The focus of this dissertation is on students' thinking regarding the function concept in the context of abstract algebra, focusing on the properties of well- and everywhere-definedness. This dissertation follows a three-paper format, where each paper has a different yet related focus. In the first paper, I analyze (non-)examples found in…

Noticing children's mathematical thinking is foundational to teaching that is responsive to children's thinking. To better understand the range of noticing expertise for teachers engaged in multiyear professional development, we assessed the noticing of 72 upper elementary school teachers using three instructional scenarios involving fraction…

Understanding and reasoning with functions permeates mathematics education standards at all grade levels. Research shows students take multiple approaches to building functions. Deep content knowledge of functions may affect how pre-service teachers link student reasoning with mathematical ideas. To extend and connect the research literature on…

Preparing graduates for employment is the goal of postsecondary education. In response to the national quantitative reasoning deficiency, institutions have joined nationwide to align college math content and learning outcomes to meet the needs of common careers and everyday life. The problem addressed in this study is that traditional college math…

The persistence of a 120 year-old mathematics curriculum despite dramatic changes in society (Dossey et al., 2016; NCTM, 2018) and the failure of the US mathematics education system to achieve many of its stated aims, especially for students from traditionally marginalized populations (Attridge & Inglis, 2013; Carnevale & Desrochers,…

In mathematics education, translation of resources from one language to another occurs in a wide range of situations. This paper explores how conceptual and technical apparatus from contemporary translation studies may be of use in guiding and analysing such translation. Key concepts--including those of source, target and intermediate text, of…

Research into didactics of calculus has maintained a long-standing interest in students' grasp of the relations between definite integrals and areas. This study comes to contribute to this line of research by unpacking how students use the concept of area to find definite integrals. Specifically, we focus on mathematical situations where the…

Recent research on teacher education has highlighted the importance of teachers' noticing of students' mathematical thinking. Much remains unknown about the ways in which teachers develop noticing abilities in professional development settings. This study examines changes in teachers' noticing in the context of peer discussions in a video club.…

This paper introduces the term "explorative mathematical argumentation" (EMA), signifying a concept for describing and analysing learners' mathematical argumentation processes. Despite multiple recent empirical evidence for argumentation promoting learning in science education, still little is known about the development of early…

Humans can learn complex functional relationships between variables from small amounts of data. In doing so, they draw on prior expectations about the form of these relationships. In three experiments, we show that people learn to adjust these expectations through experience, learning about the likely forms of the functions they will encounter.…

This descriptive and interpretive multi-case study investigated secondary prospective teachers' (PTs') beliefs about mathematical authority and how their student teaching experience(s) influenced their beliefs. The theoretical framework guiding this study was comprised of a novel definition for mathematical authority I developed, a conception of…

This article presents the findings of a case study that introduced online quizzes as a form of assessment in pure mathematics. Rather than being designed as an assessment of learning, these quizzes were designed to be an assessment for learning; they aimed to academically support students in their transition from A-Level mathematics to…

When embarking on our journey of learning in mathematics, we may envision a linear path of modules for acquiring mathematical knowledge and understanding to reach a predetermined outcome. However, this is a partial representation since the outcome and path are developing and adapting and our learning is continuously emerging. In this paper, our…

It is generally accepted that teachers' beliefs impact their instructional choices, but characterizations of that relationship are limited in college settings. Furthermore, examinations of instructor beliefs, instruction, and student beliefs together in one setting are rarely described. Based on interviews with two Abstract Algebra instructors,…

The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take…