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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
This paper will be exploring a marketing plan for a Kpop Fan artist, Jennifer Lee. Kpop is a genre of music originating from South Korea that provides a whole-package entertainment. Fan artists are producers who create produce for the consumption and purchase of other Kpop fans. The paper will consider segmentation and the products and platforms that best target them in order to maximize revenue. A survey was performed with a sample size of 314 participants to find out consumer behavior and preference as well as producer situation. Consumers come from both the United States and abroad. Customers come directly and almost exclusively from followers. Therefore, increasing the number of followers on Instagram is essential to increasing revenue. Jennifer has time, resource, and ability constraints, while the market has limited potential. The conclusion is that Jennifer should become more organized as a business. To grow her following, she should cater more towards the most popular fandoms (BTS), make art tutorials, consider collaborations, and better inform followers of her products/services available for purchase. The social media platforms key to marketing Jennifer's products are Instagram and Twitter. Other platforms to be used to increase exposure are Tumblr, Amino Apps, DeviantArt, Reddit, and YouTube. She must also declutter all of these virtual storefronts of unnecessary content to varying degrees in order to build ease of access and a trustworthy brand image. The best platforms for transaction is a personal store, RedBubble (a website that allows users to sell a variety of products with their uploaded images printed onto them), Patreon, and in-person at conventions.
ContributorsXu, Everest Christine (Author) / Eaton, Kathryn (Thesis director) / Ingram-Waters, Mary (Committee member) / Department of Marketing (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The advertising agency, in its variety of forms, is one of the most powerful forces in the modern world. Its products are seen globally through various multimedia outlets and they strongly impact culture and economy. Since its conception in 1843 by Volney Palmer, the advertising agency has evolved into the recognizable—and unrecognizable—firms scattered around the world today. In the United States alone, there are roughly 13.4 thousand agencies, many of which also have branches in other countries. The evolution of the modern advertising agency coincided with, and even preceded, some of the major inflection points in history. Understanding how and why changes in advertising agencies affected these inflection points provides a glimpse of understanding into the relationship between advertising, business, and societal values.
In the pages ahead we will explore the future of the advertising industry. We will analyze our research to uncover the underlying trends pointing towards what is to come and work to apply those explanations to our understanding of advertising in the future.
In the pages ahead we will explore the future of the advertising industry. We will analyze our research to uncover the underlying trends pointing towards what is to come and work to apply those explanations to our understanding of advertising in the future.
ContributorsHarris, Chase (Co-author) / Potthoff, Zachary (Co-author) / Gray, Nancy (Thesis director) / Samper, Adriana (Committee member) / Department of Information Systems (Contributor) / Department of Marketing (Contributor) / Herberger Institute for Design and the Arts (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The goal of our project was to determine how to create the most marketable hockey team. To do this, consumer needs, team psychology, and financing were all researched and evaluated. With this information, a business plan was designed around the next NHL expansion team. Two surveys, one for marketing distributed to the general public, and one for team psychology distributed to current and former hockey players were created and sent out, while data for the financing aspect was collected by comparing data from other NHL teams and franchises from different sports. In terms of financials, this comes in lower than average ticket prices, a nice and expensive stadium, the ideal city to generate capital, and sufficient money spent on advertising. Our ticket prices of $140 is based on having a low enough price to generate lots of demand while high enough to make a profit. The $600 million stadium (which will be fully funded) will surely draw a significant crowd. Choosing Seattle as a city is the most ideal to meet these goals and lastly, in meeting with an NHL GM, we determined $4 million in yearly advertising costs as sufficient in creating the most marketable team. Throughout this whole process, we remained data focus. We focused on data from a customized marketing survey, organizational structures, salary cap, and attendance. What our marketing survey results showed us is that our potential fans wanted three characteristics in a hockey team: speed, intensity, and scoring. In looking at organizational structures teams that exemplified these characteristics had a heavy emphasis on development and scouting. So we built our organizational tree around those two ideals. We hired GM Mike Futa, a current director of player personnel for the L.A. Kings, and Head Coach Adam Oates, a current skills development coach for top players to bring those ideals to fruition. In constructing our team we replicated the rules set forth for the Vegas Knights' expansion draft and hypothesized a likely protected list based off of last years lists. As a result we were able to construct a team that statistically out performed the Vegas Knights draft numbers by double, in goals, assists, and points, while also beating them in PIM. Based off of these numbers and an analysis of how goals translate into game attendance we are confident that we have constructed a team that has the highest potential for marketability. For the team psychology area, when creating a roster and scouting players, some of our main findings were that it is important to pursue players who get along well with their teammates and coaching staff, are aggressive, are leaders on the team, and are vocal players who communicate effectively. We also recommended avoiding players who significantly portrayed any "pet-peeve" traits, with the most emphasis placed on "disrespectful toward teammates," and the least emphasis placed on "over-aggression." By following all of these recommendations, we believe the most marketable hockey team possible can be created.
ContributorsQuinn, Colin Christopher (Co-author) / Spigel, Carlos (Co-author) / Meyer, Matt (Co-author) / Eaton, John (Thesis director) / McIntosh, Daniel (Committee member) / Department of Marketing (Contributor) / Sandra Day O'Connor College of Law (Contributor) / Department of Management and Entrepreneurship (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The town of Guadalupe, Arizona has a long history of divided residents and high poverty rates. The high levels of poverty in the town can be attributed to numerous factors, most notably high rates of drug abuse, low high school graduation rates, and teen pregnancy. The town has named one of its most pressing issues of today to be youth disengagement. There are currently a handful of residents and community members passionate about finding a solution to this issue. After working with Guadalupe's Ending Hunger Task Force and resident youth, I set out to create a program design for a Guadalupe Youth Council. This council will contribute to combating youth disengagement. The program design will assist the task force in creating a standing youth council and deciding on the structure and role the council has in the town. I will offer learning outcomes and suggestions to the Task Force, youth council staff, and the youth of the youth council. This study contains an analysis of relevant literature, youth focus group results and data, and how the information gathered has contributed to the design of the youth council. The results of this study contain recommendations about four themes within the program design of a youth council: size, recruitment, activities and engagement, and adult support. The results also explore how the youth council will impact the power, policy, and behavior of Guadalupe youth.
ContributorsBalderas, Erica Theresa (Author) / Wang, Lili (Thesis director) / Avalos, Francisco (Committee member) / School of Community Resources and Development (Contributor) / Department of Information Systems (Contributor) / W.P. Carey School of Business (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05