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This dissertation reports the research methodology and findings of three studies focused on mathematical modeling as a structured problem-solving process where the student translates relevant details from a situation, event, or phenomena in daily life into mathematical language with mathematically-defined structures and operations, which form the mathematical model. This model can be manipulated to yield a solution, description, explanation, or prediction, which are then interpreted within the context of the real-life event. Part of this process requires determining when the mathematical model can be adapted to new situations and identifying the model's boundaries and limitations in providing descriptions, explanations, and predictions. Constructing, testing, and applying mathematical models meet the needs of science, technology, engineering, and mathematics (STEM) educational goals for advancing students' knowledge and skills and for increasing students' understanding of the world around them. To capture the complexities of mathematical modeling, this research defined the Systems Theory in Mathematical Modeling Education (STiMME) theoretical framework to investigate the interactions that influence students' mathematical modeling experiences. Many stakeholders, e.g. researchers, educators, students, etc., work at various levels to deliver content standards, develop materials, implement lessons, engage in learning, and participate in a myriad of ways that affect outcomes. The STiMME framework probes these interrelated elements and their impact on students in hopes of revealing new ways of thinking about mathematical modeling teaching and learning practices and theories. This research defined three STiMME subsystems. The first study scrutinized 241 STEM literature artifacts distinguishing between models and representations, models and mathematical models, and modeling and mathematical modeling to yield definitions for laying a united foundation for investigating mathematical modeling teaching and learning in all STEM education subjects. Similarities in mathematical modeling, engineering and technology design, and scientific methodological processes were identified that may improve students' transfer of knowledge and skills within different STEM courses. The second study scrutinized 342 resources to create a taxonomy for the 14 different types of mathematical models described in professional and educational writing. The final study examined student work from the same mathematical modeling activity presented in three math and physics classes to determine the impact that student question posing, tracking, and reviewing had on the types of mathematical models they constructed. The students asked exploratory, consolidating, clarifying, and elaborating questions. It was noted that students with more exposure to mathematical modeling activities and more experience with question strategies were able to construct more sophisticated mathematical models. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]

Rimstone dams, formed in cave environments and composed primarily of calcium carbonate, are constructed through both chemical and mechanical processes. As calcium rich water flows over a sloped limestone cave floor, calcium carbonate precipitates from the solution. Standard fluid mechanics equations govern the hydrodynamics. Chemical kinetics describe movement through the boundaries and explain the bulk and surface reactions that influence precipitation. The free boundaries, gas-fluid and solid-fluid interfaces, couple the hydrodynamic and reactive transport equations. The chemical kinetics of bulk and surface reactions that result in precipitation are determined. This research focuses on the effect of changing the volumetric flow rate and altering the concentrations of calcium on the growth of the mineral-fluid boundary by developing and solving, through thin-film fluid flow approximations, appropriate hydrodynamic and reactive transport equations. Three general volumetric flow conditions are explored in this paper. The effect of slow, moderate, and fast flow rates under conditions of constant flow; exponentially decreasing flow, modeling drought conditions; and seasonal flow fluctuations about a base rate are investigated. The greatest growth was observed during seasonal fluctuations with a moderate volumetric flow rate. Greatest deposition, near the initial position, was with high concentrations of calcium flowing with a fast flow rate. Further downstream, the largest growth required small concentrations of calcium, which maintained a high pH condition in the system. From the developed equation for the mineral boundary, growth was affected by pH conditions driven from bulk reactions and influenced by degassing of carbon dioxide and surface reactions from the alterations in calcium concentration, ultimately driving the system pH.