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In the 21st century, understanding variability and developing statistical investigation skills are crucial for enhancing students' data literacy. However, these essential skills are often overlooked, limiting students' growth in numeracy whereby statistical problems are frequently disconnected from real-world or cultural contexts, reducing student engagement. To address this issue, this study integrates the culturally relevant context of Lemang Batok, which enhances students' ability to understand, apply, and analyze data through appropriate statistical concepts. The research uses an ethno-flipped classroom model that promotes flexible, collaborative learning, aiming to design a learning trajectory for teaching descriptive statistics in this context to improve numeracy skills. Utilizing design research methodology, specifically a validation study, the research followed three phases: preliminary design, experimental design, and retrospective analysis. The subjects were junior high school students from Medan and Binjai Cities, North Sumatera-Indonesia. The results indicated that the learning trajectory developed through tiered discussions significantly improved students' numeracy skills in descriptive statistics, as evidenced by increased critical thinking and enhanced abilities to analyze variability.

Mathematics is a difficult subject for elementary school students. One of the causes is errors in mathematical concepts and principles made by students. For this reason, efforts need to be made to improve the concept and principles of quadrilaterals. The aim of this research is to design student learning trajectories in finding the perimeter and area of a rectangle in the context of a fishing pond. This is research with a design research approach. The subjects of this research were 50 students divided into two groups, 25 people each. Data collection techniques through observation, interviews and documentation. The data analysis technique used is qualitative and is developed through certain relationship patterns. The results of this research show that the mathematics learning trajectory design was implemented very well. It improves the ability of understanding about quadrilaterals. The learning path is activity-1: students create a representation of the context of a fishing pond and a rectangular image. Activity-2: students determine the perimeter of a rectangular image by placing match sticks around the rectangle in sequence, holding them tightly together. Activity-3: students determine the circumference of the fishing pond. Activity-4: students determine how to calculate the circumference of the pond by drawing a rectangle using matchsticks. Activity-5: students discover the principle of perimeter of a rectangle. The conclusion of this research is that the mathematics learning trajectory design is valid, practical, and increases the ability to understand concepts and principles from quadrilaterals. [ABSTRACT FROM AUTHOR]

Purpose: During the last decade, new continuing learning pathways have been designed and implemented in the Dutch Vocational Education and Training (VET) column aiming to foster students' transitions between successive educational levels. Prototypical examples of such continuing learning pathways are the Green Lyceum (GL) and the Technical Talent Development programme (TTD). In the present exploratory study, GL and TTD were compared in terms of curriculum design aspects and students' acquired VET diplomas and transitions to higher VET levels. Methods: Ten curriculum design aspects of GL and TTD were described through curriculum description forms and focus group discussions to be able to determine similarities and differences between both learning trajectories. Moreover, acquisition of VET diplomas and transitions to higher VET levels of students from both programmes were compared. Findings: The ultimate goal of GL was to promote student transitions to higher professional bachelor (HBO) programmes while TTD mainly intended to increase student numbers in the technical domain at the secondary VET level. For GL, a new and integrated VET curriculum was built with specific ingredients to prepare students for the HBO level. For TTD, the contents of the regular VET programmes were roof tile stacked in an accelerated curriculum with a stronger focus on vocation-oriented assignments in the technical domain. Conclusion: If the ultimate goal of a continuing learning pathway is to promote students' transitions to higher VET levels, curriculum design aspects as represented in GL seem more effective. However, if the focus is on promoting students' diploma acquisition at lower VET levels for specific sectors, curriculum design aspects as represented in TTD seem more effective. [ABSTRACT FROM AUTHOR]

