay as the inverse sixth power of the pairwise distance between the atoms or molecules. In this thesis we prove the van der Waals law under the technical condition that ionization energies (energies of removing electrons) of atoms are larger than electron affinities (energies released when adding electrons). This condition is well justified experimentally as can be seen from the table, \newline \begin{tabular}{|c|c|c|c|} \hline Atomic number & Element & Ionization energy (kcal/mol)& Electron affinity (kcal/mol) \\ \hline 1 & H & 313.5 & 17.3 \\ \hline 6 & C & 259.6 & 29 \\ \hline 8 & O & 314.0 & 34 \\ \hline 9 & F & 401.8 & 79.5 \\ \hline 16 & S & 238.9 & 47 \\ \hline 17 & Cl & 300.0 & 83.4 \\ \hline \end{tabular} \newline where we give ionization energies and electron affinities for a small sample of atoms, and is obvious from heuristic considerations (the attraction of an electron to a positive ion is much stronger than to a neutral atom), however it is not proved so far rigorously. We verify this condition for systems of hydrogen atoms. With an informal definition of the cohesive energy $W(y),\ y=(y_1,...,y_M)$ between $M$ atoms as the difference between the lowest (ground state) energy, $E(y)$, of the system of the atoms with their nuclei fixed at the positions $y_1,...,y_M$ and the sum, $\sum_{j=1}^M E_j$, of lowest (ground state) energies of the non-interacting atoms, we show that for $|y_i-y_j|,\ i,j \in \{1,...,M\}, i \neq j,$ large enough, $$W(y)=-\sum_{i<j}^{1,M} \frac{\sigma_{ij}}{|y_i-y_j|^6}+O(\sum_{i<j}^{1,M} \frac{1}{|y_i-y_j|^7})$$ where $\sigma_{ij}$ are in principle computable positive constants depending on the nature of the atoms $i$ and $j$.

The van der Waals forces, which are forces between neutral atoms and molecules, play an important role in physics (e.g. in phase transitions), chemistry (e.g. in chemical reactions) and biology (e.g. in determining properties of DNA). These forces are of quantum nature and it is long being conjectured and experimentally verified that they have universal behaviour at large separations: they are attractive and decay as the inverse sixth power of the pairwise distance between the atoms or molecules. In this thesis we prove the van der Waals law under the technical condition that ionization energies (energies of removing electrons) of atoms are larger than electron affinities (energies released when adding electrons). This condition is well justified experimentally as can be seen from the table, \newline \begin{tabular}{|c|c|c|c|} \hline Atomic number & Element & Ionization energy (kcal/mol)& Electron affinity (kcal/mol) \\ \hline 1 & H & 313.5 & 17.3 \\ \hline 6 & C & 259.6 & 29 \\ \hline 8 & O & 314.0 & 34 \\ \hline 9 & F & 401.8 & 79.5 \\ \hline 16 & S & 238.9 & 47 \\ \hline 17 & Cl & 300.0 & 83.4 \\ \hline \end{tabular} \newline where we give ionization energies and electron affinities for a small sample of atoms, and is obvious from heuristic considerations (the attraction of an electron to a positive ion is much stronger than to a neutral atom), however it is not proved so far rigorously. We verify this condition for systems of hydrogen atoms. With an informal definition of the cohesive energy $W(y),\ y=(y_1,...,y_M)$ between $M$ atoms as the difference between the lowest (ground state) energy, $E(y)$, of the system of the atoms with their nuclei fixed at the positions $y_1,...,y_M$ and the sum, $\sum_{j=1}^M E_j$, of lowest (ground state) energies of the non-interacting atoms, we show that for $|y_i-y_j|,\ i,j \in \{1,...,M\}, i \neq j,$ large enough, $$W(y)=-\sum_{i<j}^{1,M} \frac{\sigma_{ij}}{|y_i-y_j|^6}+O(\sum_{i<j}^{1,M} \frac{1}{|y_i-y_j|^7})$$ where $\sigma_{ij}$ are in principle computable positive constants depending on the nature of the atoms $i$ and $j$.

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.