Discussing students’ thinking and perspectives for improving teaching: An analysis of teachers’ reflection in post-lesson discussions in lesson study cycles

Lesson study is a Japanese concept to improve teaching and learning mathematics through a particular form of activity in which a group of teachers works collaboratively to plan, implement, observe, and reflect on the proposed research lessons. Although lesson study is internationally acknowledged as providing key learning opportunities for teachers, there is limited information on post-lesson discussion and its contribution toward teachers’ professional growth. This study has the aim of describing the key features of post-lesson discussions. An entire group of teachers at a public elementary school in Japan participated in three lesson study cycles. During the post-lesson discussion in each cycle, the teachers were invited to write down what they believed was significant in the planned lesson and their comments regarding the lessons, for the discussion. The writings and comments were collected and analyzed. This method helped capture teachers’ ‘voices’ better compared to recordings of oral discussion. The results of the analyses of teachers’ writings and comments revealed that a major part of post-lesson discussions involved issues related to pedagogies, and teachers’ reflections tended to focus more on students’ thinking and mathematical topics over time as the cycles of lesson study proceeded. The results suggested that teachers’ professional growth can be related to their recognition of the importance of understanding students’ thinking and their perspectives on mathematics, in order to improve teaching.


Introduction
Lesson study (jugyou kenkyu) is a common concept in Japanese educational practice and the primary form of professional development that occurs in various contexts (Fernandez & Yoshida, 2004;Lewis & Tsuchida, 1998;Shimizu, 1999;Stigler & Hiebert, 1999). It can be characterized as an approach for improving classroom teaching and professional development, whereby a group of teachers collaboratively studies the subject matter, method of instruction, and how students think and solve problems in the classroom with a particular 'theme' to be addressed by the teachers or school.
The activity includes planning and implementing a carefully designed 'research lesson' followed by a post-lesson discussion with reflection by participants based on the observation of students' learning (Shimizu, 2002(Shimizu, , 2020Yoshida, 2008). Lesson study has been internationally acknowledged as providing key learning opportunities for teachers and is one of the major foci in the research and development of mathematics teacher education worldwide (Huang & Shimizu, 2016;Quaresma et al., 2018). However, there is limited information on post-lesson discussion and its contribution toward teachers' professional growth in lesson study. This study had the aim of exploring the key features of post-lesson discussions.
The study of professional development has various forms and methodologies. For example, the use of video-recorded classroom teaching is often incorporated into professional development programs in a variety of contexts (e.g., Maher, 2008). Considering the limited data on teaching, students' learning, and their interactions in video-recorded material, lesson study is capable of providing opportunities for 1 3 observing teaching and learning on-site in the classroom. While one teacher takes a research lesson, and the others observe students' learning and research lessons from various perspectives, they can obtain abundant information to be discussed in the post-lesson discussion. In the process as a whole, teachers have opportunities to gain a deeper understanding of curriculum topics, teaching methods, materials, and students' learning (Seino & Foster, 2020).
Lesson study is being used as a professional development tool not only for in-service teachers but also for training pre-service teachers in various forms in Japan. For example, pre-service teacher-training programs at universities and colleges include lesson study as a crucial and challenging part of the final week of student teaching practice. In-service teachers also have opportunities to participate in professional development held within their own school, outside their school, but in the same school district or city, prefecture, and even at the national level for a couple of reasons (Lewis & Tsuchida, 1998). Through these processes, teachers as learners have opportunities for their own continued learning. It has been widely recognized by Japanese teachers and policymakers that the professional development of teachers is a lifelong process (Shimizu, 2010;Takahashi, 2015).
Among the various forms of lesson study, a particular type is often held within a single school as part of an activity called "konai-kenshu" (Fernandez & Yoshida, 2004) at elementary or lower secondary school levels. The term konai-kenshu refers to an intra-school activity that includes lesson study as the core of the entire program in schoolbased professional development. This intra-school lesson study activity of lesson study provides participating teachers with learning opportunities when they work with their colleagues in the same school to improve teaching and learning in the classroom.
With the accumulated body of recent studies on teacher professional development, lesson study has been widely acknowledged for its effective engagement with teachers as learners (Huang & Shimizu, 2016;Huang et al., 2019;Quaresma et al., 2018;Stigler & Hiebert, 1999). Teachers continue to share, analyze, and reflect on ideas throughout the lesson study. Teachers reflect on the teaching and students' learning in post-lesson discussions based on their observations of research lessons. The contents of post-lesson discussions provide information on the area that teachers should focus on and what needs to be addressed to improve lessons (Seino & Foster, 2020).
In this study, an entire group of teachers in a public elementary school in Japan engaged in three cycles of lesson study, and their discussions after a research lesson were analyzed. To explore in detail what was been discussed in a post-lesson discussion, and how the objects of reflection changed in cycles, a specific method of data collection was used to capture what the participating teachers perceived and believed was significant in the implemented research lesson. The Interconnected Model of Professional Growth (IMPG) proposed by Clarke and Hollingsworth (2002) is used as a framework for conceptualizing the lesson study process, which includes a post-lesson discussion as a core activity within the lesson study cycle. In addition, an adapted version of IMPG that incorporates a "Lesson Study Domain" (Schipper et al., 2017) was used for identifying and describing the changes or shifts in the topic of post-lesson discussions.

