Developing learning trajectories with the RME of phytagorean theorem

Understanding quadrilateral and triangle material is still a problem for students. These problems occur because teachers did not design learning trajectories before they tough in the classroom, or they still did not apply innovative approaches and media in the learning process. Designing and developing a learning trajectory for the quadrilateral and triangle material as an activities series based on science, technology, engineering, and mathematics (STEM) using rubber-powered car projects through lesson study is the aim of this study. This design is referred to as a hypothetical learning trajectory in teaching quadrilateral and triangle material. This study used design research as a research method, which consists of three stages: preliminary design, design experiment, and retrospective analysis. The subject of this research was seventh-grade students at a state junior high school in Semarang City. Data were collected from various sources: pre- and post-test, student worksheets, interviews, documentation of photos, and video recording of the learning process. Data analyses through qualitative research about validity and reliability refer to the quality of the observation. This study result described the learning trajectory consisting of three activities, namely creating a rubber-powered car project, identifying the property and relation among quadrilaterals, and solving a contextual problem in the area and perimeter of quadrilaterals. This research indicated that a series of activities STEM-based designed through lesson study can make students understand material actively, joyfully, and meaningfully.

High-level mathematical thinking skills are the main goal in learning mathematics. So far, the practice of learning in schools has not facilitated students to develop these abilities. PMRI (the Indonesian version of realistic mathematics education) is a learning approach that is oriented towards the students` ability to understand the realistic situations that can be imagined and integrated with the ability to understand concepts, use progressive mathematical models, interactivity and take advantage of student contributions are expected to improve students' abilities in learning mathematics. This study aims to determine the implementation of PMRI approach in improving high-order thinking skills of mathematics in class XI SMA Mutiara Bangsa 2 Tangerang on limit function through e-learning. This research was developed based on a research design that includes three, those are preliminary design, experiment, and retrospective analysis. This study was designed with five meetings using six research subjects. The subjects of this study consisted of six students who were selected based on their evaluation test and their activeness during learning process. The research instruments analyzed were formative and summative test results based on higher-order thinking skills, findings note, student questionnaires, and the hypothetical learning trajectory (HLT). The results obtained from the retrospective analysis show that the implementation of PMRI in learning and instruction of limit function is able to develop higher-order thinking skills of class XI students of SMA Mutiara Bangsa 2 Tangerang.

This research was motivated by several studies which show that students have difficulties in understanding the material ratio, wherein one contributing factor is the lack of ability in proportional reasoning. The purpose of this study is to produce a trajectory of learning to help students develop abilities in comparison reasoning. Solution strategies in this study were the ratio tables and graphs using PMRI approach (Indonesian Realistic Mathematics Education) with OKU Timur context. This research was conducted in MTs Subulussalam 2 Sriwangi Ulu OKU Timur. The method used was design research with three stages, those are the preliminary experiment, design experiment, and retrospective analysis. At the preliminary experiment, researchers designed the Hypothetical Learning Trajectory (HLT). At the design experiment, HLT tested on students to develop informal knowledge into a formal knowledge of mathematics through activities. The trial results were analyzed in a retrospective analysis phase so that the trajectory of learning or Learning Trajectory (LT) resulted. DOI :Â http://dx.doi.org/10.22342/jme.8.2.3969.211-222

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.

This study aims to explore the learning trajectory of problem-based learning with literacy and numeracy reinforcement. This study was conducted at Junior High School and involved 32 subjects from class VIII. This study employed Gravemeijer and Cobb's design research comprising three stages: preparing the experiment of hypothetical learning trajectory (HLT), designing the experiment by implementing the designed HLT-based learning, and conducting retrospective analysis by comparing HLT with the implemented learning. The learning trajectory in this study consisted of three components: learning objectives, a series of tasks and learning activities, and students' techniques to think and learn. This research was conducted based on lesson study activities to create a lesson study for the learning community. Design research stages corresponded to lesson study stages and included studying curriculum, formulating goals and plans, conducting research lessons, and reflecting the lesson. This study has revealed that students' actual learning trajectory has four levels of modeling: situational, referential, general, and formal levels. At the situational level, non-mathematical literacy becomes the initial reference for students to collaborate and communicate with others; thus, mathematical concepts are created. The referential level leads students to represent real problems in real mathematical concepts. The representation process aims to understand the concept in different situations. Finally, the students analogize the acquired understanding and bring it to the formal level to get a full understanding of the concept of one point position to another in Cartesian coordinates.

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.

