Matching Items (10)
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
Boron concentrations and isotopic composition of phlogopite mica, amphibole, and selected coexisting anhydrous phases in mantle-derived xenoliths from the Kaapvaal Craton were measured by secondary ion mass spectrometry in an effort to better understand the B isotope geochemistry of the subcontinental lithospheric mantle (SCLM) and its implications for the global geochemical cycle of B in the mantle. These samples display a wide, and previously unrecognized, range in their boron contents and isotopic compositions reflecting a complex history involving melt depletion and metasomatism by subduction- and plume-derived components, as well as late stage isotopic exchange related to kimberlite emplacements. Micas from ancient lithospheric harzburgite metasomatized by slab-derived fluids suggest extensive B-depletion during subduction, resulting in low-B, isotopically light compositions whereas kimberlite-related metasomatic products and a sample from the 2 Ga Palabora carbonatite have boron isotopic compositions similar to proposed primitive mantle. The results suggest that subduction of oceanic lithosphere plays a limited role in the B geochemistry of the convecting mantle.
ContributorsGuild, Meghan R (Author) / Hervig, Richard L (Thesis advisor) / Bell, David R. (Committee member) / Mcnamara, Allen (Committee member) / Arizona State University (Publisher)
Created2014
Description
This study contributes to the ongoing discussion of Mathematical Knowledge for Teaching (MKT). It investigates the case of Rico, a high school mathematics teacher who had become known to his colleagues and his students as a superbly effective mathematics teacher. His students not only developed excellent mathematical skills, they also developed deep understanding of the mathematics they learned. Moreover, Rico redesigned his curricula and instruction completely so that they provided a means of support for his students to learn mathematics the way he intended. The purpose of this study was to understand the sources of Rico's effectiveness. The data for this study was generated in three phases. Phase I included videos of Rico's lessons during one semester of an Algebra II course, post-lesson reflections, and Rico's self-constructed instructional materials. An analysis of Phase I data led to Phase II, which consisted of eight extensive stimulated-reflection interviews with Rico. Phase III consisted of a conceptual analysis of the prior phases with the aim of creating models of Rico's mathematical conceptions, his conceptions of his students' mathematical understandings, and his images of instruction and instructional design. Findings revealed that Rico had developed profound personal understandings, grounded in quantitative reasoning, of the mathematics that he taught, and profound pedagogical understandings that supported these very same ways of thinking in his students. Rico's redesign was driven by three factors: (1) the particular way in which Rico himself understood the mathematics he taught, (2) his reflective awareness of those ways of thinking, and (3) his ability to envision what students might learn from different instructional approaches. Rico always considered what someone might already need to understand in order to understand "this" in the way he was thinking of it, and how understanding "this" might help students understand related ideas or methods. Rico's continual reflection on the mathematics he knew so as to make it more coherent, and his continual orientation to imagining how these meanings might work for students' learning, made Rico's mathematics become a mathematics of students--impacting how he assessed his practice and engaging him in a continual process of developing MKT.
ContributorsLage Ramírez, Ana Elisa (Author) / Thompson, Patrick W. (Thesis advisor) / Carlson, Marilyn P. (Committee member) / Castillo-Chavez, Carlos (Committee member) / Saldanha, Luis (Committee member) / Middleton, James A. (Committee member) / Arizona State University (Publisher)
Created2011
Description
Type Ia supernovae are important, but mysterious cosmological tools. Their standard brightnesses have enabled cosmologists to measure extreme distances and to discover dark energy. However, the nature of their progenitor mechanisms remains elusive, with many competing models offering only partial clues to their origins. Here, type Ia supernova delay times are explored using analytical models. Combined with a new observation technique, this model places new constraints on the characteristic time delay between the formation of stars and the first type Ia supernovae. This derived delay time (500 million years) implies low-mass companions for single degenerate progenitor scenarios. In the latter portions of this dissertation, two progenitor mechanisms are simulated in detail; white dwarf collisions and mergers. From the first of these simulations, it is evident that white dwarf collisions offer a viable and unique pathway to producing type Ia supernovae. Many of the combinations of masses simulated produce sufficient quantities of 56Ni (up to 0.51 solar masses) to masquerade as normal type Ia supernovae. Other combinations of masses produce 56Ni yields that span the entire range of supernova brightnesses, from the very dim and underluminous, with 0.14 solar masses, to the over-bright and superluminous, with up to 1.71 solar masses. The 56Ni yield in the collision simulations depends non-linearly on total system mass, mass ratio, and impact parameter. Using the same numerical tools as in the collisions examination, white dwarf mergers are studied in detail. Nearly all of the simulations produce merger remnants consisting of a cold, degenerate core surrounded by a hot accretion disk. The properties of these disks have strong implications for various viscosity treatments that have attempted to pin down the accretion times. Some mass combinations produce super-Chandrasekhar cores on shorter time scales than viscosity driven accretion. A handful of simulations also exhibit helium detonations on the surface of the primary that bear a resemblance to helium novae. Finally, some of the preliminary groundwork that has been laid for constructing a new numerical tool is discussed. This new tool advances the merger simulations further than any research group has done before, and has the potential to answer some of the lingering questions that the merger study has uncovered. The results of thermal diffusion tests using this tool have a remarkable correspondence to analytical predictions.
