Matching Items (13)
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- All Subjects: Mathematics Education
- Creators: Milner, Fabio
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
Description
ABSTRACT
The early desire for and the pursuit of literacy are often mentioned in the teeming volumes devoted to African-American history. However, stories, facts, and figures about the acquisition of numeracy by African Americans have not been equally documented.
The focus of this study was to search for the third R, this is the numeracy and mathematics experiences of African Americans who were born in, and before, 1933. The investigation of this generational cadre was pursued in order to develop oral histories and narratives going back to the early 1900s. This study examined formal and informal education and other relevant mathematics-related, lived experiences of unacknowledged and unheralded African Americans, as opposed to the American anomalies of African descent who are most often acknowledged, such as the Benjamin Bannekers, the George Washington Carvers, and other notables.
Quantitative and qualitative data were collected through the use of a survey and interviews. Quantitative results and qualitative findings were blended to present a nuanced perspective of African Americans learning mathematics during a period of Jim Crow, segregation, and discrimination. Their hopes, their fears, their challenges, their aspirations, their successes, and their failures are all tangential to their overall goal of seeking education, including mathematics education, in the early twentieth century. Both formal and informal experiences revealed a picture of life during those times to further enhance the literature regarding the mathematics experiences of African Americans.
Key words: Black students, historical, senior citizens, mathematics education, oral history, narrative, narrative inquiry, socio-cultural theory, Jim Crow
The early desire for and the pursuit of literacy are often mentioned in the teeming volumes devoted to African-American history. However, stories, facts, and figures about the acquisition of numeracy by African Americans have not been equally documented.
The focus of this study was to search for the third R, this is the numeracy and mathematics experiences of African Americans who were born in, and before, 1933. The investigation of this generational cadre was pursued in order to develop oral histories and narratives going back to the early 1900s. This study examined formal and informal education and other relevant mathematics-related, lived experiences of unacknowledged and unheralded African Americans, as opposed to the American anomalies of African descent who are most often acknowledged, such as the Benjamin Bannekers, the George Washington Carvers, and other notables.
Quantitative and qualitative data were collected through the use of a survey and interviews. Quantitative results and qualitative findings were blended to present a nuanced perspective of African Americans learning mathematics during a period of Jim Crow, segregation, and discrimination. Their hopes, their fears, their challenges, their aspirations, their successes, and their failures are all tangential to their overall goal of seeking education, including mathematics education, in the early twentieth century. Both formal and informal experiences revealed a picture of life during those times to further enhance the literature regarding the mathematics experiences of African Americans.
Key words: Black students, historical, senior citizens, mathematics education, oral history, narrative, narrative inquiry, socio-cultural theory, Jim Crow
ContributorsLaCount, Marilyn Ruth (Author) / Zambo, Ronald (Thesis advisor) / Flores, Alfinio (Committee member) / Koblitz, Ann Hibner (Committee member) / Zambo, Debby (Committee member) / Arizona State University (Publisher)
Created2014
Description
A fundamental motivation for this study was the underrepresentation of women in Science, Technology, Engineering and Mathematics careers. There is no doubt women and men can achieve at the same level in Mathematics, yet it is not clear why women are opting out. Adding race to the equation makes the underrepresentation more dramatic. Considering the important number of Latinos in the United States, especially in school age, it is relevant to find what reasons could be preventing them from participating in the careers mentioned. This study highlight the experiences young successful Latinas have in school Mathematics and how they shape their identities, to uncover potential conflicts that could later affect their participation in the field. In order to do so the author utilizes feminist approaches, Latino Critical Theory and Critical Race Theory to analyze the stories compiled. The participants were five successful Latinas in Mathematics, part of the honors track in a school in the Southwest of the United States. The theoretical lenses chosen allowed women of color to tell their story, highlighting the intersection of race, gender and socio-economical status as a factor shaping different schooling experiences. The author found that the participants distanced themselves from their home culture and from other girls at times to allow themselves to develop and maintain a successful identity as a Mathematics student. When talking about Latinos and their culture, the participants shared a view of themselves as proud Latinas who would prove others what Latinas can do. During other times while discussing the success of Latinos in Mathematics, they manifested Latinos were lazy and distance themselves from that stereotype. Similar examples about gender and Mathematics can be found in the study. The importance of the family as a motivator for their success was clear, despite the participants' concern that parents cannot offer certain types of help they feel they need. This was manifest in a tension regarding who owns the "right" Mathematics at home. Results showed that successful Latinas in the US may undergo a constant negotiation of conflicting discourses that force them to distance themselves from certain aspects of their culture, gender, and even their families, to maintain an identity of success in mathematics.
