Research over the past twenty years supports the teacher as the critical factor in the implementation of educational programs. The primary purpose of this mixed design study was to determine teachers' perceptions in implementing the required Algebra I program that was mandated by the state of Virginia in 1995. The research was examined through the lens of the recommendations of the National Council of Teachers of Mathematics (NCTM): communicating mathematically, making mathematical connections, becoming mathematical problem solvers, and reasoning mathematically. This study was limited to ninth grade Algebra I teachers in a school division in Virginia. Two Concerns Based Adoption Model (CBAM) instruments were used in this research. The Stages of Concern (SoC) Questionnaire determined the teachers' concerns regarding implementation of the required Algebra I program. The Levels of Use (LoU) focused interview data supported the teachers' areas of concern. In addition, the grounded theory method was used to analyze the observation and interview data. Results were presented as narrative descriptions from which major categories of concerns emerged. Findings revealed that the NCTM recommendations of communicating mathematically, making mathematical connections, becoming mathematical problem solvers, and reasoning mathematically were not implemented in the required Algebra I program. Teachers were unaware that the required Algebra I program was designed to meet the needs of a changing population. This study supports the need for comprehensive and ongoing training for teachers if the needs of a more diverse population are to be realized in a required Algebra I program.

The purpose of this study was to examine whether gender-based mathematics classes have a positive impact on the achievement levels of males and females. The study was conducted in order to give insight into the differentiated approach to teaching males and females separately and the effects the separate instruction has on mathematics achievement while closing the achievement gap. This study examined gender-based instruction as a strategy along with other variables that are perceived as having an impact on student achievement. The independent variables in the study were instructional leadership, teacher certification level, teacher content knowledge, teacher knowledge of primary gender learning styles, teacher expectations, gender-based instruction, teacher demographics, and student demographics. The dependent variable was student achievement in mathematics. The study was conducted in an elementary school in Atlanta, Georgia among third, fourth, and filth grade students and teachers. The 2007 Georgia Criterion Reference Competency Test (GCRCT) data were used as pretest data prior to teachers teaching students in a gender-based selling. Eleven teachers were given professional development in male/female primary learning styles. differentiated instruction, culturally responsive teaching, and data analysis. Anonymous surveys were completed by teachers and students. In addition, student demographic data were analyzed. The study was conducted in a Title I elementary school. Ninety-nine percent of the school’s students are on free or reduced lunch. The school serves 85% African- American students and 15% Hispanic, Latina/Latino. The school’s enrollment, K-5 is approximately 550 students. The results of the study indicated that there was a statistically significant relationship between a teacher’s content knowledge. years of experience, and his/her expectations relative to student achievement in mathematics. A regression analysis found that the most impacting variables on student achievement in gender-based classrooms were teacher expectancy, teaching experience and instructional leadership. A regression analysis was used to further determine which independent variables had the strongest impact on student achievement in mathematics. The regression showed that the strongest impacting variables again were teacher expectancy, teacher experience and instructional leadership; however, these variables are mediating at best. Those variables found to be the most significant in relationship to student achievement in mathematics were: gender-based instruction, teacher knowledge of student learning styles, and discipline.

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Through a focus on the intersection of mathematical modeling, ethnomathematics, and cultural and historical practices, this ethnomodeling research study intentionally connected efforts to engage and value each and every student's culture and identity while emphasizing essential geometry content. The data collection took place at a large Hispanic-serving public metropolitan research university in an undergraduate secondary mathematics education methods course intended for pre-service teachers (PSTs). Over the course of eight weeks PSTs completed a questionnaire and pre-/post-surveys, two individual assignments, three small-group tasks, three task reflections, and a group reflection which were then analyzed using quantitative and qualitative methods. In addition to this, based on the questionnaire, six PSTs were selected as cases using a maximum variation strategy. These PSTs additionally participated in two individual and one group interview. In this dissertation I share the stories and experiences of four of these case participants. Through engaging in this study, PSTs shared they became more aware of the geometry that exists in the world around them while experiencing a model for the synergistic integration of both incorporating their own as well as their future students' cultures and identities as well as teaching and learning deep and meaningful mathematics content. PSTs learned to value and respect mathematical practices of diverse cultures and traditions as well as become more prepared to engage their students in such geometry-focused ethnomodeling tasks. PSTs were able to engage in mathematical content in connected and coherent ways, and naturally find links between mathematics and other content domains and explore and share their own experiences. PSTs were empowered and inspired to engage their future students in meaningful mathematical tasks through which they can experience the wonder, joy, and beauty of mathematics in the world around them.

