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The aim of this study is to advance understading of teachers' and researchers' work, in particular its cultural specificities, from a resource perspective by exploring the issues and challenges faced during the translation of a theoretical framework, the Documentational Approach to Didactics (DAD), from Western (English and French) to East Asian languages (Chinese and Japanese). A basic assumption is that the challenges encountered while translating are driven by the cultural and linguistic differences that exist between the West and East. Adopting the perspective of the "Anthropological Theory of the Didactic" (ATD), we frame the translation work as a transposition process of research praxeology (a model of the researchers' practices and knowledge) from a Western to an Eastern institution. We investigate this process to identify cultural elements at different levels (school, society, civilization, etc.) using the ATD framework. After translating a DAD article into Chinese and Japanese separately, we then worked collaboratively to identify similar or diverse translation issues and investigated their origins. Consequently, the results revealed a considerable difference between the West and East, and ample similarities between China and Japan, especially in terms of the researchers' work and its relationship with the teachers' work.

This paper reports the results of an international comparative study conducted in Switzerland and Japan of an elementary school mathematics lesson. The principal aim of the study was to advance understanding of the cultural specificities of a mathematics lesson in its totality using concrete examples of lesson design and implementation and of how cultural factors within and beyond the classroom/school shape and produce mathematics lessons in a particular country. We analysed two Grade 4 mathematics lessons designed and implemented in Switzerland and Japan by pre-service teachers in the context of a project-based international exchange programme. The lesson, initially designed collaboratively by the pre-service teachers of the two countries, was ultimately realised in different ways in each country. Specifically, we found differences between the Japanese and Swiss lessons in the structure of the lesson and validation of solutions. To elucidate these differences and identify the cultural factors that yield them, we analysed the resources developed and used during lesson design and implementation (lesson plans, official documents, and textbooks). Furthermore, we discuss three aspects of mathematics lessons that account for the main characteristics of each lesson: collective or individualistic teaching and learning, problem-solving lessons, and distance between theory and practice.

Japanese lesson study was introduced to the West in the late 1990s. Certain widely disseminated publications and classroom videos, and also reports on international surveys of students achievement, led many to consider that Japanese mathematics teaching is highly creative and effective, with lesson study as a main explanatory factor. As a result, substantial efforts have been deployed in many countries to engage mathematics teachers in lesson study. In this paper, we argue that lesson study is but one element of a comprehensive Japanese "infrastructure" for developing teacher knowledge, both personal and shared, and that the functions and workings of any particular element in this infrastructure cannot be fully understood when viewed in isolation. In this paper, we offer a systemic (institutional) analysis of how these "infrastructure" elements interact in crucial ways. Specifically we study two episodes where a teacher's "practice research" is clearly motivated and supported by other infrastructural conditions, such as the possibility to present the outcomes at a regional teacher conference, and in a publication. Moreover, a central prerequisite for the creative part of "lesson planning" is a much more limited activity: the patient study and discussion of mathematical tasks and materials. This activity is also crucially supported by elements of the infrastructure.

The aims of the present study are two-fold. The first aim is to reveal the cultural and linguistic issues that need to be considered in the development of curricular content and sequencing for teaching mathematical proof in secondary schools in Japan. The second aim is to elaborate an epistemological perspective that may allow us to understand what constitutes proof in the curriculum of a given country, so that our results may be applied in international contexts. By doing so, we then elaborate a reference epistemological model which may allow us to understand what constitutes mathematical proof in different countries.

This paper reports the results of an international comparative study on the nature of proof to be taught in geometry. Proofs in French and Japanese lower secondary schools were explored by analyzing curricular documents: mathematics textbooks and national curricula. Analyses on the three aspects of proof--statement, proof, and theory--suggested by the notion of Mathematical Theorem showed differences in these aspects and also differences in the three functions of proof--justification, systematization, and communication--that are seemingly commonly performed in these countries. The results of analyses imply two major elements that form the nature of proof: (a) the nature of the geometrical theory that is chosen to teach and (b) the principal function of proof related to that theory. This paper suggests alternative approaches to teach proof and proving and shows that these approaches are deeply related to the way geometry is taught.

In this paper, we first present a theoretical approach to study mathematics teacher knowledge and the conditions for developing it, which is firmly rooted in a systemic approach to didactic phenomena at large, namely the anthropological theory of the didactic. Then, a case of open lesson is presented and analysed, using this theoretical approach, to show how the format of open lesson contributes to the construction and diffusion of didactic knowledge in the community of mathematics teachers in Japan. The basic idea of this format is that teachers from other schools are invited to observe a class taught by a teacher then participate in a discussion session with him on the details of the lesson. For the case study, we analyse the lesson plan prepared for an open lesson, the observed lesson and the teachers' discussion. As a result, an open lesson session has been described as a specific form of post-didactic practice related directly to an actual observed lesson, and aiming specifically at elaborating the theoretical aspects of teacher's didactic practice in the lesson and beyond it.

In this paper, we analyze and compare two didactical designs for introducing primary school pupils to proportional reasoning in the context of plane polygons. One of them is well-documented in the literature; the other one is based on our own data and is accordingly presented and discussed in more detail in this paper. The two designs come from different cultural and intellectual environments: lesson study in Japan (implicitly based on the "open approach method") and "didactical engineering" in France (based on the theory of didactical situations). The general aim of our paper is to compare these two environments and their approaches to didactical design, basing our discussion on the concrete designs mentioned above. Clear differences among them are presented, while we also identify links which hold potential for integrating research and practice.