This thesis attempts to answer the question ‘What changes in understanding occur as a student develops their way of understanding similarity using geometric transformations and what teacher interventions contribute to these changes in understanding?’ Similarity is a topic taught in school geometry usually alongside the related topic Congruence. The Common Core State Standards for Mathematics, upon which many states have based their state level educational standards, recommend teachers leverage transformational geometry to explain congruence and similarity using geometric transformations. "However, there is a lack of research studies regarding how transformational geometry can be taught as a productive way of understanding similarities and what challenges students might encounter when learning similarities via transformational geometry approaches." This study aims to further the efforts of teachers who are trying to develop their students’ transformational understandings of similarity. This study was conducted as exploratory teaching interviews in Spring 2023 at a large public university. The student was an undergraduate student who had not previously taken a transformational geometry-based Euclidean geometry at the university. I, as a teacher-researcher, designed a set of tasks for the exploratory teaching interviews, and implemented them over the course of 5 weeks. I, as a researcher, also analyzed the data to create a model for the student's understanding of similarity. Specifically, I was interested in sorting the ways of understanding expressed by the student into the categories pictorial, measurement-based, and transformational. By analyzing the videos from the interviews and tracking the students’ understandings from moment to moment, I was able to see a shift in her understanding toward a transformational understanding. Thus her way of understanding similarity using geometric transformations was strengthened and I was able to pinpoint key shifts in understanding that contribute to the strengthening of this understanding. Notably, the student developed a notion of dilation as coming from a single centerpoint, negotiated definitions from each way of understanding until eventually settling on a definition rooted in transformations, and applied similarity to an unfamiliar context using both her intuition about similarity and the definition she created. The implications of this being that a somewhat advanced understanding dilation is productive for understanding similarity using geometric transformations, and that to develop a student's way of understanding similarity using geometric transformations there must be a practical need for this created by tasks the student engages with.

Over the last several centuries, mathematicians have developed sophisticated symbol systems to represent ideas often imperceptible to their five senses. Although conventional definitions exist for these notations, individuals attribute their personalized meanings to these symbols during their mathematical activities. In some instances, students might (1) attribute a non-normative meaning to a conventional symbol or (2) attribute viable meanings for a mathematical topic to a novel symbol. This dissertation aims to investigate the relationships between students’ meanings and personal algebraic expressions in the context of one topic: infinite series convergence. To this end, I report the results of two individual constructivist teaching experiments in which first-time second-semester university calculus students constructed symbols (called personal expressions) to organize their thinking about various topics related to infinite series. My results comprise three distinct sections. First, I describe the intuitive meanings that the two students, Monica and Sylvia, exhibited for infinite series convergence before experiencing formal instruction on the topic. Second, I categorize the meanings these students attributed to their personal expressions for series topics and propose symbol categories corresponding to various instantiations of each meaning. Finally, I describe two situations in which students modified their personal expressions throughout several interviews to either (1) distinguish between examples they initially perceived as similar or (2) modify a previous personal expression to symbolize two ideas they initially perceived as distinct. To conclude, I discuss the research and teaching implications of my explanatory frameworks for students’ symbolization. I also provide an initial theoretical framing of the cognitive mechanisms by which students create, maintain, and modify their personal algebraic representations.

Authors of calculus texts often include graphs in the text with the intent that the graph depicts relationships described in theorems and formulas. Similarly, graphs are often utilized in classroom lectures and discussions for the same purpose. The author or instructor includes function graphs to represent quantitative relationships and how a pair of quantities vary. Previous research has shown that different students interpret calculus statements differently depending on their meanings of points in the coordinate plane. As a result, students' widely differing interpretations of graphs presented to them. Researchers studying how students understand graphs of continuous functions and coordinate planes have developed many constructs to explain potential aspects of students' thinking about coordinate points, coordinate planes, variation, covariation, and continuous functions. No current research investigates how the different ways of thinking about graphs correlate. In other words, are there some ways of thinking that tend to either occur together or not occur together? In this research, I investigated student's system of meanings to describe how the different ways of understanding coordinate planes, coordinate points, and graphs of functions in the coordinate planes are related in students’ thinking. I determine a relationship between students' understanding of number lines or coordinate planes containing an infinite collection of numbers and their ability to identify a graph representing a dynamic situation. Additionally, I determined a relationship between students reasoning with values (instead of shapes) and their ability to create a graph to represent a dynamic situation.

