Hypothetical Learning Trajectory Research Articles

Hypothetical Learning Trajectory Through the Problem-Based Learning in Junior High School Students

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

Designing a Hypothetical Learning Trajectory Using Prezi Presentation and GeoGebra Application on Flat-Shaped Triangle Geometry Materials

Technology plays a crucial role in enhancing the quality of math learning. In addition to adapting to the rapid advancement of ICT, technology serves as a visualization aid for abstract mathematical concepts, facilitating understanding. Prezi presentations and GeoGebra application are technological tools that can be used as a medium for learning math. Prezi Presentation is used as a presentation medium, while GeoGebra functions as a dynamic tool to enhance the understanding of geometry. They both positively impact the development of mathematical thinking, facilitate meaningful learning experiences, and can be used as alternatives in designing a hypothetical learning trajectory for concepts related to triangle types. This study aims to design a Hypothetical Learning Trajectory (HLT) using Prezi presentation and GeoGebra for the material related to triangle types. The HLT tailored to suit the characteristics of students, making it easier to construct independent understanding related to triangle types through the use of dynamic technology such as Prezi and GeoGebra. This study uses a design research approach consisting of three phases: introduction, design experiments, and retrospective analysis. This was carried out solely in the preliminary phase to develop the initial prototype of the HLT focusing on the concept of triangle types. The HLT will be further examined in subsequent research and retrospectively analyzed to develop the Learning Instructional Theory (LIT) as the final product. Keywords: flat-shaped triangle geometry materials, geogebra application, hypothetical learning trajectory, prezi presentation

Proposal for a Hypothetical Learning Path Using GeoGebra in Learning Numerical Sequences

A hypothetical learning path is proposed for the teaching-learning of numerical sequences in second grade secondary school students, it is theoretically based in a Duval’s Theory of Semiotic Representation Registers and methodologically based on GeoGebra & Hypothetical Learning Path. It is intended to contribute to this field of mathematics given that there are not many the works in this regard, a fact that motivates the facilitation of better perspectives for students in learning the mathematical object numerical sequences. According to the hypothetical learning trajectory theory, each stage of learning requires a series of preconditions and appropriate instructional approaches so that the individual can progress. These stages include: 1) Acquisition of basic knowledge 2) Application of skills 3) Automation 4) Integration. It is concluded that it must be applied in the classroom to know the effectiveness of this trajectory.

A HLT for Integrated CT and Mathematics Lessons: Supporting Students' Possible Struggles when Debugging in GeoGebra Environment

As computational thinking (CT) is new to teachers, they need support on how CT integration will work. This study examines the use of a hypothetical learning trajectory (HLT) to support teachers in integrating computational thinking into mathematics lessons utilizing GeoGebra. By employing the educational design research (EDR) methodology, the researchers, in collaboration with mathematics teachers and an educator, developed Math+CT activities in GeoGebra as part of an HLT. The HLT was meant to help teachers predict students' responses and facilitate learning. A pilot experiment was conducted with a small group of students to see how and why the designed activities work. Therefore, we collected data from students' responses (GeoGebra scripts and answers) and computer screen recordings. The HLT is triangulated by comparing the predicted responses to the actual responses. The pilot experiment helped to improve the lessons and the HLT for future experimentation. Considering students' possible mistakes in debugging would prepare teachers to anticipate and facilitate the contingency situation.

HYPOTHETICAL LEARNING TRAJECTORY BASED ON ETHNOMATHEMATICS

Ethnomathematics is the study of the relationship between mathematics and culture. The culture in question is related to the habits of human behavior in their environment. It is intended that through ethnomathematics, students can better understand mathematical concepts and their culture so as to create meaningful learning. It is important for teachers to develop a mathematics learning model that is meaningful to students from the aspects of content and process. One of the efforts that can be made is to design an ethnomathematics-based Hypothetical Learning Trajectory (HLT) in mathematics learning. The method used in writing this article is literature review. Based on supporting theories, it can be concluded that HLT is a learning design that includes student learning activities based on initial understanding abilities and student characteristics to achieve higher understanding abilities. HLT consists of three components, namely learning objectives, learning activities and learning process hypothesis. HLT in material form is in the form of context and content related to culture-based environment. Developing ethnomathematics-based HLT means using cultural values contextually by considering learning obstacles, material linkages, learning stages that are in accordance with students' thinking abilities, methods used by students and the carrying capacity of learning.

Epistemological learning obstacles on fractions in elementary school

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.

