There is a strong push from within mathematics education reform to incorporate representations in math classrooms (Behr, Harel, Post, & Lesh, 1993; Kieren, 1993; NCTM, 2000). However, questions regarding what representations should be used (for a given topic) and how representations should be used (such that students gain a deep understanding of that topic and a deep understanding of the representations) remain largely unanswered. Hence, we need a well-specified and general theoretical treatment of how students co-develop domain and representational competence.

In this dissertation study, I use design-based research (DBR) to investigate and support growth and change in students' knowledge of rational number operations. "Among all the topics in K-12 curriculum, rational numbers arguably hold the distinction of being the most protracted in terms of development, the most difficult to teach, the most mathematically complex, the most cognitively challenging, and the most essential to success in higher mathematics and science" (Lamon, 2007). In order to shed some light on the domain of rational number operations, I designed a learning environment centered on the Area Model for Fraction Multiplication (AM-FM) representation, a computer-based tool intended to help students develop a deep understanding of fraction multiplication.

Data for the dissertation were collected from an urban school with a racially and socio-economically diverse student population. I met with ten students once a week for four weeks. During the first and last session students were asked to think-aloud through a pretest and posttest. The second and third sessions consisted of semi-structured clinical interviews during which students were asked to solve fraction multiplication problems using the AM-FM representation. All sessions were videotaped and transcribed. Two students were chosen to serve as cases of knowledge growth and change.

Findings indicate that both students followed a particular learning trajectory for making sense of fraction multiplication when using the AM-FM representation and their emergent knowledge was context sensitive. Futhermore, DBR is predicated on (a) design refinement and (b) local theory development (diSessa & Cobb, 2004; Schoenfeld, 2006). With respect to design, the AM-FM representation and the clinical interview protocol was refined based on analysis of the data. With respect to local theory, I offered a decomposition of competence with fraction multiplication (i.e., domain competence) and the AM-FM representation (i.e., representational competence). Local theory was also refined based on an analysis of the data.

Discussions of race in educational research have focused primarily on performance gaps and differential access to advanced coursework. Thus, very little is known about how race mediates the learning process, particularly with respect to classroom participation and student identity formation. This dissertation examines mathematics learning as a context for illuminating the racial dynamics of learning in everyday classroom activity.

Although mathematics and race may seem strange bedfellows, a poststructural analysis reveals specific linkages between them that suggest that their discourses are actually well aligned. To conceptualize this alignment, this dissertation introduces the theoretical frame of racial-mathematical discourse, which establishes the groundwork for the empirical investigation reported here. Observations took place in four mathematics classrooms at a racially diverse high school over the course of a school year. Interviews (n=35) were conducted with students from the focal classrooms. Data were analyzed to explore how students make sense of racial-mathematical discourse, and to gauge the discourse's impact on learning.

Findings indicate that racial-mathematical narratives were central to students' sense making. All students reported awareness of the "Asians are good at math" narrative, as part of a web of racial ideology. Importantly, students linked it to narratives about other groups' mathematical inferiority (e.g., "Blacks are bad at math"). They also connected racial-mathematical narratives to broader racialized discourses outside mathematics (e.g., perceptions of intelligence). Students observed the presence of these narratives in locations outside the school setting, such as media imagery and international comparisons.

Data further suggest that racial-mathematical discourse is not a static belief system. Rather, it emerges and is reified as students engage in typical classroom practices, such as noting which classmates get asked for help. This is consequential for learning, in that the deployment of racial narratives in social interaction frames students' opportunities to build identities as capable learners. This dissertation develops a framework leveraging insights from sociocultural and poststructural theory to trace the impact of racialized classroom episodes on students' identity formation. It highlights critical issues that need to be taken into account in the design of equitable learning environments, especially for students of color from persistently marginalized backgrounds.

What happens to pedagogy when a teacher’s personal goals of supporting students’ productive dispositions toward learning collide with her professional identity as a successful teacher whose students perform well on standardized tests? This dissertation is a mixed-methods case study that shows how context shapes one teacher’s identity and decision-making, such that she seems to be two drastically different teachers in two different instructional contexts – a summer course in which she had complete flexibility over the curriculum, goals, and achievement measures and an academic year course in which she felt bounded by the state standards test. The dissertation examines the very real consequences these pedagogical decisions have for students.

Using qualitative classroom observations and quantitative survey and assessment data, this dissertation examines why, despite the teacher’s strong commitment to growth mindset instruction and equity in both contexts, the teacher implemented pedagogical moves that contributed to distinctly different opportunities for students to engage with rich mathematics in each, and what those shifts meant for students’ mathematical identities and learning.

