Matching Items (9)
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- All Subjects: Mathematics Education
- Genre: Doctoral Dissertation
- Creators: Buss, Ray
- Creators: Thompson, Patrick W.
- 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
This study contributes to the ongoing discussion of Mathematical Knowledge for Teaching (MKT). It investigates the case of Rico, a high school mathematics teacher who had become known to his colleagues and his students as a superbly effective mathematics teacher. His students not only developed excellent mathematical skills, they also developed deep understanding of the mathematics they learned. Moreover, Rico redesigned his curricula and instruction completely so that they provided a means of support for his students to learn mathematics the way he intended. The purpose of this study was to understand the sources of Rico's effectiveness. The data for this study was generated in three phases. Phase I included videos of Rico's lessons during one semester of an Algebra II course, post-lesson reflections, and Rico's self-constructed instructional materials. An analysis of Phase I data led to Phase II, which consisted of eight extensive stimulated-reflection interviews with Rico. Phase III consisted of a conceptual analysis of the prior phases with the aim of creating models of Rico's mathematical conceptions, his conceptions of his students' mathematical understandings, and his images of instruction and instructional design. Findings revealed that Rico had developed profound personal understandings, grounded in quantitative reasoning, of the mathematics that he taught, and profound pedagogical understandings that supported these very same ways of thinking in his students. Rico's redesign was driven by three factors: (1) the particular way in which Rico himself understood the mathematics he taught, (2) his reflective awareness of those ways of thinking, and (3) his ability to envision what students might learn from different instructional approaches. Rico always considered what someone might already need to understand in order to understand "this" in the way he was thinking of it, and how understanding "this" might help students understand related ideas or methods. Rico's continual reflection on the mathematics he knew so as to make it more coherent, and his continual orientation to imagining how these meanings might work for students' learning, made Rico's mathematics become a mathematics of students--impacting how he assessed his practice and engaging him in a continual process of developing MKT.
ContributorsLage Ramírez, Ana Elisa (Author) / Thompson, Patrick W. (Thesis advisor) / Carlson, Marilyn P. (Committee member) / Castillo-Chavez, Carlos (Committee member) / Saldanha, Luis (Committee member) / Middleton, James A. (Committee member) / Arizona State University (Publisher)
Created2011
Description
A teacher’s mathematical knowledge for teaching impacts the teacher’s pedagogical actions and goals (Marfai & Carlson, 2012; Moore, Teuscher, & Carlson, 2011), and a teacher’s instructional goals (Webb, 2011) influences the development of the teacher’s content knowledge for teaching. This study aimed to characterize the reciprocal relationship between a teacher’s mathematical knowledge for teaching and pedagogical goals.
Two exploratory studies produced a framework to characterize a teacher’s mathematical goals for student learning. A case study was then conducted to investigate the effect of a professional developmental intervention designed to impact a teacher’s mathematical goals. The guiding research questions for this study were: (a) what is the effect of a professional development intervention, designed to perturb a teacher’s pedagogical goals for student learning to be more attentive to students’ thinking and learning, on a teacher’s views of teaching, stated goals for student learning, and overarching goals for students’ success in mathematics, and (b) what role does a teacher's mathematical teaching orientation and mathematical knowledge for teaching have on a teacher’s stated and overarching goals for student learning?
Analysis of the data from this investigation revealed that a conceptual curriculum supported the advancement of a teacher’s thinking regarding the key ideas of mathematics of lessons, but without time to reflect and plan, the teacher made limited connections between the key mathematical ideas within and across lessons. The teacher’s overarching goals for supporting student learning and views of teaching mathematics also had a significant influence on her curricular choices and pedagogical moves when teaching. The findings further revealed that a teacher’s limited meanings for proportionality contributed to the teacher struggling during teaching to support students’ learning of concepts that relied on understanding proportionality. After experiencing this struggle the teacher reverted back to using skill-based lessons she had used before.
The findings suggest a need for further research on the impact of professional development of teachers, both in building meanings of key mathematical ideas of a teacher’s lessons, and in professional support and time for teachers to build stronger mathematical meanings, reflect on student thinking and learning, and reconsider one’s instructional goals.
Two exploratory studies produced a framework to characterize a teacher’s mathematical goals for student learning. A case study was then conducted to investigate the effect of a professional developmental intervention designed to impact a teacher’s mathematical goals. The guiding research questions for this study were: (a) what is the effect of a professional development intervention, designed to perturb a teacher’s pedagogical goals for student learning to be more attentive to students’ thinking and learning, on a teacher’s views of teaching, stated goals for student learning, and overarching goals for students’ success in mathematics, and (b) what role does a teacher's mathematical teaching orientation and mathematical knowledge for teaching have on a teacher’s stated and overarching goals for student learning?
Analysis of the data from this investigation revealed that a conceptual curriculum supported the advancement of a teacher’s thinking regarding the key ideas of mathematics of lessons, but without time to reflect and plan, the teacher made limited connections between the key mathematical ideas within and across lessons. The teacher’s overarching goals for supporting student learning and views of teaching mathematics also had a significant influence on her curricular choices and pedagogical moves when teaching. The findings further revealed that a teacher’s limited meanings for proportionality contributed to the teacher struggling during teaching to support students’ learning of concepts that relied on understanding proportionality. After experiencing this struggle the teacher reverted back to using skill-based lessons she had used before.
The findings suggest a need for further research on the impact of professional development of teachers, both in building meanings of key mathematical ideas of a teacher’s lessons, and in professional support and time for teachers to build stronger mathematical meanings, reflect on student thinking and learning, and reconsider one’s instructional goals.
ContributorsMarfai, Frank Stephen (Author) / Carlson, Marilyn P. (Thesis advisor) / Ström, April D. (Committee member) / Thompson, Patrick W. (Committee member) / Middleton, James A. (Committee member) / Zandieh, Michelle J. (Committee member) / Arizona State University (Publisher)
Created2016
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
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
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
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