ContributorsRaskin, Cody (Author) / Scannapieco, Evan (Thesis advisor) / Rhoads, James (Committee member) / Young, Patrick (Committee member) / Mcnamara, Allen (Committee member) / Timmes, Francis (Committee member) / Arizona State University (Publisher)
Created2011
Description
Stratification is a dominant feature of all planetary interiors. Fine-scale structure associated with layering, as well as heterogeneities hold important clues on a planet's compositional, thermal, and dynamical state, as well as its evolution. This research centers on using data from seismic arrays, networks of seismic sensors, and array processing methodologies to map the fine scale structure in the Earth's upper mantle and deep layering in the Moon - Earth and Moon are the only two planetary bodies with seismic available data for such analyses. Small-scale structure in the Earth's upper mantle can give rise to seismic wave scattering. I studied high frequency data from the Warramunga Array in Australia using array seismology. I developed and employed back-projection schemes to map the possible upper mantle scattering or reflection locations. Mapped scatterers show good correlation to strong lateral P-wave velocity gradients in tomography models and may be associated with the complex tectonic history beneath north of Australia. The minimum scale of scatterers relates to the seismic wavelength, which is roughly between 5 and 10 km in the upper mantle for the frequencies we study. The Apollo Passive Seismic Experiment (APSE) consisted of four 3-component seismometers deployed between 1969 and 1972 that continuously recorded lunar ground motion until late 1977. I studied the deep lunar interior with array methods applied to the legacy APSE dataset. The stack results suggest the presence of a solid inner and fluid outer core, overlain by a partially molten boundary layer, but their reflector impedance contrasts and reflector depths are not well constrained. With a rapidly increasing number of available modern broadband data, I developed a package, Discovery Using Ducttape Excessively (DUDE), to quickly generate plots for a comprehensive view of earthquake data. These plots facilitate discovery of unexpected phenomena. This dissertation identifies evidence for small-scale heterogeneities in Earth's upper mantle, and deeper lunar layering structure. Planetary interiors are complex with the heterogeneities on many scales, and discontinuities of variable character. This research demonstrates that seismic array methods are well-suited for interrogating heterogeneous phenomena, especially considering the recent rapid expansion of easily available dense network data.
ContributorsLin, Pei-Ying (Author) / Garnero, Edward J. (Thesis advisor) / Fouch, Matthew (Committee member) / Mcnamara, Allen (Committee member) / Sharp, Thomas (Committee member) / Tyburczy, James (Committee member) / Arizona State University (Publisher)
Created2011
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
This research investigates Earth structure in the core-mantle boundary (CMB) region, where the solid rocky mantle meets the molten iron alloy core. At long wavelengths, the lower mantle is characterized by two nearly antipodal large low shear velocity provinces (LLSVPs), one beneath the Pacific Ocean the other beneath Africa and the southern Atlantic Ocean. However, fine-scale LLSVP structure as well as its relationship with plate tectonics, mantle convection, hotspot volcanism, and Earth's outer core remains poorly understood. The recent dramatic increase in seismic data coverage due to the EarthScope experiment presents an unprecedented opportunity to utilize large concentrated datasets of seismic data to improve resolution of lowermost mantle structures. I developed an algorithm that identifies anomalously broadened seismic waveforms to locate sharp contrasts in shear velocity properties across the margins of the LLSVP beneath the Pacific. The result suggests that a nearly vertical mantle plume underlies Hawaii that originates from a peak of a chemically distinct reservoir at the base of the mantle, some 600-900 km above the CMB. Additionally, acute horizontal Vs variations across and within the northern margin of the LLSVP beneath the central Pacific Ocean are inferred from forward modeling of differential travel times between S (and Sdiff) and SKS, and also between ScS and S. I developed a new approach to expand the geographic detection of ultra-low velocity zones (ULVZs) with a new ScS stacking approach that simultaneously utilizes the pre- and post-cursor wavefield.. Strong lateral variations in ULVZ thicknesses and properties are found across the LLSVP margins, where ULVZs are thicker and stronger within the LLSVP than outside of it, consistent with convection model predictions. Differential travel times, amplitude ratios, and waveshapes of core waves SKKS and SKS are used to investigate CMB topography and outermost core velocity structure. 1D and 2D wavefield simulations suggest that the complicated geographic distribution of observed SKKS waveform anomalies might be a result of CMB topography and a higher velocity outermost core. These combined analyses depict a lowermost mantle that is rich in fine-scale structural complexity, which advances our understanding of its integral role in mantle circulation, mixing, and evolution.