This study investigated two undergraduate mathematics students’ meanings for derivatives of univariable and multivariable functions when creating linear approximations. Both participants completed multivariable calculus at least two semesters prior to participating in a sequence of four to five exploratory teaching interviews. One purpose of the interviews was to understand the students’ meaning of the idea of rate of change and its role in their understanding ideas of derivative, partial derivative, and directional derivative. A second purpose was to understand and advance the ways in which each student used the idea of rate of change to make linear approximations. My analysis of the data revealed (i) how a student’s understanding of constant rate of change impacted their conception of derivatives, partial derivatives, and directional derivatives, and (ii) how each student used these ideas to make linear approximations. My results revealed that conceptualizing a rate of change as the ratio of two quantities’ values as they vary together was critical for their conceptualizing partial and directional derivatives quantitatively as directional rates of change, and in particular, how they visualized these ideas graphically and constructed symbols to represent the quantities and the relationships between their values. Further, my results revealed the importance of distinguishing between conceptualizing an instantaneous rate of change assuming a constant rate of change over any amount of change in the independent quantity(s) and using this rate of change to generate an approximate amount of change in the value of the dependent quantity. Alonzo initially conceptualized rate of change and derivative as the slantiness of a line that intersected a function’s curve. John also referred to the derivative at a point as the slope of the line tangent to the curve at that point, but he appeared to conceptualize the derivative as a ratio of the changes in two quantities values and imagined (represented graphically) two changes while discussing how to make this ratio more precise and use its value to make linear projections of future function values and amounts of accumulation. John also conceptualized the derivative as the best local, linear approximation for a function.

This dissertation reports on three studies about students’ conceptions and learning of the idea of instantaneous rate of change. The first study investigated 25 students’ conceptions of the idea of instantaneous rate of change. The second study proposes a hypothetical learning trajectory, based on the literature and results from the first study, for learning the idea of instantaneous rate of change. The third study investigated two students’ thinking and learning in the context of a sequence of five exploratory teaching interviews. The first paper reports on the results of conducting clinical interviews with 25 students. The results revealed the diverse conceptions that Calculus students have about the value of a derivative at a given input value. The results also suggest that students’ interpretation of the value of a rate of change is related to their use of covariational reasoning when considering how two quantities’ values vary together. The second paper presents a conceptual analysis on the ways of thinking needed to develop a productive understanding of instantaneous rate of change. This conceptual analysis includes an ordered list of understandings and reasoning abilities that I hypothesize to be essential for understanding the idea of instantaneous rate of change. This paper also includes a sequence of tasks and questions I designed to support students in developing the ways of thinking and meanings described in my conceptual analysis. The third paper reports on the results of five exploratory teaching interviews that leveraged my hypothetical learning trajectory from the second paper. The results of this teaching experiment indicate that developing a coherent understanding of rate of change using quantitative reasoning can foster advances in students’ understanding of instantaneous rate of change as a constant rate of change over an arbitrarily small input interval of a function’s domain.

This thesis is an extension of previous research done by Roh and Lee (2018). Their research involved the design and implementation of a survey to analyze students’ cognitive inconsistencies. This thesis expands upon this research to interview students who demonstrated logical inconsistencies and evaluates the kinds of struggles students faced while evaluating statements and validating arguments. Three students who demonstrated logical inconsistencies were interviewed and asked to answer questions originally pulled from Roh and Lee’s (2018) survey. This thesis found that there were many aspects of each section of the survey that students had struggled with, including use of intuition, analyzing a proof-by-contradiction that utilized a negated statement, and distrust of alternate proving methods. Overall, these techniques the students used while evaluating statements and validating arguments gives interesting insight into the pedagogy of teaching proofs.