This research resulted in a method or learning path that is appropriate for learning mathematics about number patterns using magnetic cube media and STEM learning. The method of this research is design research with three stages, namely: preliminary design, experimental design, retrospective analysis. With the research subjects 32 students of class VIII.7 in SMP Negeri 27 Palembang. Data was collected through observation sheets, video recordings, interviews, photo documentation, and student activity sheets. The data collected was then analyzed retrospectively collected on HLT (Hypothetical Learning Trajectory). From the results of data analysis, it can be concluded that this research produces a mathematical method or trajectory for the subject of class VIII number patterns that are valid and reliable. Valid from HLT and traceable. Reliability is seen in data triangulation, namely from field notes, observation sheets, and video recordings. Where by using magnetic cube media that combines using STEM students are happier and excited in learning because they use knowledge, technology from the media, their ability to use media so that it brings out the expected mathematical knowledge

The rural area's student difficulties in learning the concept of number operation had been documented by several studies, especially for the case of multiplication. The teacher typically introduces the multiplication concepts using the formula without involving the concept itself. Furthermore, this study aims to design learning trajectory on multiplication operations in the Mathematics of GASING (Math GASING) by focusing more on the concept itself than the formula and by starting from the informal to a formal level of teaching. Design research used as the research method to solve this problem consisting of three phases, namely preliminary design, teaching experiment, and retrospective analysis. The research results show that the Math GASING has a real contribution for students to understanding and mastering in the concept of the multiplication operations. This research also explains the strategy and the model discovered by students in learning multiplication that the students used as a basic concept of multiplication. Finally, the students were able to understand the concept of multiplication more easily, and they showed interest in using this learning trajectory.

Students’ persistent difficulties in understanding three-dimensional geometric figures, particularly cylinders, due to limited spatial visualization and difficulty identifying relationships among their elements, such as cylinder nets. These difficulties are often rooted in traditional instructional practices that emphasize procedural tasks over conceptual development. Despite various interventions, there remains a lack of structured instructional models based on cognitive development frameworks to support students’ conceptual growth in geometry. Addressing this gap, the present study aims to develop and evaluate a Hypothetical Learning Trajectory (HLT) grounded in Bloom’s taxonomy to enhance students' understanding of cylinders. This study employed a design research methodology consisting of three phases: preliminary design, design experiments, and retrospective analysis. Two experimental cycles were conducted with 28 fifth-grade students, categorized into low, moderate, and high levels of understanding. Data were collected through classroom observations, student worksheets, tests, and interviews, and analyzed qualitatively. The HLT consisted of four key learning activities: modeling a cylinder, identifying its elements, constructing the net, and solving application problems, mapped to Bloom’s cognitive levels of remembering, understanding, and applying. Findings revealed that students showed significant improvement in the first three activities, with increased spatial reasoning and conceptual clarity. However, difficulties persisted in the final activity involving reasoning and problem-solving. The results indicate that the proposed Bloom’s taxonomy-based HLT offers a systematic framework for guiding geometry instruction. This study contributes a practical and theoretically grounded instructional model that can support teachers in designing adaptive learning experiences. Further research is recommended to explore its application across diverse topics and student groups.

This study produces a learning trajectory in transformation geometry, specifically on reflection using a game called “bom-boman”. This study using PMRI, the Indonesian version of realistic mathematics education. Subjects of this study are 36 seventh grade students of SMP Negeri 24 Palembang. This study using a design research approach with conducted in three stages: preliminary design, the design experiment (pilot experiment & teaching experiment), and retrospective analysis. Data were collected by observations, interviewing the student, collecting student worksheet, and documentation of classroom event and group works. Hypothetical Learning Trajectory (HLT) has been designed then compared to Actual Learning Trajectory (ALT) during the learning process. Retrospective analysis shows that student’s learning trajectories using the game help students to understand transformation geometry in reflection.

The aims this study are to produce Learning Trajectory use Rumah Limas as context. The approach used is Pendidikan Matematika Realistik Indonesia (PMRI). The subjects were 26 students of class seventh of SMPN 1 Keluang. This study was using design research approach, the hypothetical Learning Trajectory was developed from a set of activities using Rumah Limas as context. In theory the development carried through the interactive process comprise of preliminary design, teaching experiment and restrospective analysis to contribute to Local Instruction Theory (LIT) to supporting students learn of angle. Data collected by making video recording about incident in the class, collecting students worksheet, giving pretest and postest, interviewing students. The designed HLT compared with Actual Learning Trajectory (ALT) during teaching experiment to analyze whether the students learned or didn't learn from what we had designed in the instructional sequence. Retrospective analysis of teaching experiment showed that by using Rumah Limas as context can support students learning of angle.

