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Description
Research on combinatorics education is sparse when compared with other fields in mathematics education. This research attempted to contribute to the dearth of literature by examining students' reasoning about enumerative combinatorics problems and how students conceptualize the set of elements being counted in such problems, called the solution set. In particular, the focus was on the stable patterns of reasoning, known as ways of thinking, which students applied in a variety of combinatorial situations and tasks. This study catalogued students' ways of thinking about solution sets as they progressed through an instructional sequence. In addition, the relationships between the catalogued ways of thinking were explored. Further, the study investigated the challenges students experienced as they interacted with the tasks and instructional interventions, and how students' ways of thinking evolved as these challenges were overcome. Finally, it examined the role of instruction in guiding students to develop and extend their ways of thinking. Two pairs of undergraduate students with no formal experience with combinatorics participated in one of the two consecutive teaching experiments conducted in Spring 2012. Many ways of thinking emerged through the grounded theory analysis of the data, but only eight were identified as robust. These robust ways of thinking were classified into three categories: Subsets, Odometer, and Problem Posing. The Subsets category encompasses two ways of thinking, both of which ultimately involve envisioning the solution set as the union of subsets. The three ways of thinking in Odometer category involve holding an item or a set of items constant and systematically varying the other items involved in the counting process. The ways of thinking belonging to Problem Posing category involve spontaneously posing new, related combinatorics problems and finding relationships between the solution sets of the original and the new problem. The evolution of students' ways of thinking in the Problem Posing category was analyzed. This entailed examining the perturbation experienced by students and the resulting accommodation of their thinking. It was found that such perturbation and its resolution was often the result of an instructional intervention. Implications for teaching practice are discussed.
ContributorsHalani, Aviva (Author) / Roh, Kyeong Hah (Thesis advisor) / Fishel, Susanna (Committee member) / Saldanha, Luis (Committee member) / Thompson, Patrick (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2013
DescriptionAt its core, Leaning finds profound significance in unlikely moments of intimate
detail; the upkeep of a brother's gravesite, for example, is as quietly important as rummaging through a collection of sex toys. Haiku-like in their simplicity, meditation, and declaration, these poems give meaning to the smallness of our world.
detail; the upkeep of a brother's gravesite, for example, is as quietly important as rummaging through a collection of sex toys. Haiku-like in their simplicity, meditation, and declaration, these poems give meaning to the smallness of our world.
ContributorsBender, Brian, M.F.A (Author) / Ball, Sara (Thesis advisor) / Hummer, Terry (Committee member) / Dubie, Jr., Norman (Committee member) / Arizona State University (Publisher)
Created2015
Description
This dissertation report follows a three-paper format, with each paper having a different but related focus. In Paper 1 I discuss conceptual analysis of mathematical ideas relative to its place within cognitive learning theories and research studies. In particular, I highlight specific ways mathematics education research uses conceptual analysis and discuss the implications of these uses for interpreting and leveraging results to produce empirically tested learning trajectories. From my summary and analysis I develop two recommendations for the cognitive researchers developing empirically supported learning trajectories. (1) A researcher should frame his/her work, and analyze others’ work, within the researcher’s image of a broadly coherent trajectory for student learning and (2) that the field should work towards a common understanding for the meaning of a hypothetical learning trajectory.
In Paper 2 I argue that prior research in online learning has tested the impact of online courses on measures such as student retention rates, satisfaction scores, and GPA but that research is needed to describe the meanings students construct for mathematical ideas researchers have identified as critical to their success in future math courses and other STEM fields. This paper discusses the need for a new focus in studying online mathematics learning and calls for cognitive researchers to begin developing a productive methodology for examining the meanings students construct while engaged in online lessons.
