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- Creators: Roh, Kyeong Hah
- Creators: Barrett, The Honors College
- Creators: Buss, Ray
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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 study explores teacher educators' personal theories about the instructional practices central to preparing future teachers, how they enact those personal theories in the classroom, how they represent the relationship between content, pedagogy, and technology, and the function of technology in teacher educators' personal theories about the teaching of mathematics and their practices as enacted in the classroom. The conceptual frameworks of knowledge as situated and technology as situated provide a theoretical and analytical lens for examining individual instructor's conceptions and classroom activity as situated in the context of experiences and relationships in the social world. The research design employs a mixed method design to examine data collected from a representative sample of three full-time faculty members teaching methods of teaching mathematics in elementary education at the undergraduate level. Three primary types of data were collected and analyzed:
a) structured interviews using the repertory grid technique to model the mathematics education instructors' schemata regarding the teaching of mathematics methods; b) content analysis of classroom observations to develop models that represent the relationship of pedagogy, content, and technology as enacted in the classrooms; and c) brief retrospective protocols after each observed class session to explore the reasoning and individual choices made by an instructor that underlie their teaching decisions in the classroom. Findings reveal that although digital technology may not appear to be an essential component of an instructor's toolkit, technology can still play an integral role in teaching. This study puts forward the idea of repurposing as technology -- the ability to repurpose items as models, tools, and visual representations and integrate them into the curriculum. The instructors themselves became the technology, or the mediational tool, and introduced students to new meanings for "old" cultural artifacts in the classroom. Knowledge about the relationships between pedagogy, content, and technology and the function of technology in the classroom can be used to inform professional development for teacher educators with the goal of improving teacher preparation in mathematics education.
a) structured interviews using the repertory grid technique to model the mathematics education instructors' schemata regarding the teaching of mathematics methods; b) content analysis of classroom observations to develop models that represent the relationship of pedagogy, content, and technology as enacted in the classrooms; and c) brief retrospective protocols after each observed class session to explore the reasoning and individual choices made by an instructor that underlie their teaching decisions in the classroom. Findings reveal that although digital technology may not appear to be an essential component of an instructor's toolkit, technology can still play an integral role in teaching. This study puts forward the idea of repurposing as technology -- the ability to repurpose items as models, tools, and visual representations and integrate them into the curriculum. The instructors themselves became the technology, or the mediational tool, and introduced students to new meanings for "old" cultural artifacts in the classroom. Knowledge about the relationships between pedagogy, content, and technology and the function of technology in the classroom can be used to inform professional development for teacher educators with the goal of improving teacher preparation in mathematics education.
ContributorsToth, Meredith Jean (Author) / Middleton, James (Thesis advisor) / Sloane, Finbarr (Committee member) / Buss, Ray (Committee member) / Atkinson, Robert (Committee member) / Arizona State University (Publisher)
Created2014
Description
This dissertation report follows a three-paper format, with each paper having a different but related focus. In Paper 1 I discuss conceptual analysis of mathematical ideas relative to its place within cognitive learning theories and research studies. In particular, I highlight specific ways mathematics education research uses conceptual analysis and discuss the implications of these uses for interpreting and leveraging results to produce empirically tested learning trajectories. From my summary and analysis I develop two recommendations for the cognitive researchers developing empirically supported learning trajectories. (1) A researcher should frame his/her work, and analyze others’ work, within the researcher’s image of a broadly coherent trajectory for student learning and (2) that the field should work towards a common understanding for the meaning of a hypothetical learning trajectory.
In Paper 2 I argue that prior research in online learning has tested the impact of online courses on measures such as student retention rates, satisfaction scores, and GPA but that research is needed to describe the meanings students construct for mathematical ideas researchers have identified as critical to their success in future math courses and other STEM fields. This paper discusses the need for a new focus in studying online mathematics learning and calls for cognitive researchers to begin developing a productive methodology for examining the meanings students construct while engaged in online lessons.
Paper 3 describes the online Precalculus course intervention we designed around measurement imagery and quantitative reasoning as themes that unite topics across units. I report results relative to the meanings students developed for exponential functions and related ideas (such as percent change and growth factors) while working through lessons in the intervention. I provide a conceptual analysis guiding its design and discuss pre-test and pre-interview results, post-test and post-interview results, and observations from student behaviors while interacting with lessons. I demonstrate that the targeted meanings can be productive for students, show common unproductive meanings students possess as they enter Precalculus, highlight challenges and opportunities in teaching and learning in the online environment, and discuss needed adaptations to the intervention and future research opportunities informed by my results.