ContributorsGuerra Lombardi, Paula Patricia (Author) / Middleton, James (Thesis advisor) / Battey, Daniel (Committee member) / Koblitz, Ann (Committee member) / Flores, Alfinio (Committee member) / Arizona State University (Publisher)
Created2011
Description
This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet vector calculus, calculus on manifolds, linear algebra, and differential equations all rest upon the idea of functions of two (or more) variables. This dissertation contributes to understanding productive ways of thinking that can support students in thinking about functions of two or more variables as they describe complex systems with multiple variables interacting. This dissertation focuses on modeling the way of thinking of four students who participated in a specific instructional sequence designed to explore the limits of their ways of thinking and in turn, develop a robust model that could explain, describe, and predict students' actions relative to specific tasks. The data was collected using a teaching experiment methodology, and the tasks within the teaching experiment leveraged quantitative reasoning and covariation as foundations of students developing a coherent understanding of two-variable functions and their rates of change. The findings of this study indicated that I could characterize students' ways of thinking about two-variable functions by focusing on their use of novice and/or expert shape thinking, and the students' ways of thinking about rate of change by focusing on their quantitative reasoning. The findings suggested that quantitative and covariational reasoning were foundational to a student's ability to generalize their understanding of a single-variable function to two or more variables, and their conception of rate of change to rate of change at a point in space. These results created a need to better understand how experts in the field, such as mathematicians and mathematics educators, thinking about multivariable functions and their rates of change.
ContributorsWeber, Eric David (Author) / Thompson, Patrick (Thesis advisor) / Middleton, James (Committee member) / Carlson, Marilyn (Committee member) / Saldanha, Luis (Committee member) / Milner, Fabio (Committee member) / Van de Sande, Carla (Committee member) / Arizona State University (Publisher)
Created2012
Description
The primary purpose of this study is to examine the effect of knowledge for teaching mathematics and teaching practice on student mathematics achievement growth. Thirty two teachers and 299 fourth grade students in three elementary schools from one school district in urban area participated in the study. Most of them are Hispanic in origin and about forty percent is English Language Learners (ELLs). The two level Hierarchical Linear Model (HLM) was used to investigate repeated measures of teaching practice measured by Classroom Assessment Scoring System (CLASS) instrument. Also, linear regression and a multiple regression to examine the relationship between teacher knowledge measured by Learning for Mathematics Teaching (LMT) and Developing Mathematical Ideas (DMI) items and teaching practice were employed. In addition, a three level HLM was employed to analyze repeated measures of student mathematics achievement measured by Arizona Assessment Consortium (AzAC) instruments. Results showed that overall teaching practice did not change weekly although teachers' emotional support for their students improved by week. Furthermore, a statistically significant relationship between teacher knowledge and teaching practice was not found. In terms of student learning, ELLs have significantly lower initial status in mathematics achievement than non-ELLs, as were growth rates for these two groups. Lastly, teaching practice significantly predicted students' monthly mathematics achievement growth but teacher knowledge did not. The findings suggest that school systems and education policy makers need to provide teachers with the chance to reflect on their teaching and change it within themselves in order to better support student mathematics learning.