Paper 3 describes the online Precalculus course intervention we designed around measurement imagery and quantitative reasoning as themes that unite topics across units. I report results relative to the meanings students developed for exponential functions and related ideas (such as percent change and growth factors) while working through lessons in the intervention. I provide a conceptual analysis guiding its design and discuss pre-test and pre-interview results, post-test and post-interview results, and observations from student behaviors while interacting with lessons. I demonstrate that the targeted meanings can be productive for students, show common unproductive meanings students possess as they enter Precalculus, highlight challenges and opportunities in teaching and learning in the online environment, and discuss needed adaptations to the intervention and future research opportunities informed by my results.
In Paper 2 I argue that prior research in online learning has tested the impact of online courses on measures such as student retention rates, satisfaction scores, and GPA but that research is needed to describe the meanings students construct for mathematical ideas researchers have identified as critical to their success in future math courses and other STEM fields. This paper discusses the need for a new focus in studying online mathematics learning and calls for cognitive researchers to begin developing a productive methodology for examining the meanings students construct while engaged in online lessons.
Paper 3 describes the online Precalculus course intervention we designed around measurement imagery and quantitative reasoning as themes that unite topics across units. I report results relative to the meanings students developed for exponential functions and related ideas (such as percent change and growth factors) while working through lessons in the intervention. I provide a conceptual analysis guiding its design and discuss pre-test and pre-interview results, post-test and post-interview results, and observations from student behaviors while interacting with lessons. I demonstrate that the targeted meanings can be productive for students, show common unproductive meanings students possess as they enter Precalculus, highlight challenges and opportunities in teaching and learning in the online environment, and discuss needed adaptations to the intervention and future research opportunities informed by my results.
ContributorsO'Bryan, Alan Eugene (Author) / Carlson, Marilyn P (Thesis advisor) / Thompson, Patrick W (Committee member) / Milner, Fabio (Committee member) / Roh, Kyeong Hah (Committee member) / Tallman, Michael (Committee member) / Arizona State University (Publisher)
Created2018
Description
This research compares shifts in a SuperSpec titanium nitride (TiN) kinetic inductance detector's (KID's) resonant frequency with accepted models for other KIDs. SuperSpec, which is being developed at the University of Colorado Boulder, is an on-chip spectrometer designed with a multiplexed readout with multiple KIDs that is set up for a broadband transmission of these measurements. It is useful for detecting radiation in the mm and sub mm wavelengths which is significant since absorption and reemission of photons by dust causes radiation from distant objects to reach us in infrared and far-infrared bands. In preparation for testing, our team installed stages designed previously by Paul Abers and his group into our cryostat and designed and installed other parts necessary for the cryostat to be able to test devices on the 250 mK stage. This work included the design and construction of additional parts, a new setup for the wiring in the cryostat, the assembly, testing, and installation of several stainless steel coaxial cables for the measurements through the devices, and other cryogenic and low pressure considerations. The SuperSpec KID was successfully tested on this 250 mK stage thus confirming that the new setup is functional. Our results are in agreement with existing models which suggest that the breaking of cooper pairs in the detector's superconductor which occurs in response to temperature, optical load, and readout power will decrease the resonant frequencies. A negative linear relationship in our results appears, as expected, since the parameters are varied only slightly so that a linear approximation is appropriate. We compared the rate at which the resonant frequency responded to temperature and found it to be close to the expected value.
ContributorsDiaz, Heriberto Chacon (Author) / Mauskopf, Philip (Thesis director) / McCartney, Martha (Committee member) / Department of Physics (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
This paper considers what factors influence student interest, motivation, and continued engagement. Studies show anticipated extrinsic rewards for activity participation have been shown to reduce intrinsic value for that activity. This might suggest that grade point average (GPA) has a similar effect on academic interests. Further, when incentives such as scholarships, internships, and careers are GPA-oriented, students must adopt performance goals in courses to guarantee success. However, performance goals have not been shown to correlated with continued interest in a topic. Current literature proposes that student involvement in extracurricular activities, focused study groups, and mentored research are crucial to student success. Further, students may express either a fixed or growth mindset, which influences their approach to challenges and opportunities for growth. The purpose of this study was to collect individual cases of students' experiences in college. The interview method was chosen to collect complex information that could not be gathered from standard surveys. To accomplish this, questions were developed based on content areas related to education and motivation theory. The content areas included activities and meaning, motivation, vision, and personal development. The developed interview method relied on broad questions that would be followed by specific "probing" questions. We hypothesize that this would result in participant-led discussions and unique narratives from the participant. Initial findings suggest that some of the questions were effective in eliciting detailed responses, though results were dependent on the interviewer. From the interviews we find that students value their group involvements, leadership opportunities, and relationships with mentors, which parallels results found in other studies.
