Matching Items (16)
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- All Subjects: Mathematics Education
- All Subjects: Applied Mathematics
- All Subjects: Quantitative Reasoning
- Genre: Doctoral Dissertation
- Creators: Milner, Fabio
- Member of: Theses and Dissertations
Description
The need for improved mathematics education in many of America's schools that serve students from low income households has been extensively documented. This practical action research study, set in a suburban Title I school with a primarily Hispanic, non-native English speaking population, is designed to explore the effects of the progression through a set of problem solving solution strategies on the mathematics problem solving abilities of 2nd grade students. Students worked in class with partners to complete a Cognitively Guided Instruction-style (CGI) mathematics word problem using a dictated solution strategy five days a week for twelve weeks, three or four weeks for each of four solution strategies. The phases included acting out the problem using realia, representing the problem using standard mathematics manipulatives, modeling the problem using a schematic representation, and solving the problem using a number sentence. Data were collected using a five question problem solving pre- and post-assessment, video recorded observations, and Daily Answer Recording Slips or Mathematics Problem Solving Journals. Findings showed that this problem solving innovation was effective in increasing the problem solving abilities of all participants in this study, with an average increase of 63% in the number of pre-assessment to post-assessment questions answered correctly. Additionally, students increased the complexity of solutions used to solve problems and decreased the rate of guessing at answers to word problems. Further rounds of research looking into the direct effects of the MKO are suggested as next steps of research.
ContributorsSpilde, Amy (Author) / Zambo, Ronald (Thesis advisor) / Heck, Thomas (Committee member) / Nicoloff, Stephen J. (Committee member) / Arizona State University (Publisher)
Created2013
Description
ABSTRACT There is a continuing emphasis in the United States to improve student's mathematical abilities and one approach is to better prepare teachers. This study investigated the effects of using lesson study with preservice secondary mathematics teachers to improve their proficiency at planning and implementing instruction. The participants were students (preservice teachers) in an undergraduate teacher preparation program at a private university who were enrolled in a mathematics methods course for secondary math teachers. This project used lesson study to engage preservice teachers in collaboratively creating lessons, field testing them, using feedback to revise the lessons, and re-teaching the revised lesson. The preservice teachers worked through multiple cycles of the process in their secondary math methods class receiving feedback from their peers and instructor prior to teaching the lessons in their field experience (practicum). A mixed methods approach was implemented to investigate the preservice teacher's abilities to plan and implement instruction as well as their efficacy for teaching. Data were collected from surveys, video analysis, student reflections, and semi-structured interviews. The findings from this study indicate that lesson study for preservice teachers was an effective means of teacher education. Lesson study positively impacted the preservice teachers' ability to plan and teach mathematical lessons more effectively. The preservice teachers successfully transitioned from teaching in the methods classroom to their field experience classroom during this innovation. Further, the efficacy of the preservice teachers to teach secondary mathematics increased based on this innovation. Further action research cycles of lesson study with preservice teachers are recommended.
ContributorsMostofo, Jameel (Author) / Zambo, Ronald (Thesis advisor) / Elliott, Sherman (Committee member) / Heck, Thomas (Committee member) / Arizona State University (Publisher)
Created2013
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
In any instructional situation, the instructor's goal is to maximize the learning attained by students. Drawing on the adage, 'we learn best what we have taught,' this action research project was conducted to examine whether students, in fact, learned college algebra material better if they taught it to their peers. The teaching-to-learn process was conducted in the following way. The instructor-researcher met with individual students and taught a college algebra topic to a student who served as the leader of a group of four students. At the next step, the student who originally learned the material from the instructor met with three other students in a small group session and taught the material to them to prepare an in-class presentation. Students in these small group sessions discussed how best to present the material, anticipated questions, and prepared a presentation to be shared with their classmates. The small group then taught the material to classmates during an in-class review session prior to unit examinations. Quantitative and qualitative data were gathered. Quantitative data consisted of pre- and post-test scores on four college algebra unit examinations. In addition, scores from Likert-scale items on an end-of-semester questionnaire that assessed the effectiveness of the teaching-to-learn process and attitudes toward the process were obtained. Qualitative data consisted of field notes from observations of selected small group sessions and in-class presentations. Additional qualitative data included responses to open-ended questions on the end-of-semester questionnaire and responses to interview items posed to groups of students. Results showed the quantitative data did not support the hypothesis that material, which was taught, was better learned than other material. Nevertheless, qualitative data indicated students were engaged in the material, had a deeper understanding of the material, and were more confident about it as a result of their participation in the teaching-to-learn process. Students also viewed the teaching-to-learn process as being effective and they had positive attitudes toward the teaching-to-learn process. Discussion focused on how engagement, deeper understanding and confidence interacted with one another to increase student learning. Lessons learned, implications for practice, and implications for further action research were also discussed.
ContributorsNicoloff, Stephen J (Author) / Buss, Ray R (Thesis advisor) / Zambo, Ronald (Committee member) / Shaw, Phyllis J (Committee member) / Arizona State University (Publisher)
Created2011
Description
This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference.
