The purpose of this study is to describe how university honors calculus students negotiate meaning and language for conceptually important ideas through mathematical discourse. Mathematical discourse has been recognized as an important topic by mathematics education researchers of various theoretical perspectives. This study is written from a perspective that merges symbolic interactionism (Blumer, 1969) with personal agency (Walter & Gerson, 2007) to assert that human choice reflects, but is not determined by, meanings that are primarily developed through social interaction. The process of negotiation of meaning is identified, described, and analyzed in the discourse of four students and their professor as they draw conclusions about the volume of water in a reservoir based on graphs of inflow and outflow. Video data, participant work, and transcript were analyzed using grounded theory and other qualitative techniques to develop three narrative accounts. The first narrative highlights the participants' use of personal pronouns and personal experience to negotiate meaning for the conventional mathematical terms "inflection" and "concavity." The second narrative describes how the participants' choices in discourse reflect an effort to represent both their mathematical and experiential understandings correctly as they negotiate language to describe critical "zero points." The third narrative describes the participants' process of mapping analogical language and meaning from the context of motion to the context of water in a reservoir. Analysis of these three narratives from the perspective of conventional and ordinary mathematical language suggests that the contextualized study of mathematics may provide students access to mathematical discourse if the relevant mappings between mathematical language and language from other appropriate contexts are made explicit. Analysis from the perspective of social speech (Piaget 1997/1896) suggests that specific uses of personal pronouns, personal experience, and revoicing (O'Connor & Michaels, 1996) may serve to invite students to become participants in mathematical discourse. An agency-based definition of mathematical discourse is suggested for application in research and practice.

Nearly all research that seeks to assist in reforming mathematics student teaching in the United States has been limited in that it (1) does not consider student teaching models in non-Western cultures, and (2) has not sufficiently studied the unique context of mathematics in student teacher-cooperating teacher interactions. This multi-case study addresses these issues by analyzing the interactions between three cooperating teachers and two student teachers in the mathematics student teaching setting in Japan. Four conclusions are presented to generate a coherent picture of the principles of teaching and learning to teach that are emphasized during this Japanese student teaching experience.

Within the current mathematics teacher population there are teachers that want to change from traditional teaching styles to become more reform-oriented (i.e. focusing on student understanding rather than procedures). Many of these teachers do not know how to begin this change. This research looks into the tools that are most valuable for teachers as they change from traditional teaching practices to include more reform-oriented teaching practices. Through this phenomenological study, six successful reform-oriented teachers were interviewed to understand what tools they found to be most valuable in their process of change. The interviews uncovered a common guiding principle that facilitates successful change towards reform teaching—focusing on the students' mathematics. This guiding principle led all the teachers to implementing task-based lessons and improving their questioning towards their students. The two tools found to be of most value, reflection and collaboration, are identified and explored. The implications of a reform curriculum are also discussed. Limitations of the study are identified and areas of future research are explored.

Teaching mathematics is a difficult and complicated task. For student teachers, who are extremely new to the mathematics classroom, this difficulty is magnified. One of the biggest challenges for student teachers is learning how to effectively use the student thinking that emerges during mathematics lessons. I report the results of a case study of two mathematics education student teachers. I focus on how they make decisions while teaching in order to use their students' mathematical thinking. I also present analysis of the student teachers' discourse patterns, the reasons they gave to justify these patterns, and how their reasons affected how they used their students' thinking. I found that generally the student teachers used student thinking in ineffective ways. However, the reasons the student teachers gave for using student thinking always showed the best of intentions. Though given with the best of intentions, most of the reasons for using student thinking given by the student teachers were correlated with the student teachers ineffectively using their student's thinking. However, some of the reasons given by the STs for using student thinking seemed to help the student teachers more effectively use their students' thinking. I conclude with implications for preparing future student teachers to better use student thinking.

This study was conducted in order to determine how mathematics teachers allocate their time in the classroom and the factors that influence teacher priorities in that time allocation. Research has indicated that math teachers may choose not to do certain activities in their classroom because of the amount of time that they take, but other underlying reasons may exist. In order to study this idea, six math teachers were interviewed on their current time allocation and rationale for that allocation, and the results from these interviews were used to create a survey that was sent to 581 math teachers in Utah. The results from the 224 completed surveys showed that many teachers allocate their classroom time in a fairly traditional manner, with an average of about 10% of class time being spent on student-centered activities. 40.63% of teachers spent 0% of their class time on student-centered activities. There is variation in time allocation and influencing factors based on a teacher's schedule, level of teaching, experience, and how student-centered their teaching methods are. Also, the results support the claim that there are factors, other than limited class time, that affect how teachers choose whether or not to do certain activities. Some of the most significant deciding factors found are whether the activities will help students with their end of level tests, if they will keep students working hard mathematically, whether others are using those activities or not, how the activities affect classroom rowdiness. It was also found that teachers who are more teacher-centered tended to choose activities based on how easy they were to implement, including their personal comfort level, ease of preparation, and ease of management with student behavior. More student-centered teachers tended to care more about keeping the students working hard mathematically.