The different cultural contexts in the summer and academic years offered the teacher identity resources about what was valued as good teaching, which led to distinct pedagogical decisions that aligned with the salient aspects of her professional identity in each context. Despite her commitment to growth mindset instruction in both contexts, this teacher implemented pedagogical moves that contributed to distinctly different opportunities for students to engage with rich mathematics and develop productive mathematical self-concepts.

This dissertation examines the ways the institutional context shifted and practices changed subtly as a result, and uses these comparisons to unpack which elements of the whole system of teaching for a growth mindset are necessary to contribute to productive changes in student mindsets or dispositions toward mathematics, engagement, and persistence with learning. Using Ms. M as a case study, this dissertation sheds light on the ways in which school contexts - in concert with a teachers’ multifaceted identity - contribute to decision-making while setting instructional goals.

The dominant image of mathematics as an abstract, universal, disinterested, and pure body of knowledge both misrepresents disciplinary practice and alienates many students. Undergraduate proof-based courses such as real analysis, which are supposed to introduce students to contemporary academic mathematics, often contribute to such idealized notions as well. To counter this idealized image, social scientists from various disciplines have characterized academic mathematics as an embodied, socially and materially distributed, socio-historically situated, and ideologically laden practice. Mathematics educators have been drawing on such insights to develop pedagogical approaches aimed at providing students with more authentic disciplinary experiences. Yet, not enough attention has been devoted to examining how (and why) proof-based mathematics is idealized to begin with. Specifically, how is it that real analysis lectures, taught by practicing research mathematicians who have first-hand experience with the discipline, give rise to idealizations that both distort and exclude?

In this dissertation I address this question through the close examination of language. That is, I set out to understand how mundane features of lecture discourse construct potentially problematic common sense ideas about mathematics. The reported study is a video-based micro-ethnography, conducted in three iterations of data collection in a prestigious mathematics department in a public research university in the United States. The primary data are video-recordings of lectures and lecture notes of real analysis courses taught by five different instructors, one in Spring 2015 and four in Fall 2020, all delivered on Zoom due to Covid-19.

Informed by socio-cultural theories and drawing on a variety of discourse analytic techniques, the dissertation examines three features of lecture discourse: the stories instructors told about the purpose of the real analysis course and academic mathematics as a whole, the value-laden attributes they deployed to characterize and appraise mathematics, and the way lecture discourse enacted human subjectivity in the context of mathematical activity. To situate the stories told to students in real analysis lectures, I also examined the stories about the purpose of mathematics that prominent mathematicians articulated in famous meta-reflective essays about disciplinary practice.

In chapter ‎3, I identify four different stories articulated by mathematicians in meta-reflective essays, distinguished by the kind of object they construct for mathematics as an activity system. I argue that the stories are consequential in that they correspond to different grammars of evaluations and can, in particular, construe radically different educational goals for courses such as real analysis. Furthermore, the existence of distinct high-profile perspectives on practice suggests that any presentation of mathematics as a practice with a single, universally agreed upon telos can function as an unproductive idealization of the discipline.

Chapter ‎5 examines stories about mathematics told in introductory real analysis lectures, where instructors face the rhetorical challenge of ‘motivating’ the subject and its new way of doing math. I identify five distinct meta-stories mobilized by instructors, characterizing each story by the assumptions it presupposes (what one needs to buy into to find the stories compelling). Assessing each story and its underlying assumptions for faithfulness to the discipline and relatability to students’ (likely) past experiences, I found that all five meta-stories were somewhat idealizing. One notable mechanism of idealization was the stories’ reliance on metaphors that render certain goals and values self-evident (e.g. “math is a structure, so it needs solid foundations.”)

Chapter ‎6 examines a more mundane and pervasive feature of lecture discourse: appraisals using value-laden attributes (e.g. “this is a precise definition”). I flagged all instances of instructors assigning a characteristic to a mathematical object or processes in four focal lectures, coding both the exact adjective or adverb used and the stance with which it was deployed. I found that instructors tended to comment much more on precision, validity, and difficulty, than on characteristics such as utility, interest and aesthetics, and that they repeatedly framed ambiguity with a negative stance. I argue that such patterns construe disciplinary orientations at odds both with what mathematicians say is important and with what many students might value as well.

