Hypothetical Learning Trajectory for Negative Integer in Differentiated Instruction: A Prospective Analysis in Didactical Design Research

This study aims to investigate students' learning trajectory while solving comparison problems using the KoPer learning media integrated with STEM, as well as to enhance students' numerical problem-solving skills. The research involved 27 fifth-grade students from Taman KalijagaPermai Elementary School in Cirebon City. The students are divided into four grup using utilized purposive sampling to form learning groups and conduct numerical problem-solving tests. Students' learning trajectories were observed through patterns of numerical problem-solving, and the Hypothetical Learning Trajectory (HLT) was employed to develop hypotheses and serve as a research guide. This qualitative study adopted a Didactical Design Research (DDR) approach. Data collection was conducted through interviews, tests, and documentation. The results indicate that creating a good and effective learning design for a STEM-integrated learning trajectory requires effectively combining STEM aspects. During the implementation of the learning trajectory, all groups conducted practical experiments using KoPer integrated with STEM, yielding varied group results, with three groups performing well and two groups not meeting expectations. The numerical problem-solving tests revealed diverse answer patterns among students. These differences represent new findings that are hoped to contribute to future research. Through STEM media, student learning improved, demonstrating that STEM education can stimulate innovation and critical thinking in problem-solving. In conclusion, STEM-integrated numerical problem-solving in mathematics education related to comparisons is significantly effective for practical application in learning.

This study aims to determine the trajectory of students' thinking when solving proportion problems using STEM-based learning media. The participants were 27 fifth-grade students from SD Negeri 2 Pilangsari in Cirebon Regency. The students are divided into four groups using purposive sampling and receive the same treatment. The treatment involved a proportion study that utilized STEM media, and the student’s learning trajectory was monitored based on their problem-solving patterns. Hypothetical Learning Trajectory (HLT) was used to develop the hypotheses. The HLT was used as a guide for the researchers' assumptions. The data were collected through observation by researchers, student work, and documentation. The results of the HLT were used to test the assumptions related to the student's thinking processes and their learning in completing proportion operations using STEM. Based on the results obtained during the practice, some findings exceeded the researcher's expectations and hypotheses, but some did not. These differences become a new finding expected to become a subject for further research, where several groups have different ways of thinking based on mathematical disposition. Through STEM media, the electrical engineering students' high enthusiasm and creativity can be known through the electric graph. In conclusion, proportional relationships are an important mathematical concept with practical applications in various fields. The use of STEM media for teaching materials can help students acquire a better understanding of mathematical concepts and skills.

This study aims to a learning trajectory about elasticity for tenth grade students ofSMA Negeri 1 Indralaya Utara by using Hypothetical Learning Trajectory (HLT). The methodused in this study is design research. The subjects of this research were six students in the pilotexperiment and 24 students in the teaching experiment. A set of activities designed in HLT wasapplied and developed to gain the understanding of elasticity concept. The HLT designedduring the preparation phase was compared to the students’ actual learning trajectory duringthe implementation in the teaching experiment. Retrospective analysis of the teachingexperiment showed that students have diverse conjecture thinking in understanding the conceptof elasticity. This learning trajectory describes the development of student’s understanding inmaterial of elasticity during the learning process that consists of learning goals, learningactivities and conjecture of students’ thinking. Students who initially assumed that elasticmaterial only rubber and spring, after the observing and classifying objects of elastic and nonelasticactivities,startedtounderstandtheconceptofelasticityandtheelasticlimit.Inthenext activity, choosing plastic experiment, students could determine the modulus of elasticity of anobject. Some students assumed that all spring have the same constants, after Hooke’s lawexperiment and created the graphs of Hooke’s law, student understood that the elasticity of thespring depends on the spring constant. It was got by students analyzing about the relationshipof force F with the length of the spring (extension  ). Based on this research, it was foundthat the use of HLT could determine the students' learning trajectory in understanding theconcept of elasticity.

Computational thinking (CT) is a fundamental skill that needs to be developed by prospective mathematics teachers to improve problem-solving and logical reasoning. Integrating programming into mathematics learning is an effective approach to training this skill. This study aimed to design a hypothetical learning trajectory (HLT) for developing CT using Python programming on Google Colab. This study used a didactical design research (DDR) framework consisting of three stages: prospective analysis, metapedadidactic analysis, and retrospective analysis. The research participants were prospective mathematics teacher students enrolled in a computer programming course. Data were collected through observation, code artefacts, and reflective interviews. The results showed that HLT, designed in stages, improved the four main components of CT: decomposition, abstraction, pattern recognition, and algorithmic thinking. The students experienced improvements in breaking down problems, devising more efficient solutions, recognising patterns in code structures, and systematically designing algorithms. In addition, Google Colab supports learning by providing a collaborative and accessible programming environment. However, minor syntax errors and lack of attention to indentation were found. This study recommends using structured debugging strategies and project-based learning in optimizing CT development. The findings indicate that the integration of programming into the education of prospective mathematics teachers can equip them with essential CT skills to support technology-based mathematics teaching.

