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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
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
Trichloroethene (TCE) is a ubiquitous soil and groundwater contaminant. The most common bioremediation approach for TCE relies on the process of reductive dechlorination by Dehalococcoides mccartyi. D. mccartyi use TCE, dichloroethene, and vinyl chloride as electron acceptors and hydrogen as an electron donor. At contaminated sites, reductive dechlorination is typically promoted by adding a fermentable substrate, which is broken down to short chain fatty acids, simple alcohols, and hydrogen. This study explored microbial chain elongation (MCE), instead of fermentation, to promote TCE reductive dechlorination. In MCE, microbes use simple substrates (e.g., acetate, ethanol) to build medium chain fatty acids and also produce hydrogen during this process. Soil microcosm using TCE and acetate and ethanol as MCE substrates were established under anaerobic conditions. In soil microcosms with synthetic groundwater and natural groundwater, ethene was the main product from TCE reductive dechlorination and butyrate and hydrogen were the main products from MCE. Transfer microcosms using TCE and either acetate and ethanol, ethanol, or acetate were also established. The transfers with TCE and ethanol showed the faster rates of reductive dechlorination and produced more elongated products (i.e., hexanoate). The microbial groups enriched in the soil microcosms likely responsible for chain elongation were most similar to Clostridium genus. These investigations showed the potential for synergistic microbial chain elongation and reductive dechlorination of chlorinated ethenes.
ContributorsRobles, Aide (Author) / Delgado, Anca G. (Thesis advisor) / Torres, Cesar I. (Committee member) / van Paassen, Leon (Committee member) / Arizona State University (Publisher)
Created2019
Description
This research explores microbial chain elongation as a pathway for production of complex organic compounds in soils with implication for the carbon cycle. In chain elongation, simple substrates such as ethanol and short chain carboxylates such as acetate can be converted to longer carbon chain carboxylates under anaerobic conditions through cyclic, reverse β oxidation. This pathway elongates the carboxylate by two carbons. The chain elongation process is overall thermodynamically feasible, and microorganisms gain energy through this process. There have been limited insights into the versatility of chain elongating substrates, understanding the chain elongating microbial community, and its importance in sequestering carbon in the soils.
We used ethanol, methanol, butanol, and hydrogen as electron donors and acetate and propionate as electron acceptors to test the occurrence of microbial chain elongation in four soils with different physicochemical properties and microbial communities. Common chain elongation products were the even numbered chains butyrate, caproate, and butanol, the odd numbered carboxylates valerate and heptanoate, along with molecular hydrogen. At a near neutral pH and mesophilic temperature, we observed a stable and sustained production of longer fatty acids along with hydrogen. Microbial community analysis show phylotypes from families such as Clostridiaceae, Bacillaceae, and Ruminococcaceae in all tested conditions. Through chain elongation, the products formed are less biodegradable. They may undergo transformations and end up as organic carbon, decreasing the greenhouse gas emissions, thus, making this process important to study.
We used ethanol, methanol, butanol, and hydrogen as electron donors and acetate and propionate as electron acceptors to test the occurrence of microbial chain elongation in four soils with different physicochemical properties and microbial communities. Common chain elongation products were the even numbered chains butyrate, caproate, and butanol, the odd numbered carboxylates valerate and heptanoate, along with molecular hydrogen. At a near neutral pH and mesophilic temperature, we observed a stable and sustained production of longer fatty acids along with hydrogen. Microbial community analysis show phylotypes from families such as Clostridiaceae, Bacillaceae, and Ruminococcaceae in all tested conditions. Through chain elongation, the products formed are less biodegradable. They may undergo transformations and end up as organic carbon, decreasing the greenhouse gas emissions, thus, making this process important to study.