ContributorsZhao, Chunpeng (Author) / Garnero, Edward J (Thesis advisor) / Mcnamara, Allen (Committee member) / Tyburczy, James (Committee member) / Fouch, Matthew (Committee member) / Sharp, Thomas (Committee member) / Arizona State University (Publisher)
Created2012
Description
Teachers must recognize the knowledge they possess as appropriate to employ in the process of achieving their goals and objectives in the context of practice. Such recognition is subject to a host of cognitive and affective processes that have thus far not been a central focus of research on teacher knowledge in mathematics education. To address this need, this dissertation study examined the role of a secondary mathematics teacher’s image of instructional constraints on his enacted subject matter knowledge. I collected data in three phases. First, I conducted a series of task-based clinical interviews that allowed me to construct a model of David’s mathematical knowledge of sine and cosine functions. Second, I conducted pre-lesson interviews, collected journal entries, and examined David’s instruction to characterize the mathematical knowledge he utilized in the context of designing and implementing lessons. Third, I conducted a series of semi-structured clinical interviews to identify the circumstances David appraised as constraints on his practice and to ascertain the role of these constraints on the quality of David’s enacted subject matter knowledge. My analysis revealed that although David possessed many productive ways of understanding that allowed him to engage students in meaningful learning experiences, I observed discrepancies between and within David’s mathematical knowledge and his enacted mathematical knowledge. These discrepancies were not occasioned by David’s active compensation for the circumstances and events he appraised as instructional constraints, but instead resulted from David possessing multiple schemes for particular ideas related to trigonometric functions, as well as from his unawareness of the mental actions and operations that comprised these often powerful but uncoordinated cognitive schemes. This lack of conscious awareness made David ill-equipped to define his instructional goals in terms of the mental activity in which he intended his students to engage, which further conditioned the circumstances and events he appraised as constraints on his practice. David’s image of instructional constraints therefore did not affect his enacted subject matter knowledge. Rather, characteristics of David’s subject matter knowledge, namely his uncoordinated cognitive schemes and his unawareness of the mental actions and operations that comprise them, affected his image of instructional constraints.
ContributorsTallman, Michael Anthony (Author) / Carlson, Marilyn P (Thesis advisor) / Thompson, Patrick W (Committee member) / Saldanha, Luis (Committee member) / Middleton, James (Committee member) / Harel, Guershon (Committee member) / Arizona State University (Publisher)
Created2015
Description
Silicic volcanoes produce many styles of activity over a range of timescales. Eruptions vary from slow effusion of viscous lava over many years to violent explosions lasting several hours. Hazards from these eruptions can be far-reaching and persistent, and are compounded by the dense populations often surrounding active volcanoes. I apply and develop satellite and ground-based remote sensing techniques to document eruptions at Merapi and Sinabung Volcanoes in Indonesia. I use numerical models of volcanic activity in combination with my observational data to describe the processes driving different eruption styles, including lava dome growth and collapse, lava flow emplacement, and transitions between effusive and explosive activity.
Both effusive and explosive eruptions have occurred recently at Merapi volcano. I use satellite thermal images to identify variations during the 2006 effusive eruption and a numerical model of magma ascent to explain the mechanisms that controlled those variations. I show that a nearby tectonic earthquake may have triggered the peak phase of the eruption by increasing the overpressure and bubble content of the magma and that the frequency of pyroclastic flows is correlated with eruption rate. In 2010, Merapi erupted explosively but also shifted between rapid dome-building and explosive phases. I explain these variations by the heterogeneous addition of CO2 to the melt from bedrock under conditions favorable to transitions between effusive and explosive styles.
At Sinabung, I use photogrammetry and satellite images to describe the emplacement of a viscous lava flow. I calculate the flow volume (0.1 km3) and average effusion rate (4.4 m3 s-1) and identify active regions of collapse and advance. Advance rate was controlled by the effusion rate and the flow’s yield strength. Pyroclastic flow activity was initially correlated to the decreasing flow advance rate, but was later affected by the underlying topography as the flow inflated and collapsed near the vent, leading to renewed pyroclastic flow activity.
