WHAT KNOWLEDGE DO MATHEMATICS TEACHERS DEVELOP DURING THE LESSON STUDY PRACTICE? A CASE STUDY FROM SECONDARY SCHOOL

Lesson study has received significant attention as a model of professional development among mathematics teachers. Evidence highlights its effectiveness in improving pedagogical practices and student learning, however, less is known about the predispositions which may encourage teachers’ participation in Lesson Study or the impact of participation on teachers’ attitudes. Such findings are relevant considering the voluntary context of teachers’ participation in professional development in Ireland. This research investigates the motivational variables which impact teachers’ participation in Lesson Study, specifically their self-efficacy in teaching mathematics for conceptual understanding and their attitudes towards self-development in Lesson Study. Post-primary mathematics teachers (N = 64), spanning various levels of experience in Lesson Study, completed a survey using a set of pre-validated scales. Findings indicate that teachers’ mathematics teaching self-efficacy is a significant predictor of their participation in Lesson Study. Furthermore, the research finds that teachers’ familiarity with Lesson Study impacts the likelihood of their participation in this model of teacher education. These findings build upon previous knowledge in this field and demonstrate the significance of teaching self-efficacy as a presage variable for developing a positive disposition towards Lesson Study. The paper discusses the implications of these findings for teacher education in Ireland.

PurposeTeachers in the USA have been conducting lesson study for more than ten years since it was introduced from Japan in the late 1990s. Although interest in conducting lesson study in the USA is still strong and greater numbers of teachers have become involved in this professional learning, there are significant obstacles to conducting high quality and effective lesson study that enhances teachers’ content and pedagogical knowledge, as well as improving their instruction and student learning in classrooms. Because of the needs of improvement in lesson study in the USA, so it can be administered effectively and sustained, the purpose of this paper is to discuss the current status of lesson study in the USA, what high quality and effective lesson study is, and what ideas might be help to improve lesson study in the USA.Design/methodology/approachIn this paper, issues that are common barriers to conducting effective lesson study, such as: US teachers’ misunderstanding or lack of understanding of lesson study; teachers’ insufficient knowledge of content, pedagogical content, curriculum knowledge; lack of support from administrators for lesson study; non‐systematic approach to implement lesson study; and having short‐sighted vision to conduct lesson study will be discussed. The discussions are based on the author's 12 years of experience working with teachers, schools, and school districts in the USA, interactions and information exchanges with other lesson study educators and researchers and professional development coordinators in schools and districts in the USA, and existing research documents in the USA. Through this discussion, the author attempts to provide suggestions for improving lesson study in the USA.FindingsIn order for lesson study to be successful, teachers need to think of lesson study as a way to improve their own learning as well as student learning. Spending more time studying mathematical content and curriculum, developing a strong pedagogical content knowledge with colleagues, and establishing a professional community of learning through lesson study will help it to be effective for improving classroom teaching and learning.Originality/valueThe paper provides some helpful suggestions for improving quality and effectiveness of lesson study in order to improve: classroom teaching – teacher's content, pedagogical content and curriculum knowledge; and student learning. The paper is particularly valuable for lesson study practitioners, and administrators and staff developers who are implementing lesson study in schools.

Research on teachers' knowledge for teaching is an essential domain in teacher education. This article used a Japanese Lesson Study to develop prospective mathematics teachers' knowledge for teaching measures of central tendency from bar charts and histograms. More specifically, the research focuses on developing specific content knowledge (Specialized Content Knowledge, SCK), knowledge of content and student (Knowledge of Content and Students, KCS), and knowledge of content and teaching (Knowledge of Content and Teaching, KCT) for prospective mathematics teachers through a lesson study process. The research results indicated that the prospective mathematics teachers had improved their knowledge for teaching through a lesson study cycle. The results provide insights into the effectiveness of the implementation of lesson study in initial mathematics teacher education.