Research questions to be addressed
Given the findings of earlier studies, how teachers change over time while they are engaged in an intra-school lesson study in Japan is of particular interest in research on lesson study. Developing a better understanding of the mechanism of teachers' professional growth of teachers in the context of lesson study is key for theorizing lesson study as a model for professional learning. This study focused on the post-lesson discussion of teachers participating in an intra-school lesson study and analyzed their reflections and comments during the discussion.
With the research context and background described above, the following three research questions were addressed in this study.
1. What topics or themes can we find in teachers' reflections in the post-lesson discussions in a school-based lesson study in mathematics? 2. Are there any shifts or changes in participating teachers' reflections over time on classroom events in the postlesson discussions in lesson study cycles? 3. To what extent is the IMPG applicable in incorporating lesson study elements for describing teacher professional growth?
2 Theoretical framework

Interconnected model of professional growth
To explore the mechanism of teacher change and professional growth of teachers, Clarke and Hollingsworth's (2002) IMPG has been used in recent studies on teacher professional growth through collaborations with colleagues (e.g., Goldsmith et al., 2014;Witterholt et al., 2012), in general, and through cycles of lesson study (e.g., Schipper et al., 2017;Widjaja et al., 2017), in particular. The IMPG includes relationships among four different domains; External Domain (external source of information or stimulus for a change), Personal Domain (teacher's knowledge, beliefs, and attitudes), Domain of Practice (where professional experimentation takes place), and Domain of Consequence (where teachers recognize salient outcomes inferred to be the result of experimentation), as well as the two major processes of Enactment and Reflection (Fig. 1).
It should be noted that the External Domain, represented as a square, is distinguished from the other three domains in that it is located outside the teacher's personal world. Conversely, the Domain of Practice, the Personal Domain, and the Domain of Consequence in combination constitute an individual teacher's professional world of practice, encompassing professional actions, inferred consequences of those actions, and knowledge and beliefs that prompt and respond to those actions (Clarke & Hollingsworth, 2002).
IMPG represents the complexity of teacher change and growth in multiple ways. Compared to early research (e.g., Guskey, 1986), which represented teacher changes as sequential and theorized in linear models, IMPG identifies how teachers' practices have influenced teaching and students' learning and notes that teachers' change and growth are achieved through interactions called enactment and reflection across domains. What teachers have identified in professional development and as lesson evidence allows them to play a critical role in driving their own change and growth (Desimone, 2009;Evans, 2014).
For the current study, Clarke and Hollingsworth's (2002) IMPG provided a conceptual framework for identifying and describing the key features of post-lesson discussion that took place after a 'research lesson'. In addition, enactive and some reflective links (Clarke & Hollingsworth, 2002) were used to characterize the processes in the lesson study cycle, while special attention was given to the major processes of Enactment and Reflection shown as arrows between different domains. IMPG can be regarded differently when we consider the professional growth of teachers through lesson study cycles. As lesson study occurs in the school, where teachers' activities are perceived as collaborative work among colleagues, the border between activities in the External Domain and activities in the Domain of Practice can be more interactive and may be unclear. In addition, while we can have an arrow of enactment from External Domain to Domain of Practice, as Domain of Practice includes professional experimentation in lesson study, we can also think that reflecting on Domain of Practice can have certain impacts on the External Domain.

