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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.

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 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.

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.