Research interest in the professional knowledge of mathematics teachers has grown considerably in recent years. In the COACTIV project, tests of secondary mathematics teachers’ pedagogical content knowledge (PCK) and content knowledge (CK) were developed and implemented in a sample of teachers whose classes participated in the PISA 2003/04 longitudinal assessment in Germany. The present article investigates the validity of the COACTIV constructs of PCK and CK. To this end, the COACTIV tests of PCK and CK were administered to various “contrast populations,” namely, candidate mathematics teachers, mathematics students, teachers of biology and chemistry, and advanced school students. The hypotheses for each population’s performance in the PCK and CK tests were formulated and empirically tested. In addition, the article compares the COACTIV approach with related conceptualizations and findings of two other research groups.

Teachers' reflection is one important step in the implementation of Lesson Study. To enhance the professionalism of teachers, lesson study has become an important practice in the implementation of the Professional Learning Community (PLC) in Malaysia. "Video Critique" and "Lesson Study" are some of the tools in the implementation of PLC. As Lesson Study is newly introduced in 2011, teachers are still in the learning stage of implementing it. Among the problems teachers face is ways of making good reflections during classroom observation. Thus, this study aims to explore the ways in which excellent teachers reflect as well as their reflection level by looking at a video of teachers teaching at a school in Negeri Sembilan. The study was conducted using qualitative methods. The study participants were 17 excellent teachers, including two lecturers. Data was collected using reflective observations, documents of the participants, and recordings of the video critique sessions. Data was analysed using reflective dialogue session video documents analytical technique that has been transcribed. It was found that there are four stages of reflections: Descriptive, Dialogue Reflective, Descriptive Reflective and Critical Reflective. A majority of the excellent teachers reflected at the stage of Dialogue Reflective. Some of the excellent teachers made Descriptive Reflective, and only a few teachers succeeded brilliantly at Critical Reflective. Originality/value - This study implies a need for exposure to reflective practice during the process of observation so that teachers receive deep and meaningful feedback through practice in community learning.

This thesis documents the design, implementation and evaluation of a professional learning programme to support teachers in developing their ability to teach mathematics through problem solving. Teaching mathematics through problem solving is recognised as an effective way to teach mathematics. However, there are a number of different approaches with different descriptors such as problem posing and problem-based learning. In this study, teaching mathematics through problem solving is an approach where learners explore a problem or a task and their responses to the task are ‘orchestrated’ by the teacher to introduce and develop the mathematics to be learned. The development of this teaching approach involves the principles of task design and planning the sequencing of anticipated responses to be orchestrated. The professional development programme was informed by research on task design and the features of effective professional learning for teachers. The programme was built around the professional learning programme Lesson Study and was modified with a ‘pause’ in the research lesson and the use of remote visualisers to observe and analyse the work of the pupils. This latter feature was an adaption of the original programme design to respond to conditions arising in schools during the Covid-19 pandemic. Also a number of additional modifications were made to accommodate the ambitions of the programme. A qualitative approach was adopted using the methodology of design research and comprising two research cycles. The cohort in the first cycle were secondary school teachers and in the second cycle they were primary school teachers, all working in England. The aim of the study was to understand the effect of the programme on the participants’ knowledge, skills and confidence in teaching mathematics through problem solving. Existing frameworks in the literature on the known characteristics of effective professional learning were synthesised to evaluate the effectiveness of the programme. The main outcomes from this research confirm that the designed features of the PD programme were effective in supporting teachers in developing their ability to teach mathematics through problem solving. The use of task design principles in conjunction with the development of the teaching technique of orchestration resulted in gains in teacher confidence and subject and pedagogical knowledge. The pause in the research lesson and the use of visualisers to observe and analyse the pupils’ work provided the participants with a focused learning opportunity to develop the teaching approach. The evaluation of the programme using the designed analytical framework for CPD identified [a range of] known characteristics of effective CPD in the programme as designed and implemented. Three main implications emerged from the study, the first of which concerns the future development of teaching mathematics through problem solving. It was apparent that there were contextual barriers to the approach becoming an established part of teachers’ practice. To address these, the design of mathematics teaching resources would need to reflect the ambition to teach mathematics through problem solving. This is an issue for curriculum designers. There would also need to be recognition of the problem-solving skills that pupils require to access the problems, which would need to be taught either as part of teaching through problem solving or as a separate approach known as teaching for problem solving. The second implication identifies the challenges associated with introducing this type of programme into the current landscape of professional learning for teachers of mathematics in England, including the need for greater coherence between the professional learning of teachers of mathematics and their own workplace contexts. Finally, the third implication relates to the use of Lesson Study. This study described how the use of design research can develop Lesson Study variants that can make a positive contribution to professional learning environments for teachers without compromising the key principles of Lesson Study.