The background of this research is the learning obstacles that arise on the topic of determining the solution of a system of linear equations (SLE) in applying elementary row operations (ERO). From these findings, it is necessary to formulate a hypothetical learning trajectory (HLT) that can guide the flow of thinking in designing didactic situations in the learning process. HLT expect to anticipate learning obstacles (LO) that arise. This research is a preliminary design of a series of didactical design research. The steps taken in compiling the HLT are 1) reviewing the lesson plan, the methods used to convey the material to determine SST solutions, and the textbooks used; 2) reviewing students' thinking processes conceptually and procedurally regarding theoretically determining SLE solutions; 3) reviewing design plans didactic. HLT is formulated into five types of student responses. It integrates into didactic situations by explaining the relevance of the elimination method in high school with three operations on ERO and then by explaining matrix material before SPL material. From this didactic situation, it can be formulated in teaching material that can accommodate emerging LO.

Students' understanding on the concept of a circle especially the relationship between central angle, arc length, and sector area of the circle based on learning obstacle and learning trajectory need to be developed well by creating a good didactic design. Therefore, it is necessary to design hypothetical learning trajectories (HLT) for students in learning sector area and arc length using the clockwork context. The didactical design was developed through three stages, namely: (1) the preliminary design; (2) teaching experiment; and (3) retrospective analysis. This study focused on the first stage, since the main purpose of this research was to develop a series of learning activities to delve into the concept of sector area and arc length as proportional problem in the form of HLT. The HLT was designed using a realistic mathematics education (RME) approach. There are three anticipated learning obstacles in the design of the activities, including errors related to the concept of the relationship between arc length, circle circumference, sector area, and circle area; errors related to the concept of the relationship between the central angle, arc length, and sector area of a circle; as well as errors related to the connection of concepts in circle material with other mathematical material; and students' difficulties in problem solving questions.

Geometry is an essential topic things to needs to be mastered by participants, one of the topics on geometry is quadrilateral. However, in conditions in the field students Still have difficulty understanding quadrilateral topics. Research This aims to design Hypothetical Learning Trajectory (HLT) Quadrilateral Topics for Grade VII Junior High School Students according to with needs of students and appropriate with the process of thinking student with analyze the effectiveness of HLT applied. The method used in the study This is method design research consisting of three stages : (1) preparation experiment to design HLT (2) design experiment that implements HLT which has been designed, and (3) analysis retrospective that analyzes effectiveness HLT applied in learning. Deep data collection techniques study this : (1) Interviews with teachers and students, (2) Observation and observation, and (3) Documentation For analysis and clarifying HLT. Subject study This consists of 20 students of SMPN 2 Cimahi Class VII with characteristics of heterogeneous students with high, medium, and low ability. This article serves results at stage preparation experiment, design experiments, and analysis retrospective as well as analyze HLT material effectiveness quadrilateral. Track design Study includes, objectives learning material quadrilateral, activities and tasks designed learning For material quadrilateral, and hypothesis channel think a student in understanding draft material quadrilateral. The study This produced HLT which consists of a series of activity learning material rectangle on indicator understand properties quadrilateral, and find around quadrilateral. Analysis results show that HLT is designed in study This Still needs development For activities on the topic properties square, but gives change positive to student class VII in understanding draft around quadrilateral. Analysis results effectiveness reviewed from completeness Study produced by 85% of studes Already achieve KKM on the material quadrilateral, thing This show that the HLT was designed Already effective applied in learning.

Este trabajo propone y evalúa una trayectoria hipotética de aprendizaje (THA) para los conceptos de conjunto generador y espacio generado. La THA se diseñó siguiendo el planteamiento de Simon (1995), la heurística de diseño de los modelos emergentes y el mecanismo de reflexión sobre la relación actividad-efecto. Se realizó un experimento con estudiantes universitarios para evaluar si la THA contribuyó al aprendizaje esperado de los conceptos. Para ello, se contrastó la THA con la trayectoria real de aprendizaje (TRA) de dos estudiantes que mostraron aproximarse a las fases clave de la THA. Los resultados muestran que la noción de conjunto es útil para la construcción del concepto de espacio generado y que, además, se requiere incorporar nuevas interrogantes en la THA que faciliten la reflexión sobre la relación actividad-efecto para dar cuenta mejor de la progresión en el aprendizaje.

