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We aim to identify what prospective mathematics teachers indicated that they learned regarding mathematics and mathematics teaching as they participated in a lesson study. The theoretical framework is supported by models about teachers' knowledge and prospective teachers learning processes. The methodology is qualitative and interpretative, with data collected by interviews and documents made by participants. The participants indicated to understand the importance of different representations in mathematics. They also indicated to appreciate the value of the detailed planning of the lesson, the careful selection of tasks, the analysis of possible student strategies and difficulties, and the preparation of classroom communication. [ABSTRACT FROM AUTHOR]

The research, guided by the question "What professional knowledge is mobilized in a Calculus lesson study conducted with university instructors?", involved eight instructors from higher education institutions in southern Brazil who are dedicated to the training of prospective mathematics teachers. The lesson study, structured in twelve weekly two-hour meetings, addressed the topic "maxima and minima". The qualitative analysis showed that the approach and dynamics of lesson study mobilized and deepened professional knowledge regarding mathematics, mathematics didactics and the mission of the trainer of future mathematics teachers. [ABSTRACT FROM AUTHOR]

We discuss how collaborative relationships may develop in lesson study groups, paying attention to the critical incidents that may arise. This qualitative research study is based on two lesson studies with primary and middle school teachers. Data were collected through a researcher's journal, audio records of working sessions, video records of research lessons, post lesson reflections, and individual interviews. In the two cases, the collaborative relationships developed as common aims and ways of productive joint work emerged. In the development of these relationships, several critical incidents had a strong influence, in some cases positive and in other cases negative. [ABSTRACT FROM AUTHOR]

This article presents the results of an investigation inserted in an intervention cycle of a design based research (DBR) on the development of didactic knowledge of mathematics teachers, namely with respect to statistics. Its aim is to understand how the articulation of a set of design principles in the context of a teacher education experience can contribute to promote the didactic knowledge of the participants. Data analysis is based on the following design principles: use of exploratory tasks; use of authentic classroom situations; and use of technology. The results suggest that the design principles assumed in the teacher education experience contributed to the development of the teachers' knowledge regarding the potential of the exploratory approach to statistics, in order to expand their knowledge about the tasks that can make a contribution to this type of teaching. [ABSTRACT FROM AUTHOR]

This paper focuses on a professional development process framed by an exploratory curricular perspective which values an orientation towards practice and the collective reflection among the participants. Its goal is to analyze the changes that teachers refer to have felt in their perspectives about teaching as well as how they regard the work carried out. The theoretical framework is based in a conception of professional practice that assumes the central role of the concepts of task and communication and a perspective about the design of professional development frameworks based on the exploratory approach. The participants are 19 teachers from grades 1 to 6. The study is based on a researcher's journal, an open response questionnaire and interviews. The results show that the participants came to enhance exploratory learning and whole class discussions and to assume a high expectation regarding the students' abilities. Concerning the work carried out, the participants highlight the connection with their practice and the moments of sharing experiences. [ABSTRACT FROM AUTHOR]

This paper analyses grade 6 pupils' mathematical processes and difficulties in solving proportion problems before the formal teaching of this topic. Using a qualitative methodology, we examine pupils' thinking processes at four levels of performance in missing value and comparison problems. The results show that pupils tend to use scalar composition and decomposition strategies in missing value problems and functional strategies in comparison problems. Pupils' difficulties are related to a lack of recognition of the multiplicative nature of proportion relationships. [ABSTRACT FROM AUTHOR]

Prospective mathematics teachers often have learned mathematics through situations quite different from current curriculum recommendations. Such is the case of Fábio, who, at the beginning of his practicum, proposes mostly exercises and uses work at the blackboard. Assuming that didactic knowledge is reformulated from practice, this article analyzes the influence of a teacher education setting based on ICT (e-mail and forum) in developing aspects of such knowledge -- tasks proposed and ways of using technological materials. The qualitative and interpretative case study of this prospective teacher is based in interviews and in the analysis of the documentation generated in the teacher education setting. The results show that ICT enabled sharing and discussing classroom experiences and joint work with other prospective teachers, supervisor and mentor. Besides exercises, this prospective teacher begun proposing students tasks more open and challenging as well as the exploration of technological materials. His reflections regarding the tasks and the ways to use the technological materials in his classes contributed to enlarge his didactic knowledge and to regard his own practice as learning setting. [ABSTRACT FROM AUTHOR]

This study analyses the relationship between solving exploratory and investigation tasks involving functional relationships and the development of algebraic thinking in grade 8 students, giving special attention to the way they interpret and use the algebraic language. The methodology, qualitative and interpretative, is based in two case studies of students involved in a teaching unit of 16 classes which included the study of numerical sequences, functions and first degree equations. The data collection involved two interviews (one carried before and the other after the teaching unit), the participant observation of the classes by the teacher, registered in her diary, and gathering of students' written records. The results show that the emphasis in the study of functional relationships based on exploratory and investigation tasks promotes the development of meaning for the algebraic language and the construction of a wider vision regarding the use of symbols. [ABSTRACT FROM AUTHOR]

Education professionals face countless problems in their practice. Instead of waiting for external solutions, many of them investigate directly those problems. This paper discusses the meaning of such investigation for teacher education and constructing professional knowledge. To illustrate such perspective, I analyse the journey of Grupo de Trabalho de Investigação (GTI) of the Associação de Professores de Matemática (APM), presenting two cases of teachers' investigations about their own practice. I also discuss problematic issues in this kind of investigation, notably paradigmatic kinship, methodology, role and characteristic difficulties of collaboration and the institutional conditions and professional culture necessary to undertake it. [ABSTRACT FROM AUTHOR]