This paper presents the findings on mathematics teachers' pedagogical content knowledge (PCK) in statistics teaching. Six mathematics teachers were initially selected for the study based on their school’s performance over two years in the senior certificate examination in mathematics, and the four top scorers in the a conceptual knowledge exercise (CKE) in statistics were finally selected for this study . The study adopted a qualitative research method. The data on the teachers’ PCK were collected through lesson observations, questionnaires, interviews, video recordings, teachers’ written reports, and document analyses. The results of the study show that the teachers possess topic specific subject matter content knowledge and use of procedural and conceptual knowledge to teach statistics in school mathematics. The implications for mathematics education programmes are discussed. DOI: 10.5901/mjss.2013.v4n14p191

This study aims to investigate and compare mathematics teachers’ knowledge of pedagogical content, knowledge of students’ understanding and knowledge of instructional strategies, subcomponents of pedagogical content knowledge, student errors in the relationship between perimeter and areas in rectangles, squares, and parallelograms. 10 pre-secondary school in-service mathematics teachers and 10 prospective mathematics teachers participated in the study. The qualitative case research approach was used. To collect the data, an interview form consisting of four questions showing student errors related to perimeter and area was prepared. The participants were asked to comment on the questions given in the form, and their answers were recorded. Later, they were asked to write down their answers to these questions. According to the outcomes, there is a lack of knowledge of students’ understanding and knowledge of instructional strategies, which are the subcomponents of the pedagogical content knowledge of prospective mathematics teachers. Moreover, prospective teachers are found to be incompetent as regards the necessary mathematical subject matter knowledge. To prevent difficulties, when the concepts of perimeter and area are taught, instead of giving formulae initially, concrete materials or real-life examples about these concepts should be provided. Keywords: pedagogical content knowledge, perimeter, area, mathematics teachers, prospective mathematics teachers.

The purpose of this activity is to observe the learning process in relation between learning materials, students and teachers interaction and interactions among students in learning through the implementation of lesson study. In general, this activity is a collaboration among the University of Muhammadiyah Semarang, the Regional Leadership of Muhammadiyah Central Java and all Muhammadiyah schools in Central Java from July to September 2021. In particular, this activity was held at Muhammadiyah Plus Junior High School Salatiga in August 2021. Lesson study based on The school was participated by mathematics and science teachers at Muhammadiyah Plus Junior High School Salatiga based on the following stages: (1) planning (plan), (2) implementation (do), and (3) reflection (see). Mathematics and science teachers carry out lesson plans by making chapter designs and lesson designs. The implementation of lesson study at Muhammadiyah Plus Junior High School Salatiga based on integrated learning which applied in group learning through the interaction of living things with the environment. Furthermore, there is a reflection towards implementation stages and find to solve the problem happened so that the next learning can be carried out properly.

This quantitative correlational study investigated the relationship between mathematics teachers’ content knowledge and pedagogical skills and the learners’ mathematics achievement in secondary schools in Cagdianao District, Dinagat Islands. Participants included 20 secondary mathematics teachers and 326 students from Grades 7 to 11, selected through total enumeration and stratified random sampling, respectively. Data were gathered through validated survey questionnaires measuring teachers’ mathematical content knowledge, pedagogical skills, and students’ self-assessed competencies, alongside students’ final grades in mathematics. Results revealed that teachers generally possessed high content knowledge and pedagogical skills, though technology-related pedagogical skills were only moderately satisfactory. Educational attainment significantly influenced content knowledge and selected pedagogical skills, but no significant correlation was found between teachers’ content knowledge and students’ achievement. Findings underscore the importance of continuous professional development, particularly for teachers without postgraduate qualifications. Recommendations include targeted training, incentives for advanced study, and ongoing assessment of teachers’ mathematical proficiency to enhance student outcomes.

