The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.

The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.

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Mathematics education could and should benefit from technology in order to improve teaching and learning, particularly in topics where visualizations and connections to other concepts are needed. Handheld technologies such as graphing calculators can provide students with visualization, confirmation and exploration of problems and concepts they are learning. Handheld graphing technologies have been taken place widely in elementary and secondary level mathematics courses and considered to be beneficial in various means in mathematics education. Mathematics teachers have a crucial role in the use of GCs in mathematics classrooms. Therefore, pre-service teachers&rsquo / use of GCs and their views on the use of the tool in mathematics learning are considered to be valuable. The purpose of this study was to investigate the difficulties pre-service elementary mathematics teachers face, and the benefits and constraints they emphasize while learning elementary school algebra through using the Casio Classpad after receiving an instruction with graphing calculators. The graphing calculator used in the present study is the Casio Classpad 330, which is an evolved handheld device combining features of graphing calculators, dynamic geometry environment, computer algebra systems and more. The following two research problems guided the study: What are the difficulties do pre-service elementary mathematics teachers face while using Classpad in learning elementary school algebra after receiving an instruction with graphing calculators? What benefits and constraints do pre-service elementary mathematics teachers emphasize while learning elementary school algebra through using Classpad after receiving an instruction with graphing calculators? With the aim of investigating the views of a group of pre-service elementary mathematics teachers, qualitative research strategies were used. The data was collected and analyzed by means of a case study design. Classroom observations, a questionnaire, and focus group interviews were the main data sources of the existing study. The study was carried out with 21 pre-service elementary mathematics teachers. In the classroom studies elementary level algebra was taught to the participants with the use of Classpad as a main tool by giving one tool to each of the participants. Classroom observations ended in five weeks &ndash / 20 courses &ndash / including one week of a training period. After the classroom observations, participants filled out a questionnaire including five open-ended questions about the classroom studies. Finally, data collection procedure was ended with three focus group interviews. The data was analyzed with qualitative means by transcribing and analyzing the observation records, answers of the questionnaire, and records of the three interviews. Results revealed that pre-service teachers&rsquo / view Classpad in three categories / as a personal tool, as an educational tool, and the relationship between CP and motivation. They viewed CP as a personal tool that they were eager to use the tool in every level of mathematics from elementary to mastering degrees. As an educational tool, they preferred to use the tool as a teacher by giving some cautions that teachers and students should be careful with. Lastly, they considered that the tool has a positive effect on motivation when used appropriately. Pre-service elementary mathematics teachers faced some difficulties in the beginning courses of the classroom studies which was their learning period of how to use CP and they overcome most of the difficulties at the end of the classroom studies. As the new elementary school level mathematics curriculum encourages the use of various technologies in teaching and learning of mathematics, the results of this study will have useful implications for mathematics teachers and curriculum developers.

The Florida Department of Education's Next Generation Sunshine State Standards were adopted in 2008 as a means of increasing the rigor of mathematics learning and teaching mathematics to depth. In this study teachers' beliefs about developmentally appropriate tasks were examined through a web-based survey. Participants' responses to the survey were compared to the Florida Department of Education's Next Generation Sunshine State Standards. These standards were created in accordance to the 2006 National Council of Teachers of Mathematics Curriculum Focal Points in Mathematics Prekindergarten through Grade 8. This study included 25 teachers from a large metropolitan school district in Florida. Results demonstrate that a difference does exist between teachers' beliefs regarding when certain tasks should appear in the curriculum and what is advocated by the Next Generation Sunshine State Standards. Teachers' demographic information such as years of experience, employment by Title 1 school, and number of different grades taught, were not found to be significant predictors of their responses to prompts regarding developmentally appropriate tasks.

This dissertation explores elementary mathematics teachers' understanding and implementation of effective questioning techniques within the instructional context. A qualitative, multiple case study design was employed for this dissertation in practice. This study was conducted in a K-5 school in Florida, focusing on four elementary school teachers in grades 3-5. Data collected included semi-structured interviews and live observations. Analysis of transcripts from interviews and observational notes provided an insider look into teachers’ planning and implementation of questioning. The research was driven by two research questions: How do teachers implement questioning in mathematics instruction? and What factors influence teachers' planning and implementation of questioning in mathematics instruction? Through the examination of these questions, the study sought to unveil critical insights into the planning processes, the types of questions incorporated during instruction, and teachers' conceptualization of what constitutes effective questioning. The findings of this dissertation revealed three themes which were 1) factors influencing questioning, 2) teachers’ understanding of effective questioning and professional development, and 3) alignment with learning objectives and student diversity. By identifying areas that warrant further examination, the study contributes ideas tailored to enhance the effectiveness of questioning in mathematics instruction. This dissertation in practice research aspires to guide instructional practices and make significant contributions to the broader landscape of mathematics education. In conclusion, this dissertation offers a nuanced exploration of the multifaceted dimensions of questioning in mathematics instruction, with the intention of providing practical recommendations for educators and contributing to the ongoing discourse in the field of mathematics education.

This ethnographic study focused on understanding the culture and describing the experiences of elementary mathematics teachers as they used a mandated curriculum and scope and sequence at a state-supervised school. The scope and sequence was created and provided by the school district with the expectation of implementation with high fidelity by the Bureau of School Improvement within the state of Florida. This study used key informant interviews, Professional Learning Community observations, and classroom observations of mathematics teachers in grades three through five to analyze and describe how the school culture of planning and instruction influenced mathematics teaching. Findings suggest that strict expectations and frequent observations from the district and state impacted the school culture, which put pressure on teachers to follow the mandated curriculum with high fidelity. Despite the mandates and strict expectations, many teachers still perceived to have some autonomy to adapt their instruction with the goal of meeting students’ needs. Participants prioritized what they considered most important from the district resources. This led to changes related to pacing and small group instruction. The discussion of the findings connects to previous research on how mandated curriculum impacts teacher autonomy and how school culture is developed and shaped. Recommendations for future research and practice focused on increasing mathematics instructional time and creating equitable structures for students where they can build deep mathematical understanding in a flexible space of time.