Professional growth of teachers through lesson study cycles
To specify teacher growth through lesson study cycles, Widjaja et al. (2017) investigated the professional growth of a group of teachers participating in two lesson study cycles by using IMPG to identify 'change sequences and growth networks' (Fig. 2) to explain the mechanisms through which this group of teachers' professional growth occurred. In their study, among different phases, the External Domain remained almost the same (Japanese Problem-Solving, Lesson Structure, Japanese Lesson Study process, and Cross-school project as external source of information). Phase 1 included 'JLS professional development day' and both Phase 2 and 3 included 'Outside expert and observers' in addition to the first four elements. Meanwhile, the content in the Personal Domain, the Domain of Practice, and the Domain of Consequence changed and expanded significantly between phases (Widjaja et al., 2017). In the case of schoolbased lesson study, in particular, the Domain of Practice has an impact on External Domain which is located outside the teacher's personal world, but simultaneously the teachers join the activities in this domain as members of the school.
When we examine the overall changes in the domains (Widjaja et al., 2017), we see that the process of reflection can have an impact on significant changes in each domain.  (Widjaja et al., 2017, p. 366) Teachers' personal domains did not change quickly and confirmed the outcome of their practices. This delay is because teachers change only after checking the evidence and justifications for their actions by continuously enacting and reflecting on other domains. Particularly, the Domain of Practice in both Phase 2 and Phase 3 includes "Post-lesson discussion based on evidence of student work" as well as "Give and receive feedback from colleagues and outside expert," while the Domain of Consequence in Phase 2 includes "Focus on students' learning during post-lesson" and includes "Unanticipated students' solutions" and "Value of observations of students' thinking" in Phase 3 (Widjaja et al., 2017). As post-lesson discussion that is based on the evidence of student work observed in the research lesson influences the Domain of Consequence with respect to "Focus on students' learning during post-lesson" and "Value of observations of students' thinking," the mutual influence of these two domains is clear (Widjaja et al., 2017).
The observations from this study appear to be consistent with the fact that Japanese teachers value students' thinking during classroom teaching and use a particular form of lesson planning that includes anticipated students' solution methods as an indispensable element (Shimizu, 1999). Therefore, it is interesting to explore what participating teachers in lesson study consider to be significant in a planned and implemented lesson, and what they described as the critical characteristics of observed lessons as the cycles of lesson study proceeded.
Meanwhile, based on interviews with 22 lesson study participants from different school contexts, Schipper et al. (2017) examined whether the professional development approach to lesson study enhances a teacher's professional growth in terms of adaptive teaching. The study showed how the intensive focus on students' learning, collaborative professional experimentation, and the facilitators' role might contribute to growth. They set out to investigate issues related to professional growth in the lesson study with semistructured interviews with the participants, in which they asked questions directly related to the four domains.
Their research questions included, "To what extent is the IMPG applicable in analyzing teacher professional growth in the context of lesson study?" and they proposed an adapted version of the interconnected model of professional growth (Fig. 3, Schipper et al., 2017, p. 300). They noted the interchangeability of activities in the External Domain and activities that are related to professional experimentation in the Domain of Practice, and their findings showed the complexity of distinguishing these elements in a framework of analysis. They then suggested that lesson study should not be perceived as an external PD activity.
They proposed the limited adaptation of IMPTG in the context of lesson study by integrating the External Domain and the Domain of Practice into a Lesson Study Domain ( Fig. 3). In this integrated domain, the lesson study procedure, professional experimentation, and the role of the lesson study facilitator are highlighted. The lesson study facilitator is explicitly presented in this adapted model because the facilitator may play an essential role in stimulating further teacher professional growth through lesson study. In addition, the reflection and enactment patterns are presented in reciprocal directions between all domains because these patterns often overlap or intertwine.

Focus on the dynamic relationship between external domain and domain of practice
Although the adapted version of the IMPG seems to be more suitable for representing the professional growth of teachers in lesson study cycles, it was proposed in a different context where the lesson study facilitator plays an essential role. In addition, the findings of the study are based on the analyses of retrospective self-reports by teachers and interviews conducted after completion of two lesson study cycles (Schipper et al., 2017, p. 300). In the current study our aim was to describe the focus of post-lesson discussion in lesson study in situ, and to explore the dynamic relationship between External Domain and Domain of Practice by examining the details of what participating teachers enact and reflect on in lesson study cycles. Accordingly, we intended to maintain both External Domain and Domain of Practice as in the original model, but focused on considering the details in Domain of Practice in the case of lesson study. The authors are aware that Japanese teachers who engage in an intra-school lesson study share the understanding that lesson study is embedded in daily work in their school (Yoshida, 2008). They also share a particular style of teaching mathematics through the "structuredproblem solving" approach (Fujii, 2014;Stigler & Hiebert, 1999). Meanwhile, although the importance of the  (Schipper et al., 2017, p. 300) teacher's reflection and learning through lesson study has been recognized, the post-lesson discussion does not always lead to productive and critical learning opportunities, or emphasize whether the research lesson is conducted according to the lesson plan (Fujii, 2014;Huang & Shimizu, 2016;Pang, 2016).
There is limited research related to post-lesson discussion in lesson study, and there are certain challenges when lesson study is operationalized as professional development. It can be superficial, and it failed to present critical views among Korean elementary school teachers (Pang, 2016), with more focus on the teacher rather than teaching and student learning among secondary school teachers in Uganda and Malawi (Fujii, 2014). It may be limited by the use of information obtained from watching videorecorded research lessons instead of observing live lessons in the United States (Yoshida, 2012). In the British lesson study episodes, while a secondary school mathematics teacher raised the issue of using more precise mathematical expressions, elementary school teachers suggested general approaches and there were challenges in attempting to connect observations of student and teacher learning in teachers' discussions (Warwick et al., 2016). It is imperative to conduct empirical studies that describe the process of lesson study in detail in order to theorize its activity. A detailed analysis of the process of post-lesson discussion and its impact on teacher professional growth, in particular, is required to expand our understanding of how and why lesson study can be effective for enhancing teaching competencies (Murata et al., 2012).
With this backdrop, the current study has the aims of describing key features of a post-lesson discussion following a 'research lesson' with a focus on participating teachers' reflections on and their learning from the observed research lesson. Post-lesson discussions provided teachers with a critical opportunity, in that it is important to interpret students' learning through lesson study, understand and reflect more deeply on curriculum, teaching methods, and teaching materials, rather than to merely participate in or experience lesson study (Seino & Foster, 2020). However, there is limited information on how an entire group of teachers is engaged in post-lesson discussions and how the focus of the discussions may or may not change over time. The authors noted that evidence demonstrating what teachers discuss and learn through post-lesson discussions was still very scarce and needed to be explored.