The number of undergraduate students enrolled in non-credit earned mathematical courses such as Beginning Algebra and Intermediate Algebra has increased in recent decades. Students who have unresolved algebra learning from high school face significant difficulties in college mathematics courses. Organizations such as The National Council of Teachers of Mathematics' (NCTM) published mathematics education standards based on the constructivist approach and encouraged the development of constructivist curricula. New technologies, such as adaptive learning systems, Intelligent Tutoring System (ITS), Intelligent Computer Assisted Instruction System (ICAI) or Computerized Adaptive Testing (CAT) are released and new teaching practices have been developed and adopted to help students gain conceptual understanding of mathematics and build their procedural fluency. The purpose of this study was to explore student learning outcomes and compare student attitudes, perceived ease of use and perceived usefulness of two adaptive learning systems used alongside the work-and-walk-by orchestration to teach Intermediate Algebra at a large metropolitan research university. The participants of the study were all students enrolled in the only two sections of Intermediate Algebra in the semester in which the study occurred and who consented to have their data analyzed in the study. This dissertation study was a quantitative in nature and included two research questions. To answer the first research question, a repeated measure ANCOVA was conducted and found no statistically significant results between the control and the intervention group when controlling for the amount of time students spent in the work-and-walk-by orchestration (time in class), the time students spent working on the adaptive learning system (time on online platform), attitudes (AT), perceived ease of use (PE), perceived usefulness (PU), gender and ethnicity group. To answer the second research question, a MANCOVA was used as a statistical procedure and when controlling for the pre-test, it was found a positive statistical significance for students who used ALEKS compared to students who used Knewton Alta. Recommendations were made for future research when work-and-walk-by orchestration is used as well as different adaptive learning platforms.

The purpose of this study was to explore how a veteran teacher implemented a state-approved commercial math curriculum in a prekindergarten classroom. In this action research study, guided by ethnographic principles, I employed Spradley's (2016) developmental research sequence to study what resources from the curriculum the teacher used, how the teacher adapted the curriculum, and what influenced the decisions the teacher made while implementing the curriculum. Data collected for this study included: the teacher guide provided by the commercial curriculum; teacher lesson plans; photographs of student created artifacts; the teacher's reflection journal and notes; and videos of a teacher implementing the math portion of a commercial curriculum. Spradley's (2016) developmental research sequence was used to analyze the data for domains linked through semantic relationships of strict inclusion, rationale, and means-end. The domains were then linked to create a taxonomy representing resources, adaptations and influences for a teacher's choices in math curriculum implementation in a prekindergarten classroom. The teacher in this study needed to use a hidden set of resources in order to implement the curriculum, adapted the instruction from whole group to small group, and adjusted the curriculum to correct for misalignments of scope and sequence between the curriculum and the State mandated learning outcomes. Based on these findings, I recommend more rigorous teacher preparation in prekindergarten mathematics and ongoing training support while implementing a commercial curriculum. I also encourage school districts to evaluate commercial curriculum to ensure they align with state standards.

Learning calculus concepts plays a huge role in understanding phenomena in STEM-related disciplines. Those concepts tend to be dynamic in nature, and the visual exploration and representation of calculus concepts using paper and pencil is limited compared to pedagogically and intentionally using dynamic geometry software. As such, a primary component of this dissertation study involves the integration of dynamic technology. Additionally, previous studies have shown that students have difficulties constructing proofs related to calculus concepts. Despite the existing body of research on students' comprehension of proof and justification, there has not been much focus on teachers' knowledge and perception of proof and justification in connection to the ways that prospective secondary teachers can teach and learn calculus concepts. This study uses a qualitative methodology to investigate the ways in which integrating technology could help both in-service and pre-service secondary teachers gain a deeper understanding of the process of proof. Through a multiple case study approach, research participants were engaged with different mathematical tasks to explore geometric series and subsequently construct and prove conjectures through the integration of dynamic technology. This study showed that dynamic geometry software could help teachers to appreciate the value of visual representation in teaching and learning mathematics. Those technological pieces helped them with exploring different ideas, which is crucial in the process of proving. However, a lack of experience both with visual representations and constructing conjectures held participants back from using their full potential. When it comes to mathematical proofs in school mathematics, it should be considered as a process of exploring ideas, making conjectures, and checking the validity of those conjectures and not a single notion and visual representations - specifically dynamic ones that are created by technology – play a huge role in deepening teachers understanding of the process through their connection with key ideas.

