In recent decades, education researchers have recognized the need for teachers to have a nuanced content knowledge in addition to pedagogical knowledge, but very little research was conducted into what this knowledge would entail. Beginning in 2008, math education researchers began to develop a theoretical framework for the mathematical knowledge needed for teaching, but their work focused primarily on elementary schools. I will present an analysis of the mathematical knowledge needed for teaching about the regular curves and surfaces, two important concepts in differential geometry which generalize to the advanced notion of a manifold, both in a college classroom and in an on-line format. I will also comment on the philosophical and political questions that arise in this analysis.

Teachers must have specific knowledge of a subject and how to teach it to promote learning in their students (also known as pedagogical content knowledge). Research has shown that project-based curriculum can be an effective way for teachers to leverage this knowledge into deeper student understanding and application readiness, but observations about when and how this happens in the classroom have not been adequately documented. In this study we will explore teaching and learning in a middle-school boat-building curriculum focused on real-world application of math concepts. The boat-building program took place over one week, included seven students, and was taught by three teachers. The teaching phase of this study examined how the three boat-building teachers applied their pedagogical content knowledge (PCK) through a participant observation case study. The three teachers had diverse training and teaching backgrounds. At the completion of the course these teachers were interviewed on their prior teaching experiences and training to determine how they acquired their pedagogical content knowledge. The learning part of this study involved a pre- and post-test application task completed by the students. After all students completed the application task, each was interviewed to see what, if any, knowledge or approach these teachers used had an impact on the ability of the students to do the task. Analysis of the pre-post assessments showed that students were not able to make statistically significant gains over the one week of instruction. However, students did note many aspects of instruction that they thought helped them. Additionally, students showed gains in assessing importance of geometry in design, the vocabulary associated with scale, geometry, and woodworking, and creating context for prior instruction. For the teachers, higher scores on the PCK rubric did align with a greater amount of experience teaching. Also, the two more experienced teachers influenced each other and the third teacher regarding student learning and instructional approach.

When attempting to study the learning process of undergraduate chemistry student, the classroom and any interaction that take place within it constitute the social context of interest. By studying how different approaches can foster different classroom environments, it is possible to approach course design from an informed and scientifically sound perspective. Thus, it becomes necessary to identify and quantify the factors that have a positive or negative effect on the classroom environment. Social comparison concerns, comfort levels and self-efficacy have been shown to be social factors that affect each other as well as the learning process and have therefore been deemed suitable for use in this study. POGIL, a pedagogic approach to teaching chemistry based on small-group work and active learning, has been shown to lead to positive academic outcomes and is currently employed by several faculties at Virginia Commonwealth University. This study seeks to investigate differences in the learning environment observed in lecture and POGIL based chemistry courses, by adapting Micari’s survey for measuring social comparison, comfort levels and self-efficacy in small-group science learning. Reliance on the combustion of fossil-fuels, such as coal, oil and natural gas, as sources of energy has, since the industrial revolution, caused atmospheric CO2 to increase to the current level of 400ppm by volume; an increase of 25% from the 1960s when monitoring started. Climatologists predict that an increase to 450 ppm would have irreversible effects on the Earth’s environment and recommend that, in order to preserve the conditions in which civilization developed, levels be reduced to below 350 ppm. The use of porous organic polymers for capture and separation of CO2 from industrial sources has been at the forefront of research attempting to curb CO2 emission into the atmosphere. Benzimidazole based polymers have shown a high selectivity for CO2.7 To attempt to improve on the capture abilities of these polymers, we sought to synthesize sulfur containing analogs presenting thiazole moieties. Two such polymers were synthesized using a pyrene-based linker. Furthermore, the pyrene-derived fluorescence of these polymers enabled their use as chemosensors targeting nitroaromatic compounds and mercury

Recently, there has been a big surge of research and public interest in increasing the math capabilities and skills of American children. This paper serves as a literature review examining how the home environment, specifically parents, can help with that. This meta-analysis delves into the factors of maternal math talk, a parent’s own math anxiety, and the relationship between a parent and child in the context of a parent’s gender stereotypes and a parent’s perception on his or her child’s math abilities. Interventions, suggestions, and future implications are also discussed. This paper will hopefully bring needed awareness to parents about their roles in their child’s math development, abilities, and achievement.

