Most widely published books and papers on lesson study emphasise that a main attraction of this format is that it enables teachers to build and maintain a shared knowledge base for the teaching profession. This, however, clearly requires that lesson study is established as more than an occasional experiment, which is new to most teachers involved and involves only exceptional teachers. Therefore, research on lesson study increasingly focuses on the question of sustainability of the format as a practice-based form of professional development. We present and expand the recent theoretical idea of paradidactic infrastructure and analyse sustainability from the institutional perspective of the Anthropological Theory of the Didactic and especially based on the idea of paradidactic infrastructure. Our data come from the Danish context where lesson study in mathematics has been experimented with for more than 10 years, first by individual researchers and now, increasingly driven by institutions, such as schools or municipalities.

Over the last few decades mathematics education in China (refers to the Chinese mainland throughout this paper) has made important progress, along with its rapid economic and social development, and attracted increasing attention internationally. At the same time, mathematics education research in China has also experienced a noteworthy growth in a variety of aspects. To examine how mathematics education research in China has developed and hence explore what are possible and needed directions, we conducted a focused survey of research as mainly published since the early 1990s in the Journal of Mathematics Education, the sole national research journal and primary publication avenue in China particularly devoted to mathematics education research. With a focus on first research methods in terms of the general dichotomy of non-empirical methods and empirical methods and then research issues in terms of different areas in mathematics education, the results revealed that over the three decades, empirical studies have turned to be the mainstream of mathematics education research in China, while issues about mathematics teaching and learning have received the dominant attention. Issues about curriculum and textbooks have also attracted fast growing attention, particularly when the issues are related to curriculum reforms. Other topics that have received noticeable increasing attention include issues about mathematics teacher education and professional development, and issues concerning culture and history in mathematics education. Possible and needed directions of the development of mathematics education research in China are discussed at the end of this article.

The current COVID-19 pandemic has made us rethink ways to provide educational opportunities. The purpose of this discussion paper is to identify issues and challenges in online teaching and learning of mathematics to be explored in future research, and what mathematics education can contribute to this research area. I first identified recent relevant research themes from 2015 to 2020 by reviewing key literature from mathematics education journals. International mathematics education research studies have been examining various issues around designing online learning environments, social interactions and communication in online teaching and learning, and the role of resources and tools. These concerns also exist among Japanese teachers, and I argue Lesson Study can be used creatively in order to seek effective ways to deliver online teaching and learning of mathematics.