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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
Breast microcalcifications are a potential indicator of cancerous tumors. Current visualization methods are either uncomfortable or impractical. Impedance measurement studies have been performed, but not in a clinical setting due to a low sensitivity and specificity. We are hoping to overcome this challenge with the development of a highly accurate impedance probe on a biopsy needle. With this technique, microcalcifications and the surrounding tissue could be differentiated in an efficient and comfortable manner than current techniques for biopsy procedures. We have developed and tested a functioning prototype for a biopsy needle using bioimpedance sensors to detect microcalcifications in the human body. In the final prototype a waveform generator sends a sin wave at a relatively low frequency(<1KHz) into the pre-amplifier, which both stabilizes and amplifies the signal. A modified howland bridge is then used to achieve a steady AC current through the electrodes. The voltage difference across the electrodes is then used to calculate the impedance being experienced between the electrodes. In our testing, the microcalcifications we are looking for have a noticeably higher impedance than the surrounding breast tissue, this spike in impedance is used to signal the presence of the calcifications, which are then sampled for examination by radiology.
ContributorsWen, Robert Bobby (Co-author) / Grula, Adam (Co-author) / Vergara, Marvin (Co-author) / Ramkumar, Shreya (Co-author) / Kozicki, Michael (Thesis director) / Ranjani, Kumaran (Committee member) / School of Molecular Sciences (Contributor) / Electrical Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Breastfeeding has been shown by a number of studies to have numerous benefits on both the mother and the infant. Major health organizations such as the World Health Organization (WHO), now agree that breastfeeding should be encouraged and supported in all countries. But like many things, the wheels of the law are slow to catch up with scientific evident. Although breastfeeding is supported, working women do not have the option of breastfeeding without consequences. For example, in 2003, Kirstie Marshall, a then member of parliament in Australia was ejected from the lower house chamber on February 23, for breastfeeding her baby [3]. According to standing order 30 at the time, "Unless by order of the House, no Member of this House shall presume to bring any stranger into any part of the House appropriated to the Members of this House while the House, or a Committee of the whole House, is sitting" [3]. The rules did not specify the age of strangers, so the then 11-day-old baby, Charlotte Louise and her mother were shown the exit door of parliament. She had to choose between being present at times of major discussions or leaving the house to breastfeed her child, she chose to leave. More recent statistics show that developed nations like the US and Australia which also have high rates of women employment had low rates of breastfeeding. This might mean that workplace policies do not favor breastfeeding or expressing milk at work. Fortunately, laws have since been introduced in both the United States and Australia that support breastfeeding at the workplace. The next step would be to access how these laws affect breastfeeding statistics and how variation between these two countries like the paid parental leave in Australia (which is not present in all US states) would affect these numbers.
ContributorsSakala, Lydia (Author) / Alison, Alison (Thesis director) / Reddy, Swapna (Committee member) / School of Molecular Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Ketone bodies are produced in the liver from the acetyl CoA derived from fatty acids that cannot enter the Krebs cycle. This is a sub-analysis of a larger study which had numerous outcome markers. This analysis focuses on the relationship between ketone blood levels and cognition. The study looked at the relationship between Time Restricted Feeding (TRF), a method of intermittent fasting. TRF is something that can be easily adapted into an individual’s lifestyle and has been shown to have multiple advantages. This 8-week study began with 23 enrolled participants, but due to COVID-19 only 11 participants could be tested for cognition and blood ketone levels after week 4. All participants had similar ranges of weight, height, age, BMI, hip, and waist measurements at baseline. Moreover, these demographic variables were not related to ketone levels or cognition. The data indicate that ketone bodies increased in participants practicing TRF and that the increase in ketone bodies in the blood, specifically β-hydroxybutyrate was strongly correlated to increased cognitive function. This is consistent with theories that elevated ketone levels allowed for early hunter-gather communities and other mammals to survive prolonged periods of nutrient deprivation while keeping high cognitive function.
ContributorsTaha, Basel Mahmoud (Author) / Johnston, Carol (Thesis director) / Karen, Sweazea (Committee member) / School of Molecular Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
The purpose of this thesis experiment was to design and create an Acoustically Active Cannula (AAC), which is furnished by a piezoelectric crystal placed at its tip that produces an acoustic navigation signal. I tested the functionality of the cannula in vitro and demonstrated its navigational abilities in vivo in anesthetized pigs. This experiment was based upon ultrasound science and technology, and thus some practical experience with conventional (B-mode) and Doppler ultrasound was achieved as well. The results of bench and experimental animal studies indicated proper functionality of the AAC for identification and spatial navigation of its tip with color Doppler ultrasound imaging.
ContributorsShamsa, Kayvan (Author) / Tyler, William (Thesis director) / Belohlavek, Marek (Committee member) / School of Molecular Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05