This dissertation will examine the Knowledge-as-Theory-and-Elements perspective on knowledge structure. The dissertation creates a set of theoretical criteria given within a template by which lesson plans can be designed to teach mathematics and the physical sciences. The dissertation also will test the Knowledge-as-Theory and-Elements theoretical perspective by designing lesson plans to teach a branch of mathematics, graph theory, by using the new template. The dissertation will include a comparative study investigating the effectiveness of the lesson plans conforming to the new template and the lesson plans designed by the traditional theoretical perspective Knowledge-as-Elements.

If one has three sticks (lengths), when can you make a triangle with the sticks? As long as any two of the lengths sum to a value strictly larger than the third length one can make a triangle. Perhaps surprisingly, if one is given 6 sticks (lengths) there is no simple way of telling if one can build a tetrahedron with the sticks. In fact, even though one can make a triangle with any triple of three lengths selected from the six, one still may not be able to build a tetrahedron. At the other extreme, if one can make a tetrahedron with the six lengths, there may be as many 30 different (incongruent) tetrahedra with the six lengths. Although tetrahedra have been studied in many cultures (Greece, India, China, etc.) Over thousands of years, there are surprisingly many simple questions about them that still have not been answered. This thesis answers some new questions about tetrahedra, as well raising many more new questions for researchers, teachers, and students. It also shows in an appendix how tetrahedra can be used to illustrate ideas about arithmetic, algebra, number theory, geometry, and combinatorics that appear in the Common Cores State Standards for Mathematics (CCSS -M). In particular it addresses representing three-dimensional polyhedra in the plane. Specific topics addressed are a new classification system for tetrahedra based on partitions of an integer n, existence of tetrahedra with different edge lengths, unfolding tetrahedra by cutting edges of tetrahedra, and other combinatorial aspects of tetrahedra.

The purpose of this study was to verify Bandura's theory on the relationship of self-efficacy and performance particularly in mathematics among high school students. A rural school in the Philippines was selected for its homogenous student population, effectively reducing the effects of confounding variables such as race, ethnic and cultural backgrounds, socioeconomic status, and language. It was shown that self-efficacy was a positive but minor predictor of future performance only for male students who previously had higher mathematics grades. The effects were different between genders. It was not a strong predictor for women regardless of previous grades, and men with weaker mathematics skills. On the other hand, mathematics self-efficacy was predicted by previous mathematics achievement for women; and also the number of siblings and parental education for the higher performing women. The use of a second language in the mathematics classroom negatively affected confidence and performance. It was also found that there were differences in terms of academic behavior, peers, and family life between students with high and low self-efficacy. Positive behaviors were found for all female students regardless of self-efficacy levels and fewer were found among men. Negative behaviors were only found among low self-efficacy students. No differences were found in terms of the lives and families of the participants, but the interviews revealed that family members and their experiences of poverty affected educational goals and ambitions. In terms of other dispositional factors, students expressed classroom and test anxieties, concerns of being embarrassed in front of their classmates, and beliefs that mathematics was naturally difficult and not enjoyable. The students who did not talk about any of these themes were better performing and had higher self-efficacy scores.

Mathematical Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development - Young Seh Bae - This study investigated mathematical word problem solving and the factors associated with the solution paths adopted by two groups of participants (N=40), students with autism spectrum disorders (ASDs) and typically developing students in fourth and fifth grade, who were comparable on age and IQ (greater than 80). The factors examined in the study were: word problem solving accuracy; word reading/decoding; sentence comprehension; math vocabulary; arithmetic computation; everyday math knowledge; attitude toward math; identification of problem type schemas; and visual representation. Results indicated that the students with typical development significantly outperformed the students with ASDs on word problem solving and everyday math knowledge. Correlation analysis showed that word problem solving performance of the students with ASDs was significantly associated with sentence comprehension, math vocabulary, computation and everyday math knowledge, but that these relationships were strongest and most consistent in the students with ASDs. No significant associations were found between word problem solving and attitude toward math, identification of schema knowledge, or visual representation for either diagnostic group. Additional analyses suggested that everyday math knowledge may account for the differences in word problem solving performance between the two diagnostic groups. Furthermore, the students with ASDs had qualitatively and quantitatively weaker structure of everyday math knowledge compared to the typical students. The theoretical models of the linguistic approach and the schema approach offered some possible explanations for the word problem solving difficulties of the students with ASDs in light of the current findings. That is, if a student does not have an adequate level of everyday math knowledge about the situation described in the word problem, he or she may have difficulties in constructing a situation model as a basis for problem comprehension and solutions. It was suggested that the observed difficulties in math word problem solving may have been strongly associated with the quantity and quality of everyday math knowledge as well as difficulties with integrating specific math-related everyday knowledge with the global text of word problems. Implications for this study include a need to develop mathematics instructional approaches that can teach students to integrate and extend their everyday knowledge from real-life contexts into their math problem-solving process. Further research is needed to confirm the relationships found in this study, and to examine other areas that may affect the word problem solving processes of students with ASDs.