Participants and context of the study
The participants in this study included teachers at a public elementary school in Saitama prefecture, in Japan. The school, which is located in the suburbs of a city with a population of approximately 66,000, has 486 students in 18 classrooms in Grades 1 through 6, and 26 teachers with nine staff members. About one third of the teachers were beginning teachers who had teaching experiences of less than five years. The school was designated by the prefecture's board of education as a school for improving students' academic achievement in mathematics for the school years 2013-2014 and 2014-2015. In this context, the school set a research theme (goal) for a 2-year project as 'improving students' academic achievement' with a focus on developing students' thinking for its 'konai-keshu' (intra-school teacher training) of the school year. Under the leadership of the principal, all teachers in the school participated in school-based activities for professional development. The first author was invited to attend the lesson study cycles as a knowledgeable other (Seino & Foster, 2020), and to make suggestions and comments for the improvement of classroom teaching at lesson study meetings. He observed the activities of teachers in lesson study meetings, and attended research lessons with a post-lesson discussions in particular, three times a year, to facilitate the teachers' reflections on the observed lessons and to offer suggestions for the improvement of teaching.
At the school level, a typical lesson study cycle commences at the end of an academic school year, typically January or February, when the faculty decides on a research theme for the new school year that starts in April. To maintain school-wide conduct of lesson study, the school establishes a committee to promote its research activity. During the school year, once the school starts the cycle, several research lessons are scheduled, typically once each term. For the school year 2013-2014, the committee planned and implemented three cycles of research lessons (Table 1). Between the research lessons, groups of teachers discussed issues related to the improvement of students' academic achievement and developed and revised lesson plans for research lessons.
In a 2-year project at K Elementary School, an entire group of teachers participated in several lesson study cycles. In the study reported in this paper we collected data from three lesson study cycles, as shown in Table 1.

Data collection
Given that the study focused on addressing the research questions, a specific method was used to capture what the participating teachers perceived and believed to be significant in the planned and implemented lessons, and what they described as critical characteristics of the lesson they had just observed. In each post-lesson discussion, teachers were first classified into small groups of three (in June and January) or four (October), and they later met as one group for discussions. For each research lesson in June and January, the teachers were categorized into three groups according to the grade level they taught (teachers in Grades 1 and 2, Grades 3 and 4, Grades 5 and 6). The teachers were segregated into four groups only in October with roughly the same numbers. They were first invited to write down on small pieces of paper their reflections and comments on the lessons they observed. Subsequently, these comments were structured in each group by representing them on a large piece of paper (a 'summary sheet') within the group, for the discussion. Summary sheets were used in a particular format during each post-lesson discussion (see Appendices 1, 2, and 3).
After the reflections of teachers were shared in each group, the summaries were shared by the teachers for a general discussion. Accordingly, teachers' 'voices' were better captured than in the typical oral post-lesson discussions where experienced teachers' comments often dominated.
Overall, 496 comments by all the participating teachers were collected and submitted for the analyses, along with other materials such as lesson plans and copies of textbooks. The 'summary sheets' to which all the small pieces of paper were attached in each group of teachers were also collected and analyzed.

