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.
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 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 (C1 through C3) 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, C1 through C3, 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. 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.