<div><p>Learning trajectory on the topic of plane figures (i.e., perimeter and area of squares and rectangles) is necessary for elementary students to facilitate meaningful learning and predict their learning progress. This study aims to develop a learning trajectory through various tasks for the perimeter and area of squares and rectangles that can help students develop understanding and construct mathematical concepts. Study participants were fourth grade elementary students. The method used was design research, which has three stages: preparing for the experiment, teaching, and retrospective analysis. This study used data collection instruments in worksheets, observation sheets, interview guidelines, and field notes. Research results obtained included a learning trajectory that contains adequate learning activities to facilitate learning for elementary students on the perimeter and area of squares and rectangles. Adequate learning trajectory is obtained in hypothetical learning trajectory (HLT) 2, which was previously revised from HLT 1. Learning trajectory developed through a variety of tasks can significantly construct and facilitate elementary students’ knowledge to learn the concepts of perimeter and area of squares and rectangles.</p></div>

<p class="06ContentAbstract">Statistics is a staple in the curriculum because it is considered necessary in everyday life. However, students often find statistics challenging. Students' difficulties in learning statistics are caused by several factors, including a lack of understanding of mathematical concepts and principles in solving problems. This study aimed to design and develop a Hypothetical Learning Trajectory (HLT) for grade VIII statistics using the Realistic Mathematics Education (RME) approach, with Pranata Mangsa as the learning context. The method used in this research was validation-type design research. This research produced an HLT of statistics material using <em>pranata mangsa</em> context consisting of 3 activities to help students understand and master the material. The activities of the HLT are 1) collecting, presenting, and analyzing data, 2) measuring centralized data, and 3) measuring data distribution. The results showed an HLT for teaching statistics using RME that can effectively support students of all ability levels. This HLT allows students to rediscover statistics concepts through horizontal and vertical mathematical processes. Finally, the HLT facilitates students to create their models from informal to formal and increases the interaction between students and teachers.</p><p class="06ContentAbstract"> </p><p class="06ContentAbstract">Statistika menjadi materi pokok dalam kurikulum karena dianggap penting dalam kehidupan sehari-hari. Namun, pada kenyataannya statistika merupakan salah satu pelajaran yang dianggap sulit bagi siswa. Kesulitan siswa dalam belajar statistika disebabkan oleh beberapa faktor, diantaranya kurangnya pemahaman konsep dan prinsip-prinsip matematika dalam menyelesaikan soal. Tujuan penelitian untuk mendesain dan mengembangkan Hypothetical Learning Trajectory (HLT) menggunakan pendekatan Realistic Mathematics Education (RME) pada materi statistika kelas VIII menggunakan konteks Pranata Mangsa yang menjadi starting point dalam pembelajaran. Metode yang digunakan dalam penelitian ini adalah design research tipe validasi. Penelitian ini menghasilkan HLT materi statistika menggunakan konteks pranata mangsa yang terdiri dari 3 iceberg untuk membantu siswa dalam memahami dan menguasai materi. Adapun iceberg dari HLT tersebut adalah 1) mengumpulkan, menyajikan, dan menganalisi data; 2) ukuran pemusatan data; dan 3) ukuran penyebaran data. Hasil penelitian ini menunjukkan bahwa HLT mampu membantu siswa menemukan kembali konsep statistika melalui proses matematika horizontal dan vertikal. HLT memfasilitasi siswa untuk membuat model mereka sendiri dari informal ke formal dan meningkatkan interaksi antara siswa dan guru.</p>

The purpose of this study was to determine the hypothetical learning trajectory through a problem-based learning model to class VIII junior high school students. Learning mathematics requires a meaningful process and that process cannot be obtained in a relatively short time because it requires good preparation in conveying mathematical concepts to students. The preparation is designed according to the characteristics of students so that the delivery of the material goes well as expected. The preparation referred to in this case is to design a good hypothetical learning trajectory (HLT) according to the needs and subject matter. Hypothetical learning trajectory (HLT) is a temporary hypothesis about students’ activities in the learning process on certain materials. This study uses a design research method with stages referring to Plomp (2013) which consists of three phases including (1) preliminary research; (2) development or prototyping phase; and (3) assessment phase. However, in this research, it is only in the first phase, namely preliminary research. In designing the HLT, it is based on a problem-based learning model which has five stages in each process. The instrument used in this study was a teacher and student interview sheet. The results obtained are the HLT design on linear equation material that is adjusted to the stages in the problem-based learning model. The results obtained from this study can be used as an initial reference for mathematics teachers in designing learning material for linear equations that are in accordance with the real context of students. Keywords: Learning Trajectory, Problem-Based Learning, Junior High Students

