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Description
Based on poor student performance in past studies, the incoherence present in the teaching of inverse functions, and teachers' own accounts of their struggles to teach this topic, it is apparent that the idea of function inverse deserves a closer look and an improved pedagogical approach. This improvement must enhance students' opportunity to construct a meaning for a function's inverse and, out of that meaning, produce ways to define a function's inverse without memorizing some procedure. This paper presents a proposed instructional sequence that promotes reflective abstraction in order to help students develop a process conception of function and further understand the meaning of a function inverse. The instructional sequence was used in a teaching experiment with three subjects and the results are presented here. The evidence presented in this paper supports the claim that the proposed instructional sequence has the potential to help students construct meanings needed for understanding function inverse. The results of this study revealed shifts in the understandings of all three subjects. I conjecture that these shifts were achieved by posing questions that promoted reflective abstraction. The questions and subsequent interactions appeared to result in all three students moving toward a process conception of function.
ContributorsFowler, Bethany (Author) / Carlson, Marilyn (Thesis advisor) / Roh, Kyeong (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2014
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
There have been a number of studies that have examined students’ difficulties in understanding the idea of logarithm and the effectiveness of non-traditional interventions. However, there have been few studies that have examined the understandings students develop and need to develop when completing conceptually oriented logarithmic lessons. In this document, I present the three papers of my dissertation study. The first paper examines two students’ development of concepts foundational to the idea of logarithm. This paper discusses two essential understandings that were revealed to be problematic and essential for students’ development of productive meanings for exponents, logarithms and logarithmic properties. The findings of this study informed my later work to support students in understanding logarithms, their properties and logarithmic functions. The second paper examines two students’ development of the idea of logarithm. This paper describes the reasoning abilities two students exhibited as they engaged with tasks designed to foster their construction of more productive meanings for the idea of logarithm. The findings of this study provide novel insights for supporting students in understanding the idea of logarithm meaningfully. Finally, the third paper begins with an examination of the historical development of the idea of logarithm. I then leveraged the insights of this literature review and the first two papers to perform a conceptual analysis of what is involved in learning and understanding the idea of logarithm. The literature review and conceptual analysis contributes novel and useful information for curriculum developers, instructors, and other researchers studying student learning of this idea.
ContributorsKuper Flores, Emily Ginamarie (Author) / Carlson, Marilyn (Thesis advisor) / Thompson, Patrick (Committee member) / Milner, Fabio (Committee member) / Zazkis, Dov (Committee member) / Czocher, Jennifer (Committee member) / Arizona State University (Publisher)
Created2018
Description
The advancement of technology has substantively changed the practices of numerous professions, including teaching. When an instructor first adopts a new technology, established classroom practices are perturbed. These perturbations can have positive and negative, large or small, and long- or short-term effects on instructors’ abilities to teach mathematical concepts with the new technology. Therefore, in order to better understand teaching with technology, we need to take a closer look at the adoption of new technology in a mathematics classroom. Using interviews and classroom observations, I explored perturbations in mathematical classroom practices as an instructor implemented virtual manipulatives as novel didactic objects in rational function instruction. In particular, the instructor used didactic objects that were designed to lay the foundation for developing a conceptual understanding of rational functions through the coordination of relative size of the value of the numerator in terms of the value of the denominator. The results are organized according to a taxonomy that captures leader actions, communication, expectations of technology, roles, timing, student engagement, and mathematical conceptions.
ContributorsPampel, Krysten (Author) / Currin van de Sande, Carla (Thesis advisor) / Thompson, Patrick W (Committee member) / Carlson, Marilyn (Committee member) / Milner, Fabio (Committee member) / Strom, April (Committee member) / Arizona State University (Publisher)
Created2017
Description
Researchers have described two fundamental conceptualizations for division, known as partitive and quotitive division. Partitive division is the conceptualization of a÷b as the amount of something per copy such that b copies of this amount yield the amount a. Quotitive division is the conceptualization of a÷b as the number of copies of the amount b that yield the amount a. Researchers have identified many cognitive obstacles that have inhibited the development of robust meanings for division involving non-whole values, while other researchers have commented on the challenges related to such development. Regarding division with fractions, much research has been devoted to quotitive conceptualizations of division, or on symbolic manipulation of variables. Research and curricular activities have largely avoided the study and development of partitive conceptualizations involving fractions, as well as their connection to the invert-and-multiply algorithm. In this dissertation study, I investigated six middle school mathematics teachers’ meanings related to partitive conceptualizations of division over the positive rational numbers. I also investigated the impact of an intervention that I designed with the intent of advancing one of these teachers’ meanings. My findings suggested that the primary cognitive obstacles were difficulties with maintaining multiple levels of units, weak quantitative meanings for fractional multipliers, and an unawareness of (and confusion due to) the two quantitative conceptualizations of division. As a product of this study, I developed a framework for characterizing robust meanings for division, indicated directions for future research, and shared implications for curriculum and instruction.
