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Fractions are one of the most essential concepts in mathematics learning. In elementary school, a student's understanding of fractions does not fully involve the meaning of various representations, so it could cause learning obstacles. The purpose of this study was to identify and analyze students' learning obstacles on the topic of fractions in grade 5 of elementary school. This study uses a qualitative method. The participants of this study were 21 elementary school students with different characteristics. Data was collected using test techniques, interviews, observation, and document study. Data were analyzed qualitatively to identify learning obstacles experienced by students. The research results are an overview of learning obstacles with the type of epistemological obstacle in fractions, including errors in operating fractions, needing to understand the purpose of the questions, and not understanding the problem with a different context. The research results obtained can be used to develop a hypothetical learning trajectory.

Traditional games can be used as a context to understand mathematical concepts. Many studies on learning trajectory (LT), which are set in the context of traditional games, have been carried out. However, no study focuses on using the context of the traditional kempreng game to facilitate primary students in understanding integer addition and subtraction. This research aims to design an LT through traditional game kempreng to help construct students' understanding and mathematical concepts through various activities that have been developed. This study uses design research with stages, namely preliminary design, teaching experiment, and retrospective analysis. This study produces LT which consists of five activities, namely: consisting of calculating addition points through the kempreng context, calculating subtraction (difference) through the kempreng context, performing additional operations on the “same” kempreng card, performing subtraction operations on “different” kempreng, and determining the results of addition or subtraction operations of number cards with the opposing group. In this study, LT set in the traditional game of kempreng can be used as a context in preparing learning designs that could stimulate students thinking in understanding the mathematical concepts of integer addition and subtraction. The impact of the research design also occurs on teachers. Learning becomes a student center, the teacher as a facilitator. The results of this study can have implications as one of the considerations for using contexts with real and cultural backgrounds to reduce or build a meaningful understanding of the concept of integer addition and subtraction for primary students.

Statistics is one of the mathematics subjects that can be useful in everyday life, especially data presentation material. However, many students still have difficulty understanding the data presentation material. So, it is necessary to design learning using the appropriate context and approach in learning data presentation material. Therefore, the researcher aims to determine student learning trajectories in understanding the presentation of statistical data using traditional markets as a context in the learning process using the Indonesian Realistic Mathematics Education (PMRI) approach assisted by interactive video. The research method used is the design research, which has three stages: the preliminary design stage, the experimental design stage (pilot experiment and teaching experiment), and the retrospective analysis stage. This study indicates that the learning trajectory obtained can help students understand the concept of data presentation material. The learning trajectory uses traditional market contexts and consists of 5 activities, namely: observing traditional market context learning videos and determining data, presenting data in the form of various tables, presenting data in the form of bar charts, presenting data in a line chart, presenting data in the form of pie charts and solving a contextual problem. Furthermore, the results of this study are expected to be an inspiration in learning using the context of everyday life.

Reflection and translation material are critical to be mastered by students in learning transformation material. However, reflection and translation material are still difficult for students to understand. Therefore, this study aims to develop a learning trajectory that will assist ninth-grade students of Junior High School in grasping the notion of reflection and translation in the context of the Sam Poo Kong Temple in Semarang. The research used the design research method, which consisted of three stages: the preliminary design, the design experiment (pilot experiment and teaching experiment), and retrospective analysis. The PMRI approach was used to develop learning activities in this research. This study involved 32 ninth-grade students at a junior high school in Semarang city. The results of this study is a learning trajectory that includes a series of learning processes in four activities. These are observing the Sam Poo Kong Semarang video and analyzing the properties of reflection, finding the reflection formula, analyzing the properties of translation, discovering the translation formula, and solving contextual problems related to reflection and translation. The activities conducted can enable students to develop a better understanding of material reflection and translation. The research's findings indicate that using the Sam Poo Kong Semarang temple context can equip ninth-grade students to comprehend the concept of material reflection and translation. Additionally, the outcome of this study offers additional local wisdom options that can be used as a context in mathematics.

One goal of learning mathematics is to foster students’ critical thinking skills, but this has not been realized well in Indonesia. This study aims to produce a learning trajectory that can help students grow their critical thinking skills in studying quadrilaterals using the Burongko cake context. This study uses a design research method that comprises three stages, namely (1) preliminary design, (2) experimental design, and (3) retrospective analysis. This research was conducted at Madrasah Tsanawiyah As’adiyyah Bontouse with 25 students, nine males and 16 females of class VIII B. The results showed that this learning trajectory could improve students’ understanding of learning quadrilaterals with ease and fun because mathematical concepts are associated with students’ daily habits. The resulting learning trajectory is relevant to indicators of critical thinking skills, including interpretation, analysis, evaluation, and decision making. It was found that there was no significant difference in critical thinking skills between students who had high mathematical abilities and low mathematical abilities. Subjects with high mathematical abilities could carry out critical thinking processes to solve problems much better than subjects with low mathematical abilities. The results provide information about the use of local Bugis culture as a context of learning mathematics to improve students’ critical thinking.

Geometry is an essential subject for students. However, geometry, especially the cone, remains a challenging material for students. Besides, we need the proper media and context for learning mathematics on cone material. This research aims to develop a learning trajectory that utilizes the Megono Gunungan tradition as a context in the learning process using the Indonesian Realistic Mathematics Education (PMRI) approach. The research method used is Design Research, which consists of three phases of the study: preliminary design, experimental design, and retrospective study. This study's outcome is designing a learning trajectory on cone material using the Megono Gunungan tradition. This learning trajectory comprises four activities, including observing the interactive video of the Megono Gunungan tradition to discover cone elements, finding the concept surface area of a cone using origami paper, and grasping the cone concept volume with magic seeds, and solving contextual problems related to the cone. Studies have shown that using the Megono Gunungan context could help students improve their understanding of the cone concept. Furthermore, this study hoped could be an inspiration for exploring another local wisdom that can be a context in learning mathematics.