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- Genre: Doctoral Dissertation
- Member of: Theses and Dissertations
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
This action research addressed teacher effectiveness in supporting students’ critical thinking skills by implementing differentiated instructional strategies in eight 3rd- and 4th-grade, self-contained, inclusive classrooms. This study addressed how third- and fourth-grade teachers perceived their instructional effectiveness, how differentiated instructional strategies influence third- and fourth-grade teachers, and how third- and fourth-grade teachers make further use of differentiated instruction to support students’ critical thinking skills across cultures, linguistics, and achievement levels to increase student achievement. Out of the enrollment in a southwest Phoenix elementary school, there was a 35% mobility rate; 76%, free and reduced lunches; 35%, Spanish-speaking homes; 10%, ELL services; and 10%, special education. The school was comprised of 52 certified teachers, out of which there were five related arts teachers, and four teachers who served gifted and special education students. Participants included all eight third- and fourth-grade teachers, 75% female and 25% males; 75% identified as Caucasian and 25% Hispanic/Latina, middle-class citizens. Professional development training was provided to these eight individual teachers during four months on differentiated instructional strategies to support students’ critical thinking. At this study’s beginning, these teachers perceived an obstacle to supporting students’ critical thinking as they struggled to learn new curriculums. Persevering through this challenge, teachers discovered success by implementing design-thinking, developing students’ growth mindsets, and utilizing cultural responsive teaching. These teachers identified three differentiated instructional strategies which impacted students’ academic progress: instructional scaffolds, collaborative group work, and project-based learning. Building upon linguistic responsive teaching, cultural responsive teaching, and Vygotsky’s socio-cultural theory, teachers revealed how to support students’ critical thinking through the use of graphic organizers, sentence frames, explicit instructions, growth mindsets, cultural references, and grouping structures. In addition, the outcomes demonstrated teachers can make further use of differentiated instruction by focusing on instructional groups, teachers’ mindsets, and methods for teaching accelerated learners. This study’s results have implications on teachers’ perception toward using differentiated instructional strategies as a viable method to support the multiple ways all students learn.
ContributorsRamos, Richard K (Author) / Liou, Daniel Dinn-You (Thesis advisor) / Dyer, Penelope (Committee member) / Saltmarsh, Sarah (Committee member) / McIntosh, Jason (Committee member) / Arizona State University (Publisher)
Created2018
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
The ideas of measurement and measurement comparisons (e.g., fractions, ratios, quotients) are introduced to students in elementary school. However, studies report that students of all ages have difficulty comparing two quantities in terms of their relative size. Students often understand fractions such as 3/7 as part-whole relationships or “three out of seven.” These limited conceptions have been documented to have implications for understanding the quotient as a measure of relative size and when learning other foundational ideas in mathematics (e.g., rate of change). Many scholars have identified students’ ability to conceptualize the relative size of two quantities values as important for learning specific ideas such as constant rate of change, exponential growth, and derivative. However, few researchers have focused on students’ ways of thinking about multiplicatively comparing two quantities’ values as they vary together across select topics in precalculus. Relative size reasoning is a way of thinking one has developed when conceptualizing the comparison of two quantities’ values multiplicatively, as their values vary in tandem. This document reviews literature related to relative size reasoning and presents a conceptual analysis that leverages this research in describing what I mean by a relative size comparison and what it means to engage in relative size reasoning. I further illustrate the role of relative size reasoning in understanding rate of change, multiplicative growth, rational functions, and what a graph’s concavity conveys about how two quantities’ values vary together.
This study reports on three beginning calculus students’ ways of thinking as they completed tasks designed to elicit students’ relative size reasoning. The data revealed 4 ways of conceptualizing the idea of quotient and highlights the affordances of conceptualizing a quotient as a measure of the relative size of two quantities’ values. The study also reports data from investigating the validity of a collection of multiple-choice items designed to assess students’ relative size reasoning (RSR) abilities. Analysis of this data provided insights for refining the questions and answer choices for these assessment items.
ContributorsLock, Kayla Ashley (Author) / Carlson, Marilyn (Thesis advisor) / Apkarian, Naneh (Thesis advisor) / Strom, April (Committee member) / Byerley, Cameron (Committee member) / Roh, Kyeong-Hah (Committee member) / Arizona State University (Publisher)
Created2023
Description
This dissertation reports on three studies about students’ conceptions and learning of the idea of instantaneous rate of change. The first study investigated 25 students’ conceptions of the idea of instantaneous rate of change. The second study proposes a hypothetical learning trajectory, based on the literature and results from the first study, for learning the idea of instantaneous rate of change. The third study investigated two students’ thinking and learning in the context of a sequence of five exploratory teaching interviews. The first paper reports on the results of conducting clinical interviews with 25 students. The results revealed the diverse conceptions that Calculus students have about the value of a derivative at a given input value. The results also suggest that students’ interpretation of the value of a rate of change is related to their use of covariational reasoning when considering how two quantities’ values vary together.
The second paper presents a conceptual analysis on the ways of thinking needed to develop a productive understanding of instantaneous rate of change. This conceptual analysis includes an ordered list of understandings and reasoning abilities that I hypothesize to be essential for understanding the idea of instantaneous rate of change. This paper also includes a sequence of tasks and questions I designed to support students in developing the ways of thinking and meanings described in my conceptual analysis.
The third paper reports on the results of five exploratory teaching interviews that leveraged my hypothetical learning trajectory from the second paper. The results of this teaching experiment indicate that developing a coherent understanding of rate of change using quantitative reasoning can foster advances in students’ understanding of instantaneous rate of change as a constant rate of change over an arbitrarily small input interval of a function’s domain.
ContributorsYu, Franklin (Author) / Carlson, Marilyn (Thesis advisor) / Zandieh, Michelle (Committee member) / Thompson, Patrick (Committee member) / Roh, Kyeong Hah (Committee member) / Soto, Roberto (Committee member) / Arizona State University (Publisher)
Created2022
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