Development of a coding system
In this study, all the comments made by the teachers in the post-lesson discussions within the three cycles of lesson study were analyzed. In post-lesson discussions, teachers' comments on the research lesson they observed can be made on various aspects of teaching and learning mathematics in the classroom, such as the learning environment in the classroom, mathematical tasks presented to students, instructional materials, teachers' behavior, students' work, issues related to curriculum sequences, and so forth. Therefore, to analyze teachers' reflections and comments in post-lesson discussions, we needed a broader perspective on proficiency in teaching mathematics that included planning and implementing a lesson as a core.
A conceptual framework for examining proficiency in teaching mathematics (Schoenfeld & Kilpatrick, 2008) was used for developing a coding system. In the concluding chapter in the book on "tools and processes in mathematics teacher education," Schoenfeld and Kilpatrick (2008) presented a "provisional framework" for examining proficiency in the teaching mathematics that provided us with 'dimensions' for examining teachers' comments in this study. Schoenfeld and Kilpatrick (2008) discussed the nature of lesson study as follows: Lesson study is a form of "on the job" professional development, built into the work week. In pursuing lesson design, (What is the mathematics we consider central? How do we plan to have students approach it? What are students likely to do, and how will we react to it?), lesson study involves teachers in the full spectrum of knowledge issues we delineated above. (p. 328) The original framework included the following seven elements related to learning opportunities for teachers ( Table 2).
The first category corresponds to mathematical knowledge of various types that are necessary for proficient teaching. It includes broad and connected knowledge of the content at hand, deep knowledge of where the content comes from and where it might lead, and understanding of 'big ideas' or major themes, knowledge of effective ways to introduce students to particular mathematical ideas, and ways to instill understanding or help counter misunderstanding (Schoenfeld & Kilpatrick, 2008, p. 327). Examining knowledge of various types needed for planning a lesson is the core of kyozai-kenkyu, which entails analyses of teaching material for planning a mathematics lesson in a Japanese tradition (Watanabe et al., 2008).
The second and third categories are related to the planning and implementation of a lesson from a student's perspective. They pointed out the underlying belief that current student understanding is and should be the raw material Table 2 A framework for examining proficiency for teaching mathematics (Schoenfeld & Kilpatrick, 2008, p. 322) Knowing school mathematics in depth and breadth Knowing students as thinkers Knowing students as learners Crafting and managing learning environments Developing classroom norms and supporting classroom discourse as part of "teaching for understanding" Building relationships that support learning Reflecting on one's practice from which lessons are crafted, and this particular view is a core aspect of lesson study (Schoenfeld & Kilpatrick, 2008). The fourth through sixth categories encompass not only the mathematical nature of a task but also the general pedagogical considerations that teachers need to attend to. Although the framework of proficiency in teaching mathematics was not proposed for descriptive purposes in relation to lesson study, each element or dimension corresponds to the elements in professional development through lesson study.
Using the categories in the framework as a guide for examining teachers' reflections and comments, the first attempt at coding with 30 randomly selected comments was implemented. There were some difficulties in identifying categories exclusively for several comments. With some adjustments and amendments, and by sub-dividing the categories, a system of codes was developed resulting in the following seven general categories for coding (Table 3). A few examples of comments in each category are provided.
In a broader sense, Category A is directly related to mathematics, as in the original framework. In the context of lesson study, examining teaching material from a mathematical viewpoint is a core activity for planning a lesson, and teachers often comment on this particular aspect in post-lesson discussions. Category B, "Students' thinking," resulted from integrating the second and third categories with a focus on students' thinking and problem solving during the lessons. In the post-lesson discussion, teachers shared evidence of students' thinking and learning based on their observations of students' behaviors. Categories related to pedagogies (C 1 through C 3 ) included comments on teachers' behavior and/ or utterances in the classroom. Teachers in K elementary school shared the importance of "teaching mathematics through problem solving" as a "motto," and they often mentioned the term and phrases related to a particular focus, such as "task and problem," "problem solving," "looking back," and so on. In addition, when they summarized the comments in the groups, teachers used a particular format that consisted of phases of problem-solving processes in the classroom: presenting a problem or task, anticipation, problem-solving by students on their own, and discussion and summing up. All the categories, C 1 through C 3 , are codes for teachers' comments related to pedagogies with different foci. Category D came from the fifth category in the original framework, "developing classroom norms and supporting classroom discourse". Teachers' comments were based on the classroom environment and, implicit or explicit emphasis by the teacher of mathematical values of the methods students used.
It was noted in the process of analysis that particular terms and phrases were shared by the teachers. For example, the abbreviation "I, HA, KA, SE," which stands for the importance of generalizability, quickness, simplicity, and exactness of mathematical methods to determine the solution, was often found in the comments made by the teachers. The phrase is directly related to the value of mathematics, and teachers use the phrase when they discuss the efficiency of the mathematical methods that students use.
After the January lesson on the division of decimal fractions, teachers in each of the three groups independently commented on the efficiency of the method to find the answer. In this lesson, the teacher posed the following problem: Three children shared an orange juice of 7.2 L. A. Mathematics: Knowing school mathematics in depth and breadth The value of the ratio and its simplification could be further emphasized The term 'common multiple' could have been referred to, when the number 2000 was presented B. Students' thinking: Knowing students as thinkers and as learners The students were explaining their ideas using a diagram Those who found a solution were trying to find alternative methods C 1 . Problem Solving: Pedagogies related to teaching through problem solving The task and the problem were differentiated What strategies could be used? C 2 . Teaching/Learning: Pedagogies related to students' learning Teacher invited another student, when one student had trouble The teacher was trying to gauge the opinions of the children C 3 . General pedagogies The use of the color pen was effective The numbers, the given and the unknown in the problem, were highlighted by underlining D. Rules/Norms: Developing classroom norms and supporting classroom discourse An amiable atmosphere that encourages presentation even if the answer is wrong Finding ideas of "I, HA, KA, SE" (*) E. Other The role of each teacher was very clear in team-teaching The voice of the student who presented his idea was limited How much orange juice does each student consume? Here, students proposed methods such as "multiply 7.2 by 10," translating 7.2 L as 72 dL, and counting "0.1"s in the numbering "7.2." In the lesson in October on "speed," students proposed three different methods to compare the speed. The teachers then mentioned "I, HA, KA, SE," again, to reflect on the aim of the lesson to introduce the concept of speed. It is important to note that these terms and phrases are used in the discourse in a particular context embedded in the whole system to describe and improve the teaching of mathematics.

