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- All Subjects: Mathematics Education
- Creators: Thompson, Patrick
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
There have been a number of studies that have examined students’ difficulties in understanding the idea of logarithm and the effectiveness of non-traditional interventions. However, there have been few studies that have examined the understandings students develop and need to develop when completing conceptually oriented logarithmic lessons. In this document, I present the three papers of my dissertation study. The first paper examines two students’ development of concepts foundational to the idea of logarithm. This paper discusses two essential understandings that were revealed to be problematic and essential for students’ development of productive meanings for exponents, logarithms and logarithmic properties. The findings of this study informed my later work to support students in understanding logarithms, their properties and logarithmic functions. The second paper examines two students’ development of the idea of logarithm. This paper describes the reasoning abilities two students exhibited as they engaged with tasks designed to foster their construction of more productive meanings for the idea of logarithm. The findings of this study provide novel insights for supporting students in understanding the idea of logarithm meaningfully. Finally, the third paper begins with an examination of the historical development of the idea of logarithm. I then leveraged the insights of this literature review and the first two papers to perform a conceptual analysis of what is involved in learning and understanding the idea of logarithm. The literature review and conceptual analysis contributes novel and useful information for curriculum developers, instructors, and other researchers studying student learning of this idea.
ContributorsKuper Flores, Emily Ginamarie (Author) / Carlson, Marilyn (Thesis advisor) / Thompson, Patrick (Committee member) / Milner, Fabio (Committee member) / Zazkis, Dov (Committee member) / Czocher, Jennifer (Committee member) / Arizona State University (Publisher)
Created2018
Description
The concept of distribution is one of the core ideas of probability theory and inferential statistics, if not the core idea. Many introductory statistics textbooks pay lip service to stochastic/random processes but how do students think about these processes? This study sought to explore what understandings of stochastic process students develop as they work through materials intended to support them in constructing the long-run behavior meaning for distribution.
I collected data in three phases. First, I conducted a set of task-based clinical interviews that allowed me to build initial models for the students’ meanings for randomness and probability. Second, I worked with Bonnie in an exploratory teaching setting through three sets of activities to see what meanings she would develop for randomness and stochastic process. The final phase consisted of me working with Danielle as she worked through the same activities as Bonnie but this time in teaching experiment setting where I used a series of interventions to test out how Danielle was thinking about stochastic processes.
My analysis shows that students can be aware that the word “random” lives in two worlds, thereby having conflicting meanings. Bonnie’s meaning for randomness evolved over the course of the study from an unproductive meaning centered on the emotions of the characters in the context to a meaning that randomness is the lack of a pattern. Bonnie’s lack of pattern meaning for randomness subsequently underpinned her image of stochastic/processes, leading her to engage in pattern-hunting behavior every time she needed to classify a process as stochastic or not. Danielle’s image of a stochastic process was grounded in whether she saw the repetition as being reproducible (process can be repeated, and outcomes are identical to prior time through the process) or replicable (process can be repeated but the outcomes aren’t in the same order as before). Danielle employed a strategy of carrying out several trials of the process, resetting the applet, and then carrying out the process again, making replicability central to her thinking.
I collected data in three phases. First, I conducted a set of task-based clinical interviews that allowed me to build initial models for the students’ meanings for randomness and probability. Second, I worked with Bonnie in an exploratory teaching setting through three sets of activities to see what meanings she would develop for randomness and stochastic process. The final phase consisted of me working with Danielle as she worked through the same activities as Bonnie but this time in teaching experiment setting where I used a series of interventions to test out how Danielle was thinking about stochastic processes.
My analysis shows that students can be aware that the word “random” lives in two worlds, thereby having conflicting meanings. Bonnie’s meaning for randomness evolved over the course of the study from an unproductive meaning centered on the emotions of the characters in the context to a meaning that randomness is the lack of a pattern. Bonnie’s lack of pattern meaning for randomness subsequently underpinned her image of stochastic/processes, leading her to engage in pattern-hunting behavior every time she needed to classify a process as stochastic or not. Danielle’s image of a stochastic process was grounded in whether she saw the repetition as being reproducible (process can be repeated, and outcomes are identical to prior time through the process) or replicable (process can be repeated but the outcomes aren’t in the same order as before). Danielle employed a strategy of carrying out several trials of the process, resetting the applet, and then carrying out the process again, making replicability central to her thinking.