Eleven years after being put into practice, the Common Core State Standards for Mathematics still take a back seat as traditional approaches drive many secondary geometry classrooms, specifically in regard to congruence. This thesis explores how university students reason about congruence based on their high school learning experience, as well as how in-service geometry teachers reason about and teach congruence. During the Summer of 2020, two distinct surveys were distributed to 33 undergraduate students at Arizona State University and two in-service geometry teachers in Arizona to characterize the ways they understand congruence and reflect on their experiences in secondary geometry classrooms. The results of the survey indicate that students who understood congruence either in terms of corresponding measurements or transformations were successful in identifying congruent shapes, while only students who understood congruence in terms of transformations were successful in constructing congruent shapes. Transformational reasoning was both the most productive and the least prominent way of understanding congruence among students. Their responses to activities and reflections on their experiences also suggested that deductive reasoning is not practiced or prioritized in many secondary geometry classrooms. Teacher understandings of congruence varied, and reflections suggested that development of materials and training that are aligned with the goals of CCSSM for both pre-service and in-service teachers would help teachers create an environment conducive to a transformational understanding of congruence and that promotes deductive reasoning.

This study investigates several students’ interpretations and meanings for negations of various mathematical statements with quantifiers, and how their meanings for quantified variables impact their interpretations and denials of these quantified statements. Eight students participated in three separate exploratory teaching interviews and were selected from Transition-to-Proof and advanced mathematics courses beyond Transition-to-Proof. In the first interview, students were asked to interpret mathematical statements from Calculus contexts and provide justifications and refutations for why these statements are true or false in particular situations. In the second interview, students were asked to negate the same set of mathematical statements. Both sets of interviews were analyzed to determine students’ meanings for the quantified variables in the statements, and then these meanings were used to determine how students’ quantifications influenced their interpretations, denials, and evaluations for the quantified statements. In the final interview, students were also be asked to interpret and negation statements from different mathematical contexts. All three interviews were used to determine what meanings comprised students’ interpretations and denials for the given statements. Additionally, students’ interpretations and negations across different statements in the interviews were analyzed and then compared within students and across students to determine if there were differences in student denials across different moments.

Construction is a defining characteristic of geometry classes. In a traditional classroom, teachers and students use physical tools (i.e. a compass and straight-edge) in their constructions. However, with modern technology, construction is possible through the use of digital applications such as GeoGebra and Geometer’s SketchPad.
Many other studies have researched the benefits of digital manipulatives and digital environments through student completion of tasks and testing. This study intends to research students’ use of the digital tools and manipulatives, along with the students’ interactions with the digital environment. To this end, I conducted exploratory teaching experiments with two calculus I students.
In the exploratory teaching experiments, students were introduced to a GeoGebra application developed by Fischer (2019), which includes instructional videos and corresponding quizzes, as well as exercises and interactive notepads, where students could use digital tools to construct line segments and circles (corresponding to the physical straight-edge and compass). The application built up the students’ foundational knowledge, culminating in the construction and verbal proof of Euclid’s Elements, Proposition 1 (Euclid, 1733).
The central findings of this thesis are the students’ interactions with the digital environment, with observed changes in their conceptions of radii and circles, and in their use of tools. The students were observed to have conceptions of radii as a process, a geometric shape, and a geometric object. I observed the students’ conceptions of a circle change from a geometric shape to a geometric object, and with that change, observed the students’ use of tools change from a measuring focus to a property focus.
I report a summary of the students’ work and classify their reasoning and actions into the above categories, and an analysis of how the digital environment impacts the students’ conceptions. I also briefly discuss the impact of the findings on pedagogy and future research.

This study sought to replicate previous work in student conceptions of formal proofs based on informal arguments, originally explored by Zazkis et al. (2016). Additional tasks were added to the experiment to produce new data that could further verify the analysis of Zazkis et al. (2016) as well as provide more insight into how students comprehend proofs, what types of mistakes occur, and why. Results from one-on-one interviews confirmed that some students were not able to make accurate informal to formal comparisons because they were not considering multiple facets of the problem. Additionally, patterns in the students’ analysis introduced more questions concerning the motivations behind what students choose to think about when they read and dissect proofs.