This dissertation study used multinomial logistic and ordinary least squares regression models to investigate to what extent students' mathematical sense of belonging predicted their participation and persistence in their mathematical college and career coursework. Mathematical participation and persistence were operationally defined as courses completed in high school and postsecondary, undergraduate mathematics college and careers credits. Framed on extant research regarding equitable mathematics of education and various learning theories, this study offers mathematics domain-specific statistical analyses of belongingness based on Mahar and colleagues' (2012) five transdisciplinary themes of belongingness. Data from the High School Longitudinal Study 2009 (HSLS:09) were used for analyses. First, an exploratory factor analysis (EFA) was conducted to create a mathematical belongingness construct. Then resulting mathematical belongingness subscales, a criterion-referenced mathematics assessment, and demographic variables were analyzed to find statistically significant predictors of students' participation and persistence in the high school mathematics pipeline and in postsecondary undergraduate, mathematics credit hours. Predictors varied in statistical significance within and between the generated models. However, one theme of the transdisciplinary belongingness subscale in the EFA, subjectivity/mathematics identity, was the only affective variable which was statistically significant in each model generated. Interpretations of these results suggest this study be used as support for the field to begin empirically defining and refining conceptions of mathematics-specific belongingness. Implications for research and practice with respect to mathematics identity are shared.

A qualitative multiple case study was conducted to reveal the sense-making processes third-grade students who struggle in mathematics used to build an understanding of fraction concepts. Purposive sampling identified three participants who were struggling in a local school's third grade mathematics classes. This research describes how these participants made sense of fraction concepts through their strengths and struggles while engaged in 15 small-group intervention sessions. Vygotsky's (1934/1986/2012) theory that children's optimal learning is supported by teacher-student interactions was used as an interpretive framework. Tasks were developed over the course of the intervention sessions with consideration of a model developed by Lesh, Post, and Behr (1987) for connecting mathematical representations and the Common Core State Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Data, including transcripts, tapes, and artifacts, were analyzed using two frameworks. These were Geary's (2003) classification of three subtypes of learning disabilities in mathematics and Anghileri's (2006) descriptions of social-constructivist scaffolding techniques. The first analysis resulted in a description of each participant's strengths and struggles, including alignment with Geary's subtypes, and how these strengths and struggles interacted with participant's construction of knowledge about fractions. The second analysis described episodes of learning that were supported by social-constructivist scaffolding techniques and revealed how participants made sense of fractions through their interactions with each other, the researcher, and intervention tasks. The researcher found that each participant's learning process, including struggles, was unique, with each interacting in different ways with tasks, manipulatives, pictorial representations, and questioning. For each participant, however, scaffolding techniques oriented around prompting and probing questions, participant verbalizations, and interactions with connected fraction representations were critical in their learning process.

While we are in the midst of a renaissance of interest in augmented reality (AR), there remain a small number of application domains that have seen significant development. One domain that often benefits from additional visualization capabilities is education, specifically physics and other sciences. This paper summarizes interviews with secondary school educators about their experience with AR and their most desired features. Three prototypes were created which were used to collect usability information from students and educators about their preferences for AR applications in their physics courses. Additionally, we introduce the concept of Environmental Integration, a novel method of defining mixed reality applications based on three properties: Visualization, Input Fidelity, and Spatial Understanding. Several examples are presented to illustrate different levels of environmental integration. The results of the studies conducted point towards interesting areas for further exploration for AR content creation for physics education.