In chapter ‎7, I tackle yet another mechanism of idealization in lectures: the discursive obfuscation of human agency. This phenomenon has already received significant attention in the literature. Here I focused on positive possibilities: I identify and characterize the discursive means by which instructors humanized the language of mathematics, which I operationalized as enacting aspects of human experience not typically reflected in the textual register. I argue that Bakhtin's (1981, 2010) construct of literary chronotope (discursive space-time configuration) is a useful conceptual tool for distinguishing different ways of enacting human agency in mathematical discourse, and propose a framework of three chronotopes for humanizing the language of mathematics: the here-and-now experience of doing mathematics, the socio-historical context of mathematical activity, and discursive hybridity. The framework also sheds light on how dominant epistemologies of mathematics are enacted in discourse: by omitting non-cognitive experiences, references to socio-historical context, and any allusions to other spheres of human activity, mathematical discourse constructs itself as transcendental, universal and pure.

Collectively, the analyses and findings in this dissertation show how mainstream idealizations of mathematics are discursively constructed in real analysis lectures. The identification of concrete mechanisms – such as stories, attributions, representation of human experience – is important. When mechanisms are no longer invisible, they can be used as levers for change.

This dissertation explores the roles of students' intuitive knowledge in learning formal mathematics. The formal definition of a limit, or the epsilon-delta definition, is a critical topic in calculus for mathematics majors' development. It is typically the first occasion when students engage with rigorous, formal mathematics. Research has documented that the formal definition is a roadblock for most students in calculus, but has also de-emphasized the productive role of their prior knowledge and sense making processes. The temporal order of delta and epsilon has been suggested as a conceptual obstacle for students in understanding the structure of the formal definition. The dissertation investigates the nature of and the degree to which the temporal order of delta and epsilon is a difficulty for students. The fine-grained analysis of semi-structured interviews with elementary calculus students reveals a large repertoire of reasoning patterns about the temporal order. A microgenetic study of one student shows the diversity of knowledge resources and the complex process of reasoning. Knowledge in Pieces (diSessa, 1993) and Microgenetic Learning Analysis (Parnafes & diSessa, 2013, Schoenfeld, Smith & Arcavi, 1993) provide frameworks to explore the details of the structure of students' prior knowledge and their role in learning the topic. The study offers and examines the impact of an instructional treatment called the Pancake Story, designed specifically to productively link to students' intuitive knowledge. Leveraging the notion of quality control, the instructional treatment offers an alternative to the idea of functional dependence in reasoning about the temporal order of delta and epsilon. A detailed case study shows the process by which a student incorporates resources from the story into her existing knowledge about the temporal order. The findings in this dissertation support the claim that understanding the process of learning requires serious accounting for student's prior knowledge.

Students with mathematical learning disabilities (MLDs) experience persistent challenges learning even the most elementary mathematics. While prior research on MLDs has classified students on the basis of test scores and documented performance differences between groups, this dissertation focuses on the qualitative differences in individual students with MLDs as each student attempted to learn. This study extends the understanding of MLDs by (1)focusing on the mathematical domain of fractions, (2)selecting for study students whose difficulties with mathematics are clearly due to an MLD and not other factors, (3)conducting detailed diagnostic analyses of video-taped tutoring sessions, and (4)contrasting the difficulties that the students with MLDs experience to data collected with five typically achieving fifth grade control students. The two case study students, "Lisa" (19-year-old community college student) and "Emily" (18-year-old high school student) each participated in a pretest, four weekly tutoring sessions, and a posttest focused on fraction concepts. Achievement test scores, interviews, and videotaped tutoring session data were used to establish that Lisa and Emily met classic MLD qualifications, and more stringent response-to-intervention criteria. Both students demonstrated unexplained persistent low math achievement and neither student benefited from a tutoring protocol that had been effective for typically achieving fifth grade students. The tutoring sessions were analyzed using microgenetic methods. Analysis indicated that each student relied upon a unique collection of atypical understandings, which reoccurred across the sessions, were resistant to standard instructional approaches, and proved to be highly consequential for the student's ability to understand more complex fraction concepts. A cross case analysis revealed surprising similarities in the atypical understandings displayed by both case study students. These atypical understandings stemmed from and contributed to the student's inability to conceptualize and manipulate representations of fractional quantity. These atypical understandings were not similarly problematic for the fifth grade control students, but did appear in one additional student with an MLD. This suggests that there are qualitative differences in the difficulties experienced by students with MLDs and that it may be possible to design screening measures to identify these indicators of atypicality. In addition, remediation approaches should take into account and specifically target these atypicalities.

