<p> This study investigates the development of flexibility and reversibility in a calculus environment that attends to linking multiple representations. Reversibility was studied through Krutetskii&rsquo;s framework of reversibility of two-way processes and reversibility of the mental process in reasoning. The study was conducted over approximately four months in a high school calculus classroom in an urban school district in a mid-Atlantic state. Instruction attended to linking multiple representations whenever possible. Four types of data were collected: 1) a pre-test, 2) a post-test, 3) daily assessments, and 4) clinical interviews. Twenty-one students completed a pretest and post-test that together assessed development of flexibility over the course of the study. They also completed daily assessments that were collected to provide evidence of the development of reversibility during the course of the study. Six students participated in four clinical interviews each, spread throughout the study. Inferential statistics were used to compare the results of the pre-test and post-test for significant differences and to determine significant differences in the presence of reversibility on the daily assessments over the course of the study. The clinical interviews were analyzed for evidence of students&rsquo; thought processes while solving reversible questions. Analysis revealed that over the course of the study, students demonstrated significant increases in both flexibility and reversibility. Two-way reversibility seemed to develop with relative ease for most students and often developed simultaneously with learning a forward process. Developing reversibility of the mental process in reasoning was difficult and tended to develop simultaneously with learning in a forward direction for students with high levels of flexibility. For students who did not develop reversibility simultaneously with forward learning, both two-way reversibility and reversibility of the mental process in reasoning were able to develop through multiple opportunities to solve reversible tasks of similar content. Analysis of the clinical interviews indicated that students typically followed a 4-step thought process when using reversibility to solve problems. Implications and limitations of the study and areas of further research were discussed.</p>

<p> The purpose of this study was to determine the correlation between participation of low achieving math students in a before-school computer math lab and how their participation affected the fourth grade California Standardized Test (CST) math scale scores at San Joaquin Elementary School. Students participated in the lab based their below proficient performance on multiplication facts tests. The lab was available from January through April in both 2010 and 2011. Students used interactive computer software called IXL. The focus in the lab was for students to master their multiplication tables (0 through 12). To evaluate if there was a statistically significant change in the CST scores because of the treatment, an Abbreviate Interrupted Time Series analysis was applied. The two years before the lab was implemented, the two years with the lab in place and the year after the lab was discontinued were compared. A t-test evaluation of the mean scores in a year-by-year comparison was run. The Levene Test using Homogeneity of Variances test was run which showed similar variances. For this reason, the ANOVA was then used along with the Tukey HSD post-hoc test to identify which groups showed a significant difference from each other. There was a significant improvement in the CST mean scores for fourth grade at San Joaquin Elementary School during the years the treatment was in place. By extending the school day and using interactive computer software together they showed a positive impact on student's CST scores.</p>

<p> Developmental math education continues to be a rising challenge for many post-secondary two-year institutions. In response to this challenge, many colleges are adapting their developmental math education programs by implementing best institutional and instructional practices as defined by recent literature. This descriptive and correlational study used data from the Developmental Education Program Survey (DEPS) and the selected state's Department of Education to explore how community colleges within the state have implemented these research-based, promising instructional and institutional practices in the area of developmental mathematics education. The quantitative study through multiple linear regression analysis determined if the implementation of certain practices predicted the success of the students. Findings indicated that developmental math education programs were implementing most of the research-based best practices for developmental math education. In addition, the results of this study indicated that changes in the instructional aspect of the program increased student success in developmental math more than changes in institutional policy. This study found that implementation of best institutional practices as defined in the literature had no relationship to student success while implementation of best research-based instructional practices had a significant relationship to student success. The researcher recommends that institutions utilize cooperative, collaborative and mastery learning in their developmental math courses and that they consider offering developmental math instruction as math refresher workshops, bridge programs, and co-requisite courses to college-level math. </p>

<p> According to statistics provided by the California State University Public Affairs Office, approximately 1/3 of incoming freshman need remediation in mathematics. However, there is little information available on which specific mathematical skills they lack. The CSU Entry Level Mathematics Test does not allow for disaggregation of scores by topic, so it is of limited use in studying students&rsquo; pre-existing skills. However, the Mathematics Diagnostic Testing Project Second Year Algebra Readiness Test is well correlated with the ELM, and can provide information regarding student deficiencies. </p><p> This study administered a student survey on prior preparation in mathematics and the MDTP to students enrolled in a required remedial mathematics course. The results were analyzed for patterns among the incorrect answers, and the correlation between students&rsquo; preparation and test results was studied. Results indicate that students can access only a small subset of the mathematical skills required for success in postsecondary courses.</p>

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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.