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Student autonomy has been an important object of study for mathematics educators for many years. Over time, framings of autonomy have moved from a focus on the individual to considerations of how an individual's autonomy is entangled in classroom-level interactions. What has been less closely studied is how classroom interactions provide uneven access to autonomy for individuals. This study uses a communicational perspective to clarify Piaget's intellectual autonomy and examine students' mathematical interactions. The findings describe barriers and supports to autonomous activity for three students. Students were prevented from engaging in autonomous activity when they were seen as less capable than others, when they felt the need to manage the activities of their peers, or when they focused on being seen as knowledgeable. In contrast, students acted with autonomy when they took up the teacher's request for explanations, noticed a contrast between their answer and the right answer, and worked on making connections across different representations. [ABSTRACT FROM AUTHOR]

This paper concerns proof presentation at the university level. We report on a study in which we observed ten mathematicians constructing or revising proofs for pedagogical purposes. We highlight the factors that they claimed to consider when completing these tasks. We found that intended audience and medium (lecture or textbook) influenced proof presentation. We also found that, although mathematicians generally valued pedagogical proofs featuring diagrams and emphasizing main ideas, these mathematicians did not always incorporate these aspects in the proofs they constructed or revised. [ABSTRACT FROM AUTHOR]

This paper provides a microanalysis of one Algebra I teacher's instruction to explore the advantages that are afforded us by coordinating two perspectives to document and account for the teacher's mathematical understandings. We use constructs associated with Stein, Grover and Henningsen's domain of mathematical didactics and Realistic Mathematics Education's instructional design theory to infer what the teacher might understand to effectively implement her instructional goals and, more importantly, support student learning. By coordinating these perspectives, we developed a working framework for analyzing the teacher's classroom practice retrospectively. For example, we illustrate how the mathematical possibilities related to one student's question might inform the teacher's decisions as she initiates shifts in students' self-generated models. Additionally, we illustrate how the teacher's decision to capitalize on particular students' models contributes in part to the kinds of mathematical ideas that can be explored and the connections students can make among those ideas. More generally, we explore the utility of coordinating these two perspectives to understand the landscape of ideas that teachers might traverse to align their practices with reform recommendations in the United States. [ABSTRACT FROM AUTHOR]

This paper offers for discussion and critique a conceptual framework that applies a situative perspective on learning to the study of learning to teach mathematics. From this perspective, such learning occurs in many different situations -- mathematics and teacher preparation courses, pre-service field experiences, and schools of employment. By participating over time in these varied contexts, mathematics teachers refine their conceptions about their craft -- the big ideas of mathematics, mathematics-specific pedagogy, and sense of self as a mathematics teacher. This framework guides a research project that traces the learning trajectories of teachers from two reform-based teacher preparation programs into their early teaching careers. We provide two examples from this research to illustrate how this framework has helped us understand the process of learning to teach. [ABSTRACT FROM AUTHOR]

The purpose of this cross-national study is to understand teachers' views about the meaning of instructional coherence and the ways to achieve instructional coherence. With respect to the meaning of instructional coherence, whereas the majority of U.S. teachers paid attention to connections between teaching activities, lessons, or topics, the majority of Chinese teachers emphasized the interconnected nature of mathematical knowledge beyond the teaching flow. U.S. teachers expressed their views about ways to achieve instructional coherence through managing a complete lesson structure. In contrast, Chinese teachers emphasized pre-design of teaching sequences, transitional language and questioning based on the study of textbooks and students beforehand. Moreover, they emphasized addressing student thinking and dealing with emerging events in order to achieve 'real' coherence. The findings of the study contribute to our understanding about the meaning of instructional coherence and ways to achieve instructional coherence in different cultural contexts. [ABSTRACT FROM AUTHOR]

This qualitative study examined perspectives of two key stakeholder groups, instructors and students, on mathematics content courses for prospective elementary teachers (Mathematics for Teachers [MFT] courses). A collective case study approach, which drew from the data of two cases in different but comparable settings, contributed to the robustness of the findings. Cross-case analysis of the interview data revealed several convergent themes: the role of affect in student learning, pedagogy and instructor disposition, connections to the elementary classroom, and mathematics content. The findings included both conflicting and complementary perspectives between the two key stakeholder groups. When juxtaposed, the multiple viewpoints offer insights into some of the central issues related to teaching and learning in MFT courses and suggest potential avenues for improving experiences in these courses. [ABSTRACT FROM AUTHOR]

In the United States, undergraduates-regardless of their field of study-generally must complete a mathematics course to meet breadth-of-study requirements. This report is aimed at providing a research foundation for practical efforts to improve teaching and learning in such college mathematics service courses (e.g., college algebra, liberal arts mathematics, business calculus). The case participant, Professor Kale, was a PhD mathematician with 12 years of college teaching experience, 6 years as a graduate student and 6 years after the doctorate. He and students in both of his classes agreed to the daily video recording of their meetings for an entire semester. Through constant-comparative analysis of videos and extensive interviews with Professor Kale, as well as brief interviews with his students and other members of the department, we derived a description of discursive patterns in Kale's classes. We conclude with possible implications for future work in college mathematics service course research and teaching. [ABSTRACT FROM AUTHOR]

The mixed method sequential nested study examines whether and how the cognitive type of teachers' content knowledge is associated with student achievement, and correlated with teaching practice. In the context of this study, the cognitive type refers to the kind of teacher content knowledge and thinking processes required to accomplish a task successfully, in terms of knowledge of facts and procedures (Type 1), knowledge of concepts and connections (Type 2), and/or knowledge of models and generalizations (Type 3). A sample of 102 middle school mathematics teachers (grades 6-8) was assigned to the study from a number of school districts in an urban area in the Southwestern US. Teachers were tested using the Teacher Content Knowledge Survey. Student level data of about 2,400 middle grades students' standardized test passing rates including percentage of students meeting the state standards by objectives were collected. The type of teachers' content knowledge was assessed and tested for association with student achievement on the state-mandated standardized test using multivariate methods including tests for variance and independence. The nested research consisted of three phases. Substudy-1 focused on quantitative analysis of the association between cognitive type of teacher content knowledge and student achievement. Substudy-2 aimed at the correlation between cognitive type of teacher content knowledge and teaching practice. Finally, substudy-3 was a case study on examining middle grades mathematics teachers' knowledge and understanding of fraction division. [ABSTRACT FROM AUTHOR]

In this study, we seek to describe how the meaning of a tool was co-constructed by the students and their teacher and how the students used the tool to construct mathematical meaning out of particular tasks. We report the results of a qualitative, classroom-based study that examined (1) the role, knowledge and beliefs of a pre-calculus teacher, (2) how students used graphing calculators in support of their learning of mathematics, (3) the relationship and interactions between the teacher's role, knowledge and beliefs and the students' use of the graphing calculator in learning mathematics, and (4) some constraints of the graphing calculator technology that emerged within the classroom practice. We found five patterns and modes of graphing calculator tool use emerged in this practice: computational tool, transformational tool, data collection and analysis tool, visualizing tool, and checking tool. The results of this study suggest that nature of the mathematical tasks and the role, knowledge and beliefs of the teacher influenced the emergence of such rich usage of the graphing calculator. We also found that the use of the calculator as a personal device can inhibit communication in a small group setting, while its use as a shared device supported mathematical learning in the whole class setting. [ABSTRACT FROM AUTHOR]