Bootstrap is a computer-programming curriculum that teaches students to program video games using Racket, a functional programming language based on algebraic syntax. This study investigated the relationship between learning to program video games from a Bootstrap course and the resulting effect on students' understanding of algebra. Courses in three different schools, lasting about six weeks each, were studied. Control and treatment groups were given a pre and post algebra assessment. A qualitative component consisting of observations and interviews was also used to further triangulate findings. Statistical analysis revealed that students who completed the Bootstrap course gained a significantly better understanding of variables and a suggestive improvement in understanding functions. In the assessments, students failed to demonstrate a transfer of the advanced concepts of function composition and piecewise functions from programming to algebraic notation. Interviews with students demonstrated that with coaching, students were able to relate functions written in Racket to functions written in algebraic notation, but were not yet able to transfer their experience of function composition from programming to algebra.

This study examines teacher questioning with the purpose of identifying what types of mathematical questions are being modeled by the teacher. Teacher questioning is important because it is the major source of mathematical questioning discourse from which students can learn and copy. Teacher mathematical questioning discourse in a conceptually-oriented classroom is important to study because it is helpful to promote student understanding and may be useful for students to adopt in their own mathematical questioning discourse. This study focuses on the types of questions that comprise the mathematical questioning discourse of a university teacher in a conceptually-oriented mathematics classroom for preservice elementary teachers. I present a categorization of the types of questions, an explanation of the different categories and subcategories of questions, and an analysis and count of the teacher's use of the questions. This list of question types can be used (1) by conceptually-oriented teachers to explicitly teach the important mathematical questions students should be asking during mathematical activity, (2) by teachers who wish to change their instruction to be more conceptually-oriented, and (3) by researchers to understand and improve teachers' and students' mathematical questioning.

Research has shown that preservice teachers participating in traditional student teaching programs tend to focus on classroom management, with very little focus on student mathematical thinking. The student teaching program at BYU has been redesigned in the hopes of shifting the focus of student teachers away from classroom management toward student mathematical thinking. This study compared conversations between student teachers and cooperating teachers before and after the redesign of the program to work towards determining the effectiveness of the refocusing of the new student teaching program. The study found that STs and CTs in the different student teaching structures were talking about different things. Not only were the frequencies of conversations about pedagogy, students, and mathematics different, but the ways those individual topics were discussed was also different.

The tasks with which students engage in their mathematics courses determine, for a large part, what students learn. Therefore, it is essential that teachers are able to design tasks that are worthwhile for developing mathematical understanding. Since practicing teachers seldom incorporate worthwhile mathematical tasks in their lessons, we would expect that they did not become proficient at designing worthwhile tasks while in their teacher education programs. This thesis describes a study that investigated what preservice secondary teachers attend to as they attempt to design worthwhile mathematical tasks. Three participants were selected from a course at a large private university where preservice teachers are taught and practice the skill of task design. This "Task Design" course was observed, and the three participants were interviewed to determine what they attend to while designing tasks. There were seven main characteristics that the main participants in the study attended to the most often and thought were the most important: sound and significant mathematics, reasoning, appropriateness, clarity, communication, engagement, and openness. How the participants attended to these characteristics is described. Some implications for teacher education, such as requiring preservice teachers to explain how their tasks embody certain characteristics, are given based on the results.

Teacher telling is underrepresented in the mathematics education reform literature and deserves additional scrutiny. This case study examined a skilled teacher's telling practices during a university mathematics content course for pre-service elementary teachers. I identified telling practices through discourse analysis, attending to the presence of mathematics and the contribution of new structure or ideas from the teacher. The teacher utilized seven unique types of mathematical telling while supporting collaborative group work on tasks. The study identified subtle telling, implicit telling, and explicit telling. The results suggest that mathematical telling is an integral part of the teacher's role in inquiry-based instruction and should not be overlooked. Researchers can use the telling types to identify and describe telling practices more transparently. Practitioners can more consciously incorporate and discriminate between telling practices.