This dissertation employed a mixed-methods approach to examine the relationship between spatial reasoning ability and understanding of chemistry content for both middle school students and their science teachers. Spatial reasoning has been linked to success in learning STEM subjects (Wai, Lubinski, & Benbow, 2009). Previous studies have shown a correlation between understanding of chemistry content and spatial reasoning ability (e.g., Pribyl & Bodner, 1987; Wu & Shah, 2003: Stieff, 2013), raising the importance of developing the spatial reasoning ability of both teachers and students. Few studies examine middle school students’ or in-service middle school teachers’ understanding of chemistry concepts or its relation to spatial reasoning ability. The first paper in this dissertation addresses the quantitative relationship between mental rotation, a type of spatial reasoning ability, and understanding a fundamental concept in chemistry, the particulate nature of matter. The data showed a significant, positive correlation between scores on the Purdue Spatial Visualization Test of Rotations (PSVT; Bodner & Guay, 1997) and the Particulate Nature of Matter Assessment (ParNoMA; Yezierski, 2003) for middle school students prior to and after chemistry instruction. A significant difference in spatial ability among students choosing different answer choices on ParNoMA questions was also found. The second paper examined the ways in which students of different spatial abilities talked about matter and chemicals differently. Students with higher spatial ability tended to provide more of an explanation, though not necessarily in an articulate matter. In contrast, lower spatial ability students tended to use any keywords that seemed relevant, but provided little or no explanation. The third paper examined the relationship between mental reasoning and understanding chemistry for middle school science teachers. Similar to their students, a significant, positive correlation between scores on the PSVT and the ParNoMA was observed. Teachers who used consistent reasoning in providing definitions and examples for matter and chemistry tended to have higher spatial abilities than those teachers who used inconsistent reasoning on the same questions. This is the first study to explore the relationship between spatial reasoning and understanding of chemistry concepts at the middle school level. Though we are unable to infer cause and effect relationship from correlational data, these results illustrate a need to further investigate this relationship as well as identify the relationship between different spatial abilities (not just mental rotation) and other chemistry concepts.

This curriculum project reviews the research on students who have been labeled drop-outs and/or low-achievers. Several different types of teaching models were reviewed to determine the best model to be used for drop-out and/or low-achieving students. The project includes curriculum materials that correspond to the Minimum Level Skills objectives for the General Math II course designated by the Duval County School System in Florida. The curriculum also corresponds to the required textbook for the General Math II course. This project strives to increase the ability of the students in the Graduation Enhancement Program to pass the Minimum Level Skills Test and increase their knowledge in the area of basic and common sense mathematic concepts. III

The integration of digital technologies in K-12 education is ubiquitous. Web 2.0 technologies enable students who were once passive consumers to become active participants in, and even creators of, dynamic digital experiences. Social media, in particular, can connect disparate populations, minimizing traditional barriers such as time, space and geography. Similarly, science communication has also been influenced by an expanding array of media through which scientists can now connect directly with the public. #Scistuchat, the focus of this study, uses the social media platform Twitter to bring together scientists, secondary science students and teachers outside of school in monthly, science-focused Twitter chats. Using a multiple-case (embedded) design, this study sought to answer the question "How do #scistuchat participants perceive Twitter as a tool for learning and communicating science?" Thematic, cross-case analysis of four #scistuchats revealed themes specific to the #scistuchat experience, as well as the broader use of Twitter for science learning and communication. In addition to real-time observations of each chat and later analysis of the archived tweets, videoconferencing technology was used to conduct individual interviews with participating scientists (n=16) and teachers (n=6), as well as focus groups with students (n=17). Notable #scistuchat-specific findings include a recognition of the experience as dynamic and student-focused. Regarding student outcomes, although gains in science content knowledge were limited, an evolving understanding of scientists and the nature of their work was prominent. Findings regarding the broader use of Twitter for science purposes highlighted its multidimensional, professional utility and its unique contributions when leveraged in classroom settings.