ABSTRACT EFFECTS OF GRAPHING CALCULATORS ON EIGHTH GRADE STUDENTS&amp / #8217 / ACHIEVEMENT IN GRAPHS OF LINEAR EQUATIONS AND CONCEPT OF SLOPE &Ouml / n&uuml / r, Yurdag&uuml / l M.S., Department of Elementary Science and Mathematics Education Supervisor: Assist. Prof. Dr. Ayhan K&uuml / rSat ERBAS May 2008, 76 pages The purpose of this study was to investigate the effects of graphing calculators on eight grade students&amp / #8217 / achievement in graphing linear equations and concept of slope. Pretest-posttest experimental-control group design was utilized in the study. While the students in experimental group (EG) received instruction about graphs of linear equations and concept of slope with graphing calculators, the students in control group (CG) was taught the same topics without using graphing calculators. There were 27 students (13 girls and 14 boys) in each group. Students in both EG and CG was administered an achievement test (i.e., MAT) consisting of questions related to graphing linear equations and slope concept before and after the instruction. Additionally, the teacher and six students from the EG were interviewed. The data obtained from students&amp / #8217 / post test scores of MAT were analyzed by Analysis of Variance (ANOVA). A statistically significant difference was found between the achievements of students in experimental and control groups. However, gender had no statistically significant effect on students&amp / #8217 / post test scores of MAT. Additionally, students&amp / #8217 / pre-test scores of MAT and their mathematics grades of the second semester of the seventh grade (MGS) were analysed by independent samples t-test. The results showed no statistically significant difference. On the other hand, the analysis of interview data revealed that graphing calculators affected students&amp / #8217 / attitudes towards mathematics in a positive way. Students had no considerable difficulty while using graphing calculators and they found studying with graphing calculators enjoyable. In summary, the results of the study showed that when graphing calculators used at elementary school level, they had positive effects on students&amp / #8217 / achievement and in some respects to their attitude. Consequently, integration of graphing calculators to elementary mathematics curriculum may be beneficial for students and teachers.

The purpose of this study was to trace the reflections of critical issues, such as neo-liberalism, cultural differences based on social class, gender stereotyping, and nationalism in the elementary mathematics education in Turkey. Critical discourse analysis was conducted to examine these possible reflections. By researching mathematics education from a critical perspective, this study aimed to contribute constructing a starting point for socially responsible mathematics education. There were four main data sources in the study: elementary mathematics curriculum, 6th, 7th, and 8th grade elementary mathematics textbooks, workbooks and teacher&rsquo / s guide books, 7th grade mathematics classroom observations, and pre- and post-interviews with participant teacher. The discourse analysis of mathematics education contexts implied that elementary mathematics discourse: (i) oriented students to use their mathematical abilities and skills for the benefit of private corporations instead of public welfare / (ii) replaced the &lsquo / real life&rsquo / in mathematics problems with the life of middle and upper middle classes / (iii) included sexist expressions / and (iv) fostered nationalism via ignoring ethnic and non-Muslim groups living in Turkey. It appeared that teachers might not be aware of such discourse. Findings have addressed that policy makers and textbook writers should consider these critical issues in order to reach all students and teachers&rsquo / awareness should be increased. Future research should clarify these issues in a broad sense including pre-service teachers, teachers, students, and mathematics instruction in schools.

This longitudinal study tracks how 5- to 7-year-olds perform with early number skills. The aim of this study is to diagnose at-risk mathematics students by distinguishing the skills that, if not mastered by the end of Kindergarten, lead to greater difficulty in mathematics in 1st grade. This study’s methodology is mixed as it follows an exploratory and inductive path in light of its use of a hypothesis, an interpretive path in light of its interest in the individual student, and a positivist path in light of its focus on developing rules from analyzed data. An oral diagnostic test based on a comprehensive collection of early number skills was used to test students as Kindergarteners and again as 1st graders. The test results created benchmarks, revealing how the majority of the students performed with early number skills. The test results also revealed that each early number skill is highly, moderately, or minimally predictive in terms of student placement by the end of 1st grade. When comparing the individual skill scores of each Kindergarten student to his/her total test results of 1st grade, the predictive power of each skill emerged. Performing poorly with skills that are minimally predictive did not seem to have an impact on how the Kindergarten student finished in 1st grade; performing poorly with moderately predictive skills had a greater impact on 1st grade placement; performing poorly with highly predictive skills in Kindergarten increased the likelihood that the student would finish in the lower attaining group in 1st grade. A third result of the test showed that certain skills serve as preconditions for other skills; success with certain skills usually meant success with other skills. These connections between skills point to a learning model called in this study “simultaneous pathways,” indicating that there are connections between certain skills, and that students can be learning on several pathways simultaneously. The impact of the predictive power of early number skills is that diagnosis becomes more effective. Early diagnosis means early remediation which may prevent at-risk students from falling further behind their peers. The benchmarks developed by this research will help teachers assess their students because they will know the general skill level of Kindergarteners and 1st graders. This oral diagnostic test informs curriculum development. If test results show that students are missing the skills that are highly predictive, teachers can address those gaps in order to insure mastery.