Evaluating a Hypothetical Learning Trajectory for nets of rectangular prisms: A teaching experiment

The ability to recognize and construct nets is a fundamental aspect of spatial reasoning skills; however, there is a lack of understanding regarding effective instructional sequences for fostering this ability. The aim of this study is to test a Hypothetical Learning Trajectory for enhancing sixth-grade students’ understanding of rectangular prism nets. Conducted over four weeks, this teaching experiment involved 12 sixth-grade students from a public school in Turkey, selected based on varying mathematics achievement levels. Data were collected through clinical interviews, video recordings, and student worksheets. Ongoing and retrospective analyses were employed to describe students’ understanding during the implementation of HLT. The findings detail the results of the teaching experiment and the HLT verified following the experiment. This study contributes to the existing research on students’ understanding of rectangular prism nets and provides valuable insights for mathematics educators aiming to design effective instructional sequences that promote spatial reasoning skills.

PENGEMBANGAN PEMBELAJARAN DENGAN PENDEKATAN PMRI UNTUK MENINGKATKAN KEMAMPUAN LITERASI MATEMATIS BERBUDAYA LOKAL JAKARTA

The results of the preliminary study show that students' mathematical literacy skills on the topic probability was low. This is caused by teachers' conservative teaching methods and students who are not familiar with literacy matters. The presentation of opportunity material is not fully adequate to guide students in learning. This research aims to develop learning designs with a Realistic Mathematic Education (RME) approach with local Jakarta culture on opportunity material using the Gravemeijer and Cobb model which consists of three stages, namely experimental preparation), experimental implementation), and retrospective analysis. The research subject was Health Vocational School Mulia Karya Husada class XI. The learning design in question is a learning path known as Hypothetical Learning Trajectory (HLT). HLT is a learning flow consisting of learning objectives, learning activities, and learning process hypotheses to predict how students think and understanding will develop in the context of learning activities. The research results show that there was an increase in students mathematical literacy abilities.

DESAIN PEMBELAJARAN SISTEM PERSAMAAN LINEAR DUA VARIABEL MELALUI CONTEXTUAL TEACHING AND LEARNING

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

Hypothetical Learning Trajectory of Sector Area and Arc Length Using the Clockwork Context

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.

Hypothetical learning trajectory trigonometry based on problem-based learning with Jambi Malay ethnomathematics nuances

Background: Trigonometry is a challenging subject for many high school students. Incorporating cultural relevance, such as Jambi Malay ethnomathematics, into teaching can be a way to facilitate understanding. This study focuses on developing a hypothetical learning trajectory (HLT) tailored to Class X high school students on the topic of trigonometry.Aim: The primary goal of this research is to create a valid, practical, and effective HLT for trigonometry based on problem-based learning, integrating the nuances of Jambi Malay culture.Method: This study employed a combination of two design research models: the Plomp model and the Gravemeijer and Cobb model. The research was conducted in three stages: 1) initial investigation, 2) product development, and 3) assessment. Expert validation and practical testing were carried out to assess the quality of the HLT.Result: Validation by experts indicated that the developed HLT achieved a score of 3.69, falling into the "very valid" category. Practicality testing revealed a score of 80.14, categorizing it as practical. Moreover, the HLT proved effective in enhancing students' mathematical communication skills.Conclusion: The developed Trigonometry HLT based on problem-based learning and integrated with Jambi Malay ethnomathematics is valid, practical, and effective. It not only adheres to quality benchmarks but also significantly improves students' mathematical communication abilities. Therefore, it fulfills the criteria for being a suitable and effective teaching approach.

Obstacle to Learning Algebra in Elementary Schools

This research is based on the results of a preliminary study which identified students' learning barriers in algebraic multiplication and division material in elementary schools. Students do not fully understand the concept of algebraic multiplication and division due to learning obstacles related to limited didactic designs made by teachers. Learning obstacles experienced by students must be anticipated through a learning process that is able to overcome these problems. The researcher developed a didactic learning design using a Realistic Mathematics Education (PMR) approach. The aim of this research is to determine the learning obstacles experienced by students, develop didactical designs for algebraic multiplication and division, describe, test, and describe the designs that have been created. The research method used in this research is Didactical Design Research (DDR). This method consists of three stages, namely didactic situation analysis before learning in the form of Hypothetical Learning Trajectory (HLT) and Pedagogical Didactic Anticipation (ADP), metapedadidactic analysis and retrospective analysis. The data collection technique used is triangulation which combines observation, interviews and documentation. The design development process was carried out in class V with a total of 18 students, and the preliminary study stage was carried out in class IV at SDN 1 Sukamanah with a total of 22 students, and class IV at SDN Sukamulya with a total of 16 students. This research produces data regarding student barriers to multiplication and division material, didactical design in the form of teaching materials, student worksheets (LKPD), to minimize students' learning barriers to multiplication and division material in algebra.

