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- All Subjects: Mathematics Education
- Creators: Carlson, Marilyn P
- Creators: Thompson, Patrick W.
- Member of: Theses and Dissertations
- Resource Type: Text
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
This is a report of a study that investigated the thinking of a high-achieving precalculus student when responding to tasks that required him to define linear formulas to relate covarying quantities. Two interviews were conducted for analysis. A team of us in the mathematics education department at Arizona State University initially identified mental actions that we conjectured were needed for constructing meaningful linear formulas. This guided the development of tasks for the sequence of clinical interviews with one high-performing precalculus student. Analysis of the interview data revealed that in instances when the subject engaged in meaning making that led to him imagining and identifying the relevant quantities and how they change together, he was able to give accurate definitions of variables and was usually able to define a formula to relate the two quantities of interest. However, we found that the student sometimes had difficulty imagining how the two quantities of interest were changing together. At other times he exhibited a weak understanding of the operation of subtraction and the idea of constant rate of change. He did not appear to conceptualize subtraction as a quantitative comparison. His inability to conceptualize a constant rate of change as a proportional relationship between the changes in two quantities also presented an obstacle in his developing a meaningful formula that relied on this understanding. The results further stress the need to develop a student's ability to engage in mental operations that involve covarying quantities and a more robust understanding of constant rate of change since these abilities and understanding are critical for student success in future courses in mathematics.
ContributorsKlinger, Tana Paige (Author) / Carlson, Marilyn (Thesis director) / Thompson, Pat (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-05
Description
This 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
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
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 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
Teachers must recognize the knowledge they possess as appropriate to employ in the process of achieving their goals and objectives in the context of practice. Such recognition is subject to a host of cognitive and affective processes that have thus far not been a central focus of research on teacher knowledge in mathematics education. To address this need, this dissertation study examined the role of a secondary mathematics teacher’s image of instructional constraints on his enacted subject matter knowledge. I collected data in three phases. First, I conducted a series of task-based clinical interviews that allowed me to construct a model of David’s mathematical knowledge of sine and cosine functions. Second, I conducted pre-lesson interviews, collected journal entries, and examined David’s instruction to characterize the mathematical knowledge he utilized in the context of designing and implementing lessons. Third, I conducted a series of semi-structured clinical interviews to identify the circumstances David appraised as constraints on his practice and to ascertain the role of these constraints on the quality of David’s enacted subject matter knowledge. My analysis revealed that although David possessed many productive ways of understanding that allowed him to engage students in meaningful learning experiences, I observed discrepancies between and within David’s mathematical knowledge and his enacted mathematical knowledge. These discrepancies were not occasioned by David’s active compensation for the circumstances and events he appraised as instructional constraints, but instead resulted from David possessing multiple schemes for particular ideas related to trigonometric functions, as well as from his unawareness of the mental actions and operations that comprised these often powerful but uncoordinated cognitive schemes. This lack of conscious awareness made David ill-equipped to define his instructional goals in terms of the mental activity in which he intended his students to engage, which further conditioned the circumstances and events he appraised as constraints on his practice. David’s image of instructional constraints therefore did not affect his enacted subject matter knowledge. Rather, characteristics of David’s subject matter knowledge, namely his uncoordinated cognitive schemes and his unawareness of the mental actions and operations that comprise them, affected his image of instructional constraints.
ContributorsTallman, Michael Anthony (Author) / Carlson, Marilyn P (Thesis advisor) / Thompson, Patrick W (Committee member) / Saldanha, Luis (Committee member) / Middleton, James (Committee member) / Harel, Guershon (Committee member) / Arizona State University (Publisher)
Created2015
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
This dissertation reports three studies of students’ and teachers’ meanings for quotient, fraction, measure, rate, and rate of change functions. Each study investigated individual’s schemes (or meanings) for foundational mathematical ideas. Conceptual analysis of what constitutes strong meanings for fraction, measure, and rate of change is critical for each study. In particular, each study distinguishes additive and multiplicative meanings for fraction and rate of change.
The first paper reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topics of targeted professional development.
The second paper reports the quantitative part of a mixed method study of 153 calculus students at a large public university. The majority of calculus students not only have weak meanings for fraction, measure, and constant rates but that having weak meanings is predictive of lower scores on a test about rate of change functions. Regression is used to determine the variation in student success on questions about rate of change functions (derivatives) associated with variation in success on fraction, measure, rate, and covariation items.
The third paper investigates the implications of two students’ fraction schemes for their understanding of rate of change functions. Students’ weak measurement schemes obstructed their ability to construct a rate of change function given the graph of an original function. The two students did not coordinate three levels of units, and struggled to relate partitioning and iterating in a way that would help them reason about fractions, rate of change, and rate of change functions.
Taken as a whole the studies show that the majority of secondary teachers and calculus students studied have weak meanings for foundational ideas and that these weaknesses cause them problems in making sense of more applications of rate of change.
The first paper reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topics of targeted professional development.
The second paper reports the quantitative part of a mixed method study of 153 calculus students at a large public university. The majority of calculus students not only have weak meanings for fraction, measure, and constant rates but that having weak meanings is predictive of lower scores on a test about rate of change functions. Regression is used to determine the variation in student success on questions about rate of change functions (derivatives) associated with variation in success on fraction, measure, rate, and covariation items.
The third paper investigates the implications of two students’ fraction schemes for their understanding of rate of change functions. Students’ weak measurement schemes obstructed their ability to construct a rate of change function given the graph of an original function. The two students did not coordinate three levels of units, and struggled to relate partitioning and iterating in a way that would help them reason about fractions, rate of change, and rate of change functions.
Taken as a whole the studies show that the majority of secondary teachers and calculus students studied have weak meanings for foundational ideas and that these weaknesses cause them problems in making sense of more applications of rate of change.
ContributorsByerley, Cameron (Author) / Thompson, Patrick W (Thesis advisor) / Carlson, Marilyn P (Committee member) / Middleton, James A. (Committee member) / Saldanha, Luis (Committee member) / Mcnamara, Allen (Committee member) / Arizona State University (Publisher)
Created2016
Description
Past research has shown that students have difficulty developing a robust conception of function. However, little prior research has been performed dealing with student knowledge of function composition, a potentially powerful mathematical concept. This dissertation reports the results of an investigation into student understanding and use of function composition, set against the backdrop of a precalculus class that emphasized quantification and covariational reasoning. The data were collected using task-based, semi-structured clinical interviews with individual students outside the classroom. Findings from this study revealed that factors such as the student's quantitative reasoning, covariational reasoning, problem solving behaviors, and view of function influence how a student understands and uses function composition. The results of the study characterize some of the subtle ways in which these factors impact students' ability to understand and use function composition to solve problems. Findings also revealed that other factors such as a students' persistence, disposition towards "meaning making" for the purpose of conceptualizing quantitative relationships, familiarity with the context of a problem, procedural fluency, and student knowledge of rules of "order of operations" impact a students' progress in advancing her/his solution approach.
ContributorsBowling, Stacey (Author) / Carlson, Marilyn P (Thesis advisor) / Thompson, Patrick W (Committee member) / Moore, Kevin C (Committee member) / Milner, Fabio (Committee member) / Van de Sande, Carla (Committee member) / Arizona State University (Publisher)
Created2014