In recent years, probability has become an important content strand in high school mathematics curriculum around the world. Consequently, research on the knowledge for teaching probability has attracted significant interest. However, studies addressing the knowledge for teaching conditional probability remain limited. This study investigated the changes in high school mathematics teachers’ knowledge for teaching conditional probability in Vietnam through lesson study. Lesson plans, classroom observations, reflection discussions from a convenience sample of three teachers were used as data in the current study. The data was analyzed using qualitative methods. The findings indicated changes in the teachers’ knowledge for teaching conditional probability across four domains: Common Conditional Probability Knowledge (CCPK), Specialized Conditional Probability Knowledge (SCPK), Knowledge of Conditional Probability and Students (KCPS), and Knowledge of Conditional Probability and Teaching (KCPT). Specifically, the results revealed significant shifts in teachers’ conceptual understanding of the topic, including its nature, meaning, and related formulas. The study further documented significant progress in the teachers’ pedagogical knowledge: improving proficiency in visual representations; enhancing capacity for anticipating and monitoring student thinking and misconceptions; and focusing on leveraging real-world contexts for conceptual introduction. Collectively, these advancements illustrated the positive impact of lesson study on developing knowledge required for teaching conditional probability. Suggestions about future research to support teachers with various experiences in learning to teach new mathematical topics are included.

Existing studies conducted on education have aimed to contribute to educational reforms by focusing mostly on students. Both in Turkey and throughout the world, further studies are required to analyze teachers in addition to students, particularly teachers' knowledge base needs further investigation. Educational research should not necessarily be associated only with those conducted by collecting data in the field. Meta-synthesis studies that offer general outlooks on previous studies are also required in the field of education. To address this gap in the literature, this article aims to analyze studies in Turkey that have investigated pedagogical content knowledge (PCK) in the field of mathematics education.The concept of pedagogical content knowledge was first introduced by Lee S. Shulman at the annual meeting of the American Educational Research Association in 1985. Shulman (1986) argued that there was an imbalance between content knowledge and general pedagogical knowledge and that no relation had been established between these two types of knowledge; he described this deficiency as the missing paradigm. With an effort to eliminate this deficiency, Shulman (1987) defined PCK as the special combination of content knowledge and pedagogy. Shulman conceptualized PCK in two categories. The first category is the knowledge of teaching strategies and representations that suggest how to organize, represent, and adapt the subjects that are taught. The second category is the knowledge of students' subject understanding at different levels. These two components are the unique professional expertise of teachers who act as the bridge connecting content and pedagogical knowledge.Despite the fact that Shulman's conceptualization of PCK is considered a cornerstone in the literature, he was later criticized by scholars working in this field. Depaepe, Verschaffel, and Kelchtermans (2013) collected the criticisms directed at Shulman's PCK model under five categories. The first was the lack of a theoretical and empirical basis for the presence of PCK as a separate category in the knowledge base of teachers. The second was related to his static view of PCK as a type of factual knowledge that could be acquired and applied independently from the classroom context. The third criticism arose from the concern of researchers about the possibility of theoretically and empirically distinguishing PCK from content knowledge. The fourth was that Shulman conceptualized PCK in a very narrow framework under two categories. The fifth one relied on the argument that PCK cannot be normative, as it can vary according to culture.In accordance with the criticisms in the literature, several scholars from different subject areas restructured Shulman's PCK model in different ways in an attempt to clarify the borders between PCK and other types of knowledge. In this context, the first studies on PCK were conducted by Shulman's colleagues (Grossman, 1990; Marks, 1990). Grossman, one colleague of Shulman, expressed PCK, which Shulman had framed too narrowly, by widening it. Grossman (1990) divided PCK into four components: (a) knowledge of students' understanding, (b) knowledge of teaching strategies, (c) knowledge of teaching purposes, and (d) knowledge of curriculum. Grossman expanded Shulman's PCK model by adding knowledge of teaching strategies and knowledge of curriculum as separate components in PCK.Marks (1990), another colleague of Shulman, divided PCK into four components in light of his empirical study's findings: (a) knowledge of subject matter, (b) knowledge of students' understanding, (d) knowledge of teaching media, such as materials, books, and so on, and (d) knowledge of teaching processes. Marks argued that these four components were not independent from each other but that they intensively interacted with each other. He advocated his argument, expressing that a mathematics teacher, when deciding whether or not the examples of mathematical operations in a course book are sufficient, reflects to a certain extent his/her knowledge of media for content education, knowledge of teaching processes, and knowledge of students' content comprehension. …