ContributorsWeber, Matthew Barrett (Author) / Strom, April D (Thesis advisor) / Thompson, Patrick W (Thesis advisor) / Carlson, Marilyn (Committee member) / Middleton, James (Committee member) / Tzur, Ron (Committee member) / Arizona State University (Publisher)
Created2019
Description
The concept of distribution is one of the core ideas of probability theory and inferential statistics, if not the core idea. Many introductory statistics textbooks pay lip service to stochastic/random processes but how do students think about these processes? This study sought to explore what understandings of stochastic process students develop as they work through materials intended to support them in constructing the long-run behavior meaning for distribution.
I collected data in three phases. First, I conducted a set of task-based clinical interviews that allowed me to build initial models for the students’ meanings for randomness and probability. Second, I worked with Bonnie in an exploratory teaching setting through three sets of activities to see what meanings she would develop for randomness and stochastic process. The final phase consisted of me working with Danielle as she worked through the same activities as Bonnie but this time in teaching experiment setting where I used a series of interventions to test out how Danielle was thinking about stochastic processes.
My analysis shows that students can be aware that the word “random” lives in two worlds, thereby having conflicting meanings. Bonnie’s meaning for randomness evolved over the course of the study from an unproductive meaning centered on the emotions of the characters in the context to a meaning that randomness is the lack of a pattern. Bonnie’s lack of pattern meaning for randomness subsequently underpinned her image of stochastic/processes, leading her to engage in pattern-hunting behavior every time she needed to classify a process as stochastic or not. Danielle’s image of a stochastic process was grounded in whether she saw the repetition as being reproducible (process can be repeated, and outcomes are identical to prior time through the process) or replicable (process can be repeated but the outcomes aren’t in the same order as before). Danielle employed a strategy of carrying out several trials of the process, resetting the applet, and then carrying out the process again, making replicability central to her thinking.
I collected data in three phases. First, I conducted a set of task-based clinical interviews that allowed me to build initial models for the students’ meanings for randomness and probability. Second, I worked with Bonnie in an exploratory teaching setting through three sets of activities to see what meanings she would develop for randomness and stochastic process. The final phase consisted of me working with Danielle as she worked through the same activities as Bonnie but this time in teaching experiment setting where I used a series of interventions to test out how Danielle was thinking about stochastic processes.
My analysis shows that students can be aware that the word “random” lives in two worlds, thereby having conflicting meanings. Bonnie’s meaning for randomness evolved over the course of the study from an unproductive meaning centered on the emotions of the characters in the context to a meaning that randomness is the lack of a pattern. Bonnie’s lack of pattern meaning for randomness subsequently underpinned her image of stochastic/processes, leading her to engage in pattern-hunting behavior every time she needed to classify a process as stochastic or not. Danielle’s image of a stochastic process was grounded in whether she saw the repetition as being reproducible (process can be repeated, and outcomes are identical to prior time through the process) or replicable (process can be repeated but the outcomes aren’t in the same order as before). Danielle employed a strategy of carrying out several trials of the process, resetting the applet, and then carrying out the process again, making replicability central to her thinking.
ContributorsHatfield, Neil (Author) / Thompson, Patrick (Thesis advisor) / Carlson, Marilyn (Committee member) / Middleton, James (Committee member) / Lehrer, Richard (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2019
Description
Aside from uplifting and tearing down the mood of a young LGBTQ+ kid, journalistic media has the potential to alter the way audiences understand and react to individuals of the LGBTQ+ community. Looking at the rhetorical approaches, frameworks, and expanded narratives of news sources, this project engages with the concepts of same-sex marriage, lifestyles, bans, and children in education in order to attain an understanding of what media messages are being shared, how they are being communicated, and what the implications of such rhetoric are. Summary of the findings:
• Same-sex marriage as the win that cannot be repeated.
Infamously known as the central legal battle for the LGBTQ+ community, same-sex marriage finds itself in many political speeches, campaigns, and social commentaries. Interestingly, after being legalized through a Supreme Court decision in the United States, Same-Sex Marriage finds itself framed as the social inevitability that should not be repeated in politics or any legal shift. In other words, “the gays have won this battle, but not the war.”
• There are risks around the “LGBTQ+ lifestyle” and its careful catering to an elite minority and the mediation through bans.
The risks of the LGBTQ+ “lifestyle” date back far, with many connotations being attached to being LGBTQ+ (AIDS epidemics, etc.). In modern journalism, many media outlets portray LGBTQ+ individuals to be a tiny minority (.001% according to some) that demands the whole society to adhere to their requests. This framework portrays the LGBTQ+ community as oppressors and obsessed advocates that can never “seem to get enough” (ex: more than just marriage). The bans are framed as the neutralizing factor to the catering.