A current trend in social justice oriented education research is the promotion of certain intellectual virtues that support epistemic responsibility, or differently put, the dispositions necessary to be a good knower. On the surface, the proposition of epistemically responsible teaching, or teaching students to be responsible knowers is innocuous, even banal. In the mathematics classroom, however, it is patently at odds with current practice and with the stated goals of mathematics education. This dissertation begins by detailing the extant paradigm in mathematics education, which characterizes mathematics as a body of skills to be mastered, and which rewards ways of thinking that are highly procedural and mechanistic. It then argues, relying on a wide range of educational thinkers including John Dewey, Maxine Greene, Miranda Fricker, and a collection of scholars of white privilege, that an important element in social justice education is the eradication of such process-oriented thinking, and the promotion of such intellectual virtues as courage and humility. Because the dominant paradigm is supported by an ideology and mythology of mathematics, however, changing that paradigm necessitates engaging with the underlying conceptions of mathematics that support it. The dissertation turns to naturalist philosophers of education make clear that the nature of mathematics practice and the growth of mathematical knowledge are not characterized by mechanistic and procedural thinking at all. In these accounts, we can see that good mathematical thinking relies on many of the same habits and dispositions that the social justice educators recommend. In articulating an isomorphism between good mathematical thinking and socially responsive thinking, the dissertation aims to offer a framework for thinking about mathematics education in and for a democratic society. It aims to cast the goals of mathematically rigorous education and socially responsible teaching not only as not in conflict, but also overlapping in meaningful ways.

The Florida Legislature passed a bill that changed the placement methods for some incoming students to the Florida State College System in 2013. This analysis of state policy looks at Senate Bill 1720 as the treatment in an interrupted time-series trend study at one state college in Florida. This research attempts to answer three questions: (1) What is the enrollment trend over the last 10 years in first-level math courses, such as Intermediate Algebra (MAT1033), Liberal Arts Math (MGF1106), and Elementary Statistics (STA2023) for first time in college (FTIC) students? (2) What is the trend of course passing rates in the above listed gateway math courses before and after the developmental education requirements changed? (3) What are the trends in student success rates before and after the changes to developmental education requirements in these courses for various demographics, such as race, age, and gender, for FTIC students in these math courses? This study looked at one college’s gateway math sequence in terms of enrollment and student success. The observed benefits to this institution were the gains in FTIC student enrollment in the gateway math courses. There were observed decreases in FTIC passing rates in the three gateway math courses, yet the total share of FTIC students taking and passing gateway math courses increased. This study should be shared with the Mathematics Department with the hope that it will continue to track student success in its courses and investigate other research in the area of gateway math instruction for younger post-secondary students as their enrollment continues to follow a decrease in the average student’s age. / A Dissertation submitted to the Department of Educational Leadership and Policy Studies in partial fulfillment of the requirements for the degree of Doctor of Education. / Spring Semester 2019. / April 1, 2019. / Includes bibliographical references. / Toby Park, Professor Directing Dissertation; Elizabeth M. Jakubowski, University Representative; Shouping Hu, Committee Member; Stacey Rutledge, Committee Member.

Understanding mathematics and teaching mathematics involve numerous factors, one of which may be an individual's spatial ability. This paper examines research conducted on the relationship between spatial abilities and mathematics, gender differences in the area of spatial ability, the types of experiences that may affect one's spatial ability, and issues surrounding the teaching of spatial skills. Researchers have found that spatial ability does relate to mathematics and males tend to have greater spatial ability than females. Instruction has also been shown to be successful in helping individuals learn spatial skills. This paper also reports the results of a study that examined the differences in spatial ability among 98 participants (males, females, faculty, and students in the sciences and non-sciences) at a Pacific Northwest university. Although not all the results were statistically significant, they tend to agree with earlier studies that found gender advantages in spatial abilities favoring males over females. They also provide evidence of the existence of greater spatial abilities among participants who are engaged in scientific rather than non-scientific pursuits. The participants in this study also reported experiences that they believed influenced their success or failure in tasks requiring spatial ability. Such experiences were success in math and art classes, computer modeling, drafting, puzzles/games, Legos, construction, woodworking, and playing with blocks as a child. Participants also stated their belief that spatial ability related to success or lack of success in mathematics. Over half of the students felt that spatial ability would help in a math class. This study reveals that spatial ability does differ in individuals; that there exist experiences that individuals feel are important for developing spatial ability; and that spatial ability relates to mathematics. This information can be beneficial for both teachers and researchers. / Graduation date: 2002

