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- Member of: Theses and Dissertations
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 honors thesis outlines a method for teaching argument writing in the secondary classroom, including the elements of an argument based upon the Toulmin method, and diverse ways to help students who are all types of learners become engaged and receive the support they need. It includes all elements of argument, including evidence, warrants, backing, counterargument, claims, theses, the rhetorical triangle and the rhetorical appeals, including definitions and how they fit together in an argumentative essay. The largest portion of the project is dedicated to activities and resources for teachers based upon all of those elements, along with activities for the writing process as a whole. These activities are based upon the student's individual experience as well as various scholarly resources from leading professionals in the curriculum development field for English Language Arts. This is not meant to be an end-all be-all solution for teaching argument writing, but rather one of many resources that teachers can use in their classroom. This 30-page paper, including references, are condensed into an accessible website for teachers to use more easily. Each tab on the website refers to a different element or focus of the argument writing process, with both a definition and introduction as well as one or more activities for teachers to implement into the classroom. The activities are versatile and general for the purpose of teachers being able to include them into whatever curriculum they are currently teaching. The goal is that they can add argument instruction into what they are already either willingly or being required to teach in an easy and logical way. The website is available for any secondary teachers to use as they see fit at www.teachingargumentwriting.weebly.com.
ContributorsBrooks, Jenna Nicole (Author) / Blasingame, James (Thesis director) / Barnett, Juliet (Committee member) / Division of Teacher Preparation (Contributor) / School of Film, Dance and Theatre (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
DescriptionThis project is designed to generate enthusiasm for science among refugee students in hopes of inspiring them to continue learning science as well as to help them with their current understanding of their school science subject matter.
ContributorsSipes, Shannon Paige (Author) / O'Flaherty, Katherine (Thesis director) / Gregg, George (Committee member) / School of Molecular Sciences (Contributor) / Division of Teacher Preparation (Contributor) / Barrett, The Honors College (Contributor)
Created2017-12
Description
This creative project explores the concept of how music is like a language and how, as a teacher, I plan to enforce this concept through my teaching. The aim of this project is to highlight the importance of completing research and acquiring knowledge of aspects, such as the composer's life, historical background and literary references, when learning a piece of music. Through this project, I address connections between the brain and music pertaining to memorization, the components of language, the similarities between language and music, the role of the teacher and the development of a "toolbox" of knowledge for studying a piece of music. I present my own research on Schumann's Novelette Op. 21, No. 8 in f-sharp minor as well as my own experiences of learning the piece to demonstrate an example of the process and discoveries I hope my students will make in their own studies of repertoire.
ContributorsLee, Kara Jia-Shan (Author) / Pagano, Caio (Thesis director) / Creviston, Hannah (Committee member) / Division of Teacher Preparation (Contributor) / School of Music (Contributor) / Barrett, The Honors College (Contributor)
Created2017-12
Description
The SolarSPELL is an offline, ruggedized, digital library, created by Dr. Laura Hosman for the U.S. Peace Corps. It has thousands of pieces of educational content that can be accessed through a self-contained Wi-Fi hotspot on the device itself. Currently, there are more than 200 deployed in several Pacific Island nations. After visiting one of these nations, Tonga, in December of 2016, I learned that almost all of the Peace Corps volunteers stationed around the Pacific Islands suffered from a lack of resources due to a variety of reasons. While the SolarSPELL helps to remedy that, the device is lacking classroom activities and resources for creative work and educational drama. Furthermore, for many students in these environments, schools are for learning information and producing high scores on exams, not for learning about creative strengths and identity. After researching curriculum development and the use of drama in an educational setting, I compiled over 50 pieces of content to include on the SolarSPELL involving art, drama, music, movement, and most importantly, imagination. These resources will allow Peace Corps volunteers to explore additional ways to teach English in their schools, while also creating a classroom environment that allows for creative expression. All the content is compiled into one folder as "Teaching Resources", and is then broken down into seven sub- categories. In the first sub-category, Art Projects, there is a collection of several hands-on projects, many of which involve recyclable or readily available materials. These projects will allow for a greater understanding of conservation and "green" living, concepts that are crucial to the stability of these island nations. The next 5 categories are Drama Readings, Music, Movement, and Video, Group Exercises, Creative Writing, and Worksheets. The second sub- category is a collection of beginner-level "Reader's Theater" scripts. The third sub-category involves music and video to engage students in movement activities. The fourth sub-category is a compilation of group games and activities to help students express themselves and learn social skills. The fifth sub-category includes a collection of activities such as fill-in-the-blank story worksheets and journal prompts which will aid in creative thinking and the practice of the English language. The sixth sub-category involves a collection of worksheets that mainly focus on self-reflection and identity. The seventh and final sub-category, Content Guide and Information, works to explain the benefits of using of drama and creative play in the classroom, as well as strategies teachers can implement in order to further engage their students in dramatic learning and play. Overall, these pieces of content are meant to be used as resources for the Peace Corps volunteers in order to provide alternative ways to practice reading, writing, and speaking the English language, a critical part of education in the Pacific Islands.
ContributorsTaylor, Amanda Nicole (Author) / Hosman, Laura (Thesis director) / McAvoy, Mary (Committee member) / School of Film, Dance and Theatre (Contributor) / Division of Teacher Preparation (Contributor) / Barrett, The Honors College (Contributor) / School for the Future of Innovation in Society (Contributor)
Created2017-12