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This research aims to understand how prospective teachers develop their knowledge about whole-class discussions during a lesson study. Following a qualitative approach, we focus on the work of three prospective mathematics teachers preparing, leading, and reflecting on wholeclass discussions. Data was collected through participant observation with field notes, audio recordings, and document collection. The results show that a detailed analysis of a lesson plan, previously prepared autonomously by a prospective teacher based on a guide, promotes a focused discussion and a collaborative environment. Additionally, being involved in each other's formative process, through lesson study micro-cycles, created opportunities for prospective teachers to analyze and discuss different classroom situations, which promoted the development of their knowledge. Finally, classroom experiences seemed to be fundamental for the development of prospective teachers' knowledge. They were able to put into practice the work carried out during the lesson study, which allowed them to link theory and practice, and reflect on their own practice. This research also underlines the importance of the teacher educator's role in creating fruitful opportunities for the development of prospective teachers' knowledge.

This investigation adjusted the lesson study (LS) model for the initial teacher education aiming to examine its effect in the development of the pre-service teachers' pedagogical content knowledge (PCK) namely, both their knowledge and understanding on how to transform a science topic to make it more understandable for the students. The LS topic was the speed of sound that is taught in the 8th grade. The PCK construct was evaluated, as well as and in terms of both PCK-topic specific knowledge and PCK-pedagogical knowledge. The participants on this investigation were three pre-service teachers. The data were collected through video recording of LS sessions. The qualitative data collection was combined with quantitative and qualitative analysis procedures. The results showed that the LS promoted an improvement on pre-service teachers' PCK as well as in most of the aspects of both PCK-topic specific knowledge and PCK-pedagogical knowledge. PCK's aspect' 'Strategies for pupil self-evaluation' has not improved satisfactorily. These findings suggest that LS should be carried out during the training programs of the science PSTs.

A teaching practice consistent with the development of students' mathematical reasoning requires teachers to hold a profound understanding of mathematical reasoning. The aim of this research is to study the development of middle and secondary mathematics teachers' understanding about the processes of generalizing and justifying in a professional development course. Data collection included observation of the course sessions, with video recordings, and document collection of teachers' and their students' productions. A qualitative in-depth analysis was carried out focusing on the ways and complexity of generalizing identified by the teachers. The results indicate that teachers progressively focus their attention on these reasoning processes, identifying and characterizing them. The way the course was designed and enacted appears to have contributed to enable the teachers to deepen their understanding of two central mathematical reasoning processes, generalizing and justifying, in addition to their ability to foster and identify such processes in the work with their students.

This article discusses how two beginning mathematics teachers mobilize and develop their didactics knowledge, specifically concerning knowledge of statistical content and teaching practice, during a teacher education experience. This work was developed as part of a specialization course for elementary school mathematics teachers, in the subjects Data Analysis I and II. The teacher education course took place in the West Zone of Rio de Janeiro, Brazil. This is a qualitative study with an interpretive paradigm, based on a case study with two beginning teachers. The data collected was obtained from sessions by a logbook, audio recordings, and interviews with the teachers after the teacher education course. The course was structured to articulate didactics and statistics knowledge from activities directed to teaching practice. Concerning statistical knowledge, the results show the development of knowledge about different concepts associated with carrying out statistical investigations. Regarding teaching practice, teacher's learning is highlighted to carry out statistical activities of exploratory nature and statistical investigations in the school context.

We examine aspects of professional collaboration in a lesson study conducted with mathematics teachers from the final years of basic education belonging to public schools in the State of Rio Grande do Sul, Brazil. The lesson study, conceived as a professional development approach focused on collaboration and reflection, took place in the 2nd semester of the 2019 school year, organized in 12 meetings of 2.5 hours each, and involved 8 tenured teachers in schools belonging to the 15th Regional Education Coordination of that State. The empirical material of the study consisted of the field notes produced at each meeting of the lesson study, the data from an initial questionnaire administered to the teachers, as well as the transcription of the audio recordings of the interviews with the participants and of the sessions of the lesson study. The qualitative and interpretative analysis showed that the dynamics of the lesson study enhanced several basic aspects of collaboration, namely mutual encouragement and support, cooperation and shared reflection.

Purpose: This paper seeks to understand how teachers' mathematical knowledge for teaching (MKT) is developed during a lesson study regarded as a professional development process that places particular emphasis on teachers' knowledge of tasks, representations and students' learning. Design/methodology/approach: This study is based on the networking of two theories, namely the Interconnected Model of Teacher Professional Growth (IMTPG) and MKT. The methodology is qualitative and interpretative. The participants are three primary school teachers and team members involved in a lesson study. Data collection was conducted by means of participant observation, with audio recordings of the sessions and a final individual interview. Findings: The participant teachers developed their knowledge of tasks and representations, which are essential elements of content and teaching knowledge, as well as knowledge of the curriculum, content and students. Additionally, they developed specialized knowledge of mathematics, the meaning of fraction as a measure and its representation on the number line. This development arose from activities conducted in the Domain of Practice, in many cases prompted by the External Domain, and was particularly important when these activities gave rise to unforeseen consequences. Connections among the various domains were established through enactment and reflection processes. Originality/value: The study shows how the networking of two different theories, in this case the IMTPG and MKT, may enhance further understanding of educational phenomena. This networking involved the coordination of these two theories, which were superimposed in the Group Domain.

