Abstract: 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 sociohistorical 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. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]
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