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.

This project is centered around a decade-old video game called League of Legends, which is one of the most popular video games in esports. Due to its nature of being a complex team-based strategy game, intuitive human predictions of the game’s outcome are relatively unreliable. Many approaches have been adopted to assist intuitive human predictions in traditional team-based sports, such as the Least Squares Method and various supervised machine learning algorithms. These methods have been significantly outperforming human predictions. The objective of this research is, hence, to test whether the predictive models generated using these methods can achieve a similar level of reliability in a more complex game like League of Legends.

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 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.

While most of the media attention given to video games focuses on those geared towards the entertainment industry, a less covered topic is the role of serious games. Also known as “educational” games, serious games are designed with the intent to teach the player a particular skill or topic. These games have gradually been working their way into our educational environments. Children are often taught to type, perform simple math, and correctly spell through a variety of games that have been widely adopted by teachers. However, teaching multiplication is one thing; teaching college-level advanced mathematics is another beast altogether. Can video games actually be used as an educational tool in higher education?
This is a difficult question for a variety of reasons. A major issue to consider is whether the students who play this game are actually learning the material, or simply improving at the game itself. If the game is not designed correctly, one could potentially learn to exploit game mechanics without applying knowledge of the material. While this person’s efficiency at completing the game quickly would suggest mastery of the topic, they may not actually be prepared to take a test on the subject. As such, it is important to thoroughly study the effectiveness of serious games before they are deployed to actual classrooms. This study will do just that with the game Vector Unknown, which was designed to help college students learn linear algebra.

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 dissertation reports three studies of the relationships between meanings teachers hold and meanings their students construct.

The first paper reports meanings held by U.S. and Korean secondary mathematics teachers for teaching function notation. This study focuses on what teachers in U.S. and Korean are revealing their thinking from their written responses to the MMTsm (Mathematical Meanings for Teaching secondary mathematics) items, with particular attention to how productive those meanings would be if conveyed to students in a classroom setting. This paper then discusses how the MMTsm serves as a diagnostic instrument by sharing a teacher’s story. The data indicates that many teachers name rules instead of constructing representations of functions through function notation.

The second paper reports the conveyance of meaning with eight Korean teachers who took the MMTsm. The data that I gathered was their responses to the MMTsm, what they said and did in the classroom lessons I observed, pre- and post-lesson interviews with them and their students. This paper focuses on the relationships between teachers’ mathematical meanings and their instructional actions as well as the relationships between teachers’ instructional actions and meanings that their students construct. The data suggests that holding productive meanings is a necessary condition to convey productive meanings to students, but not a sufficient condition.

The third paper investigates the conveyance of meaning with one U.S. teacher. This study explores how a teacher’s image of student thinking influenced her instructional decisions and meanings she conveyed to students. I observed 15 lessons taught by a calculus teacher and interviewed the teacher and her students at multiple points. The results suggest that teachers must think about how students might understand their instructional actions in order to better convey what they intend to their students.

The studies show a breakdown in the conveyance of meaning from teacher to student when the teacher has no image of how students might understand his or her statements and actions. This suggests that it is crucial to help teachers improve what they are capable of conveying to students and their images of what they hope to convey to future students.

Functions represented in the graphical register, as graphs in the Cartesian plane, are found throughout secondary and undergraduate mathematics courses. In the study of Calculus, specifically, graphs of functions are particularly prominent as a means of illustrating key concepts. Researchers have identified that some of the ways that students may interpret graphs are unconventional, which may impact their understanding of related mathematical content. While research has primarily focused on how students interpret points on graphs and students’ images related to graphs as a whole, details of how students interpret and reason with variables and expressions on graphs of functions have remained unclear.

This dissertation reports a study characterizing undergraduate students’ interpretations of expressions in the graphical register with statements from Calculus, its association with their evaluations of these statements, its relation to the mathematical content of these statements, and its relation to their interpretations of points on graphs. To investigate students’ interpretations of expressions on graphs, I conducted 150-minute task-based clinical interviews with 13 undergraduate students who had completed Calculus I with a range of mathematical backgrounds. In the interviews, students were asked to evaluate propositional statements about functions related to key definitions and theorems of Calculus and were provided various graphs of functions to make their evaluations. The central findings from this study include the characteristics of four distinct interpretations of expressions on graphs that students used in this study. These interpretations of expressions on graphs I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. The findings from this study suggest that different contexts may evoke different graphical interpretations of expressions from the same student. Further, some interpretations were shown to be associated with students correctly evaluating some statements while others were associated with students incorrectly evaluating some statements.

I report the characteristics of these interpretations of expressions in the graphical register and its relation to their evaluations of the statements, the mathematical content of the statements, and their interpretation of points. I also discuss the implications of these findings for teaching and directions for future research in this area.

Today, there is a gap between the effectiveness of learning online and learning in person. Online educational videos such as ones found on Youtube mimic more of a lecture style of learning, which is easy ignore without a teacher nearby to engage the viewer. Furthermore, there is a lack of educational videos on the topic of Euclid’s Elements geometry proofs. This project remedies both accounts by offering a new approach on interactive online learning videos and exercises for the topic of Euclid’s Elements Book One, Propositions One and Two. This is accomplished by combining interactive videos, exercises, questions, and sketchpads into one online worksheet. The interactive videos are made using traditional methods of audio and visual elements, with an emphasis on having more dynamic visuals to engage with the viewer. The exercises are made using a program called Geogebra, and consist in having a question to solve, and diagram the use can manipulate to help solve the question. The questions consist in ensuring the viewer understands the material, as well as potential questions to gauge general understanding before and after using the worksheet. The sketchpads consist in stating the proposition being proved, and giving the user all the tools they need to construct or prove the Euclidean proposition in the online interactive environment offered by Geogebra. All of these components are then ordered into the worksheet to make an interactive online learning experience for the viewer. This way the viewer may both watch and actively use the material being presented to promote learning through engagement in a teacher-less environment.