• LGBTQ+ children and topics in academic and social spaces are the extreme degree.
When it comes to LGBTQ+ issues and conversations as they revolve around children, media outlets have some of the most passionate opinions about them. Often portrayed as “the line that shouldn’t be crossed,” LGBTQ+ issues, as they find themselves in schools and other spaces, are thus portrayed as bearable to a certain degree, never completely. Claims of indoctrination are also presented prominently even when institutional efforts are to protect LGBTQ+ kids.
• Same-sex marriage as the win that cannot be repeated.
Infamously known as the central legal battle for the LGBTQ+ community, same-sex marriage finds itself in many political speeches, campaigns, and social commentaries. Interestingly, after being legalized through a Supreme Court decision in the United States, Same-Sex Marriage finds itself framed as the social inevitability that should not be repeated in politics or any legal shift. In other words, “the gays have won this battle, but not the war.”
• There are risks around the “LGBTQ+ lifestyle” and its careful catering to an elite minority and the mediation through bans.
The risks of the LGBTQ+ “lifestyle” date back far, with many connotations being attached to being LGBTQ+ (AIDS epidemics, etc.). In modern journalism, many media outlets portray LGBTQ+ individuals to be a tiny minority (.001% according to some) that demands the whole society to adhere to their requests. This framework portrays the LGBTQ+ community as oppressors and obsessed advocates that can never “seem to get enough” (ex: more than just marriage). The bans are framed as the neutralizing factor to the catering.
• LGBTQ+ children and topics in academic and social spaces are the extreme degree.
When it comes to LGBTQ+ issues and conversations as they revolve around children, media outlets have some of the most passionate opinions about them. Often portrayed as “the line that shouldn’t be crossed,” LGBTQ+ issues, as they find themselves in schools and other spaces, are thus portrayed as bearable to a certain degree, never completely. Claims of indoctrination are also presented prominently even when institutional efforts are to protect LGBTQ+ kids.
ContributorsNieto Calderon, Ramon Antonio (Author) / Himberg, Julia (Thesis director) / Sturges, Robert (Committee member) / Department of English (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Waltz, is a collection of poems written to play along the boundaries between sound, language, and meaning. As a vehicle for exploration, the poems in Waltz, commandeer themes of nostalgia, love, loss, and abstraction, all of which build up and break each other down to create something of a nonlinear narrative, and concomitant sketch of the poet.
ContributorsAieta, Joseph (Author) / Ball, Sally (Thesis director) / Liston, Chelsea (Committee member) / Department of English (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The primary purpose of this thesis is two-fold: (1) to understand the resources presently available for Native American college student leaders at Predominantly White institutions (PWIs), and; (2) to consider ways to develop their leadership abilities and knowledge of how experience with college leadership contributes to becoming successful leaders with/in their Indigenous communities. The secondary purpose of this thesis is to propose additional resources for PWIs that can inform Native American leadership practices across academic disciplines and fields through the creation of the Indigenous & Innovative Leadership course syllabus and conference. This Honor's Thesis Project begins by exploring leadership development opportunities for Native American undergraduate students at Arizona State University, a predominantly White institution. Also explored are conceptions of Indigenous leadership as it applies to engagement in or with on-campus student organizations, tribal governments, and within surrounding Indigenous communities. This project has implications for thinking about American Indian student success beyond graduation and the role leadership and organization development has for the success of tribal communities.
ContributorsTom, Megan Joyce (Author) / Brayboy, Bryan (Thesis director) / Solyom, Jessica (Committee member) / Department of English (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Genocide studies have traditionally focused on the perpetrator’s intent to eradicate a particular identity-based group, using the Holocaust as their model and point of comparison. Although some aspects of the Holocaust were undoubtedly unique, recent scholars have sought to challenge the notion that it was a singular phenomenon. Instead, they draw attention to a recurring pattern of genocidal events throughout history by shifting the focus from intent to structure. One particular branch of scholars seeks to connect the ideology and tactics of imperialism with certain genocidal events. These anti-imperialist genocide scholars concede that their model cannot account for all genocides, but still claim that it creates meaningful connections between genocides committed by Western colonialist powers and those that have occurred in a neoimperialist world order shaped according to Western interests. The latter includes genocides in postcolonial states, which these scholars believe were shaped by the scars of their colonial past, as well as genocides in which imperial hegemons assisted local perpetrators. Imperialist and former colonial powers have contributed meaningfully to all of these kinds of genocides, yet their contributions have largely been ignored due to their own influence on the creation of the current international order. Incorporating the anti-imperialist perspective into the core doctrine of genocide studies may lead to breakthroughs in areas of related policy and practice, such as prevention and accountability.
ContributorsParker, Ashleigh Mae (Author) / Thies, Cameron (Thesis director) / Sivak, Henry (Committee member) / School of Politics and Global Studies (Contributor) / School of Social Transformation (Contributor) / Department of English (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05