The chapter starts by explaining how the United Kingdom (UK) Lesson Study model emerged and how learning difficulties are understood in a UK context. This leads to a programme summary of design-based research over the past four years that has applied and developed this model of Lesson Study in the field of teaching and learning of pupils with learning difficulties. In the first two phases of this programme of design-based research, secondary school subject teachers used a UK model of Lesson Study to develop their teaching and the learning of pupils with moderate learning difficulties (MLD). This model involved a focus on 2–3 case-students around whom the research lesson planning and review takes place. The evaluation led to changes in the Lesson Study model in terms of the use of a relevant knowledge base to inform the Lesson Study review and planning meetings and the role of senior teachers in supporting the Lesson Study process. The research showed how teachers developed and intensified their general teaching strategies to meet the needs of pupils and illustrated the mechanisms perceived by teachers as underpinning positive Lesson Study outcomes. There was evidence of case-students’ gains in academic skills and learning behaviour, while participating teachers developed relevant pedagogic knowledge about teaching pupils with learning difficulties. In the third phase, primary and secondary Lesson Study teams used an adaptation of Lesson Study to assess the learning needs of pupils with learning difficulties. This use of Lesson Study as a form of response to teaching or dynamic assessment arose from experiences in the first two phases. The evaluation stage of the Lesson Study for assessment (LSfA) led to changes in the Lesson Study procedures for this purpose. There was also evidence of changes in teacher assessment of case-pupils’ needs, with a greater focus on the conditions that support learning than on child functioning factors. The chapter concludes with a brief reference to the further on-going development of Lesson Study that involves dispersed inter-professional teams using video-conference links.

The Perception of Teachers and Students toward Lesson Study Implementation at Rural School of Sabah: A Pilot Study

Lesson study (LS), a teacher-oriented, student-focused professional development approach that originated in Asia, has spread globally. However, findings related to the effect of LS on teacher learning and growth have varied widely. This suggests a need to review current research on LS in the field of mathematics education. This paper focuses on mathematics teachers’ learning and professional growth through LS using Clarke and Hollingworth’s interconnected model of professional growth (IMPG) that contains three components: domains of change, mediating mechanisms, and change environment. With regard to domains of change, we found that although previous LS studies—often small scaled, qualitative studies—attended to all domains of change, they have more frequently reported changes in the personal (or collective) domain (e.g., teacher knowledge) but have relatively less frequently reported changes in other domains. In fact, there is a lack of research on systematic changes across domains in the IMPG. Regarding the mediating mechanisms, research has well explored both enactment (planning and teaching) and reflection (debriefing). For enactment, the planning process is less appreciated, and there is an inconsistent understanding of the role of the knowledgeable others. For reflection, there is a lack of focus on student thinking and constructive feedback during this process. Finally, research has suggested that LS has received support from the change environment but that tensions (e.g., time limits) have remained. These challenges call for systemic support and cross-cultural collaborations to develop sustainable and large-scale LS. We suggest future directions for continued research and practice.