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Although CLS has been implemented in China for over a century, it is barely known to educators internationally. The purpose of this paper is to synthesize the salient characteristics of Chinese lesson study (CLS), introduce the major themes of this special issue, and invite dialogues about the theories and practices of CLS.

The authors of this editorial paper conducted an extensive literature review on CLS, analyzed the contents and methods of the existing research categorically, compared CLS with other models of LS globally, and present this special issue articles and their major contributions thematically. The theoretical framework of the paper relies mainly on cultural theories and theories on research paradigms such as improvement science, which explain why and how CLS functions in Chinese education system over time.

Existing studies suggest that CLS is a deliberate practice for developing instructional expertise, a research methodology for linking research and practice, and an improvement science for instruction and school improvement system wide. In addition to the theorization of CLS, this special issue also introduces some adaptations of CLS outside of China such as the USA and Italy.

This paper, for the first time, spells out some salient features of CLS, and discusses issues in adapting CLS in other parts of the world. It will enrich the understanding of LS theories and practices in China and promotes trans-cultural development of LS internationally.

This study aimed to explore how an adapted theoretical framework by networking two theories could help document teachers' collective learning through lesson study.

Interconnected Model of Teacher Professional Growth (IMPG) and Documentational Approach to Didactics (DAD) has been used individually to document teachers' professional learning from different aspects. The IMPG captures teachers’ growth through the iterative process and dynamics (i.e. enactment and reflection) among four domains (personal domain, external domain, practice domain and consequence domain). At the same time, the DAD primarily focuses on teacher learning through the interactions between resources and teachers. To deepen understanding of teachers' learning through lesson study, in this study, a networked theoretical framework through which the DAD model is enriched by incorporating some ideas from IMPG is proposed. A lesson study, including five stages of study, plan, enact, reflect and refine, facilitated by a researcher was conducted in Shanghai China. The data sets including all videotaped meetings and research lessons and lesson plans of the lesson study are analyzed based on the adapted framework qualitatively.

The results show that the teachers' document evolved from adopting the traditional teaching materials to adapting both traditional ones and e-resources with careful consideration of student learning through the lesson study process.

This study advances our understanding of networking strategies and their usefulness for deepening teacher professional learning as document development through lesson study. However, the sustainability of this type of professional learning needs to be further explored.

This study expands teacher learning through lesson study as the document development and enriches the DAD theory by illuminating the process of the documentational genesis.

Finding ways in which technology can be used to modify, strengthen, scale up and sustain lesson study (LS) is an emerging field of research. It has become even more important due to a pandemic leading to teacher and student learning being delivered online. The purpose of this paper is to present research findings about experiences of online LS and identify issues for further research.

A systematic search of articles from 2010 to 2020 identified 13 relevant papers, and through analysis, some major themes were identified. All papers in the special issue were synthesized from the lens of the identified themes; finally, further directions are discussed.

In general, various online LS models were found to have features that resulted in positive effects on teaching and learning, but, whilst several characteristics of effective online learning were identified, some studies also highlighted issues.

There is a need for larger scale projects over an extended period to assess the effectiveness of online LS. Future research focused on carrying out learning study online as well as consideration of equity issues associated with online LS are also suggested.

The studies presented in this issue address the opportunities and challenges of conducting online LS during a pandemic and beyond. Together, the literature review and contributory papers provide an international perspective of using online LS and identify important issues for further research.

This study aimed to explore teachers' learning processes through a hybrid cross-cultural lesson study (LS) because little is known about the learning process through this novel and promising LS approach.

This cross-cultural LS lasted over six months focusing on developing a research lesson (RL) related to linear functions/equations by addressing a commonly concerned student learning difficulty. The data collected were lesson plans, videos of RLs, cross-culture sharing meetings and post-lesson study teacher interviews. A cultural-history activity theory (CHAT) perspective (Engeström, 2001) was used as a theoretical and analytical framework, and contradictions were viewed as driving forces of teachers' learning. The data were analyzed to identify contradictions and consequent teachers' learning by resolving these contradictions.

The results revealed four contradictions occurring during the hybrid cross-cultural LS that are related to the preferred teaching approach, culturally relevant tasks, making sense of the specific topic and enactment of the RL. By addressing these contradictions, the participating teachers perceived their learning in cultural beliefs, pedagogical practice and organization of the lesson.

This study details teachers' collaborative learning processes through hybrid cross-cultural LS and provides implications for effectively conducting cross-cultural LS. However, how the potential learning opportunity revealed from this case could be actualized at a larger scale in different cultures and the actual impact on local practices by adapting effective practices from another culture are important questions to be investigated further.

This study expands teacher learning through cross-cultural LS by focusing on contradictions cross-culturally as driving forces.

This paper examines how mathematics teacher educators (MTEs) learn to enact equitable mathematics instruction using technology through lesson study (LS).

A LS team with three MTEs conducted three iterations of LS on teaching the Pythagorean Theorem in an in-person, technology-mediated environment. Many forms of data were collected: Desmos activities, videos of research lessons (RLs), videos of MTE RL debriefings, artifacts of student learning in the Desmos Dashboard, and MTEs' written self-reflection. The authors investigate the teacher educators' learning through LS by analyzing the MTE debriefings of the RLs using Bannister’s (2015) framework for teacher learning in communities of practice.

