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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
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