Lesson Study for Learning Community (LSLC) is one of the efforts that can be done to improve teacher learning abilities. LSLC is an activity that can solve problems faced by teachers by applying various methods or appropriate learning strategies according to the demands of the 2013 curriculum and 21st century skills. PMRI of Indonesian Realistic Mathematics Education is a learning approach that is associated with real things or real situations that have been experienced by students so that the role of PMRI strongly supports the implementation of LSLC. The method used is descriptive qualitative research. While data collection is done by observation, documentation, field notes, and lesson plan and students' worksheet. This research was conducted on eleventh grade students at a state secondary school in South Sumatra Indonesia with the subject matter of three dimensions. There are four stages of LSLC; Plan, Do, See, and Re-design, and the teacher designs the lesson plans and worksheets based on PMRI. The media used is student worksheet. The results of this study indicate that learning with the application of PMRI-based LSLC can help students understand the concept of three-dimensional material, can build good collegiality between mathematics teachers, can solve problems faced by teachers, implementation of learning is in favor of students, and teachers gain experience about learning with PMRI based LSLC.

This study aims to determine the impact of implementing mathematics learning with LSLC in schools so that it can be a consideration for teachers so that they can be applied in the next learning process. The research method used is a literacy study with a meta-analysis type of research. After analyzing, the implementation of LSLC in mathematics learning in schools has a good impact on mathematics learning itself. Collaborative learning appears in learning, mathematical concepts are easily understood by students, and indicators of mathematical processes appear in learning. Students are also more active because learning is centered on students, while the teacher only serves as a facilitator. Most of the findings explain the effectiveness of implementing LSLC in learning mathematics in schools. So, Research on the implementation of LSLC in learning mathematics in schools should be increased in order to reduce problems in learning mathematics.

Teaching learning process in math hasn't been requiring student to construct their knowledge and to exercise their critical thinking. So need to develop constructivism instruction materials for exercise student's ability in critical thinking. This research aim to: (1) Produce the valid and practice constructivism instruction materials to exercise student's ability in critical learning, (2) Know effectiveness it to exercise student's ability in critical learning. Instruction materials were developed in this research are plan learning, student worksheet and test of student's ability in critical thinking. Development research methods are used by this research. The methods consist of analysis, design, evaluation and revision. Collecting data use Observation and test. Observation is used to know the student's activity in teaching learning process and to know the teacher's ability in learning constructivism approach management. Test is used to know the result of student's ability in critical thinking after teaching learning process. Subjects in this research are students from the first semester in class X.7 at SMA Negeri 10, Palembang . The total subjects are 40 students. The conclusions are: 1. Instruction materials process development is for quadrate equivalence only that begins from analysis design, and than evaluation until revision. 2. Instruction materials prototype is develope, fulfill in valid and practice criteria also have effect potential. 3. Some aspects of student's activity in learning process do not effective; because restructure (57.92 %) and review (55.00 %) just in enough category. The result learning test to evaluate student's ability in critical thinking is 24.38. It means the student's ability in critical thinking is good category. The final conclusion is constructivism instruction materials can practice to exercise student's ability in critical thinking for math learning.

Teachers' academic and professional competencies significantly improve the quality of their learning. An ongoing process is needed to support and develop their quality. This study developed a learning environment through the Realistic Mathematics Education (RME)-based Lesson Study for Learning Community (LSLC) for high school mathematics teachers. The model is valid and practical and potentially affects the learning quality of high school mathematics teachers. The research employed a design research method of development studies was conducted in three stages: the preliminary stage, the development or prototyping stage, and the assessment stage. Prototyping development is a formative evaluation in which the phases include self-evaluation, expert review, one-to-one, small group, and field tests. The research subjects were 15 high school mathematics teachers from four schools in Palembang. Data was collected through questionnaires, observation, and documentation. The research has resulted in a valid and practical teachers’ working group-learning community-class model that potentially affects high school, mathematics teachers. The learning environment is in the form of training in working groups for mathematics teachers, teacher mentoring in learning communities in schools, and teacher assessment learning processes in the classroom. The learning tools were produced using the RME-based LSLC system. The data analysis shows that the learning environment using the RME-based LSLC model can make high school mathematics teachers significantly understand learning, design learning tools, carry out learning, and evaluate learning. Consequently, the teachers’ academic competence and professionalism significantly improve their learning.

There have been many government policies to improve the quality of teachers. However, the results of the 2017 Teacher Competency Test (UKG) show that the national average professional ability is at 55.91 and the national average pedagogic ability is at 50.71. This shows that all these scores are below the minimum standard, namely 70. In the process of learning Mathematics in schools, the quality of teachers must be continuously fostered. Therefore, this study aims to develop the professionalism of high school mathematics teachers, through the 3P Model (Training, Guidance and Assessment) with the Lesson Study For Learning Community (LSLC) system. This study involved 25 teachers from four high schools in Palembang City. This research uses two types of design research, namely development studies and validation studies. The result of this research is a learning environment using the 3P model (Training, Assistance, and Assessment) with the LSLC system. So that with the 3P model and the LSLC system can be a bridge that connects teachers by focusing on learning strategies in improving professional and pedagogical mathematics teachers comprehensively. These results show a significant increase in the professional development of mathematics teachers in the city of Palembang.