Coding the comments and analysis of the summary sheet
All comments made by the teachers in the post-lesson discussions within the three cycles of lesson study were analyzed by applying the codes developed in this study. For the analysis of teachers' comments, two research assistants independently coded all the comments (n = 496). The match of categories between two coders was with 399 items (80.4%). As a measure of consensus, Cohen's kappa was estimated for inter-rater reliability, showing substantial agreement (κ ≥ 0.76).
Discrepancies in the remaining items were resolved one by one through discussions between the two coders and the first author. For instance, there was a comment by a teacher: "It is nice for the students to be able to express that they did not understand what classmates said" (October, Group 2). While one of the coders categorized the comment as "Students' Thinking," another coder judged it as "Rules/Norms." Following the discussion, which took into account the situation in which the comment was made, they agreed that the comment could be classified in the category of "Rules/Norms" for the comment refers to the nature of classroom discourse in a general way.
The comments made by teachers in groups at the beginning of the post-lesson discussion were summarized in each group using a "summary sheet" (See Appendices 1, 2, and 3 for examples) and then presented in a whole group discussion. The comments that the teachers considered significant were highlighted and used as a source for the entire discussion. In the following discussion, topics related to reflection are directly related to the classified comments. The original summary sheet included three components of lessons for organizing teachers' comments: (1) presenting a problem or task, (2) prospect and problem-solving by students on their own, and (3) discussion and summing up. The teachers reflected on and discussed issues related to each component individually. The discussion covered broader issues, as Table 4 shows, but a major focus was on improving teaching in practice. Table 4 shows the number of comments made by the teachers in the post-lesson discussion of the three cycles of lesson study. As demonstrated in Table 4, the teachers tended to focus directly on the improvement of teaching in each lesson context by commenting on general pedagogy (27.0%), pedagogies related to students' learning (14.9%), and pedagogies related to problem solving (21.6%). Therefore, a majority of teachers provided comments (approximately two-thirds of the total comments) that were related to pedagogies (63.5%). Teachers' comments on mathematical aspects of the task posed in the lessons were less than expected (8.1%), while comments on students' thinking were third in their frequency (15.3%). The results also revealed that the teachers showed interest in the development of classroom norms and in supporting classroom discourse with comments on rules and norms (9.3%).

An overview of teachers' reflections and comments
There was a change in the format of the 'summary sheet' of the written reflections in the post-lesson discussion. It is interesting to observe that the classification of the small papers differed within the three cycles of lesson study, and the form of the summary sheet was changed by the teachers themselves to better grasp the observations they made on research lessons between the second and third cycles of lesson study. They revised the format to be more problem-solving oriented by aligning the following five components of lessons with the progress of teaching and learning; (1) understanding (problem/task), (2) prospect (looking forward), (3) solving, (4) kneading-up (discussion), and (5) looking back (what was found). This Table 4 Comments made by teachers in the post-lesson discussion *The numbers in parentheses denote the percentage of each category **'Pedagogies related to students learning' is represented as Teaching/Learning hereafter change appeared along with changes in the focus of the teachers' observation and discussion. Table 5 shows the number of comments made by the teachers, with the percentages of each post-lesson discussion in the three cycles of the lesson study. As demonstrated in Table 5, the focus and composition of teachers' comments differed among the three opportunities. Table 5 illustrates the composition of the comments in different categories for each of the three opportunities, and we observe certain changes in the proportion of the categories. A majority of comments in the first opportunity were, for example, pedagogies related to students' learning ('Teaching/Learning'), while general pedagogy dominated in the second opportunity with pedagogies related to problem solving ('Problem Solving') in the second place. It is interesting to observe that teachers' comments on students' thinking and pedagogies related to problem-solving accounted for more than half of the comments in the third opportunity. Figure 4 shows the changes in the percentages of each category across the three opportunities. The percentage of comments related to mathematics, students' thinking, and pedagogy-related problem solving increased as the cycles proceeded. Meanwhile, teachers' comments on pedagogy related to students' learning and rules/norms in the classroom decreased. Taking the categories of 'Students' thinking' and 'Pedagogies related to students' learning' into consideration together, we notice a constant focus on students' thinking was maintained during lesson study in this school.

