In the past, frameworks for what it means for students to understand elementary mathematical concepts like addition have been well-researched. These frameworks are useful for identifying what students must understand to have a good grasp of the concept. Few such research-based frameworks exist for secondary mathematical topics. The intent of this study was to create such a framework for what it means for students to understand logarithms, a topic that has been under-researched up to this point. Four task-based interviews were conducted with each of four different preservice secondary mathematics teachers in order to test a preliminary framework I had constructed to describe what it means for students to understand logarithms. The framework was adjusted according to the findings from the interviews to better reflect what it means for students to have a good understanding of logarithms. Also, a common practice taught to students learning logarithms, switching from logarithmic form to exponential form, was found to possibly have negative effects on student understanding of logarithms. The refined, research-based framework for what it means for students to understand logarithms is described in full in this document. The implications of the results of this study for mathematics teachers as well as for mathematics education researchers are also discussed.

The purpose of this study was to describe how manipulatives facilitated the growth of one group of high school students' mathematical understanding of combinatorics and Pascal's Triangle. The role of manipulatives in mathematics education has been extensively studied, but much of the interest in manipulatives is focused on the general uses of manipulatives to support student learning. Unfortunately, there is a lack of research that explicitly defines how manipulatives can help students develop mathematical understanding. I have chosen to examine mathematical understanding through the lens of the Pirie-Kieren Theory for Growth of Mathematical Understanding. Through analysis of the students' explorations of the Towers Task, I identified ways in which manipulatives facilitated students' understanding of combinatorics and Pascal's Triangle. It was found that the properties and arrangements of the manipulatives were significant in prompting students' progression through levels of understanding and helped students to reason abstractly and develop mathematical generalizations and theories. From this study we can gain insights into explicit ways in which manipulatives facilitate mathematical understanding. These results have implications for research, teaching and teacher education.

Mathematics students and teachers are familiar with the difficulty of learning and teaching concepts of continuity and limits. Research has expanded our knowledge of how students think about these concepts, including different conceptions and metaphors students use to reason about continuity and limits at a point. From the literature I have identified four potentially problematic conceptions (PPCs) students may use when reasoning about limit and continuity at a point. Questionnaires were administered to 861 BYU students in various mathematics courses to determine how prevalent and persistent the PPCs are among the students in each course. Interviews were conducted with nine first semester calculus to get an idea of how students reason about continuity and limit at a point and how that influences whether they use the PPCs. Students showed evidence of holding the four PPCs with a decrease in these conceptions typically after they took a course in analysis. Participants also did not understand the Formal definition of a Limit until they took a course in Analysis. Students were able to reason appropriately using many different conceptions of continuity. Considering limit conceptions, students using a Dynamic conception of Limit tended to be better able to reason about continuity and limit at a point. Students who did not use a Dynamic conception of limit tended to use the PPCs in general and incorrectly more often.

This qualitative study analyzes a professional development course and its associated influence on the induction of a beginning mathematics teacher from a sociocultural perspective. Specifically, it examines whether a specific high school mathematics professional development course formed a community of practice through the elements of mutual engagement, joint enterprise, and shared repertoire. A community of practice is an inherently sociocultural framework. The results show how each element was present in the professional development, indicating that a community of practice had formed. Using those three elements of community of practice, the study further analyzes the induction of one first-year teacher, Sarah, who was a participant in the community of practice. Sarah's induction is framed as consisting of her conformance to the school mathematics department accepted by both her colleagues and herself, as well as her transformation of the system. The results of the study demonstrate how each element of the community of practice influences each aspect of Sarah's induction into the mathematics department at her school.

Research has suggested that although mathematics is a universal language, the notation, algorithms, and language in which it is taught varies from country to country. The literature has addressed the challenges associated with understanding the language of mathematics in English. Variable is a topic of mathematics where the literature has documented students' difficulty understanding the language of mathematics. Part of that difficulty is attributed to the fact that mathematics and English each have their own register and at times the differences in these registers conflict. This study examined the introduction and use of variable in the middle school textbooks of three English speaking countries and three Spanish speaking countries. The results of this study are that textbooks rarely make any attempt to use bridging language, and do not explain to students how they are using variable when they use it. The results of this study also show that the language used to talk about variable is similar English (Spanish) speaking countries.

