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.

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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.

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.

The problem of this study is to analyze and evaluate methods of making sixth grade arithmetic meaningful. In the accomplishment of the purpose, the writer made an intensive investigation into teaching methods of arithmetic as set forth by writers from the beginning of the twentieth century to the present day, with special interest centered upon the place of the teacher in the development and utilization of meaningful situations.

The problem under consideration attempts to answer the question, Will different methods of teaching the solution of problems in finding areas and perimeters of plane surfaces produce different results; and if so, how great is the difference?

The purpose of this study is to make a survey of some changes in the aims, techniques, and methods of teaching mathematics in the secondary schools and to some philosophical bases underlying the changes.