Matching Items (10)
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- All Subjects: Mathematics Education
- Genre: Doctoral Dissertation
- Creators: Buss, Ray
- Creators: Zazkis, Dov
- Member of: Theses and Dissertations
- Status: Published
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
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
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
Description
Studies of discourse are prevalent in mathematics education, as are investigations on facilitating change in instructional practices that impact student attitudes toward mathematics. However, the literature has not sufficiently addressed the operationalization of the commognitive framework in the context of Calculus I, nor considered the inevitable impact on students’ attitudes of persistence, confidence, and enjoyment of mathematics. This study presents an innovation, founded, designed, and implemented, utilizing four frameworks. The overarching theory pivots to commognition, a theory that asserts communication is tantamount to thinking. Students experienced a Calculus I class grounded on four frames: a theoretical, a conceptual, a design pattern, and an analytical framework, which combined, engaged students in discursive practices. Multiple activities invited specific student actions: uncover, play, apply, connect, question, and realize, prompting calculus discourse. The study exploited a mixed-methods action research design that aimed to explore how discursive activities impact students’ understanding of the derivative and how and to what extent instructional practices, which prompt mathematical discourse, impact students’ persistence, confidence, and enjoyment of calculus.
This study offers a potential solution to a problem of practice that has long challenged practitioners and researchers—the persistence of Calculus I as a gatekeeper for Science, Technology, Engineering, and Mathematics (STEM). In this investigation it is suggested that Good and Ambitious Teaching practices, including asking students to explain their thinking and assigning group projects, positively impact students’ persistence, confidence, and enjoyment. Common calculus discourse among the experimental students, particularly discursive activities engaging word use and visual representations of the derivative, warrants further research for the pragmatic utility of the fine grain of a commognitive framework. For researchers the work provides a lens through which they can examine data resulting from the operationalization of multiple frameworks working in tandem. For practitioners, mathematical objects as discursive objects, allow for classrooms with readily observable outcomes.
ContributorsChowdhury, Madeleine Perez (Author) / Judson, Eugene (Thesis advisor) / Buss, Ray (Committee member) / Reinholz, Daniel (Committee member) / Arizona State University (Publisher)
Created2022
Description
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.
ContributorsEckman, Derek (Author) / Roh, Kyeong Hah (Thesis advisor) / Carlson, Marilyn (Committee member) / Martin, Jason (Committee member) / Spielberg, John (Committee member) / Zazkis, Dov (Committee member) / Arizona State University (Publisher)
Created2023
Description
The purpose of this action research study was to implement and analyze an intervention designed to improve perceptions of working with others as well as practice and improve emotional tools related to such interactions through the systematic development of ability emotional intelligence (EI) related skills. The present study sought to: (1) explore high school students’ perceptions of their role as part of a team during teamwork; (1a) investigate how perceptions differed by EI level; (2) examine how students’ perceptions of their role in teamwork were influenced by being paired with more advanced (ability EI) peers or less advanced peers, based on ability emotional intelligence test scores; (3) determine if ability emotional intelligence related skills could be developed over the course of a 7-week intervention.
The intervention took place in a 12th grade US Government & Economics classroom with 34 participants for examination of general trends, and 11 focal participants for focused and in-depth analysis. Students were taught about emotion theory and engaged in two weeks of ability emotional intelligence skills training, followed by a five-week project cycle in which students were required to work together to achieve a common goal. The research design was mixed methods convergent parallel. Quantitative data were collected from post- and retrospective pre-intervention surveys regarding student perceptions about working with others and their ability EI related skills. Qualitative data were collected through on-going student reflective journal entries, observational field notes, and interviews with the focal group of participants.
Results suggested the intervention had a significant effect on students’ perceptions of working with others and perceived ability emotional intelligence related skills. Significant positive change was found through quantitative data analysis, revealing students’ perceptions about working with others in teams had improved as a result of the intervention as had their perceptions about their ability EI related skills. Qualitative analysis revealed rich, thick descriptions exploring this shift in perception among the 11 focal students, providing the evidence necessary to support the effectiveness of the intervention. Results suggested the possibilities for improved teamwork in the classroom.
The intervention took place in a 12th grade US Government & Economics classroom with 34 participants for examination of general trends, and 11 focal participants for focused and in-depth analysis. Students were taught about emotion theory and engaged in two weeks of ability emotional intelligence skills training, followed by a five-week project cycle in which students were required to work together to achieve a common goal. The research design was mixed methods convergent parallel. Quantitative data were collected from post- and retrospective pre-intervention surveys regarding student perceptions about working with others and their ability EI related skills. Qualitative data were collected through on-going student reflective journal entries, observational field notes, and interviews with the focal group of participants.
Results suggested the intervention had a significant effect on students’ perceptions of working with others and perceived ability emotional intelligence related skills. Significant positive change was found through quantitative data analysis, revealing students’ perceptions about working with others in teams had improved as a result of the intervention as had their perceptions about their ability EI related skills. Qualitative analysis revealed rich, thick descriptions exploring this shift in perception among the 11 focal students, providing the evidence necessary to support the effectiveness of the intervention. Results suggested the possibilities for improved teamwork in the classroom.