By creating opportunities for participation and intellectual engagement, standardized classroom routines are large determinants of the conceptual meaning students make. It is through repeated engagement in patterns of talk and intellectual practices that students are socialized into ways of thinking and habits of mind. The focus of this study is on moment-to-moment interactions between teachers and students in order to describe, identify and operationalize meaningful regularities in their discourse. Using classroom-level measures, I investigate the robustness of relationships between students’ mathematics achievement and discursive patterns across multiple classrooms with the statistical methods of Hierarchical Linear Modeling. Specifically, I investigated two theoretically significant constructs reflected in teacher’s follow-up moves -- responsiveness and intellectual work. Responsiveness is an attempt to understand what another is thinking displayed in how she builds, questions, clarifies, takes up or probes that which another says. Intellectual work reflects the cognitive work requested from students with a given turn of talk. After developing coding schemes to measure and quantify these discursive constructs, statistical analyses revealed positive relationships between the responsiveness and intellectual work of teachers’ follow-up and student learning of rate and proportionality (p=.01 and .08, respectively). Additionally, classroom communities with higher levels of responsiveness and intellectual work moderate the effect of prior knowledge on student learning by decreasing the degree to which pretest scores predict students’ post-test achievement (though neither are statistically significant). Based on these results, I conclude that classroom discourse and normative interaction patterns guide and influence student learning in ways that improve achievement. Recommendations are primarily concerned with ways the educational community can support and encourage teachers to develop responsive, intellectually demanding discursive patterns in their classrooms. In particular, we need to increase the awareness of the power of discourse, provide appropriate and sustained support for teachers to change current patterns, re-examine the design of teacher preparation programs, and develop ways to thoughtfully integrate responsiveness and intellectual work with core mathematics content. There is tremendous and often unrealized power in the ways teachers talk with their students; it is our obligation to help teachers learn how to recognize and leverage this power. / text

People use metaphors to describe or understand one thing in terms of another. The central idea of this thesis is that metaphors can be used to teach mathematics, particularly abstract topics such as directed numbers. Using directed numbers as a context, this study develops a framework and a coding scheme that can be used as a tool for analysing the use of metaphors in the teaching of mathematics. The part of the theoretical framework of the coding scheme is based on the work of Lakoff and Johnson (1980) and Lakoff and Nunez (2000). In those studies, the authors classify metaphors used for teaching mathematics into one of three categories: ontological, orientational and structural metaphors. By considering the source domain of metaphors, they can be classified into either grounding or linking metaphors. Similarly, the target domain of the metaphors can be categorized by the intended learning outcomes and by the functions of the metaphors. One of the primary contributions of this thesis is the development of a coding scheme that is specifically designed to analyze the use of metaphors in mathematics lessons. The scheme was then used and validated through the analysis of mathematics lessons taught by two teachers with contrasting academic backgrounds and teaching experiences. Three lessons taught by each teacher on the topic of directed numbers at Secondary One level were recorded and analysed. The metaphors used by each teacher were identified, coded and analyzed in order to determine how metaphors can be extended and transformed into other metaphors.. Finally, this thesis compared how the two teachers differed in their use of metaphors, particularly in terms of the selection, sequencing and organization of the metaphors used. This can be indicative the level of conceptual learning that is made available for students in their classes. The research questions: 1. What kinds of metaphors did the teachers use to introduce and explain the concepts and computational processes of directed numbers? 2. What functions did these metaphors serve? 3. What is the developmental path of these metaphors within and across the lessons? 4. What were the differences in the selecting, sequencing, and organization of the metaphor used by the two teachers? Findings This thesis designed and tested an original coding scheme. The findings revealed that the two teachers had used many kinds of metaphors in their lessons. They were used for classifying different kinds of numbers, constructing concepts, and explaining the properties and computational processes of directed numbers. Most of the metaphors found in this study were used to provide a cognitive function that facilitates the introduction of new mathematical concepts and helps the students make sense of the operational processes; only a few metaphors served a memorable function. When comparing the use of metaphors by the two teachers, we can analyze their teaching philosophies. Teacher 1’s use of metaphors demonstrated a linear development path from a simple to a more advance perspective, whereas Teacher 2’s use of metaphors revealed more comprehensive, sophisticated and multi-layered perspective. Significance of the study This study provides insights into the meaning and implication of using metaphors in teaching mathematical concepts. At research level, this study extends the existing work of Lakoff and develops an analytical tool specifically designed to understand the pedagogical values of using metaphors to teach abstract mathematical concepts such as directed numbers. At pedagogical level, the metaphor coding scheme can act as an initial foray into how metaphors can be used in and for teaching. Moreover, the Metaphor-Concept Development Chart developed in this study is a practical tool that can help teachers to analyze and improve their own use metaphors, thereby furthering their professional development and teaching effectivenss. / published_or_final_version / Education / Doctoral / Doctor of Education

This exploratory case study examines an alignment-focused coaching model of mathematics professional development during a school district's second-year implementation of the coaching model. Specifically, the study describes the content of coach-teacher talk as five coach-teacher pairs, grades K-8, engage in planning and analyzing mathematics lessons. Using an alignment framework designed around the components of curriculum, instruction, and assessment to analyze talk, four patterns unfold. Issues of curriculum, instruction, and assessment were more often discussed in isolation than interconnected, mathematics was most often the content focus when teacher and/or coach were using the state standards document to plan, student thinking and learning were most often a focus when students were struggling, and teachers often talked about instruction as actions isolated from student thinking and learning. In addition, teachers reported changes to instruction as an outcome of participating in coaching. Self-reported benefits to teachers' practice included planning lessons that focused on student learning, that is, considering the mathematics in the standards and ways students would learn the content. Teachers also reported asking "better questions" more often and in different ways, using models such as manipulatives and representations for connecting mathematics ideas, thinking more about student learning, and analyzing and scrutinizing textbooks to align with the state standards.