In this research, I utilize the theoretical perspective Knowledge In Pieces to identify the knowledge resources students utilize while in the process of completing various differential equations tasks. In addition I explore how this utilization changes over the course of a semester, and how resources related to the concepts of function and rate of change supported the students in completing the tasks. I do so using data collected from a series of task-based individual interviews with two students enrolled in separate differential equations courses. The results provide a fine-grained description of the knowledge students consider to be productive with regard to completing various differential equations tasks. Further the analysis resulted in the identification of five ways students interpret differential equations tasks and how these interpretations are related to the knowledge resources students utilize while completing the various tasks. Lastly, this research makes a contribution to mathematics education by illuminating the knowledge concerning function and rate of change students utilize and how this knowledge comes together to support students in drawing connections between differential equations and their solutions, structuring those solutions, and reasoning with relationships present in the differential equations. / Ph. D.

Women of color in general and Black women in particular who pursue undergraduate and graduate degrees in mathematics are nearly invisible in the mathematics education research literature (Borum & Walker, 2012). The majority of research published in the mid-to-late twentieth century that explored the mathematics education of women was limited not only by failing to explore the unique mathematical experiences of women of color but also by employing quantitative methodologies in positivist frames (see, e.g., Benbow & Stanley, 1980; Fennema & Sherman, 1977; Hyde, Fennema, Ryan, Frost, & Hopp, 1990). Therefore, the purpose of this narrative inquiry project was to come alongside Black women who earned an undergraduate degree in mathematics and conduct an inquiry into their mathematics teaching and learning experiences. Specifically, the study explored the life and schooling experiences of mathematically talented Black women who attended Spelman College from the 1980s to 2000s. While theoretical and methodological elements from both Black feminist standpoint theory (e.g., Collins, 1986) and womanist theory (e.g., Phillips, 2006) have framed my thinking, in the end, both theoretically and methodologically, narrative inquiry grounded the project, affording my participants (and me) the opportunity to tell stories of their (our) mathematical experiences. Initially, three central questions guided the research: (1) What were the life and schooling experiences of Black women who pursued their undergraduate degree in mathematics at Spelman College from the 1980s to 2000s? (2) How did larger socio-historical and -cultural contexts and life experiences (on and off campus) affect their image of themselves as mathematicians? and (3) How did relationships with other Spelman students, faculty, and staff influence their short- and long-term goals in the field of mathematics? As I employed narrative inquiry and developed my research puzzle, I focused on particular moments in my participants’ mathematical lives—their sacred stories—identifying common threads across experiences. I share my participants lived experiences in the hope that readers will engage in “resonant remembering” as they “rethink and reimagine” relationships and “wonder alongside” my participants and me (Clandinin, 2013, p. 51). My participants’ stories highlight the importance of familial support and influence on education, the role and academic experience of Black women mathematics majors, and mentorship of caring faculty and staff and positive peer relationships. Implications for mathematically talented Black women are discussed.

This dissertation examines the relationship between an instructional style called Interactive-Engagement (IE) and gains on a measure of conceptual knowledge called the Calculus Concept Inventory (CCI). The data comes from two semesters of introductory calculus courses (Fall 2010 and Spring 2011), consisting of a total of 482 students from the first semester and 5 instructors from the second semester. The study involved the construction and development of a videocoding protocol to analyze the type of IE episodes which occurred during classes. The counts of these episodes were then studied along with student gains, measured in a number of different ways. These methods included a traditionally used measure of gain, called normalized gain, which is computed at the instructor level. Additionally, gains were further investigated by constructing hierarchical linear models (HLMs) which allowed us to consider individual student characteristics along with the measures of classroom interactivity. Another framework for computing ability estimates, called Item Response Theory (IRT), was used to compute gains, allowing us to determine whether the method of computing gains affected our conclusions. The initial investigation using instructor-level gain scores indicated that the total number of interactions in a classroom and a particular type of interaction called "encouraging revisions" were significantly associated with normalized gain scores. When individual-level gain scores were considered, however, these instructor-level variables were no longer significantly associated with gains unless a variable indicating whether a student had taken calculus or precalculus in high school or in college was included in the model. When IRT was used to create an alternative measure of gain, the IE variables were not significant predictors of gains, regardless of whether prior mathematics courses were included, suggesting that the method of calculating gain scores is relevant to our findings.

In this research, I set out to elucidate the relationships that might exist between students' conceptual understanding upon which they draw in their proof activity. I explore these relationships using data from individual interviews with three students from a junior-level Modern Algebra course. Each phase of analysis was iterative, consisting of iterative coding drawing on grounded theory methodology (Charmaz, 2000, 2006; Glaser and Strauss, 1967). In the first phase, I analyzed the participants' interview responses to model their conceptual understanding by drawing on the form/function framework (Saxe, et al., 1998). I then analyzed the participants proof activity using Aberdein's (2006a, 2006b) extension of Toulmin's (1969) model of argumentation. Finally, I analyzed across participants' proofs to analyze emerging patterns of relationships between the models of participants' understanding of identity and inverse and the participants' proof activity. These analyses contributed to the development of three emerging constructs: form shifts in service of sense-making, re-claiming, and lemma generation. These three constructs provide insight into how conceptual understanding relates to proof activity. / Ph. D.