Purpose: To understand how lesson study (LS) can promote the development of pedagogical content knowledge (PCK) of prospective primary teachers. More specifically, to know what PCK prospective primary teachers develop during LS and how this development occurs. Design/methodology/approach: Following a qualitative approach, this study took place in a teacher education institution where a LS was carried out during the last semester of the academic year with the participation of two prospective teachers, a teacher educator, a cooperating teacher and a researcher. Findings: The results suggest that prospective teachers may develop PCK when they participate in LS, regarding lesson planning (goals and lesson plan), task design, students' difficulties and solving strategies, whole-class discussions and observation of student learning. This development occurs through the engagement in LS activities that allow prospective teachers to deepen their knowledge. Originality/value: This study investigates how the development of prospective teachers' PCK occurs during LS, providing knowledge about how different activities of the LS help develop different aspects of PCK.

Purpose: This article aims to analyse how a teacher leads whole-class discussions during and after participating in lesson studies and to what extent that participation influences her teaching practice. Design/methodology/approach: This is a qualitative/interpretative research with a case study design, carried out with a secondary school mathematics teacher who participated in two lesson studies. Data were collected from participant observation, audio recording of lesson study (LS) sessions and discussions with the teacher, video recording of lessons and semi-structured interviews. Frameworks regarding the teachers' actions are used in the analysis. Findings: The results suggest that in her teaching practice, the teacher led students to explain their strategies with supporting/guiding actions, but she also challenged the students to justify their productions, ensuring that the students' ideas were clear. Additionally, the teacher explored incorrect strategies and disagreements, inviting and challenging other students to intervene or react and involved students in drawing connections, as discussed in the LS. Therefore, the teacher put into practice several actions teachers can do in leading whole-class discussions to promote students' learning. Participating in LS was an opportunity to rethink her teaching practice, as the teacher pointed out, bringing her a new perspective on leading discussions in which students play an active role in learning mathematics, creating opportunities for the students to explain and react to their colleagues' ideas. Originality/value: This article examines an under-researched issue: the influence of LS on the way a teacher leads whole-class discussions, during and after participating in lesson studies.

Purpose: This research aims to understand the learning with regard to didactic knowledge of an early childhood teacher who participated in a Lesson Study (LS) on the topic of patterns, as well as her perspective about LS as a teacher education process. Design/methodology/approach: This is a qualitative case study of an early childhood teacher who participated in a LS alongside three other colleagues and a facilitator. Data collection was made by participant observation with elaboration of a research journal, audio recording of sessions and a collective interview. Data analysis was made by content analysis based on the conceptual framework. Findings: This case study shows that early childhood teachers may become aware of the importance of working with their children mathematical ideas. It also shows that they may develop their didactic knowledge in several dimensions such as school mathematics, curriculum, children and their learning, and teaching practice and that they may appreciate LS as a means of professional development. Originality/value: This study shows the potential of LS for the professional development of early childhood teachers, a scarcely researched area. It also shows that their learning may include dimensions related to specific curricular areas such as mathematics.

Teachers' professional knowledge has been an active area in the field of mathematics education research in the last decades, but we still know very little regarding how teachers learn. This article aims to identify learning opportunities experienced by two teachers regarding mathematical and didactical knowledge for the teaching of function, in a context of practice-based learning. To this end, we analyze data from a practice-based teacher education program based on a cycle that incorporates three segments of teachers' practice: planning, enacting and reflecting on a high school mathematics lesson, conducted from the perspective of inquiry-based teaching. We analyze two strands of the professional knowledge mobilized by the teachers, namely (i) the mathematics knowledge for teaching function, (ii) the knowledge for teaching practice, regarding function. The results indicate learning opportunities that emerged from the work done, regarding the use of the classroom as a teaching professional learning environment: the search for ways to encourage students to participate in the discussions; the recognition of the potential of the chosen lesson plan for the covariational approach, the importance of organizing mathematical discussion practices for students' learning, organizing the time devoted to whole-class discussions and preparing for leading them and the potential of a class developed from the perspective of inquiry-based teaching.