ContributorsZuniga, Alison Kara (Author) / Bertrand, Melanie (Thesis advisor) / Buss, Ray (Committee member) / Caruso, David (Committee member) / Arizona State University (Publisher)
Created2019
Description
The extent of students’ struggles in linear algebra courses is at times surprising to mathematicians and instructors. To gain insight into the challenges, the central question I investigated for this project was: What is the nature of undergraduate students’ conceptions of multiple analytic representations of systems (of equations)?
My methodological choices for this study included the use of one-on-one, task-based clinical interviews which were video and audio recorded. Participants were chosen on the basis of selection criteria applied to a pool of volunteers from junior-level applied linear algebra classes. I conducted both generative and convergent analyses in terms of Clement’s (2000) continuum of research purposes. The generative analysis involved an exploration of the data (in transcript form). The convergent analysis involved the analysis of two student interviews through the lenses of Duval’s (1997, 2006, 2017) Theory of Semiotic Representation Registers and a theory I propose, the Theory of Quantitative Systems.
All participants concluded that for the four representations in this study, the notation was varying while the solution was invariant. Their descriptions of what was represented by the various representations fell into distinct categories. Further, the students employed visual techniques, heuristics, metaphors, and mathematical computation to account for translations between the various representations.
Theoretically, I lay out some constructs that may help with awareness of the complexity in linear algebra. While there are many rich concepts in linear algebra, challenges may stem from less-than-robust communication. Further, mathematics at the level of linear algebra requires a much broader perspective than that of the ordinary algebra of real numbers. Empirically, my results and findings provide important insights into students’ conceptions. The study revealed that students consider and/or can have their interest piqued by such things as changes in register.
The lens I propose along with the empirical findings should stimulate conversations that result in linear algebra courses most beneficial to students. This is especially important since students who encounter undue difficulties may alter their intended plans of study, plans which would lead them into careers in STEM (Science, Technology, Engineering, & Mathematics) fields.
My methodological choices for this study included the use of one-on-one, task-based clinical interviews which were video and audio recorded. Participants were chosen on the basis of selection criteria applied to a pool of volunteers from junior-level applied linear algebra classes. I conducted both generative and convergent analyses in terms of Clement’s (2000) continuum of research purposes. The generative analysis involved an exploration of the data (in transcript form). The convergent analysis involved the analysis of two student interviews through the lenses of Duval’s (1997, 2006, 2017) Theory of Semiotic Representation Registers and a theory I propose, the Theory of Quantitative Systems.
All participants concluded that for the four representations in this study, the notation was varying while the solution was invariant. Their descriptions of what was represented by the various representations fell into distinct categories. Further, the students employed visual techniques, heuristics, metaphors, and mathematical computation to account for translations between the various representations.
Theoretically, I lay out some constructs that may help with awareness of the complexity in linear algebra. While there are many rich concepts in linear algebra, challenges may stem from less-than-robust communication. Further, mathematics at the level of linear algebra requires a much broader perspective than that of the ordinary algebra of real numbers. Empirically, my results and findings provide important insights into students’ conceptions. The study revealed that students consider and/or can have their interest piqued by such things as changes in register.
The lens I propose along with the empirical findings should stimulate conversations that result in linear algebra courses most beneficial to students. This is especially important since students who encounter undue difficulties may alter their intended plans of study, plans which would lead them into careers in STEM (Science, Technology, Engineering, & Mathematics) fields.
ContributorsSipes, Janet (Author) / Zandieh, Michelle J (Thesis advisor) / Milner, Fabio A (Committee member) / Roh, Kyeong H (Committee member) / Wawro, Megan (Committee member) / Zazkis, Dov (Committee member) / Arizona State University (Publisher)
Created2019
Description
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.
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.
ContributorsDavid, Erika Johara (Author) / Roh, Kyeong Hah (Thesis advisor) / Thompson, Patrick W (Committee member) / Zandieh, Michelle (Committee member) / Dawkins, Paul C (Committee member) / Zazkis, Dov (Committee member) / Arizona State University (Publisher)
Created2019
Description
This dissertation is on the topic of sameness of representation of mathematical entities from a mathematics education perspective. In mathematics, people frequently work with different representations of the same thing. This is especially evident when considering the prevalence of the equals sign (=). I am adopting the three-paper dissertation model. Each paper reports on a study that investigates understandings of the identity relation. The first study directly addresses function identity: how students conceptualize, work with, and assess sameness of representation of function. It uses both qualitative and quantitative methods to examine how students understand function sameness in calculus contexts. The second study is on the topic of implicit differentiation and student understanding of the legitimacy of it as a procedure. This relates to sameness insofar as differentiating an equation is a valid inference when the equation expresses function identity. The third study directly addresses usage of the equals sign (“=”). In particular, I focus on the notion of symmetry; equality is a symmetric relation (truth-functionally), and mathematicians understand it as such. However, results of my study show that usage is not symmetric. This is small qualitative study and incorporates ideas from the field of linguistics.
ContributorsMirin, Alison (Author) / Zazkis, Dov (Thesis advisor) / Dawkins, Paul C. (Committee member) / Thompson, Patrick W. (Committee member) / Milner, Fabio (Committee member) / Kawski, Matthias (Committee member) / Arizona State University (Publisher)
Created2021