In Paper 2 I argue that prior research in online learning has tested the impact of online courses on measures such as student retention rates, satisfaction scores, and GPA but that research is needed to describe the meanings students construct for mathematical ideas researchers have identified as critical to their success in future math courses and other STEM fields. This paper discusses the need for a new focus in studying online mathematics learning and calls for cognitive researchers to begin developing a productive methodology for examining the meanings students construct while engaged in online lessons.
Paper 3 describes the online Precalculus course intervention we designed around measurement imagery and quantitative reasoning as themes that unite topics across units. I report results relative to the meanings students developed for exponential functions and related ideas (such as percent change and growth factors) while working through lessons in the intervention. I provide a conceptual analysis guiding its design and discuss pre-test and pre-interview results, post-test and post-interview results, and observations from student behaviors while interacting with lessons. I demonstrate that the targeted meanings can be productive for students, show common unproductive meanings students possess as they enter Precalculus, highlight challenges and opportunities in teaching and learning in the online environment, and discuss needed adaptations to the intervention and future research opportunities informed by my results.
ContributorsO'Bryan, Alan Eugene (Author) / Carlson, Marilyn P (Thesis advisor) / Thompson, Patrick W (Committee member) / Milner, Fabio (Committee member) / Roh, Kyeong Hah (Committee member) / Tallman, Michael (Committee member) / Arizona State University (Publisher)
Created2018
Description
Mathematics is an increasingly critical subject and the achievement of students in mathematics has been the focus of many recent reports and studies. However, few studies exist that both observe and discuss the specific teaching and assessment techniques employed in the classrooms across multiple countries. The focus of this study is to look at classrooms and educators across six high achieving countries to identify and compare teaching strategies being used. In Finland, Hong Kong, Japan, New Zealand, Singapore, and Switzerland, twenty educators were interviewed and fourteen educators were observed teaching. Themes were first identified by comparing individual teacher responses within each country. These themes were then grouped together across countries and eight emerging patterns were identified. These strategies include students active involvement in the classroom, students given written feedback on assessments, students involvement in thoughtful discussion about mathematical concepts, students solving and explaining mathematics problems at the board, students exploring mathematical concepts either before or after being taught the material, students engagement in practical applications, students making connections between concepts, and students having confidence in their ability to understand mathematics. The strategies identified across these six high achieving countries can inform educators in their efforts of increasing student understanding of mathematical concepts and lead to an improvement in mathematics performance.
ContributorsAnglin, Julia Mae (Author) / Middleton, James (Thesis director) / Vicich, James (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-12
Description
The focus of this study was to examine how a student's understanding of function notation impacted their approaches to problem solving. Before this question could be answered, students' understandings about function notation had to be determined. The goal of the first part of the data was to determine the norm of understanding for function notation for students after taking a college level pre-calculus class. From the data collected, several ideas about student understanding of notation emerged. The goal of the second data set was to determine if student understanding of notation impacted their reasoning while problem solving, and if so, how it impacted their reasoning. Collected data suggests that much of what students "understand" about function notation comes from memorized procedures and that the notation may have little or no meaning for students in context. Evidence from this study indicates that this lack of understanding of function notation does negatively impact student's ability to solve context based problems. In order to build a strong foundation of function, a well-developed understanding of function notation is necessary. Because function notation is a widely accepted way of communicating information about function relationships, understanding its uses and meanings in context is imperative for developing a strong foundation that will allow individuals to approach functions in a meaningful and productive manner.
ContributorsLe, Lesley Kim (Author) / Carlson, Marilyn (Thesis director) / Greenes, Carole (Committee member) / Barrett, The Honors College (Contributor) / Division of Teacher Preparation (Contributor)
Created2015-05
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
Public Mathematics Education is not at its best in the United States and technology is often seen as part of the solution to address this issue. With the existence of high-speed Internet, mobile technologies, ever-improving computer programming and graphing, the concepts of learning management systems (LMS’s) and online learning environments (OLE’s), technology-based learning has elevated to a whole new level. The new generation of online learning enables multi-modal utilization, and, interactivity with instant feedback, among the other precious characteristics identified in this study. The studies that evaluated the effects of online learning often measured the immediate impacts on student achievement; there are very few studies that have investigated the longer-term effects in addition to the short term ones.
In this study, the effects of the new generation Online Learning Activity Based (OLAB) Curriculum on middle school students’ achievement in mathematics at the statewide high-stakes testing system were examined. The results pointed out that the treatment group performed better than the control group in the short term (immediately after the intervention), medium term (one year after the intervention), and long term (two years after the intervention) and that the results were statistically significant in the short and long terms.