In the research reported in this paper, using a modified version of the interconnected model of teacher professional growth (IMTPG) proposed by Clarke and Hollingsworth (Teaching Teacher Educ 18:947-967, 2002), we aimed to understand the learning dynamics in a lesson study of a group of five teachers of grades 5-6, during their work around mathematical tasks and in the elaboration and analysis of a diagnostic test of students' knowledge. We draw on examples from a lesson study that took place in a school in Lisbon, involving five teachers that constituted the whole mathematics group of the school. The main feature of our adaptation is the consideration of a Group Domain instead of the Personal Domain of the original model. In the original model, the Domain of Practice had a broad definition but usually it was restricted to classroom experimentation. In the research reported in this paper, the lesson study sessions were taken as part of the Domain of Practice. For data collection, there was audio and video recording of the lesson study sessions and of the research lesson and the writing of a research journal. Data analysis was done using the modified IMTPG. The results suggest that this model is helpful in understanding the processes that make lesson study a valuable professional development setting, contributing to responding to the question of 'how lesson study works' (Lewis et al., Theory and practices of lesson study in mathematics: an international perspective, Springer, New York, 2019). In addition, the use of this model showed that the lesson study promoted the teachers' development regarding knowledge of didactics, especially knowledge of students and their learning processes and of teaching practice.

This article investigates the pedagogical content knowledge (PCK) of a teacher educator who teaches 5th-grade mathematics in a school in the context of a university-school partnership project. PCK is analyzed in a qualitative way through video-taped classroom episodes with focus on interactions between the teacher and the students as well as the teacher's reflections and pedagogical reasoning on the interactions. The analyses indicate examples of PCK development during teaching, especially refinement in the domain of knowledge of instructional strategies and representations. This knowledge improved as a result of reflection on student questioning and analysis of students' misconceptions. Different roles of being teacher, teacher educator, and researcher afforded opportunities to gain insights on how to develop knowledge required for teaching and analyze it in order to facilitate future teachers' learning.

A proof is a connected sequence of assertions that includes a set of accepted statements, forms of reasoning and modes of representing arguments. Assuming reasoning to be central to proving and aiming to develop knowledge about how teacher actions may promote students' mathematical reasoning, we conduct design research where whole-class mathematical discussions triggered by exploratory tasks play a key role. We take mathematical reasoning as making justified inferences and we consider generalizing and justifying central reasoning processes. Regarding teacher actions, we consider inviting, informing/suggesting, supporting/guiding and challenging actions can be identified in whole-class discussions. This paper presents design principles for an intervention geared to tackle such reasoning processes and focuses on a whole-class discussion on a grade 7 lesson about linear equations and functions. Data analysis concerns teacher actions in relation to design principles and to the sought mathematical reasoning processes. The conclusions highlight teacher actions that lead students to generalize and justify. Generalizations may arise from a central challenging action or from several guiding actions. Regarding justifications, a main challenging action seems to be essential, while follow-up guiding actions may promote a further development of this reasoning process. Thus, this paper provides a set of design principles and a characterization of teacher actions which enhance students' mathematical reasoning processes such as generalization and justification.

In teacher education, a key issue is how prospective teachers learn. At the University of Lisbon, based on an inquiry-based approach to mathematics learning, we developed a secondary school mathematics teacher education program, in which a central feature is the elaboration of a final investigative report based on teaching practice. In this paper, our aim is to understand the professional learning opportunities and difficulties recognized by prospective teachers (PT) in their final reports, by addressing the following research questions: (1) What didactical choices do PT mention regarding the approach to teaching, the use of tasks, resources and assessment strategies in their teaching practices? (2) What transversal elements of teaching practices do they recognize as enabling their professional development? and (3) What elements of the investigative work do PT refer to as major professional learning outcomes? Using content analysis, we reviewed, coded, and analyzed all 38 reports produced so far in the program. The results suggest that the PT embraced an inquiry-based approach to learning, valuing the role of suitable tasks and of whole class discussions. They also valued reflection and collaboration as practices that support professional development. Prospective teachers also indicated some difficulties and challenges in doing this investigation based on their teaching practice, but even so they tended to regard it a learning opportunity. We conclude that the final report, by its content, structure, and working processes as framed in this teaching education program, supports participants' development as teachers who hold research in positive regard.

This paper seeks to identify actions that can be regarded as building elements of teachers' classroom practice in mathematical discussion and how these actions may be combined to provide fruitful learning opportunities for students. It stands on a framework that focuses on two key elements of teaching practice: the tasks that teachers propose to students and the way teachers handle classroom communication. Tasks are appraised concerning their level of challenge. Teachers' actions in discussions are classified as informing/suggesting, guiding, and challenging. The methodology is qualitative with data collected from video recording of the classroom. The analysis of classroom episodes dealing with rational numbers but with different agendas, such as providing students opportunities for learning about representations, concepts, connections, and procedures and for developing reasoning suggests that some degree of challenge promotes fruitful learning situations. However, such situations tend to require preparation and follow-up with guiding and even informing/suggesting actions so that the students can learn what has been set in the teacher's agenda.

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]

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]