This work describes previously poorly understood mechanisms of silicic lava emplacement, including multiple causes of pyroclastic flows, and improves the understanding, monitoring capability, and hazard assessment of silicic volcanic eruptions.
Both effusive and explosive eruptions have occurred recently at Merapi volcano. I use satellite thermal images to identify variations during the 2006 effusive eruption and a numerical model of magma ascent to explain the mechanisms that controlled those variations. I show that a nearby tectonic earthquake may have triggered the peak phase of the eruption by increasing the overpressure and bubble content of the magma and that the frequency of pyroclastic flows is correlated with eruption rate. In 2010, Merapi erupted explosively but also shifted between rapid dome-building and explosive phases. I explain these variations by the heterogeneous addition of CO2 to the melt from bedrock under conditions favorable to transitions between effusive and explosive styles.
At Sinabung, I use photogrammetry and satellite images to describe the emplacement of a viscous lava flow. I calculate the flow volume (0.1 km3) and average effusion rate (4.4 m3 s-1) and identify active regions of collapse and advance. Advance rate was controlled by the effusion rate and the flow’s yield strength. Pyroclastic flow activity was initially correlated to the decreasing flow advance rate, but was later affected by the underlying topography as the flow inflated and collapsed near the vent, leading to renewed pyroclastic flow activity.
This work describes previously poorly understood mechanisms of silicic lava emplacement, including multiple causes of pyroclastic flows, and improves the understanding, monitoring capability, and hazard assessment of silicic volcanic eruptions.
ContributorsCarr, Brett B (Author) / Clarke, Amanda B (Thesis advisor) / Arrowsmith, Ramon (Committee member) / Mcnamara, Allen (Committee member) / Shirzaei, Manoochehr (Committee member) / Williams, Stanley (Committee member) / Arizona State University (Publisher)
Created2016
Description
This dissertation reports three studies of students’ and teachers’ meanings for quotient, fraction, measure, rate, and rate of change functions. Each study investigated individual’s schemes (or meanings) for foundational mathematical ideas. Conceptual analysis of what constitutes strong meanings for fraction, measure, and rate of change is critical for each study. In particular, each study distinguishes additive and multiplicative meanings for fraction and rate of change.
The first paper reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topics of targeted professional development.
The second paper reports the quantitative part of a mixed method study of 153 calculus students at a large public university. The majority of calculus students not only have weak meanings for fraction, measure, and constant rates but that having weak meanings is predictive of lower scores on a test about rate of change functions. Regression is used to determine the variation in student success on questions about rate of change functions (derivatives) associated with variation in success on fraction, measure, rate, and covariation items.
The third paper investigates the implications of two students’ fraction schemes for their understanding of rate of change functions. Students’ weak measurement schemes obstructed their ability to construct a rate of change function given the graph of an original function. The two students did not coordinate three levels of units, and struggled to relate partitioning and iterating in a way that would help them reason about fractions, rate of change, and rate of change functions.
Taken as a whole the studies show that the majority of secondary teachers and calculus students studied have weak meanings for foundational ideas and that these weaknesses cause them problems in making sense of more applications of rate of change.
The first paper reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topics of targeted professional development.
The second paper reports the quantitative part of a mixed method study of 153 calculus students at a large public university. The majority of calculus students not only have weak meanings for fraction, measure, and constant rates but that having weak meanings is predictive of lower scores on a test about rate of change functions. Regression is used to determine the variation in student success on questions about rate of change functions (derivatives) associated with variation in success on fraction, measure, rate, and covariation items.
The third paper investigates the implications of two students’ fraction schemes for their understanding of rate of change functions. Students’ weak measurement schemes obstructed their ability to construct a rate of change function given the graph of an original function. The two students did not coordinate three levels of units, and struggled to relate partitioning and iterating in a way that would help them reason about fractions, rate of change, and rate of change functions.
Taken as a whole the studies show that the majority of secondary teachers and calculus students studied have weak meanings for foundational ideas and that these weaknesses cause them problems in making sense of more applications of rate of change.
ContributorsByerley, Cameron (Author) / Thompson, Patrick W (Thesis advisor) / Carlson, Marilyn P (Committee member) / Middleton, James A. (Committee member) / Saldanha, Luis (Committee member) / Mcnamara, Allen (Committee member) / Arizona State University (Publisher)
Created2016