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- Creators: Roh, Kyeong Hah
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 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