The National Science Education Standards (1996) support inquiry-based instruction. According to the National Science Education Standards. When children or scientists inquire into the natural world they: ask questions, plan investigations and collect relevant data, and organize and analyze collected data. The purpose of this study was to investigate the effects of inquiry-based instruction on third-grade students' attitudes and participation in an elementary science classroom. Students were encouraged to ask and answer their own questions. In this study, analysis of data gathered form: pre and post survey, student journals, teacher field notes, and student interviews were triangulated to provide the support for findings reported in this study. Findins showed that inquiry-based science experiences positively affected students' attitudes in science and their participation. In addition, student worked collaboratively, made connections to other experiences, and demonstrated confidence in their ability to ask and answer their own questions through inquiry-based experiences.

This study summarizes research conducted in a second grade classroom at a rural elementary school in the fall of 2004. This study investigated the practice of using writing activities in mathematics to improve student attitudes and performance in problem solving. The classroom teacher supplemented traditional mathematics instruction with daily problem solving activities and affective journal writing. Students were asked to complete daily problem solving prompts and write about their problem-solving solutions. Attitude data was collected using a pre and post attitude survey as well as affective journal writing assignments. Performance data was collected using a performance based problem-solving rubric. Results of this study showed change in students' attitudes towards problem solving in the areas of willingness to participate and perseverance in completing problem solving tasks. Student performance gains were recorded and analyzed throughout the six-week study period. Thirteen out of the 17 students who participated in this study showed performance growth in problem solving.

The purpose of this study is to describe the effect of a curriculum rich in spatial reasoning activities and experiences on the ability of my fifth grade students to spatially reason. The study was conducted to examine 1) the effects of my practice of incorporating spatial reasoning lessons and activities in my fifth-grade mathematics classroom on the students' ability to spatially reason and 2) the effects of my practice of incorporating spatial reasoning lessons and activities on my students' ability to problem solve. Data were collected over a ten-week period through the use of student interviews, anecdotal records, photos of student work, student journals, pre- and posttests and a post-study survey. In this study, students demonstrated a statistically significant increase on all pre- and posttests. The student interviews, anecdotal records, photos of student work, and student journals all revealed spatial reasoning was used in mathematics problem solving. The study suggests that spatial reasoning can be taught and spatial reasoning skills can be used in problem solving.

The purpose of this study was to investigate how my use of discourse and journal writing affected students' attitudes and conceptions of mathematics in a 5th grade mathematics classroom. The nature of students' strengths and weaknesses with mathematics using discourse and journal writing were described. To show students' attitudes towards the two teaching methods transcription of teacher and student discourse on digital audio recordings, observational notes, journal writing, and pre and post attitude surveys were used. The research approach was qualitative and quantitative. The participants in this twelve-week study were twenty fifth grade students from a private school in central Florida. Through analysis of the data collected, the students showed positive improvement in attitude towards discourse and journal writing in the mathematics classroom. The study supported that discourse and journal writing are important to student learning.

This qualitative study examined how my practice of using manipulatives to teach multiplying and dividing fractions to 8th grade students facilitated their conceptual understanding of those operations. The students who participated in the study were enrolled in one of my intensive mathematics classes. Before the lessons began, I interviewed the students and gave them a pre-assessment to determine their content knowledge and comfort level with manipulatives. The students engaged in activities that included solving problems using various manipulatives. During the activities, I made observations of their problem solving techniques and how they used the manipulatives. At the conclusion of the unit I gave them a post assessment and conducted post interviews to determine any change in their content knowledge and comfort level with using manipulatives. I concluded through my research that by giving the students a hands-on, minds-on approach to learning they were able to develop an understanding of the concepts and apply that knowledge to multiplying and dividing fractions.