The MTEs learned to enact equitable mathematics instruction using technology through addressing emerging issues related to intellectual authority and use of student thinking. Throughout the LS, the MTEs sought ways of promoting students' mathematical authority and using student thinking through features of the Desmos platform.

This study focuses on MTEs' learning without examining participating preservice teachers' learning. It demonstrates the benefits of LS for MTEs' professional learning.

This study showcases how a research-based Desmos activity is used and refined to promote MTE learning how to implement equitable mathematics instruction.

The study contributes to better understanding of how LS could be used to develop MTEs' professional learning. Moreover, the dual process of participation and reification was concretized through diagnostic and prognostic frames in the LS context, which enriches the concept of community of practice.

This paper presented the learning process of a group of primary mathematics teachers who participated in two iterations of lesson design, enactment and reflection in a Chinese Lesson Study.

An expansive learning theory was employed to examine the teachers’ learning process in lesson study (LS) on representing fractions on a number line. The evolution of a germ cell was utilized to feature the transformation of the object of activity from abstract to concrete through resolving contradictions among LS members. The videos of lesson planning, research lessons (RLs) and debriefing meetings were collected and analyzed to reveal the expansive learning process.

The analysis showed that the teachers expanded their learning through transforming the object from diffuse to concrete and expanded through consciously articulating the germ cell. The outcomes of object-oriented activity include improving the enacted lesson which promoted students’ conceptual understanding.

This study made a unique contribution to understanding the learning process of teachers in Chinese LS from the perspective of expansive learning.

The purpose of this paper is to explore what professional knowledge and competence (PKC) that knowledgeable others, namely, mathematics teaching research specialists (MTRS) in China, need to know, and how they may develop their PKC.

This study adopts mixed methods. A survey on PKC with 549 MTRS is utilized to examine MTRS’ perceived held and ideal PKC and perceived effective ways of developing their PKC. The responses to the open-ended questions on the survey were used to identify additional dimensions of PKC and ways of developing PKC. Multiple techniques of quantitative data analysis were employed to feature the characteristics of PKC and structure of the survey, and the relationship between background variables and perception of PKC. Collectively, this study paints a rich and comprehensive picture about Chinese knowledgeable others’ knowledge and competence, and its development.

The data analysis reveals that the participants appreciated the six-dimension model of MTRS’ PKC. They were confident with their PKC in general, but varied in different aspects. The factor analysis showed the six-dimension model could be further clustered into two components: knowledge about mathematics teaching and learning and competence in mentoring and educational leadership, and knowledge about content, assessing student learning, and use of technology. The participants perceived their learning through multiple ways including: learning through reading, attending specific training programs, attending and mentoring teaching research activities both school-based and across regions, observing and debriefing lessons, sharing within online learning communities. All these venues jointly contribute to developing MTRS’ PKC.

This is the first empirical study on MTRS’ PKC and its development in China based on such a large sample. The findings of this study not only contribute to an understanding of knowledgeable others in Chinese lesson study and providing suggestions for support of their development, but also provide implications for studies of practice-based mathematics teacher-educators globally.

This study aimed to explore how a group of Chinese primary mathematics teachers learned through conducting an online cross-cultural lesson study between China and Australia.

An expansive learning theory was adopted to examine teachers' learning through collective activities across different activity systems. Multiple data sets including videos of research lessons, debriefings and audios of interviews were collected. From the expansive learning perspective, based on a fine-grained qualitative data analysis, various contradictions (as driving forces of learning) were identified and the ways of resolving the contradictions (as enactment of learning) were located to feature teacher learning throughout the online lesson study process.

Teachers' expansive learning includes enhancing teachers' MKT and Mathematics TPACK, developing instructional design skills and capabilities in addressing challenges occurring in the virtual environment were identified.

Theoretically, the study illustrated how expansive learning theory could be utilized to examine teacher collaborative learning in the online cross-cultural lesson study. Practically, this study showed that reiterative cycles and experts' facilitation are crucial to expansive learning for linking research to classroom practice. However, this study did not focus on student learning in the virtual environment. Australian teachers' reciprocal learning through the online lesson study also requires further exploration.

Both online lesson study and cross-cultural collaboration are innovative. The expansive learning lens are creatively used to examine the complexity of teacher learning in such a novel environment.

This study aims to explore how a Chinese lesson study (LS) supports a teacher's learning of transforming curriculum reform ideas into classroom practices.

Data analyzed in this study included lesson plans, three videotaped research lessons, three audio-taped group meetings, interviews, the teacher's reflection journals and other related materials. Clarke and Hollingsworth's interconnected model of professional growth (IMPG) was adopted as the theoretical and analytical framework for examining the teacher's learning process.

This study found that teachers can deepen their understanding of innovative curriculum ideas and implement them through an iterative cycle of planning, enactment and reflection in LS. Involving knowledgeable others in the process and reflecting on evidence-based evaluation of students' learning outcomes is crucial for changing the teacher's knowledge, beliefs, attitudes and teaching practices. Theoretical tools, such as learning trajectory, are useful for transforming curriculum reform ideas that teachers find abstract and overly broad into concrete and actionable lesson designs.

Theoretically, this study can broaden our knowledge about how the under-studied mechanism of Chinese LS can contribute to helping teachers to transform reform ideas into classroom practices. Practically, this study provides suggestions for researchers and educators to reflect on and improve the effectiveness of teachers' professional development programs in a reform context.