The purpose of this study was to assess the impact of a school division policy on early algebra on students' course taking patterns in high school. Over the past two decades, there has been significant growth in the number of students taking Algebra 1 in middle school. Research about the advantages and drawbacks to completing Algebra 1 prior to high school have mixed conclusions, with some suggesting that students benefit from the opportunity to take more advanced mathematics and science courses in high school and others concluding that students are more likely to fail and need to repeat courses if they take Algebra 1 early (Stein et al., 2011). Most of the research has focused on students taking Algebra 1 in eighth grade. At the same time, there is an ever-growing group of students seeking to take Algebra 1 even earlier, as evidenced by expansive growth in the number of students accessing Advanced Placement Calculus prior to twelfth grade (College Board, 1997; College Board, 2017). To assess the impact of early Algebra 1, the researcher considered transcript data for two cohorts of students in a large, suburban school district who took Algebra 1 in seventh or eighth grade. Statistical analysis was performed to assess whether students were likely to access the highest level mathematics courses available to them, whether they were staying in mathematics courses throughout all years of high school, and what patterns might emerge in mathematics and science course taking for students based on when they took Algebra 1. The findings indicated that students in this cohort who took Algebra 1 in eighth grade were more likely to complete the highest level mathematics courses available to them than those who took Algebra 1 in seventh grade, but they also took, on average, fewer total mathematics and science courses. For all students taking middle school Algebra 1, there were sharp declines in students accessing honors-level mathematics coursework as they advanced through the mathematics sequence. / Doctor of Education / The purpose of this study is to assess the impact of a school division policy on early algebra on students' course taking patterns in high school. Over the past two decades, there has been significant growth in the number of students taking Algebra 1 in middle school. Research about the advantages and drawbacks to completing Algebra 1 prior to high school have mixed conclusions, with some suggesting that students benefit from the opportunity to take more advanced mathematics and science courses in high school and others concluding that students are more likely to fail and need to repeat courses if they take Algebra 1 early (Stein et al., 2011). Most of the research has focused on students taking Algebra 1 in eighth grade. At the same time, there is an ever-growing group of students seeking to take Algebra 1 even earlier, as evidenced by expansive growth in the number of students accessing Advanced Placement Calculus prior to twelfth grade (College Board, 1997; College Board, 2017). To assess the impact of early Algebra 1, the researcher considered transcript data for two cohorts of students in a large, suburban school district who took Algebra 1 in seventh or eighth grade. Statistical analysis was performed to assess whether students were likely to access the highest level mathematics courses available to them, whether they were staying in mathematics courses throughout all years of high school, and what patterns might emerge in mathematics and science course taking for students based on when they took Algebra 1. The findings indicate that students in this cohort who took Algebra 1 in eighth grade were more likely to complete the highest-level mathematics courses available to them than those who took Algebra 1 in seventh grade, but they also took on average fewer total mathematics and science courses. For all students taking middle school Algebra 1, there were sharp declines in students accessing honors-level mathematics coursework as they advanced through the mathematics sequence.

This thesis examines the potential of a change analysis of student beliefs and attitudes about mathematics to inform teachers' reflective practice and provide the basis for modifying classroom practice. The author and two colleagues were involved in Math4theNines, a collaborative inquiry project for Ontario Grade 9 Mathematics. As part of that project, they developed an online survey to track the impact of their classroom practice on how students felt about mathematics and how students felt about themselves as learners of mathematics. The teachers reported that the before-and-after course survey and the accompanying change analysis that indicated any shifts in their students' attitudes and beliefs toward mathematics provided some unique and revealing perspectives on their practice. This study is a retrospective of that experience and an attempt to reproduce the results with three volunteer teachers. Although the results were mixed, there is some evidence to suggest that this approach has the potential to enhance teachers' focus on the new social-emotional strand in Ontario Grades 1 to 9 mathematics which presently is neither evaluated nor reported on. There is also potential to inform teachers in their efforts to develop the positive attitudes that have been shown to improve academic achievement and encourage entry into STEM fields.

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This textbook is intended for college juniors or seniors majoring in mathematics, who plan to become high school teachers. It seeks to provide a deeper perspective on secondary mathematics, showing the interplay between plane geometry and algebra. A distinctive characteristic of the book is the frequent discussion of multiple paths to the solution of a problem or the proof of a theorem. Practically none of the topics covered in the book overlap with the content of courses taken by mathematics majors, say real analysis, abstract algebra, differential equations, combinatorics, probability and statistics, number theory, etc. These courses, and several others, provide indispensable mathematical maturity but are rather distant from the core of high school mathematics. Precisely, one of our main objectives is to bridge the gap between the latter and college-level mathematics. / https://dc.etsu.edu/etsu_books/1084/thumbnail.jpg