This study was conducted to investigate the effects of learning trajectories-based coding (LTs) and LTs-based program on preschoolers' length, area, volume, and angle measurement skills. A quasi-experimental research design was utilized with a quantitative approach. The study's participants were 47 children between the ages of 55–71 months who attended a state kindergarten. The children were randomly assigned to three groups: experimental 1 (n = 15), experimental 2 (n = 17), and control (n = 15). The current preschool education curriculum was implemented in the control group, LTs-based coding activities were implemented in experimental 1, and LTs-based activities were implemented in experimental 2. Data regarding the children's measurement skills were collected using the Early Measurement Assessment Tool (EMAT) before the implementation of programs, immediately after, and four weeks after the final session. The EMAT scores were analyzed using Hierarchical Linear Modeling (HLM), and the results indicated that both programs had a significant and permanent effect on preschoolers' measurement skills when compared to the control group. However, there was no significant difference between the experimental groups. The results were consistent across length, area, volume, and angle and turn subtests. The study contributes to the existing literature on the effectiveness of the LTs approach in improving preschoolers' mathematics skills and highlights the potential benefits of incorporating the LTs approach and coding into preschool education. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Purpose: During the last decade, new continuing learning pathways have been designed and implemented in the Dutch Vocational Education and Training (VET) column aiming to foster students' transitions between successive educational levels. Prototypical examples of such continuing learning pathways are the Green Lyceum (GL) and the Technical Talent Development programme (TTD). In the present exploratory study, GL and TTD were compared in terms of curriculum design aspects and students' acquired VET diplomas and transitions to higher VET levels. Methods: Ten curriculum design aspects of GL and TTD were described through curriculum description forms and focus group discussions to be able to determine similarities and differences between both learning trajectories. Moreover, acquisition of VET diplomas and transitions to higher VET levels of students from both programmes were compared. Findings: The ultimate goal of GL was to promote student transitions to higher professional bachelor (HBO) programmes while TTD mainly intended to increase student numbers in the technical domain at the secondary VET level. For GL, a new and integrated VET curriculum was built with specific ingredients to prepare students for the HBO level. For TTD, the contents of the regular VET programmes were roof tile stacked in an accelerated curriculum with a stronger focus on vocation-oriented assignments in the technical domain. Conclusion: If the ultimate goal of a continuing learning pathway is to promote students' transitions to higher VET levels, curriculum design aspects as represented in GL seem more effective. However, if the focus is on promoting students' diploma acquisition at lower VET levels for specific sectors, curriculum design aspects as represented in TTD seem more effective.

Background:Understanding fractions in mathematics often poses greater complexity compared to integral numbers. The primary difficulty lies in students' inadequate grasp of fractional basics, notably in comparing and sequencing fractions. Aim: This research seeks to create and evaluate a learning pathway's effect on fourth graders' comprehension of fractions, with a focus on their comparison and sequencing. Method: Employing a design research validation study approach, the research progresses through preliminary design, experimental design, and retrospective analysis. Conducted in a Palembang school involving six fourth graders recommended by their teacher, data collection encompassed essay-based pre and post-tests, Student Activity Sheets, observations, interviews, and documentations. The data analysis was retrospectively carried out, anchored on HLT as the benchmark. Result: Findings reveal a student learning trajectory encompassing three principal activities. Initially, students used folding and gluing of paper to discern fraction values. Subsequently, they engaged in coloring and illustrating folds for fraction comparison. The final activity involved drawing, coloring boxes, and fraction comparison and sequencing. Overall, students showed proficiency in understanding and determining fraction values and comparing them, yet struggled with ordering certain fractions. Conclusion: The structured learning path facilitated students' understanding of basic fraction concepts, especially in comparing them. Nevertheless, reinforcing methods or approaches for teaching fraction ordering is essential.

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. [ABSTRACT FROM AUTHOR]

According to Bandura's theory of reciprocal determinism, self-efficacy and academic achievement can have a mutual influence over one another. While empirical research generally supports this position, little focus has been given to within- and between-person factors that may moderate this relationship. The present study explored how initial performance and task difficulty impact learners' performance and self-efficacy trajectory over subsequent tasks. A diverse university sample of 261 students; 118 females aged 18–67 and 143 males, aged 18–64, was used. Significant positive pathways were found in the achievement-self-efficacy direction, but not for pathways in the self-efficacy-performance direction, failing to support the reciprocal hypothesis. Repeated measures ANOVA also revealed a moderating influence of initial task performance on the interaction between task difficulty and achievement over time. The findings highlight the importance of enhancing academic performance through scaffolded mastery, particularly for those with initial low performance outcomes, to build self-efficacy for learnt skills. [ABSTRACT FROM AUTHOR]