ContributorsAbrams, Sara (Author) / Hartwell, Lee (Thesis director) / Correa, Kevin (Committee member) / Department of Psychology (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
This study estimates the capitalization effect of golf courses in Maricopa County using the hedonic pricing method. It draws upon a dataset of 574,989 residential transactions from 2000 to 2006 to examine how the aesthetic, non-golf benefits of golf courses capitalize across a gradient of proximity measures. The measures for amenity value extend beyond home adjacency and include considerations for homes within a range of discrete walkability buffers of golf courses. The models also distinguish between public and private golf courses as a proxy for the level of golf course access perceived by non-golfers. Unobserved spatial characteristics of the neighborhoods around golf courses are controlled for by increasing the extent of spatial fixed effects from city, to census tract, and finally to 2000 meter golf course ‘neighborhoods.’ The estimation results support two primary conclusions. First, golf course proximity is found to be highly valued for adjacent homes and homes up to 50 meters way from a course, still evident but minimal between 50 and 150 meters, and insignificant at all other distance ranges. Second, private golf courses do not command a higher proximity premia compared to public courses with the exception of homes within 25 to 50 meters of a course, indicating that the non-golf benefits of courses capitalize similarly, regardless of course type. The results of this study motivate further investigation into golf course features that signal access or add value to homes in the range of capitalization, particularly for near-adjacent homes between 50 and 150 meters thought previously not to capitalize.
ContributorsJoiner, Emily (Author) / Abbott, Joshua (Thesis director) / Smith, Kerry (Committee member) / Economics Program in CLAS (Contributor) / School of Sustainability (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The objective of this paper is to provide an educational diagnostic into the technology of blockchain and its application for the supply chain. Education on the topic is important to prevent misinformation on the capabilities of blockchain. Blockchain as a new technology can be confusing to grasp given the wide possibilities it can provide. This can convolute the topic by being too broad when defined. Instead, the focus will be maintained on explaining the technical details about how and why this technology works in improving the supply chain. The scope of explanation will not be limited to the solutions, but will also detail current problems. Both public and private blockchain networks will be explained and solutions they provide in supply chains. In addition, other non-blockchain systems will be described that provide important pieces in supply chain operations that blockchain cannot provide. Blockchain when applied to the supply chain provides improved consumer transparency, management of resources, logistics, trade finance, and liquidity.
ContributorsKrukar, Joel Michael (Author) / Oke, Adegoke (Thesis director) / Duarte, Brett (Committee member) / Hahn, Richard (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Department of Economics (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
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
The Super Catalan numbers are a known set of numbers which have so far eluded a combinatorial interpretation. Several weighted interpretations have appeared since their discovery, one of which was discovered by William Kuszmaul in 2017. In this paper, we connect the weighted Super Catalan structure created previously by Kuszmaul and a natural $q$-analogue of the Super Catalan numbers. We do this by creating a statistic $\sigma$ for which the $q$ Super Catalan numbers, $S_q(m,n)=\sum_X (-1)^{\mu(X)} q^{\sigma(X)}$. In doing so, we take a step towards finding a strict combinatorial interpretation for the Super Catalan numbers.
ContributorsHouse, John Douglas (Author) / Fishel, Susanna (Thesis director) / Childress, Nancy (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05