ContributorsJoshi, Sayalee (Author) / Delgado, Anca G (Thesis advisor) / Torres, César I (Committee member) / van Paassen, Leon (Committee member) / Arizona State University (Publisher)
Created2018
Description
Laterally-loaded short rigid drilled shaft foundations are the primary foundation used within the electric power transmission line industry. Performance of these laterally loaded foundations is dependent on modulus of the subsurface, which is directly measured by the Pressuremeter (PMT). The PMT test provides the lateral shear modulus at intermediate strains, an equivalent elastic modulus for lateral loading, which mimics the reaction of transmission line foundations within the elastic range of motion. The PMT test, however, is expensive to conduct and rarely performed. Correlations of PMT to blow counts and other index properties have been developed but these correlations have high variability and may result in unconservative foundation design. Variability in correlations is due, in part, because difference of the direction of the applied load and strain level between the correlated properties and the PMT. The geophysical shear wave velocity (S-wave velocity) as measured through refraction microtremor (ReMi) methods can be used as a measure of the small strain, shear modulus in the lateral direction. In theory, the intermediate strain modulus of the PMT is proportional to the small strain modulus of S-wave velocity. A correlation between intermediate strain and low strain moduli is developed here, based on geophysical surveys conducted at fourteen previous PMT testing locations throughout the Sonoran Desert of central Arizona. Additionally, seasonal variability in S-wave velocity of unsaturated soils is explored and impacts are identified for the use of the PMT correlation in transmission line foundation design.
ContributorsEvans, Ashley Elizabeth (Author) / Houston, Sandra (Thesis advisor) / Zapata, Claudia (Thesis advisor) / van Paassen, Leon (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 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
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
Description
Construction is a defining characteristic of geometry classes. In a traditional classroom, teachers and students use physical tools (i.e. a compass and straight-edge) in their constructions. However, with modern technology, construction is possible through the use of digital applications such as GeoGebra and Geometer’s SketchPad.
Many other studies have researched the benefits of digital manipulatives and digital environments through student completion of tasks and testing. This study intends to research students’ use of the digital tools and manipulatives, along with the students’ interactions with the digital environment. To this end, I conducted exploratory teaching experiments with two calculus I students.
In the exploratory teaching experiments, students were introduced to a GeoGebra application developed by Fischer (2019), which includes instructional videos and corresponding quizzes, as well as exercises and interactive notepads, where students could use digital tools to construct line segments and circles (corresponding to the physical straight-edge and compass). The application built up the students’ foundational knowledge, culminating in the construction and verbal proof of Euclid’s Elements, Proposition 1 (Euclid, 1733).
The central findings of this thesis are the students’ interactions with the digital environment, with observed changes in their conceptions of radii and circles, and in their use of tools. The students were observed to have conceptions of radii as a process, a geometric shape, and a geometric object. I observed the students’ conceptions of a circle change from a geometric shape to a geometric object, and with that change, observed the students’ use of tools change from a measuring focus to a property focus.
I report a summary of the students’ work and classify their reasoning and actions into the above categories, and an analysis of how the digital environment impacts the students’ conceptions. I also briefly discuss the impact of the findings on pedagogy and future research.
Many other studies have researched the benefits of digital manipulatives and digital environments through student completion of tasks and testing. This study intends to research students’ use of the digital tools and manipulatives, along with the students’ interactions with the digital environment. To this end, I conducted exploratory teaching experiments with two calculus I students.
In the exploratory teaching experiments, students were introduced to a GeoGebra application developed by Fischer (2019), which includes instructional videos and corresponding quizzes, as well as exercises and interactive notepads, where students could use digital tools to construct line segments and circles (corresponding to the physical straight-edge and compass). The application built up the students’ foundational knowledge, culminating in the construction and verbal proof of Euclid’s Elements, Proposition 1 (Euclid, 1733).
The central findings of this thesis are the students’ interactions with the digital environment, with observed changes in their conceptions of radii and circles, and in their use of tools. The students were observed to have conceptions of radii as a process, a geometric shape, and a geometric object. I observed the students’ conceptions of a circle change from a geometric shape to a geometric object, and with that change, observed the students’ use of tools change from a measuring focus to a property focus.
I report a summary of the students’ work and classify their reasoning and actions into the above categories, and an analysis of how the digital environment impacts the students’ conceptions. I also briefly discuss the impact of the findings on pedagogy and future research.
ContributorsSakauye, Noelle Marie (Author) / Roh, Kyeong Hah (Thesis director) / Zandieh, Michelle (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05