ContributorsHatfield, Neil (Author) / Thompson, Patrick (Thesis advisor) / Carlson, Marilyn (Committee member) / Middleton, James (Committee member) / Lehrer, Richard (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2019
Description
Previous research has examined difficulties that students have with understanding and productively working with function notation. Function notation is very prevalent throughout mathematics education, helping students to better understand and more easily work with functions. The goal of my research was to investigate students' current ways of thinking about function notation to better assist teachers in helping their students develop deeper and more productive understandings. In this study, I conducted two separate interviews with two undergraduate students to explore their meanings for function notation. I developed and adapted tasks aimed at investigating different aspects and uses of function notation. In each interview, I asked the participants to attempt each of the tasks, explaining their thoughts as they worked. While they were working, I occasionally asked clarifying questions to better understand their thought processes. For the second interviews, I added tasks based on difficulties I found in the first interviews. I video recorded each interview for later analysis. Based on the data found in the interviews, I will discuss the seven prevalent ways of thinking that I found, how they hindered or facilitated working with function notation productively, and suggestions for instruction to better help students understand the concept.
ContributorsMckee, Natalie Christina (Author) / Thompson, Patrick (Thesis director) / Zazkis, Dov (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
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
Previous research discusses students' difficulties in grasping an operational understanding of covariational reasoning. In this study, I interviewed four undergraduate students in calculus and pre-calculus classes to determine their ways of thinking when working on an animated covariation problem. With previous studies in mind and with the use of technology, I devised an interview method, which I structured using multiple phases of pre-planned support. With these interviews, I gathered information about two main aspects about students' thinking: how students think when attempting to reason covariationally and which of the identified ways of thinking are most propitious for the development of an understanding of covariational reasoning. I will discuss how, based on interview data, one of the five identified ways of thinking about covariational reasoning is highly propitious, while the other four are somewhat less propitious.
ContributorsWhitmire, Benjamin James (Author) / Thompson, Patrick (Thesis director) / Musgrave, Stacy (Committee member) / Moore, Kevin C. (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / T. Denny Sanford School of Social and Family Dynamics (Contributor)
Created2014-05
Description
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.
ContributorsVillatoro, Barbara (Author) / Thompson, Patrick (Thesis advisor) / Carlson, Marilyn (Committee member) / Moore, Kevin (Committee member) / Roh, Kyeong Hah (Committee member) / Draney, Karen (Committee member) / Arizona State University (Publisher)
Created2023
Description
Over the past thirty years, research on teachers’ mathematical knowledge for teaching (MKT) has developed and grown in popularity as an area of focus for improving mathematics teaching and students’ learning. Many scholars have investigated types of knowledge teachers use when teaching and the relationship between teacher knowledge and student performance. However, few researchers have studied the sources of teachers’ pedagogical decisions and actions and some studies have reported that advances in teachers’ mathematical meanings does not necessarily lead to a teacher conveying strong meanings to students. It has also been reported that a teacher’s ways of thinking about teaching an idea and actions to decenter can influence the teacher’s interactions with students.This document presents three papers detailing a multiple-case study that constitutes my dissertation. The first paper reviews the constructs researchers have used to investigate teachers’ knowledge base. This paper also provides a characterization of the first case’s mathematical meaning for teaching angle measure and the impact of her meaning on her interactions with students while teaching her angle measure lessons. The second paper examines another instructor’s meaning for an angle and its measure and illustrates the symbiotic relationship between the teacher’s mathematical meanings for teaching and decentering actions. This paper also characterizes how an instructor’s commitment to quantitative reasoning influences the teacher’s instructional orientation and instructional actions. Finally, the third paper includes a cross-case analysis of the two instructors’ mathematical meanings for teaching sine function and their enacted teaching practices, including their choice of tasks, interactions with students, and explanations while teaching their sine function lessons.
ContributorsRocha, Abby (Author) / Carlson, Marilyn P (Thesis advisor) / Thompson, Patrick (Committee member) / Tallman, Michael (Committee member) / O'Bryan, Alan (Committee member) / Strom, April (Committee member) / Apkarian, Naneh (Committee member) / Arizona State University (Publisher)
Created2023
Description
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.
ContributorsYu, Franklin (Author) / Carlson, Marilyn (Thesis advisor) / Zandieh, Michelle (Committee member) / Thompson, Patrick (Committee member) / Roh, Kyeong Hah (Committee member) / Soto, Roberto (Committee member) / Arizona State University (Publisher)
Created2022