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African American students' severe underachievement in mathematics in comparison to their peers has been framed as an achievement gap that continues to widen despite the efforts of many education scholars and leaders. Throughout history in the United States, mathematics education has been designed, developed, and delivered within a Eurocentric philosophy. Consequently, African American students have been at a systemic disadvantage in terms of perceiving the cultural relevance of mathematics; which has served as a detriment to their academic success. By merging ethnomathematics and culturally relevant pedagogy (CRP) into a theoretical framework, this study investigates these issues and proposes a shift in mathematics education toward a more culturally aware approach. In this study, it is argued that implementing a multicultural education approach such as ethnomathematics into the mathematics curriculum coupled with employing culturally relevant pedagogical practices will increase relevance in the mathematics education for African American students. The purpose of this study was to gain African American high school students’ perception of mathematics, as well as their cultural awareness and its relation to mathematics education. To gain students’ perceptions about mathematics education from a cultural respect, 375 students in grades 9-12 completed three online surveys which were (1) a four-item demographic questionnaire (age, gender, grade, ethnicity), (2) the 40-item Attitude Towards Mathematics Inventory (ATMI), and (3) the 12-item Students Perception about Cultural Awareness (SPCA) survey. This study incorporated a quantitative, correlational research design. To address research questions one and two, Pearson correlations were conducted to examine the associations between the variables of interest which were (1) Value, (2) Enjoyment, (3) Sense of Security, (4) Motivation, and (5) Cultural Awareness. Variables (1), (2), (3), (4) were derived from the ATMI survey through factor analysis while variable (5) was constructed from the SPCA survey. To address research question three, a MANOVA was conducted to assess for differences in attitudes toward mathematics and perceptions of cultural awareness by ethnicity. For research questions one and two, it was found that there was a statistically significant correlation between the variables of interest. For research question three, it was found that there was not a statistically significant difference in the variables of interest by ethnicity. In further analysis of the data, it was found that many African American students have a substandard attitude of value, enjoyment, sense of security, and motivation toward mathematics. Nonetheless, these students had a high sense of cultural awareness and cultural pride. Generally, the students felt that the incorporation of culture into mathematics would assist in raising their achievement to some degree. This study highlights recommendations to educational leaders to learn about the culture of their students, allow that data to inform policy decisions, and lead a shift to the approach of mathematics education toward the theories of ethnomathematics and CRP.

A small group of profit seeking publishers dominates the American textbook market and guides the learning of the majority of our nation’s calculus students. The College Board’s AP Calculus curriculum is a de facto national standard for this gateway course that is critically important to 21st century STEM careers. A multi-representational understanding of calculus is a central pillar of the AP curriculum. This dissertation asks whether this multi-representational vision is manifest in popular calculus textbooks. This dissertation began with a survey of all AP Calculus AB Examination free response items, 2002-2011, and found that students score worse on items characterized by numerical anchors or verbal targets. Based on previously elucidated models, a new cognitive model of five levels and six principles is developed for the purpose of calculus textbook task analysis. This model explicates complexity as a function of representational input and output. Eight popular secondary calculus textbooks were selected for study based on Amazon sales rank data. All verbally anchored mathematical tasks (n=555) from sections of those books concerning the mean value theorem and all AP Calculus AB prompts (n=226) were analyzed for cognitive complexity and representational diversity using the model. The textbook study found that calculus textbooks underrepresented the numerical anchor and verbal target. It found that the textbooks were both explicitly and implicitly less cognitively complex than the AP test. The article suggested that textbook tasks should be less dense, avoid cognitive attenuation, move away from the stand-alone item, juxtapose anchor representations, scaffold student solutions, incorporate previously considered overarching concepts and include more profound follow-up questions. To date there have been no studies of calculus textbook content based on established research on cognitive learning. Given the critical role that their calculus course plays in the lives of hundreds of thousands of students annually, it is incumbent upon the College Board to establish a textbook review process at the very least in the same vain as the teacher syllabus auditing process established in recent years.