Developing a Hypothetical Learning Trajectory with Problem-Based Learning and a Learning Medium for Middle School

A teacher must consider the learning trajectory that will emerge during the learning process while designing an instructional plan. In mathematics, the degree to which students comprehend the learning context the teacher applies indicates the teacher's ability to construct a learning framework. This study aims to design the Hypothetical Learning Trajectory (HLT) for learning linear equations in two variables using the graphical method, problem-based learning, and the WolframAlpha application (as a learning medium). The design research methodology was used to construct the HLT design. This study begins with interviews with teachers who have taught similar material and students who have received this learning material. The findings from these interviews serve as the foundation for this research. As a result, this study provided an HLT design employing the graphical method for teaching mathematics in middle schools. It was discovered that the order in which students solved problems on the HLT corresponded to their cognitive states. Using the proposed learning trajectory design, students will find it less scary to advance through the current mathematical learning stages. Students might follow each procedural step without teacher pressure using this method. The developed learning strategy is expected to be able to solve the provided mathematical problem. Moreover, this HLT design indicates that the contextual qualities of the problem-based learning model make it best suited for relevant material.

The Learning Trajectory Based on STEM of Elementary School Pupils’ in Solving Proportion Material: Didactical Design-Research

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.

Learning design of quadrilateral STEM-based through lesson study

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.

HYPOTHETICAL LEARNING TRAJECTORY ON INFORMAL HYPOTHESIS TESTING IN A PROBABILITY CONTEXT

A design experiment where students in Grade 5 (11–12 years old) play the Color Run game constitutes the context for investigating how students can be introduced to informal hypothesis testing. The result outlines a three-step hypothetical learning trajectory on informal hypothesis testing. In the first step, students came to favor sample space reasoning over idiosyncratic reasoning when the sample space was changed between color runs. In the second and third steps, students used degrees of variation in the distribution of the mode across samples to infer whether an unknown sample space was uniform. Students’ reasoning disclosed the logic: the larger the variation, the greater the reason for rejecting a uniform sample space.

Development of TPACK with Web 2.0 tools: Design-based study

This mixed-method study, which aimed to examine the development of teacher candidates, was organized as a design-based study. During the 14-week instructional sequence, pre-service teachers were given training based on the designed hypothetical learning trajectory. The data collected through the lesson plans before and after the training and interviews were organized and analyzed by using the rubric prepared within the scope of Web 2.0 tools by the researchers. Moreover, the pre-training and post-training scores obtained using this performance rubric were analyzed with the related sample t-test. In addition, learning packages and interviews were analyzed using content analysis technique. The findings showed that there was an improvement in the technological pedagogical content knowledge of the pre-service mathematics teachers with the training provided for the use of Web 2.0 tools in mathematics education. In the study, an alternative educational content that can be used in lessons related to instructional technologies in undergraduate programs that train mathematics teachers has been prepared.

Hypothetical Learning Trajectory: Bagaimana Perannya dalam Pembelajaran Matematika di Sekolah Dasar?

In today's modern era, students are required to become learning centers. Thus, students must have prior experience and knowledge in order to be able to connect with the next concept (material). However, because the differences between the nature of elementary schooler and mathematics, teacher needs to structure learning that can be a bridge to neutralize these differences. One way to structure the learning is make a learning design such as Hypothetical Learning Trajectory (HLT). HLT is an alleged learning process as an anticipation of things that might happen, both students' thinking processes and other things in the learning process. The term HLT refers to the teacher's lesson plan which is designed not only to consider the material, but also the level of understanding, knowledge and student characteristics of the material being studied, so that students experience the learning process from the beginning to achieving the expected mathematics learning goals for students. This article is a literature study that aims to discuss HLT in elementary mathematics learning.

표집변이성에 대한 추론 능력 함양을 위한 학습 경로 설계 연구

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.

STEM-based approach: A learning design to improve critical thinking skills

Background: The STEM approach in learning integrates the concepts of science, the use of technology, engineering, and mathematics, which support enthusiastic, creative, and active learning activities by students. In this context, it is necessary to develop ways to improve the understanding of the concept of a three-variable system of linear equations in high school students.Aim: The purpose of this study was to produce a learning trajectory in the three-variable linear equation system material using the STEM approach for high school students, with the main objective of increasing students' critical thinking skills.Method: The method used is 'design research', which includes three stages: preliminary, teaching experiment, and retrospective analysis. The research was conducted at a public high school in Palembang City, using qualitative descriptive data analysis techniques based on observation, interviews, and tests designed to measure students' critical thinking skills.Result: The results of this study are STEM-based learning trajectories that are effective in helping students understand the system of three-variable linear equations. The hypothetical learning trajectory (HLT) that has been implemented is a successful tool in helping students understand the concept and can improve students' critical thinking skills.Conclusion: The STEM-based mathematics learning design produced in this study has the potential to serve as a reference for teachers in improving students' critical thinking skills, especially in the subject matter of a system of three-variable linear equations. It is hoped that this approach will become a valuable tool in helping students understand mathematical concepts in greater depth in the future.