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- Genre: Doctoral Dissertation
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
Four Souvenirs for Violin and Piano was composed by Paul Schoenfeld (b.1947) in 1990 as a showpiece, spotlighting the virtuosity of both the violin and piano in equal measure. Each movement is a modern interpretation of a folk or popular genre, re- envisioned over intricate jazz harmonies and rhythms. The work was commissioned by violinist Lev Polyakin, who specifically requested some short pieces that could be performed in a local jazz establishment named Night Town in Cleveland, Ohio. The result is a work that is approximately fifteen minutes in length. Schoenfeld is a respected composer in the contemporary classical music community, whose Café Music (1986) for piano trio has recently become a staple of the standard chamber music repertoire. Many of his other works, however, remain in relative obscurity. It is the focus of this document to shed light on at least one other notable composition; Four Souvenirs for Violin and Piano. Among the topics to be discussed regarding this piece are a brief history behind the genesis of this composition, a structural summary of the entire work and each of its movements, and an appended practice guide based on interview and coaching sessions with the composer himself. With this project, I hope to provide a better understanding and appreciation of this work.
ContributorsJanczyk, Kristie Annette (Author) / Ryan, Russell (Thesis advisor) / Campbell, Andrew (Committee member) / Norton, Kay (Committee member) / Arizona State University (Publisher)
Created2015
Description
Samuel Máynez Prince (1886-1966), was a prolific and important Mexican musician. Prince’s musical style followed the trends of the nineteenth-century salon music genre. His compositions include lullabies, songs, dances, marches, mazurkas, waltzes, and revolutionary anthems. Prince’s social status and performances in the famed Café Colón in Mexico City increased his popularity among high-ranking political figures during the time of the Mexican Revolution as well as his status in the Mexican music scene.
Unfortunately there is virtually no existing scholarship on Prince and even basic information regarding his life and works is not readily available. The lack of organization of the manuscript scores and the absence of dates of his works has further pushed the composer into obscurity. An investigation therefore was necessary in order to explore the neglected aspects of the life and works of Prince as a violinist and composer. This document is the result of such an investigation by including extensive new biographical information, as well as the first musical analysis and edition of the complete recovered works for violin and piano.
In order to fill the gaps present in the limited biographical information regarding Prince’s life, investigative research was conducted in Mexico City. Information was drawn from archives of the composer’s grandchildren, the Palacio de Bellas Artes, the Conservatorio Nacional de Música de México, and the Orquesta Sinfónica Nacional. The surviving relatives provided first-hand details on events in the composer’s life; one also offered the researcher access to their personal archive including, important life documents, photographs, programs from concert performances, and manuscript scores of the compositions. Establishing connections with the relatives also led the researcher to examining the violins owned and used by the late violinist/composer.
This oral history approach led to new and updated information, including the revival of previously unpublished music for violin and piano. These works are here compiled in an edition that will give students, teachers, and music-lovers access to this unknown repertoire. Finally, this research seeks to promote the beauty and nuances of Mexican salon music, and the complete works for violin and piano of Samuel Máynez Prince in particular.
Unfortunately there is virtually no existing scholarship on Prince and even basic information regarding his life and works is not readily available. The lack of organization of the manuscript scores and the absence of dates of his works has further pushed the composer into obscurity. An investigation therefore was necessary in order to explore the neglected aspects of the life and works of Prince as a violinist and composer. This document is the result of such an investigation by including extensive new biographical information, as well as the first musical analysis and edition of the complete recovered works for violin and piano.
In order to fill the gaps present in the limited biographical information regarding Prince’s life, investigative research was conducted in Mexico City. Information was drawn from archives of the composer’s grandchildren, the Palacio de Bellas Artes, the Conservatorio Nacional de Música de México, and the Orquesta Sinfónica Nacional. The surviving relatives provided first-hand details on events in the composer’s life; one also offered the researcher access to their personal archive including, important life documents, photographs, programs from concert performances, and manuscript scores of the compositions. Establishing connections with the relatives also led the researcher to examining the violins owned and used by the late violinist/composer.
This oral history approach led to new and updated information, including the revival of previously unpublished music for violin and piano. These works are here compiled in an edition that will give students, teachers, and music-lovers access to this unknown repertoire. Finally, this research seeks to promote the beauty and nuances of Mexican salon music, and the complete works for violin and piano of Samuel Máynez Prince in particular.
ContributorsEkenes, Spencer Arvin (Author) / McLin, Katherine (Thesis advisor) / Feisst, Sabine (Committee member) / Jiang, Danwen (Committee member) / Arizona State University (Publisher)
Created2016
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
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