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- All Subjects: Calculus
- Creators: Roh, Kyeong Hah
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
Description
Functions represented in the graphical register, as graphs in the Cartesian plane, are found throughout secondary and undergraduate mathematics courses. In the study of Calculus, specifically, graphs of functions are particularly prominent as a means of illustrating key concepts. Researchers have identified that some of the ways that students may interpret graphs are unconventional, which may impact their understanding of related mathematical content. While research has primarily focused on how students interpret points on graphs and students’ images related to graphs as a whole, details of how students interpret and reason with variables and expressions on graphs of functions have remained unclear.
This dissertation reports a study characterizing undergraduate students’ interpretations of expressions in the graphical register with statements from Calculus, its association with their evaluations of these statements, its relation to the mathematical content of these statements, and its relation to their interpretations of points on graphs. To investigate students’ interpretations of expressions on graphs, I conducted 150-minute task-based clinical interviews with 13 undergraduate students who had completed Calculus I with a range of mathematical backgrounds. In the interviews, students were asked to evaluate propositional statements about functions related to key definitions and theorems of Calculus and were provided various graphs of functions to make their evaluations. The central findings from this study include the characteristics of four distinct interpretations of expressions on graphs that students used in this study. These interpretations of expressions on graphs I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. The findings from this study suggest that different contexts may evoke different graphical interpretations of expressions from the same student. Further, some interpretations were shown to be associated with students correctly evaluating some statements while others were associated with students incorrectly evaluating some statements.
I report the characteristics of these interpretations of expressions in the graphical register and its relation to their evaluations of the statements, the mathematical content of the statements, and their interpretation of points. I also discuss the implications of these findings for teaching and directions for future research in this area.
This dissertation reports a study characterizing undergraduate students’ interpretations of expressions in the graphical register with statements from Calculus, its association with their evaluations of these statements, its relation to the mathematical content of these statements, and its relation to their interpretations of points on graphs. To investigate students’ interpretations of expressions on graphs, I conducted 150-minute task-based clinical interviews with 13 undergraduate students who had completed Calculus I with a range of mathematical backgrounds. In the interviews, students were asked to evaluate propositional statements about functions related to key definitions and theorems of Calculus and were provided various graphs of functions to make their evaluations. The central findings from this study include the characteristics of four distinct interpretations of expressions on graphs that students used in this study. These interpretations of expressions on graphs I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. The findings from this study suggest that different contexts may evoke different graphical interpretations of expressions from the same student. Further, some interpretations were shown to be associated with students correctly evaluating some statements while others were associated with students incorrectly evaluating some statements.
I report the characteristics of these interpretations of expressions in the graphical register and its relation to their evaluations of the statements, the mathematical content of the statements, and their interpretation of points. I also discuss the implications of these findings for teaching and directions for future research in this area.
ContributorsDavid, Erika Johara (Author) / Roh, Kyeong Hah (Thesis advisor) / Thompson, Patrick W (Committee member) / Zandieh, Michelle (Committee member) / Dawkins, Paul C (Committee member) / Zazkis, Dov (Committee member) / Arizona State University (Publisher)
Created2019
Description
This study investigates several students’ interpretations and meanings for negations of various mathematical statements with quantifiers, and how their meanings for quantified variables impact their interpretations and denials of these quantified statements. Eight students participated in three separate exploratory teaching interviews and were selected from Transition-to-Proof and advanced mathematics courses beyond Transition-to-Proof. In the first interview, students were asked to interpret mathematical statements from Calculus contexts and provide justifications and refutations for why these statements are true or false in particular situations. In the second interview, students were asked to negate the same set of mathematical statements. Both sets of interviews were analyzed to determine students’ meanings for the quantified variables in the statements, and then these meanings were used to determine how students’ quantifications influenced their interpretations, denials, and evaluations for the quantified statements. In the final interview, students were also be asked to interpret and negation statements from different mathematical contexts. All three interviews were used to determine what meanings comprised students’ interpretations and denials for the given statements. Additionally, students’ interpretations and negations across different statements in the interviews were analyzed and then compared within students and across students to determine if there were differences in student denials across different moments.
ContributorsSellers, Morgan Early (Author) / Roh, Kyeong Hah (Thesis advisor) / Kawski, Matthias (Committee member) / Keene, Karen (Committee member) / Thompson, Patrick (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2020