Many U.S. students possess only a weak knowledge of fraction arithmetic. I hypothesize that textbooks are a critical reason for students’ poor performance on fraction arithmetic. This is not because of what textbooks contain but rather because of what they lack. Distributions of fraction arithmetic problems in textbooks are imbalanced, with certain types of problems almost never presented (Braithwaite et al., 2017). As a result, students often err when they attempt to solve those rare types of problems (Siegler & Pyke, 2013). Two experiments, a pilot study (n = 40 students in grades 5 through 7) and a larger study (n = 127 students in grades 6 and 7), were conducted. These experiments utilized a pretest-intervention-posttest design to empirically test the benefits of providing either a complement of the typical textbook distribution of problems (hyperbalanced practice) or an equal distribution of problems (balanced practice) compared to the benefits of providing a practice set that followed the typical distribution found in math textbooks. A MANCOVA and follow-up ANCOVAs revealed significant differences between students in the textbook condition and students in the balanced and hyperbalanced conditions.For items involving adding fractions with unequal denominators, students who received typical textbook practice showed greater improvement and made fewer strategy errors than students who received hyperbalanced practice. For items involving multiplying two fractions with equal denominators, the opposite was true. Students who received hyperbalanced practice showed greater improvement and made fewer strategy errors than students who received typical textbook practice on items involving multiplying two fractions with equal denominators. Finally, students who received fully balanced practice showed greater improvement and made fewer strategy errors than students who received typical textbook practice on problems involving multiplying one whole number and one fraction. This last finding was of particular interest since none of the practice conditions included practice with that item type. The results of this study demonstrate that even a brief intervention in which students received extra practice with rare item types could improve performance. It also showed that gains in one type of item often resulted in decrements in others, which must be considered when making recommendations to textbook publishers and educators. In sum, this dissertation seeks to make a scholarly contribution to the field by discussing the role that textbooks play in student performance and by analyzing the benefits of supplementing typical textbook instruction with differently balanced fraction arithmetic practice.

The objective of this dissertation was to explore the paradoxical nature of Combinatorics as both a difficult and accessible domain in Mathematics, particularly for K-12 students. This paradox in Combinatorics' nature raised questions about how students interact with problems in this domain and the factors influencing their understanding and engagement with mathematics. To investigate these aspects, the study utilized a mixed-methods approach. Quantitative data was derived from the Exame Nacional do Ensino Médio (ENEM), a large-scale nationwide assessment in Brazil. The analysis focused on 28 Combinatorics problems identified across 12 years of the exam, comparing them with non-Combinatorics problems. The study also involved qualitative methods, specifically task-based interviews with Brazilian students, primarily from disadvantaged school backgrounds. These interviews aimed to provide deeper insights into how students approach, understand, and engage with Combinatorics problems. The findings revealed that while the combinatorial domain is notably accessible in the sense that it allows students with varied backgrounds to understand what problems ask, this accessibility does not necessarily translate into students consistently arriving at correct solutions. The study also found that achievement gaps between students of private and public schools in Brazil are smaller in Combinatorics is than in other mathematical domains. Together, these findings point to Combinatorics as a domain that can contribute to issues of equity in mathematics teaching and learning. Furthermore, the research underscored the importance of considering both the 'product' (correct answers) and 'process' (mathematical thinking) aspects in mathematics education, especially in contexts aiming to promote equitable learning opportunities.

No description available.

No description available.

No description available.

