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.

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.

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.

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.

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.

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.