Matching Items (26)
Description
Research on combinatorics education is sparse when compared with other fields in mathematics education. This research attempted to contribute to the dearth of literature by examining students' reasoning about enumerative combinatorics problems and how students conceptualize the set of elements being counted in such problems, called the solution set. In particular, the focus was on the stable patterns of reasoning, known as ways of thinking, which students applied in a variety of combinatorial situations and tasks. This study catalogued students' ways of thinking about solution sets as they progressed through an instructional sequence. In addition, the relationships between the catalogued ways of thinking were explored. Further, the study investigated the challenges students experienced as they interacted with the tasks and instructional interventions, and how students' ways of thinking evolved as these challenges were overcome. Finally, it examined the role of instruction in guiding students to develop and extend their ways of thinking. Two pairs of undergraduate students with no formal experience with combinatorics participated in one of the two consecutive teaching experiments conducted in Spring 2012. Many ways of thinking emerged through the grounded theory analysis of the data, but only eight were identified as robust. These robust ways of thinking were classified into three categories: Subsets, Odometer, and Problem Posing. The Subsets category encompasses two ways of thinking, both of which ultimately involve envisioning the solution set as the union of subsets. The three ways of thinking in Odometer category involve holding an item or a set of items constant and systematically varying the other items involved in the counting process. The ways of thinking belonging to Problem Posing category involve spontaneously posing new, related combinatorics problems and finding relationships between the solution sets of the original and the new problem. The evolution of students' ways of thinking in the Problem Posing category was analyzed. This entailed examining the perturbation experienced by students and the resulting accommodation of their thinking. It was found that such perturbation and its resolution was often the result of an instructional intervention. Implications for teaching practice are discussed.
ContributorsHalani, Aviva (Author) / Roh, Kyeong Hah (Thesis advisor) / Fishel, Susanna (Committee member) / Saldanha, Luis (Committee member) / Thompson, Patrick (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2013
Description
This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet vector calculus, calculus on manifolds, linear algebra, and differential equations all rest upon the idea of functions of two (or more) variables. This dissertation contributes to understanding productive ways of thinking that can support students in thinking about functions of two or more variables as they describe complex systems with multiple variables interacting. This dissertation focuses on modeling the way of thinking of four students who participated in a specific instructional sequence designed to explore the limits of their ways of thinking and in turn, develop a robust model that could explain, describe, and predict students' actions relative to specific tasks. The data was collected using a teaching experiment methodology, and the tasks within the teaching experiment leveraged quantitative reasoning and covariation as foundations of students developing a coherent understanding of two-variable functions and their rates of change. The findings of this study indicated that I could characterize students' ways of thinking about two-variable functions by focusing on their use of novice and/or expert shape thinking, and the students' ways of thinking about rate of change by focusing on their quantitative reasoning. The findings suggested that quantitative and covariational reasoning were foundational to a student's ability to generalize their understanding of a single-variable function to two or more variables, and their conception of rate of change to rate of change at a point in space. These results created a need to better understand how experts in the field, such as mathematicians and mathematics educators, thinking about multivariable functions and their rates of change.
ContributorsWeber, Eric David (Author) / Thompson, Patrick (Thesis advisor) / Middleton, James (Committee member) / Carlson, Marilyn (Committee member) / Saldanha, Luis (Committee member) / Milner, Fabio (Committee member) / Van de Sande, Carla (Committee member) / Arizona State University (Publisher)
Created2012
Description
This is a report of a study that investigated the thinking of a high-achieving precalculus student when responding to tasks that required him to define linear formulas to relate covarying quantities. Two interviews were conducted for analysis. A team of us in the mathematics education department at Arizona State University initially identified mental actions that we conjectured were needed for constructing meaningful linear formulas. This guided the development of tasks for the sequence of clinical interviews with one high-performing precalculus student. Analysis of the interview data revealed that in instances when the subject engaged in meaning making that led to him imagining and identifying the relevant quantities and how they change together, he was able to give accurate definitions of variables and was usually able to define a formula to relate the two quantities of interest. However, we found that the student sometimes had difficulty imagining how the two quantities of interest were changing together. At other times he exhibited a weak understanding of the operation of subtraction and the idea of constant rate of change. He did not appear to conceptualize subtraction as a quantitative comparison. His inability to conceptualize a constant rate of change as a proportional relationship between the changes in two quantities also presented an obstacle in his developing a meaningful formula that relied on this understanding. The results further stress the need to develop a student's ability to engage in mental operations that involve covarying quantities and a more robust understanding of constant rate of change since these abilities and understanding are critical for student success in future courses in mathematics.
ContributorsKlinger, Tana Paige (Author) / Carlson, Marilyn (Thesis director) / Thompson, Pat (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-05
Description
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.
ContributorsDziszuk, Kathryn Elizabeth (Author) / Roh, Kyeong Hah (Thesis director) / Parr, Erika David (Committee member) / School of Mathematical and Statistical Sciences (Contributor, Contributor) / Department of Psychology (Contributor) / Barrett, The Honors College (Contributor)
Created2021-05
Description
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.
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.
ContributorsSakauye, Noelle Marie (Author) / Roh, Kyeong Hah (Thesis director) / Zandieh, Michelle (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
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.
ContributorsPeng, Tina (Author) / Zazkis, Dov (Thesis director) / Roh, Kyeong Hah (Committee member) / School of Mathematical and Statistical Sciences (Contributor, Contributor) / Computer Science and Engineering Program (Contributor) / Computing and Informatics Program (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
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.
ContributorsFischer, Quinn (Co-author, Co-author) / Roh, Kyeong Hah (Thesis director) / Zandieh, Michelle (Committee member) / Computer Science and Engineering Program (Contributor, Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-12
Description
This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference.
ContributorsJoshua, Surani Ashanthi (Author) / Thompson, Patrick W (Thesis advisor) / Carlson, Marilyn (Committee member) / Roh, Kyeong Hah (Committee member) / Middleton, James (Committee member) / Culbertson, Robert (Committee member) / Arizona State University (Publisher)
Created2019
Description
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.
ContributorsCombs, Nicole (Author) / Roh, Kyeong Hah (Thesis director) / Zandieh, Michelle (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor)
Created2023-12
Description
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.
ContributorsBettersworth, Zachary S (Author) / Carlson, Marilyn (Thesis advisor) / Harel, Guershon (Committee member) / Roh, Kyeong Hah (Committee member) / Thompson, Patrick W. (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2023