ContributorsJoshua, Surani Ashanthi (Author) / Thompson, Patrick W (Thesis advisor) / Carlson, Marilyn (Committee member) / Roh, Kyeong Hah (Committee member) / Middleton, James (Committee member) / Culbertson, Robert (Committee member) / Arizona State University (Publisher)
Created2019
Description
This three-article dissertation considers the pedagogical practices for developing statistically literate students and teaching data-driven decision-making with the goal of preparing students for civic engagement and improving student achievement. The first article discusses a critical review of the literature on data-driven decision-making project conditions in K-12 educational settings. Upon reviewing the literature, I synthesized and summarized the current practices into three distinct models. The models serve to clarify the pedagogical choices of the teacher and the degree at which students' views are involved and incorporated into the projects. I propose an alternative model/framework and discuss possible implications in the classroom. In the second article, I use the framework developed in the first article as the basis for an educational research intervention. I describe a study where I developed a handbook based on the framework and implemented a sample of professional development sessions from the handbook. Advisors and teachers provided feedback on the handbook and professional development. This feedback served as the subject of analysis while I continued to refine the handbook and the professional learning sessions. I describe the refinement process and the implications in terms of design decisions of educational interventions and statistical knowledge for teaching. The final article performs a secondary data analysis of school, teacher, and student level data using the Trends in International Mathematics and Science Study (TIMSS) database. The paper seeks to answer the research question: “Which aspects of teacher professional knowledge measures predict student achievement in the mathematical domain of data and statistical topics?” The results indicate that when controlling for school level wealth index, teacher characteristics are not as influential as the school level wealth index. I discuss future research as well as school policy and curriculum implications of these results.
ContributorsRiske, Amanda Katherine (Author) / Zuiker, Steven (Thesis advisor) / Milner, Fabio (Thesis advisor) / Middleton, James (Committee member) / Pivovarova, Margarita (Committee member) / Arizona State University (Publisher)
Created2022
Description
There is a need in the ecology literature to have a discussion about the fundamental theories from which population dynamics arises. Ad hoc model development is not uncommon in the field often as a result of a need to publish rapidly and frequently. Ecologists and statisticians like Robert J. Steidl and Kenneth P Burnham have called for a more deliberative approach they call "hard thinking". For example, the phenomena of population growth can be captured by almost any sigmoid function. The question of which sigmoid function best explains a data set cannot be answered meaningfully by statistical regression since that can only speak to the validity of the shape. There is a need to revisit enzyme kinetics and ecological stoichiometry to properly justify basal model selection in ecology. This dissertation derives several common population growth models from a generalized equation. The mechanistic validity of these models in different contexts is explored through a kinetic lens. The behavioral kinetic framework is then put to the test by examining a set of biologically plausible growth models against the 1968-1995 elk population count data for northern Yellowstone. Using only this count data, the novel Monod-Holling growth model was able to accurately predict minimum viable population and life expectancy despite both being exogenous to the model and data set. Lastly, the elk/wolf data from Yellowstone was used to compare the validity of the Rosenzweig-MacArthur and Arditi-Ginzburg models. They both were derived from a more general model which included both predator and prey mediated steps. The Arditi-Ginzburg model was able to fit the training data better, but only the Rosenzweig-MacArthur model matched the validation data. Accounting for animal sexual behavior allowed for the creation of the Monod-Holling model which is just as simple as the logistic differential equation but provides greater insights for conservation purposes. Explicitly acknowledging the ethology of wolf predation helps explain the differences in predictive performances by the best fit Rosenzweig-MacArthur and Arditi-Ginzburg models. The behavioral kinetic framework has proven to be a useful tool, and it has the ability to provide even further insights going forward.
ContributorsPringle, Jack Andrew McCracken (Author) / Anderies, John M (Thesis advisor) / Kuang, Yang (Committee member) / Milner, Fabio (Committee member) / Arizona State University (Publisher)
Created2022
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
Balancing temporal shortages of renewable energy with natural gas for the generation of electricity is a challenge for dispatchers. This is compounded by the recent proposal of blending cleanly-produced hydrogen into natural gas networks. To introduce the concepts of gas flow, this thesis begins by linearizing the partial differential equations (PDEs) that govern the flow of natural gas in a single pipe. The solution of the linearized PDEs is used to investigate wave attenuation and characterize critical operating regions where linearization is applicable. The nonlinear PDEs for a single gas are extended to mixtures of gases with the addition of a PDE that governs the conservation of composition. The gas mixture formulation is developed for general gas networks that can inject or withdraw arbitrary time-varying mixtures of gases into or from the network at arbitrarily specified nodes, while being influenced by time-varying control actions of compressor units. The PDE formulation is discretized in space to form a nonlinear control system of ordinary differential equations (ODEs), which is used to prove that homogeneous mixtures are well-behaved and heterogeneous mixtures may be ill-behaved in the sense of monotone-ordering of solutions. Numerical simulations are performed to compute interfaces that delimit monotone and periodic system responses. The ODE system is used as the constraints of an optimal control problem (OCP) to minimize the expended energy of compressors. Moreover, the ODE system for the natural gas network is linearized and used as the constraints of a linear OCP. The OCPs are digitally implemented as optimization problems following the discretization of the time domain. The optimization problems are applied to pipelines and small test networks. Some qualitative and computational applications, including linearization error analysis and transient responses, are also investigated.
ContributorsBaker, Luke Silas (Author) / Armbruster, Dieter (Thesis advisor) / Zlotnik, Anatoly (Committee member) / Herty, Michael (Committee member) / Platte, Rodrigo (Committee member) / Milner, Fabio (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