ContributorsKim, Seong Hee (Author) / Sloane, Finbarr (Thesis advisor) / Middleton, James (Committee member) / Flores, Alfinio (Committee member) / Arizona State University (Publisher)
Created2012
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
The advancement of technology has substantively changed the practices of numerous professions, including teaching. When an instructor first adopts a new technology, established classroom practices are perturbed. These perturbations can have positive and negative, large or small, and long- or short-term effects on instructors’ abilities to teach mathematical concepts with the new technology. Therefore, in order to better understand teaching with technology, we need to take a closer look at the adoption of new technology in a mathematics classroom. Using interviews and classroom observations, I explored perturbations in mathematical classroom practices as an instructor implemented virtual manipulatives as novel didactic objects in rational function instruction. In particular, the instructor used didactic objects that were designed to lay the foundation for developing a conceptual understanding of rational functions through the coordination of relative size of the value of the numerator in terms of the value of the denominator. The results are organized according to a taxonomy that captures leader actions, communication, expectations of technology, roles, timing, student engagement, and mathematical conceptions.
ContributorsPampel, Krysten (Author) / Currin van de Sande, Carla (Thesis advisor) / Thompson, Patrick W (Committee member) / Carlson, Marilyn (Committee member) / Milner, Fabio (Committee member) / Strom, April (Committee member) / Arizona State University (Publisher)
Created2017
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
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
Researchers have documented the importance of seeing a graph as an emergent trace of how two quantities’ values vary simultaneously in order to reason about the graph in terms of quantitative relationships. If a student does not see a graph as a representation of how quantities change together then the student is limited to reasoning about perceptual features of the shape of the graph.
This dissertation reports results of an investigation into the ways of thinking that support and inhibit students from constructing and reasoning about graphs in terms of covarying quantities. I collected data by engaging three university precalculus students in asynchronous teaching experiments. I designed the instructional sequence to support students in making three constructions: first imagine representing quantities’ magnitudes along the axes, then simultaneously represent these magnitudes with a correspondence point in the plane, and finally anticipate tracking the correspondence point to track how the two quantities’ attributes change simultaneously.
Findings from this investigation provide insights into how students come to engage in covariational reasoning and re-present their imagery in their graphing actions. The data presented here suggests that it is nontrivial for students to coordinate their images of two varying quantities. This is significant because without a way to coordinate two quantities’ variation the student is limited to engaging in static shape thinking.
I describe three types of imagery: a correspondence point, Tinker Bell and her pixie dust, and an actor taking baby steps, that supported students in developing ways to coordinate quantities’ variation. I discuss the figurative aspects of the students’ coordination in order to account for the difficulties students had (1) constructing a multiplicative object that persisted under variation, (2) reconstructing their acts of covariation in other graphing tasks, and (3) generalizing these acts of covariation to reason about formulas in terms of covarying quantities.
This dissertation reports results of an investigation into the ways of thinking that support and inhibit students from constructing and reasoning about graphs in terms of covarying quantities. I collected data by engaging three university precalculus students in asynchronous teaching experiments. I designed the instructional sequence to support students in making three constructions: first imagine representing quantities’ magnitudes along the axes, then simultaneously represent these magnitudes with a correspondence point in the plane, and finally anticipate tracking the correspondence point to track how the two quantities’ attributes change simultaneously.
Findings from this investigation provide insights into how students come to engage in covariational reasoning and re-present their imagery in their graphing actions. The data presented here suggests that it is nontrivial for students to coordinate their images of two varying quantities. This is significant because without a way to coordinate two quantities’ variation the student is limited to engaging in static shape thinking.
I describe three types of imagery: a correspondence point, Tinker Bell and her pixie dust, and an actor taking baby steps, that supported students in developing ways to coordinate quantities’ variation. I discuss the figurative aspects of the students’ coordination in order to account for the difficulties students had (1) constructing a multiplicative object that persisted under variation, (2) reconstructing their acts of covariation in other graphing tasks, and (3) generalizing these acts of covariation to reason about formulas in terms of covarying quantities.
ContributorsFrank, Kristin Marianna (Author) / Thompson, Patrick W (Thesis advisor) / Carlson, Marilyn P (Thesis advisor) / Milner, Fabio (Committee member) / Roh, Kyeong Hah (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2017