What does it mean to be “good at math”? Traditionally, schools have valued getting the right answer quickly—a perspective that excludes important aspects of mathematics, as well as many students. This multi-site case study investigates how teachers work together to redefine mathematics and mathematical competence. The study involved more than a year of ethnographic observations and interviews at two diverse urban high schools on the West Coast of the United States, where teachers expressed strong commitments to serving all students, especially students from non-dominant backgrounds.

The dissertation tells a complex story of teacher learning, as viewed through the lenses of classroom instruction (Chapter 2), collegial conversation (Chapter 3), and the organization of teachers’ professional support networks (Chapter 4). Drawing on scholarship that takes learning as a negotiation of meaning through engagement in social practices (Vygotsky, 1986; Wenger, 1998; Saxe, 2012), the dissertation examines the relationships between extra-local systems of meaning and moment-to-moment interactions. Extending beyond prior work, the dissertation elucidates the negotiation of intensely conflicting meanings—namely, culturally dominant definitions of mathematics as a discipline, students as learners, and teachers as professionals, and non-dominant definitions that attempt to expand both teachers’ and students’ opportunities to engage with rich, challenging, and rewarding learning experiences.

In each of the contexts studied, navigating tensions between dominant, restrictive meanings and non-dominant, expansive meanings was a challenge for all of the teachers. Dominant discourses frame mathematical activity as consisting primarily of computation and memorization; mathematical ability as innate, fixed, and distributed along a bell-shaped curve; and the work of teaching as private, autonomous, and grounded in personal style and preference. In contrast, equity- and reform-oriented discourses frame mathematical activity as inclusive of a wide variety of skills and practices; position all students as capable learners; and position teachers as learners who benefit from ongoing collaboration and support. Dominant discourses are restrictive: they limit students’ opportunities to learn rich mathematics and teachers’ opportunities to negotiate equity- and reform-oriented shifts in their practice. But as teachers engage with non-dominant meanings that potentially expand learning opportunities, commonsense meanings do not simply disappear. Rather, they interact with non-dominant meanings in messy and complex ways that require careful study in order to understand how and what teachers learn.

The theme of negotiating meaning is laid out in Chapter 1, with a discussion of the dissertation’s underlying theoretical perspective. Research sites are introduced; the dissertation’s structure is presented; and major findings and contributions are highlighted.

Chapter 2, “(Re)Framing Mathematical Competence in Everyday Instruction: Struggles and Successes of Equity-Oriented Teachers,” examines tensions and contradictions in teachers’ classroom practice. It shows that despite the best intentions of the teachers in this study, many of their efforts to support all students position some students as capable of engaging with challenging mathematics and others as just the opposite. Conversely, teacher moves that are counterintuitive within dominant frames of teaching, employed by two of the teachers in the study, are shown to expand students’ opportunities to develop positive mathematical identities. The chapter thus contributes to conversations about what equitable mathematics instruction looks like, while illuminating obstacles—cultural as well as technical—that teachers face as they attempt to enact classroom practices that support all students.

Chapter 3, “Tensions in Equity- and Reform-Oriented Learning in Teachers’ Collaborative Conversations,” examines how collaborative conversations open up and close down opportunities for teachers to navigate the tensions between restrictive and expansive discourses of mathematical competence, through close analysis of a 9½-minute segment of a routine meeting of mathematics teachers. Although the group appeared to be an ideal professional learning community in many ways, and the focal interaction and others like it were generative in a number of respects, teacher talk enacted both restrictive and expansive discourses. The existence of tensions between these discourses presented opportunities for the teachers to negotiate non-dominant meanings for themselves, i.e., to learn; but the ways that teachers framed their own collaborative work interfered with these opportunities. By highlighting conversational norms that impede collaborative learning, the chapter contributes to the field’s understanding of the challenges of equity- and reform-oriented learning in teachers’ professional communities.

Ways of supporting teachers to negotiate expansive meanings are examined in Chapter 4, “Supporting Teachers’ Equity-Oriented Learning and Identities: A Resource-Centered Perspective.” The chapter investigates two cases of ongoing teacher engagement with non-dominant practice and two cases of relative disengagement, illustrating how various resources come together to support teachers’ learning and identity development (or not). Four types of resources are found to be critical, and learning and identity processes are shown to intertwine in mutually informing ways as teachers interact with these different resources.

In elucidating both challenges and supports associated with making sense of non-dominant meanings, this dissertation contributes to the field’s understanding of equity- and reform-oriented teacher learning and why it is so difficult. It also points to ways that the contexts in which teachers work might be constructed to support their engagement with non-dominant, expansive meanings, so that they can support all of their students to engage with rich, challenging mathematics and to develop identities as powerful learners and doers of mathematics.