Matching Items (4)
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
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