This study investigated secondary school students’ errors and misconceptions in algebra with a view to expose the nature and origin of those errors and to make suggestions for classroom teaching. The study used a mixed method research design. An algebra test which was pilot-tested for its validity and reliability was given to a sample of grade 11 students in an urban secondary school in Ontario. The test contained questions from four main areas of algebra: variables, algebraic expressions, equations, and word problems. A rubric containing the observed errors was prepared for each conceptual area. Two weeks after the test, six students were interviewed to identify their misconceptions and their reasoning. In the interview process, students were asked to explain their thinking while they were doing the same problems again. Some prompting questions were asked to facilitate this process and to clarify more about students’ claims. The results indicated a number of error categories under each area. Some errors emanated from misconceptions. Under variables, the main reason for misconceptions was the lack of understanding of the basic concept of the variable in different contexts. The abstract structure of algebraic expressions posed many problems to students such as understanding or manipulating them according to accepted rules, procedures, or algorithms. Inadequate understanding of the uses of the equal sign and its properties when it is used in an equation was a major problem that hindered solving equations correctly. The main difficulty in word problems was translating them from natural language to algebraic language. Students used guessing or trial and error methods extensively in solving word problems. Some other difficulties for students which are non-algebraic in nature were also found in this study. Some of these features were: unstable conceptual models, haphazard reasoning, lack of arithmetic skills, lack or non-use of metacognitive skills, and test anxiety. Having the correct conceptual (why), procedural (how), declarative (what), and conditional knowledge (when) based on the stage of the problem solving process will allow students to avoid many errors and misconceptions. Conducting individual interviews in classroom situations is important not only to identify errors and misconceptions but also to recognize individual differences.

This study investigated secondary school students’ errors and misconceptions in algebra with a view to expose the nature and origin of those errors and to make suggestions for classroom teaching. The study used a mixed method research design. An algebra test which was pilot-tested for its validity and reliability was given to a sample of grade 11 students in an urban secondary school in Ontario. The test contained questions from four main areas of algebra: variables, algebraic expressions, equations, and word problems. A rubric containing the observed errors was prepared for each conceptual area. Two weeks after the test, six students were interviewed to identify their misconceptions and their reasoning. In the interview process, students were asked to explain their thinking while they were doing the same problems again. Some prompting questions were asked to facilitate this process and to clarify more about students’ claims. The results indicated a number of error categories under each area. Some errors emanated from misconceptions. Under variables, the main reason for misconceptions was the lack of understanding of the basic concept of the variable in different contexts. The abstract structure of algebraic expressions posed many problems to students such as understanding or manipulating them according to accepted rules, procedures, or algorithms. Inadequate understanding of the uses of the equal sign and its properties when it is used in an equation was a major problem that hindered solving equations correctly. The main difficulty in word problems was translating them from natural language to algebraic language. Students used guessing or trial and error methods extensively in solving word problems. Some other difficulties for students which are non-algebraic in nature were also found in this study. Some of these features were: unstable conceptual models, haphazard reasoning, lack of arithmetic skills, lack or non-use of metacognitive skills, and test anxiety. Having the correct conceptual (why), procedural (how), declarative (what), and conditional knowledge (when) based on the stage of the problem solving process will allow students to avoid many errors and misconceptions. Conducting individual interviews in classroom situations is important not only to identify errors and misconceptions but also to recognize individual differences.

Despite the vast research on the effects of acceleration programs on student achievement there is little quantitative confirmation of the benefits of these programs and there is no research that investigates the effects of acceleration on students’ VCE Mathematics study scores. / This research attempts to fill this gap by considering four years of data provided by the Victorian Curriculum and Assessment Authority (VCAA) relating to achievement in mathematics. Acceleration in this study means the completion of the Year 12Mathematical Methods study during Year 11. The data constitutes experimental data for content acceleration and the results of students from schools without such acceleration programs provide the corresponding control data. However, the acceleration decision is not taken randomly by schools, so this data is only quasi-experimental in nature. The measures of mathematical achievement (Mathematical Methods and Specialist Mathematics study scores) are carefully audited, and are accepted as reliable and valid by the Victorian education system. Controlling for individual characteristics such as gender and prior knowledge, and allowing for moderation effects due to school sector (Government, Catholic and Independent) and school class setting (single-sex or coeducational), the effects of content acceleration are measured using multi-level modelling. / This study examines the effects of acceleration on the VCE Mathematics study scores of students who completed both Mathematical Methods (Units 3&4) and Specialist Mathematics (Units 3&4) in Victoria, over a four-year period (2001-2004). On average this involved 5341 students from 341 schools in each year with 829 students included in a content accelerated program. / The results suggest that content acceleration is beneficial, especially for students with higher prior knowledge scores. The quasi-experimental nature of the data means that a causal relationship between acceleration and students’ mathematical performance can be claimed. In particular, this study showed that the effect of acceleration on students’ Mathematical Methods (the Year 12 study taken in Year 11 by accelerated students) study score was not significant. However, the effect of acceleration on students’ Specialist Mathematics study scores was significant. Accelerated students performed, on average,2.7 points higher (on a 50 point scale) than equal ability age-peers who were not accelerated. Interestingly, for accelerated students who scored in the top 2% for their General Achievement Test, in the mathematics, science and technology component, their Specialist Mathematics study scores were on average, almost 5 points higher (on a 50point scale) than their equal ability age-peers. The statistical control of other factors means that these results can also be generalised to other states, other countries and, probably, to other subjects.