Our understanding of statistical inference is based on our understanding of the concept of sampling variability. In this study, we aimed to explore learning trajectory that would help students meaningfully construct concepts related to sampling variability under this assumption. For this purpose, in this study, based on hypothetical learning trajectory theory, we designed a learning trajectory that is expected to develop students’ reasoning about sampling variability. We qualitatively analyzed the results of a teaching experiment conducted for a group of preparatory teachers according to the guidance of the learning trajectory designed in this study. As a result, The participants were able to generate the hypothesis that sample medians obtained through a random sampling process were normally distributed around the population median. The participants were then able to generate the hypothesis that the larger the sample size, the smaller the range of distribution of the sample median. Furthermore, it is possible to generate the hypothesis that the center of the distribution of the sample median is determined by the population median, and the range of the distribution of the sample median is determined by the sample size and the interquartile range of the population. In addition, based on the results of retrospective analysis of the difference between the hypothetical learning trajectory and the actual learning trajectory, we proposed a modification of the initially designed hypothetical learning trajectory.

Difficulty in understanding the concept of angle because of learning of angle is only explain, give examples, and practice, so there are still many students who have difficulty understanding the concept of angle. Angle is one of the most basic concepts to understanding geometry and should have been understood and mastered since elementary school in terms of length and angle so as not to cause students difficulties in learning angles in high school. PMRI was chosen because it can lead students to understand mathematical concepts, which in this study is the concept of angles. The aims of this study is to design a hypothetical learning trajectory in angle by using context of the Museum Timah Pangkalpinang as a tool in understanding the concept of angles. This study was using design research approach, with three stages of research are preparing for the experiment, the experimental design, retrospective analysis. The Hypothetical Learning Trajectory (HLT) was developed from a series of corner material learning activities using context of the Museum Timah Pangkalpinang. HLT is generated with 4 activities, that are observe museum buildings and mining equipment,determine the types of angle, determine the angle using the steps outlined in LKPD, and completed the questions.Keywords: Angle, Museum Timah, PMRI, HLT

This study is being conducted to identify the learning disabilities in math literacy at SMPN 3 Kalipucang kelas VIII during the second semester of the 2023–2024 school year. The method that is used is a qualitative approach called Didactical Design Research. Study focus is on three study subjects: students with weak mathematical skills (S1), students with strong mathematical skills (S2), and students with weak mathematical skills (S3). The results of the study indicate that students in S1 experience difficulties learning epistemological obstacles, students in S2 experience difficulties learning both didactic and epistemological obstacles, and students in S3 experience difficulties learning ontogenic, didactic, and epistemological obstacles. In order to overcome this obstacle, researchers have developed learning materials that are oriented toward numerical literacy with the goal of raising student achievement standards Researchers used Learning Obstacle Identification as a basis to create a Hypothetical Learning Trajectory (HLT) that focuses on literacy numeration of material circles.

This research aims to comprehend how the context of buying and selling activities and a hypothetical learning trajectory (HLT) can aid in learning two-variables linear equation system through contextual teaching and learning (CTL) in class 8.1 and in class 8.3 at Madrasah Sanawiah al-Mubarok. The research method used is design research. The essential characteristic of design research is the cycle. The research subjects for the first cycle (preliminary teaching) included 28 students from class 8.1. In this cycle the first HLT is being tested to observe students as they progress through the learning trajectory. The revised HLT will be implemented in the second cycle (teaching experiment). The research subjects for the second cycle included 33 students from class 8.3. The obtained data are analyzed qualitatively. The results of the retrospective analysis indicate that the use of buying and selling activity contexts can facilitate students’ understanding of how the two-variables linear equation system is applied to solve related problems. According to the retrospective analysis results, 1) using context of buying and selling activities can aid students in comprehending how to apply the two-variables linear equation system to solve related problems; 2) the HLT in learning the two-variables linear equation system through CTL includes learning objectives, activities that occur during learning, and student answer hypotheses. The learning trajectory used includes making meaningful connections; engaging in significant activities; achieving high standards; self-regulated learning; nurturing and fostering student personalities; critical and creative thinking; collaboration; authentic assessment