The shifts in focus of teachers' comments
The results of the analyses revealed that the focus of postlesson discussions changed in a certain way as the cycles of lesson study proceeded. In particular, teachers tended to focus more on students' thinking and mathematics, and on teaching through problem-solving. The change in increasing attention to student thinking and problem-solving appeared in the new form of a summary sheet. Accordingly, the topics discussed by teachers during the post-lesson discussion focused more on examining students as thinkers and facilitating problem-solving in the classroom.

Teachers' reflections in post-lesson discussions
Although lesson studies are increasingly practiced around the world (Huang & Shimizu, 2016;Huang et al., 2019;Quaresma et al., 2018), there was limited data on the mechanisms and structure of post-lesson discussion and its contribution to teacher learning and growth at the school level. In the current study we aimed to provide a deeper understanding of one specific aspect of lesson study, namely, postlesson discussion as practiced in a Japanese school-based lesson study. In this section, we discuss the findings of the study in a broader context.
In addressing the first research question, 'What topics or themes can we find in teachers' reflection in the post-lesson discussion in a school-based lesson study in mathematics?' we tried to characterize all the comments made by teachers by taking the "dimensions" in the "proficiency for teaching mathematics" (Schoenfeld & Kilpatrick, 2008) into consideration. Consequently, we developed a coding system, and all comments were classified into seven categories. Fernandez and Yoshida (2004) described in detail the process of how a particular group of teachers conducted lesson study, how they collaborated to develop a lesson, what they talked about during the process, and what they considered in order to understand deeply how students were learning. Among other elements, they reported the manner in which a group of teachers in an elementary school collaboratively planned and implemented lessons and how much content knowledge the teachers manifested. As Lewis et al. (2006) identified to be one of the three critical needs for research on the functions of lesson study, we need to expand the descriptive knowledge base of the various processes involved in lesson study. In particular, there were only a few empirical studies on what occurs during post-lesson reflection and discussion in lesson studies. Warwick et al. (2016) investigated teachers' learning within group discussions among teachers during the lesson study process. Looking for evidence that teachers learned something about teaching, students' learning, or mathematics content, they identified the discussion episodes that were most productive. The nature of all these episodes was similar; the discussions were about the way in which their observations of students shaped the kind of instruction they hypothesized to be most effective. The current study provides further evidence on what teachers reflect on and discuss in post-lesson discussions. The results revealed that the teachers tended to focus directly on the improvement of teaching in each lesson context, with approximately two-thirds of the total comments related to pedagogies. Also, the importance of observation of students' activities in research lessons appeared, as teachers' reflections and discussions focused on pedagogy related to students' learning, teaching, and to problem solving.

Changes in teachers' reflection over time
As for the second research question, 'Are there any shifts and changes in participating teachers' reflection over time on classroom events in post-lesson discussions in lesson study cycles', we found that the focus of post-lesson discussions changed in a certain way as the cycles of the lesson study proceeded. The topics discussed by the teachers during the post-lesson discussion focused more on recognizing students as thinkers. In this context, the adapted version of the IMPG proposed by Schipper et al. (2017), which was proposed based on the study of retrospective self-report by teachers and the interviews that were conducted after completion of lesson study cycles, can help shed new light when we look the into 'Lesson Study Domain' with a dynamic process of lesson study cycles to see how teachers reflect on and discuss the observed lessons.
There is a shared view of teachers' excellence in Japan in terms of their ability to interpret topics taught in relation to mathematical background, and anticipate students' thinking about a topic (Shimizu, 1999). Raising teachers' awareness of students' mathematical thinking provides teachers with a basis for their instruction and also for their own continued learning (Llinares & Krainer, 2006). The results of this study revealed that a constant focus on students' thinking and learning was maintained during a school-based lesson study. The theme of the lesson study for K Elementary School was set in response to the designation of the school as a model school for the improvement of students' academic achievement by the prefecture. The theme of the lesson study was not directly mentioned in the post-lesson reflection and discussion. Therefore, teachers' motivation to conduct lesson study seemed derived partly and directly from their own views on and interests in student thinking and learning.
Interaction in the post-lesson discussion with writing can shape the development of discourse on learning mathematics. Beginner teachers in general tend to be silent in postlesson discussions in lesson study cycles due to their inexperience. For those beginner teachers, opportunities to express their (naïve) reflections seemed to be raised by writing in this particular type of post-lesson reflection and discussion, as opposed to the dominance of experienced teachers' oral comments in post-lesson discussion. It is noted that particular terms and phrases, such as the abbreviations 'I, HA, KA, SE', emerged in the analysis process. The Japanese tradition of lesson study created a teaching community in which observation and discussion of mathematics lessons are an integral part of professional practice. In the discourse of lesson study particular pedagogical terms and phrases can be found to have specific significance to describe particular roles of teachers and students (Shimizu et al., 2021).
In earlier studies, lesson study as an improvement process was considered a form of professional development with the goal of improving the competencies of teachers. In addition, lesson study can be seen as a research methodology, that is, a way of conducting research that may be especially suited toward improving our knowledge of teaching and the quality of teaching (as opposed to teachers) (Stigler & Hiebert, 2016). Focusing on a post-lesson discussion within lesson study cycles as a 'tool' for improving our knowledge of teaching and the quality of teaching can be explored in future studies.