Within the context of this study, the researcher also examined some of the factors affecting student achievement while using the OLE as a supplemental resource, namely, the time and frequency of usage, professional development of the facilitators, modes of instruction, and fidelity of implementation. While the researcher detected positive correlations between all of the variables and student achievement, he observed that school culture is indeed a major feature creating the difference attributed to the treatment group teachers.
The researcher discovered that among the treatment group teachers, the ones who spent more time on professional development, used the OLE with greater fidelity and attained greater gains in student achievement and interestingly they came from the same schools. This verified the importance of school culture in teachers’ attitudes toward making the most of the resources made available to them so as to achieve better results in terms of student success in high stakes tests.
In this study, the effects of the new generation Online Learning Activity Based (OLAB) Curriculum on middle school students’ achievement in mathematics at the statewide high-stakes testing system were examined. The results pointed out that the treatment group performed better than the control group in the short term (immediately after the intervention), medium term (one year after the intervention), and long term (two years after the intervention) and that the results were statistically significant in the short and long terms.
Within the context of this study, the researcher also examined some of the factors affecting student achievement while using the OLE as a supplemental resource, namely, the time and frequency of usage, professional development of the facilitators, modes of instruction, and fidelity of implementation. While the researcher detected positive correlations between all of the variables and student achievement, he observed that school culture is indeed a major feature creating the difference attributed to the treatment group teachers.
The researcher discovered that among the treatment group teachers, the ones who spent more time on professional development, used the OLE with greater fidelity and attained greater gains in student achievement and interestingly they came from the same schools. This verified the importance of school culture in teachers’ attitudes toward making the most of the resources made available to them so as to achieve better results in terms of student success in high stakes tests.
ContributorsMeylani, Rusen (Author) / Bitter, Gary G. (Thesis advisor) / Legacy, Jane (Committee member) / Buss, Ray (Committee member) / Arizona State University (Publisher)
Created2016
Description
The purpose of this study was to identify the algebraic reasoning abilities of young students prior to instruction. The goals of the study were to determine the influence of problem, problem type, question, grade level, and gender on: (a) young children’s abilities to predict the number of shapes in near and far positions in a “growing” pattern without assistance; (b) the nature and amount of assistance needed to solve the problems; and (c) reasoning methods employed by children.
The 8-problem Growing Patterns and Functions Assessment (GPFA), with an accompanying interview protocol, were developed to respond to these goals. Each problem presents sequences of figures of geometric shapes that differ in complexity and can be represented by the function, y = mf +b: in Type 1 problems (1 - 4), m = 1, and in Type 2 problems (5 - 8), m = 2. The two questions in each problem require participants to first, name the number of shapes in the pattern in a near position, and then to identify the number of shapes in a far position. To clarify reasoning methods, participants were asked how they solved the problems.
The GPFA was administered, one-on-one, to 60 students in Grades 1, 2, and 3 with an equal number of males and females from the same elementary school. Problem solution scores without and with assistance, along with reasoning method(s) employed, were tabulated.
Results of data analyses showed that when no assistance was required, scores varied significantly by problem, problem type, and question, but not grade level or gender. With assistance, problem scores varied significantly by problem, problem type, question, and grade level, but not gender.
The 8-problem Growing Patterns and Functions Assessment (GPFA), with an accompanying interview protocol, were developed to respond to these goals. Each problem presents sequences of figures of geometric shapes that differ in complexity and can be represented by the function, y = mf +b: in Type 1 problems (1 - 4), m = 1, and in Type 2 problems (5 - 8), m = 2. The two questions in each problem require participants to first, name the number of shapes in the pattern in a near position, and then to identify the number of shapes in a far position. To clarify reasoning methods, participants were asked how they solved the problems.
The GPFA was administered, one-on-one, to 60 students in Grades 1, 2, and 3 with an equal number of males and females from the same elementary school. Problem solution scores without and with assistance, along with reasoning method(s) employed, were tabulated.
Results of data analyses showed that when no assistance was required, scores varied significantly by problem, problem type, and question, but not grade level or gender. With assistance, problem scores varied significantly by problem, problem type, question, and grade level, but not gender.
ContributorsCavanagh, Mary Clare (Author) / Greenes, Carole E. (Thesis advisor) / Buss, Ray (Committee member) / Surbeck, Elaine (Committee member) / Arizona State University (Publisher)
Created2016