Complex systems abound on this planet, in the composition of the human body, in ecosystems, in social interaction, in political decision-making, and more. Analytical methods allowing us to better understand how these systems operate and, consequently, to have a chance to intervene and change the undesirable behavior of some of the more pernicious systems have developed and continue to be enhanced via quickly changing technology. Some of these analytical methods are accessible by pre-college students, but have not been widely used at that level of education. Jay Forrester, the founder of one of the methodologies, System Dynamics (SD), used to study complex system behavior involving feedback, laments the lack of understanding of complex systems evident in short-sited decisions made by legislators -- global climate change and fiscal policies being cases in point. In order to better prepare future decision makers with tools that could allow them to make more informed decisions about issues involving complex systems efforts have been underway to increase pre-college teacher understanding of the SD method. The research described in this dissertation introduces the mathematics education community to the value of System Dynamics modeling in pre-college algebra classes, indicates a path by which a traditional mathematics curriculum could be enhanced to include small SD models as a new representation for elementary functions studied in algebra classes, and provides an empirical study regarding conceptual understanding of functions by students. Chapter 2 indicates the numerous beneficial learning outcomes that empirical studies have shown accompany model-building activities. Chapter 3 indicates the need for students to become familiar with complex systems analysis, how SD modeling (one method of complex systems analysis) aligns with the Common Core State Standards in Mathematics, and the work that has transpired over the past two decades using SD in K-12. Chapter 4 focuses on the importance of the concept of function in high school mathematics, some limitations of exclusive reliance on the closed form equation representation for mathematizing problems and the SD stock/flow representations of some of the elementary functions that are studied in algebra classes. Chapter 5 looks at the issues affecting two traditional teachers and the challenges they faced when trying to reintroduce SD modeling into their algebra classes. Chapter 6 explains the student component of the classroom experiment that was conducted by the teachers who are highlighted in Chapter 5. The analysis of the results of student model-building activities in the two classroom studies that are part of the third paper did not indicate a statistical difference between the two experimental groups and the two control groups. Many environmental and scheduling issues conspired to adversely affect the experiment. However, positive outcomes were evident from the two pairs of students who were videotaped while they built the final multi-function drug model, the final student lesson in the experiment. Research focused on student outcomes is needed to further assess the strengths and weakness of the SD approach for student learning in mathematics.

Making the transition from calculus to advanced calculus/real analysis can be challenging for undergraduate students. Part of this challenge lies in the shift in the focus of student activity, from a focus on algorithms and computational techniques to activities focused around definitions, theorems, and proofs. The goal of Realistic Mathematics Education (RME) is to support students in making this transition by building on and formalizing their informal knowledge. There are a growing number of projects in this vein at the undergraduate level, in the areas of abstract algebra (TAAFU: Larsen, 2013; Larsen & Lockwood, 2013), differential equations (IO-DE: Rasmussen & Kwon, 2007), geometry (Zandieh & Rasmussen, 2010), and linear algebra (IOLA: Wawro, et al., 2012). This project represents the first steps in a similar RME-based, inquiry-oriented instructional design project aimed at advanced calculus. The results of this project are presented as three journal articles. In the first article I describe the development of a local instructional theory (LIT) for supporting the reinvention of formal conceptions of sequence convergence, the completeness property of the real numbers, and continuity of real functions. This LIT was inspired by Cauchy's proof of the Intermediate Value Theorem, and has been developed and refined using the instructional design heuristics of RME through the course of two teaching experiments. I found that a proof of the Intermediate Value Theorem was a powerful context for supporting the reinvention of a number of the core concepts of advanced calculus. The second article reports on two students' reinventions of formal conceptions of sequence convergence and the completeness property of the real numbers in the context of developing a proof of the Intermediate Value Theorem (IVT). Over the course of ten, hour-long sessions I worked with two students in a clinical setting, as these students collaborated on a sequence of tasks designed to support them in producing a proof of the IVT. Along the way, these students conjectured and developed a proof of the Monotone Convergence Theorem. Through this development I found that student conceptions of completeness were based on the geometric representation of the real numbers as a number line, and that the development of formal conceptions of sequence convergence and completeness were inextricably intertwined and supported one another in powerful ways. The third and final article takes the findings from the two aforementioned papers and translates them for use in an advanced calculus classroom. Specifically, Cauchy's proof of the Intermediate Value Theorem is used as an inspiration and touchstone for developing some of the core concepts of advanced calculus/real analysis: namely, sequence convergence, the completeness property of the real numbers, and continuous functions. These are presented as a succession of student investigations, within the context of students developing their own formal proof of the Intermediate Value Theorem.

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