It is well documented that undergraduate students struggle with the more formal and abstract concepts of vector space theory in a first course on linear algebra. Some of these students continue on to classes in abstract algebra, where they learn about algebraic structures such as groups. It is clear to the seasoned mathematician that vector spaces are in fact groups, and so linear transformations are group homomorphisms with extra restrictions. This study explores the question of whether or not students see this connection as well. In addition, I probe the ways in which students’ stated understandings are the same or different across contexts, and how these differences may help or hinder connection making across domains. Students’ understandings are also briefly compared to those of mathematics professors in order to highlight similarities and discrepancies between reality and idealistic expectations. The data for this study primarily comes from clinical interviews with ten undergraduates and three professors. The clinical interviews contained multiple card sorts in which students expressed the connections they saw within and across the domains of linear algebra and abstract algebra, with an emphasis specifically on linear transformations and group homomorphisms. Qualitative data was analyzed using abductive reasoning through multiple rounds of coding and generating themes. Overall, I found that students ranged from having very few connections, to beginning to form connections once placed in the interview setting, to already having a well-integrated morphism schema across domains. A considerable portion of this paper explores the many and varied ways in which students succeeded and failed in making mathematically correct connections, using the language of research on analogical reasoning to frame the discussion. Of particular interest were the ways in which isomorphisms did or did not play a role in understanding both morphisms, how students did not regularly connect the concepts of matrices and linear transformations, and how vector spaces were not fully aligned with groups as algebraic structures.

ABSTRACT STANDARDS-BASED INSTRUCTION: A CASE STUDY OF A COLLEGE ALGEBRA TEACHER by Anthonia Ekwuocha The lecture method has dominated undergraduate mathematics education (Bergsten, 2007). The lecture method promotes passive learning instead of active learning among students, thus contributing to attrition in undergraduate mathematics. Standards-based instruction has been found to be effective in reducing students’ attrition in undergraduate mathematics (Ellington, 2005). College algebra is gatekeeper for higher undergraduate mathematics courses (Thiel, Peterman & Brown, 2008). Research indicates that if college algebra is taught with standards-based teaching strategies, it will help reduce students’ attrition and encourage more students to take higher level mathematics courses (Burmeister, Kenney, & Nice, 1996). Standards-based instructional strategies include but are not limited to real life applications, cooperative learning, proper use of technology, implementation of writing, multiple approaches, connection with other experiences, and experiential teaching (American Mathematical Association of Two-Year Colleges (AMATYC), 2006). Despite all effort to improve undergraduate mathematics instruction, there are still limited empirical studies on standards-based instruction in college algebra. Research in undergraduate mathematics education is a new field of study (Brown & Murphy, 2000). Research reported that overall students’ attrition in college algebra could be as high as 41% in a community college (Owens, 2003). This high attrition rate in college algebra may impact students’ continuation in higher mathematics courses and their interest in the field of mathematics. As a result more research efforts must be focused on ways to improve college algebra instruction. Thus, the purpose of this study was to explore the teaching practices of a college algebra teacher who adopts standards-based techniques in his classroom. The research questions that guided the study were: What teaching practices are used in the mathematics classroom of a college algebra teacher? How are the teaching practices of the teacher aligned with the characteristics of standards-based instruction? The participant of the study was a college algebra teacher who was identified as a standards-based teacher. The teaching practices of the teacher were analyzed and presented using a qualitative single case study method. Data were collected from interviews with the teacher, classroom observations, and artifacts. The research project was drawn from the frameworks of culturally relevant pedagogy theory, symbolic interaction theory, experiential teaching theory, and standards-based instruction. Analysis of the data showed that the teaching practices of the participant were mathematical communication, proper use of technology in instruction and assessment, building mathematical connections, multiple representations, motivating students to learn mathematics, and repetition of key terms. The teaching practices aligned with the characteristics of standards-based instruction. Findings from the study suggest that standards-based instruction strategies should be used in undergraduate mathematics education, especially in teaching college algebra to alleviate some of the problems. Moreover, university administrators at college level should organize workshops and professional development about standards-based instruction strategies for their teachers.

In this paper I present a theoretical analysis (genetic decomposition) of the cognitive constructions for the concept of infinite Riemann sums following Piaget\'s model of epistemology. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my continual observations of students in the process of learning. Based on this analysis I plan to suggest instructional procedures that motivate the mental activities described in the proposed genetic decomposition. In a later study, I plan to present empirical data in the form of informal interviews with students at different stages of learning. The analysis of those interviews may suggest a review of my initial genetic decomposition.

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In this paper I present a theoretical analysis (genetic decomposition) of the cognitive constructions for the concept of infinite Riemann sums following Piaget\''s model of epistemology. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my continual observations of students in the process of learning. Based on this analysis I plan to suggest instructional procedures that motivate the mental activities described in the proposed genetic decomposition. In a later study, I plan to present empirical data in the form of informal interviews with students at different stages of learning. The analysis of those interviews may suggest a review of my initial genetic decomposition.