In an era where data is increasingly prevalent, statistical literacy skills are essential for active citizenship and informed decision-making. For future generations, prospective teachers play a role in developing this skill. However, current instructional approaches often overlook the integration of practical applications and local wisdom, limiting students' ability to connect abstract statistical concepts with real-world experiences. The objective of this research is to design a Learning Trajectory (LT) that supports the statistical literacy skills of prospective teachers by utilizing a hybrid learning strategy that integrates local knowledge from South Sumatra as context. A total of 60 prospective teachers from a mathematics education study program participated in this study. A design research method was employed, specifically utilizing a validation study. The research unfolded in three stages: preparation for the experiment, the experimental design, and the retrospective analysis. Data collection techniques included student activity sheet assessments, classroom observations, and interviews. Data analysis involved comparing the Hypothetical Learning Trajectory (HLT) with the Actual Learning Trajectory (ALT) in the retrospective analysis stage to develop the LT. The results indicate that the designed LT guided students through five activities that support statistical literacy: reading and interpreting data tables using statistical situations with local wisdom from South Sumatra as context, interpreting graphs, analyzing and reflecting, exploring outliers, and making conclusions and presenting findings. These findings highlight the importance of integrating local wisdom contexts into statistical education, as well as the relevance and applicability of mathematical concepts for prospective teachers. This research contributes to the design of a learning trajectory based on a local wisdom context that can be applied in statistical literacy learning.

This study aims to support students’ proportional reasoning in comparison problems using Palembang culture as context and PMRI approached. Six students of 7th grade were participated. We used design research as research approach, a Hypothetical Learning Trajectory (HLT) was develop and a set of activities using Palembang culture as context, such as ketek’ fare, empek-empek’ price, lempok’ price. Theoretical development is driven by an iterative process of designing instructional activities, performing teaching experiments and conducting retrospective analysis in order to contribute to Local Instruction Theory (LIT) to support students’ proportional reasoning in comparison problem. Data collected by making video recording about actual teaching and learning in classroom, collecting students’ worksheet, giving pretest and posttest, interviewing students. The designed HLT compared Actual Learning Trajectory (ALT) during pilot experiment to analyze whether the students learned or did not learn from what we had designed in the instructional sequence. Retrospective analysis of pilot experiment showed that by using Palembang culture as context can support students’ proportional reasoning in middle school to solve comparison problems.

This study aims to develop a learning trajectory by utilizing the local wisdom of the Megono Gunungan tradition to help students grasp the sphere's notion. This research applied the approach of the Indonesian Realistic Mathematical Approach (PMRI). Ninth-grade students at SMP 38 Semarang, Central Java, Indonesia, were the subjects of this study. The methodology used is based on Graveimeijer and Cobb's three steps of design research: preliminary design, design experiment, and retrospective analysis. Numerous data collection methods were used, involving video, photography, the outcome of students' worksheets, and student interviews. The learning trajectory comprises four activities: identifying sphere components using the Megono Gunungan tradition interactive video context, discovering the sphere surfaces area using orange fruit, finding the sphere's volume through rice media, and solving problems of the sphere. This study's findings signify that the utilization of the Megono Gunungan tradition context could encourage learners to recognize the concept of the sphere through a sequence of designed activities.

Learning place values from 2 digits to multi-digit requires various forms of representation. The representation will bring up a variety of Hypothetical Learning Trajectory (HLT) in the form of predictions of possible student responses, which will later produce the actual student learning trajectory. This study aims to find students' learning trajectory patterns and determine the factors that influence students' learning trajectory patterns in the 3-digit place value material. The research design uses qualitative research. The learning trajectory pattern was obtained through analysing student responses in the field based on the previously designed HLT. The subjects of this study were 50 students of grade II from three elementary schools in the Kebumen Regency. The research instrument is a written test and field notes. The written test contains four place value questions with a horizontal representation strategy. Field notes in the form of student responses by watching videos during the application of lesson design accompanied by making transcripts to further clarify the word for word that comes out as a form of response during learning so that the suitability of the response with HLT can be known. The results of this study indicate that the student's learning trajectory pattern takes place through the application of a didactic design consisting of 4 types of lesson designs that place the number 0 in different positions. The application of didactic design can also direct students to learn independently gradually. Meanwhile, the learning trajectory supporting factors consist of students' prerequisite abilities, planning and learning processes, and the use of learning resources.