In this dissertation, I utilize the post-structural philosophy of Gilles Deleuze and Fe´lix Guattari as a lens for investigating the proof process. Deleuze and Guattari were both post- structural philosophers who, like many in this tradition, troubled traditional notions related to stable identities, meaning, language, and mathematics. For…

Harmful and inaccurate narratives about the nature of mathematics, and about the mathematical incapability of historically marginalized groups, are pervasive and longstanding. The persistence of these narratives, and their documented negative impacts in school, the discipline, and beyond, have led some scholars to conclude that there is something…

As students transition from the calculus sequence to upper-level mathematics classes, many struggle to understand the purpose of proofs and how to produce them on their own. One tool students and mathematicians alike frequently use is examples to aid their understanding and production of proof. Using examples productively has been argued to be…

Creativity and collaboration are key components of a mathematician's work, and thus in the preparation of future mathematicians, undergraduate mathematics courses should aim to develop students who can effectively work collaboratively and creatively with one another. Despite this need, research on mathematical creativity has primarily investigated…

Mathematical justifications offer an approach to which students deepen and extend their mathematical thinking. This process has the potential to enhance students' conceptual understanding. Mathematical practices and process standards ask students to use critical thinking skills to justify their reasoning, but this is usually seen as proofs in high…

In this dissertation, I report results from a study that investigated the impact of students' engagement in a series of tasks in an introductory proof course that was designed to stimulate their learning of reading strategies to support their comprehension of mathematics proofs. Specifically, I investigated the impact of the design of five…

Knowledge assessments in undergraduate mathematics education commonly evaluate response correctness to determine learner proficiency. However, simultaneous evaluation of learner metacognition more accurately assesses the multiple dimensions of knowledge and has been shown to increase assessment validity and reliability. Research into…

This study sought to explore whether access to definitions and general representations influences the construction of general direct arguments. Data was collected in college mathematics courses for prospective elementary school teachers. Participant arguments were analyzed along two variables: the generality of the representations and the…

Proof and proving are a core component of the discipline of mathematics and proving is a required exercise in many upper-level courses in mathematics at the undergraduate level. Writing proofs remains difficult for many students (Moore, 1994; Stylianides et al., 2017). To address this difficulty many universities have began offering Introduction…

This study uses narrative analysis to track undergraduates' proof schemes throughout a transition proofs course at a research 1 university in the Midwest, with a focus on student example use and how it influences their schemes. Assessment data and cognitive interviews were conducted at four points throughout the semester in order to track the…

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…

Analogical reasoning has played a significant role in the development of modern mathematical concepts. Although some perspectives in mathematics education have argued against the use of analogies and analogical reasoning in instructional contexts, some attempts have been made to leverage the pedagogical power of analogies. I assert that with a…

Geometric proof-writing is a widely known cause of stress for secondary school students and teachers alike. As the textbook is the primary curricular tool utilized by novice teachers, a two-part qualitative study was conducted to determine (a) the types of proofs presented in a typical high school geometry textbook and (b) teachers' preparedness…

Engaging students in mathematical justification is extremely important from a learning-focused and a disciplinary perspective. Prior research has commonly found that middle grades students struggle to produce viable justifications, arguments, and proofs, yet these studies often consider students' justification in only one modality (oral or…

Students struggle with proving, a fundamental activity in upper-level undergraduate mathematics courses. Learning how to prove is a difficult transition for students, as they shift from largely computation-based to argument-based work. In response, mathematics departments have instituted courses, introduction or transition to proof, designed to…