The border between external domain and domain of practice in IMPG
With regard to the third research question, 'To what extent is the IMPG applicable in incorporating lesson study elements for describing teacher professional growth', the current study suggests that the External Domain is represented as a square in the original model, located outside the teacher's personal world (Clarke & Hollingsworth, 2002, p. 951), can be reconsidered as a more dynamic and influential domain in the case of lesson study, mostly with interaction and conversations with colleagues. Whereas the External Domain stays almost the same in the study by Widjaja et al. (2017), who attempted to incorporate the Japanese approach to teaching mathematics through problem solving, the shifts found in the topics of post-lesson discussion suggest that activities in the External Domain and the Domain of Practice seemed to be related and interchangeable. Furthermore, both the reflection and enactment between the two domains can have an impact. The major processes of Enactment and Reflection between the External Domain and the other three domains, Domain of Practice, the Personal Domain, and the Domain of Consequence, must have an impact on teachers' professional growth in terms of a deeper understanding and knowledge of students as learners. The observed change in the form of the 'summary sheet' can be considered as having been provided by the External Domain, but it resulted from teachers' activities in the Domain of Practice.
After describing lesson study as providing a good structure or system for teachers to collaborate and investigate how to improve classroom instruction and learning, Yoshida (2008) noted that "another feature of lesson study is that it puts students at the heart of the professional learning activity." He added, "Improving students learning and understanding is the focus of lesson study so it is important for the teachers to have as many opportunities as possible to carefully observe and discuss changes in student learning and understanding through classroom practices (p. 98). Therefore, the quality of activities in the Domain of Practice as related to and interchangeable with the External Domain is the key for professional growth of teachers in lesson study.

Concluding remarks
Given the previous lack of an empirical study on the reflections of participating teachers in the post-lesson discussion, in the current study we aimed to address the important gaps in the research by offering some insights to deepen our understanding of the critical process of post-lesson discussion in lesson study cycles. The use of written reflections by teachers and a summary by the teachers themselves provided a window through which researchers could see the focus of post-lesson discussion that has potential impacts on the process of teacher professional growth.
The finding of changes in teachers' reflections and comments reveals that a particular shift over time in the topics among teachers focuses more on students' thinking but is less explicit in teaching and learning and classroom rules and norms. In addition, the findings suggest that activities in the External Domain and the Domain of Practice are related and interchangeable, with both reflection and enactment between the two domains as facilitators of teachers' changes. A further study is needed to explore the dynamics in activities across domains over time.
For more than two decades, since the publication of The Teaching Gap (Stigler & Hiebert, 1999) in particular, researchers in the field of education have been interested in lesson study as a promising endeavor for teachers' professional growth in various contexts. Given the growing popularity of lesson study and the widespread interest in learning about lesson study around the world, Japanese researchers have felt compelled to become more explicit about the theory behind the 'ordinary' activity like konaikenshuu. A deeper understanding of the empirical findings of lesson study helps researchers understand which parts of the lesson study are critical for a given purpose and which are not.
By constructing explicit theories of lesson study, it is possible to distinguish between essential cultural practices and routines that can be applied to a new setting. Clarke and Hollingsworth's (2002) IMPG provides a conceptual framework for seeing the 'ordinary' activity through the lens of theories. For a Japanese mathematics educator who has been deeply involved in lesson study for more than two decades in local contexts, this model has provided an opportunity for reflecting on how lesson study as a cultural activity works as a system embedded in the community of teachers with shared values and beliefs.