A growing body of literature seeks to challenge the deficit-oriented narratives around Black students in mathematics by exploring the perspectives and experiences of successful Black students and professionals in the field—in doing so, aiming to provide counternarratives to the dominant discourse that center their participants’ voices, as well as to understand the factors that may influence Black students’ decisions to pursue advanced mathematics degrees and careers. Many such studies highlight racialized mathematics identity as a crucial factor for their participants’ long-term mathematics engagement; such studies seek to identify the sociostructural factors that contribute to its development, but generally do not examine the roles of different psychological factors. Despite evidence of the existence of relationships between mathematics identity development and specific cognitive factors such as mathematics conceptions, as well as affective factors such as mathematics self-efficacy, few studies have examined how these various constructs interact with identity development, and in turn, students’ mathematics engagement and learning outcomes. Fewer have examined these factors’ interactions for Black participants. A comprehensive understanding of how and why Black students decide to persist in mathematics necessitates an integrated approach that simultaneously examines the connections between participants’ psychological and sociostructural factors. In light of past findings around the significant role of mathematics self-efficacy in shaping students’ long-term mathematics engagement; prior findings around relationships between mathematics self-efficacy and identity; and the influences of mathematics conceptions on other affective mathematics beliefs, I chose for the present study to investigate the contributing factors for and interactions between mathematics self-efficacy, identity, and conceptions throughout the academic trajectories of several contemporary Black mathematicians. In doing so, I hoped to amplify my participants’ perspectives in the broader conversation around how to support Black students and students belonging to other historically excluded groups as they navigate mathematics learning environments that, at times, fail to stimulate, nurture, or uplift them. Using thematic analysis of several extensive semi-structured interviews, this narrative inquiry investigated these questions through the narratives of five contemporary Black mathematicians who were selected from a prior study on the impacts of sharing diverse mathematicians’ stories with students. Data sources included participants’ extant interview transcripts from both this study and other prior studies; as well as extensive follow-up interviews, online public engagement materials such as lectures and media publications, and my own memos. Data was coded using thematic analysis for both deductive themes related to the conceptual framework for this study, as well as emerging themes suggesting the existence of potential interactions between these constructs. Findings, which are reported in narrative form, suggest the importance of early and ongoing engagement with open-ended and reasoning-based mathematics tasks as a means of promoting broad conceptions of mathematics, self-efficacy for completing more challenging nonroutine tasks, as well as robust mathematics identities. Additionally, and in contrast to much of the literature around self-efficacy sources, findings from the present study highlight the particular importance of vicarious experiences of success by other Black mathematicians in shaping not only participants’ mathematics self-efficacies but their conceptions of mathematics, and accordingly, their dispositions toward mathematics.

Ed. D.

Effort was made to identify teaching competencies of value to the pre-algebra mathematics teacher, to construct a teaching competencies diagnostic inventory based on pre-algebra mathematics teaching competencies selected from those identified, to determine if such a diagnostic inventory could be validated using student observations on an instrument similar to the diagnostic inventory, and to investigate the responses on the two inventories using correlational analysis and factor analysis. Nine educators involved in the in-service and pre-service education of pre-algebra mathematics teachers identified 28 competencies of particular value to the pre-algebra mathematics teacher. From this list of competencies, 10 competencies were selected for use in the study. The teacher diagnostic inventory is composed of teaching situation descriptors and personal data items. The teacher is asked to indicate how closely his classroom behavior is described by the situation description. The student inventory consists of situations similar to those described in the teacher inventory and personal data items regarding the students' sex and selected attitudes. The student is asked to indicate his perception of his teacher1s behavior in each situation described. The teacher diagnostic inventory was distributed to a sample of pre-algebra mathematics teachers employed in 28 schools located in southeastern Oklahoma. Copies of the student inventory were distributed by each participating teacher to the students in one of his pre-algebra mathematics classes. Based on the analyses of the responses of 73 teachers and 1602 students, it was concluded that the teacher's responses could not be validated using the students' observations. Factor analytic studies of the teacher responses and the student responses revealed the presence of eight factors among the teacher inventory items and the presence of six factors among the student inventory items. These factors were identified according to the items with loadings of .35 and more on each factor. A discussion of the relationships between the item scores and responses and the factors contained in the two inventories is included. / Ed. D.

M.S.

This study describes the use of spreadsheets in business computer applications (BCA) courses and identifies and evaluates the use of spreadsheets for emphasizing and teaching mathematics in BCA courses. Survey instruments in the form of questionnaires were sent to all of the Virginia high schools with the BCA course to attain the BCA teachers' perceptions of spreadsheet use. Responses were received from 257 teachers, or 70% of the total assumed population. Findings reveal that 95% of the Virginia BCA teachers include spreadsheet instruction in their BCA courses. The mean number of class periods used for spreadsheet instruction is 27; however, the standard deviation is relatively high. The basic mathematics operations received extensive emphasis. The more complex operations only some emphasis. Accounting and Payroll computation receive the most amount of spreadsheet instructional time. Over half of the BCA teachers who teach spreadsheets include increasing their students' mathematics skills as one of the course objectives. At the same time, however, only 22% of the respondents evaluate their students' mathematics skills independently from spreadsheet application evaluation. Although this independent mathematics evaluation seldom occurs, 72% of the respondents believe that teaching spreadsheets enhances students' mathematics skills to some extent, and 42% believe that teaching spreadsheets enhances students' mathematics skills extensively. Considering the amount of material to be taught in spreadsheet instruction, not enough of the BCA class time is devoted to spreadsheet instruction. Much spreadsheet instruction precedes emphasis on mathematics as part of the instruction. Also, considering that students emphasize in their learning what they are evaluated on, the importance the BCA teachers placed on mathematics appears to be minimal. / Master of Science