The difficulty of learning to build polyhedron for middle students makes experts lecturers collaboratively create learning trajectories through the Lesson Study for Learning Community (LSLC). This study aims to produce a learning trajectory that can assist students in learning polyhedron using collaborative learning through LSLC. The research method used was design research with the first stage, preparing for experiment, experimental design, and retrospective analysis. This learning approach uses collaborative LSLC is used in the process of making teaching materials and along with research methods. This study involved 47 lecturers and 24 mathematics education students. Data collection was carried out through observing student activities, recording video events in class, collecting student work results, and student test results. The collected data were then analyzed retrospectively with the results showing that collaborative and LSLC used in the acquired learning paths can help students in learning flat-sided shapes. The learning trajectory includes (1) students can collaboratively understand the surface area of the cube (2) collaboratively, students can understand the space diagonal of a block. With the existence of LSLC, researchers can form learning communities with research subjects, collaborate in making learning designs and teaching materials, and produce multilevel effective learning through sharing tasks and jumping tasks Â

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.

A tourism route is a sequence of tourist attractions visited. Students are familiar with tour activities and often plan tourism routes to visit several tourist attractions. Students' experience and knowledge about tourism routes can be used as a context for teaching the rule of product. Therefore, this research aims to design a learning trajectory for the rule of product using the Indonesian Realistic Mathematics Education (PMRI) approach in the Kerinci nature tourism route context. The research method used is design research with a validation study type, which consists of three stages: preparation and design, teaching experiment, and retrospective analysis. The subjects involved in this research were 11 third-semester mathematics education students at a university in Jambi, Indonesia. The instruments include learning video recordings, student work results, field notes, and test sheets. The research results show that learning trajectories designed in the context of tourism routes can help students discover, understand, and use the rule of product in solving relevant mathematical problems. The learning tracjectory consists of four activities: making a list of Kerinci tourist attractions, determining the location of several tourist attractions on a sketch map, arranging tourist attractions in tourism routes, and solving problems regarding possible tourism routes. This research shows that the designed learning trajectory is vital in helping students discover and understand the rule of product and apply them to solving relevant problems.

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. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Bayburt Education Faculty / Bayburt Eğitim Fakültesi Dergisi is the property of Bayburt University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Cumhuriyet International Journal of Education is the property of Cumhuriyet University, Faculty of Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This study aims to describe the learning trajectories of male and female students in solving fraction problems. The research subjects consisted of a male student and a female student who had the same mathematical ability. Learning trajectory data for both subjects were taken through fractional problem solving tests and in-depth interviews. Observations focused on the four stages of problem solving strategies, namely understand the problem, divide a plan, carry out the plan, and look back. Data analysis begins with examining all data, reducing data, grouping data, categorizing, coding, and finally checking data through time triangulation. The results of the study show that in understanding the problem, boys identify goals and information to solve the problem to state its completeness, and girls identify information to solve the problem to state its completeness. In making plans, girls complete in detail and involve many concepts, while boys only have one concept, so girls' learning trajectories are longer. While carrying out the plan, the two solve problems according to the plan. Looking back, girls used more than one method alternative while boys only used one alternative.

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.

This study aims to analyze a learning trajectory for social arithmetic using the culinary context of Yogyakarta and the results of the retrospective analysis as an application of the hypothetical learning trajectory. The research method used was design research with three stages, namely preliminary design, teaching experiment, and retrospective analysis. The research subjects who contributed to this study were sixth elementary school graders (11-12 years old) using the purposive sampling technique. The data collection techniques were documents (written data on students’ work), and participating-interview about students' thinking in mathematics during the learning process. Data analysis used retrospective analysis. The findings of this study indicate that this learning trajectory is beneficial in achieving learning objectives. The learning trajectory in this study can be considered as an alternative way or as a reference for teachers to design RME-based learning that supports the development of students' conceptual understanding, self-confidence using their strategies, communication, and reasoning skills about social arithmetic concepts. In addition, the learning process and the results of the retrospective analysis contain characteristics of Realistic Mathematics Education (RME).

There are still many students who have difficulty understanding the concept of comparing and sorting fractions. This is because learning is emphasized only on procedural knowledge and is separated from students' daily life experiences. The research objective was to determine the role of the context of school activities and produce learning trajectories to assist students in understanding the concept of comparing and ordering fractions through geogebra-assisted discovery learning. The research method uses design research with the stages of preliminary design, teaching experiment and retrospective analysis. The results showed that: (1) the use of the context of school activities through geogebra-assisted discovery learning can help students' understanding of concepts in the material of comparing and ordering fractions; and (2) the learning trajectories that are generated in studying the material of comparing and ordering fractions, namely: (a) understanding the concept of fractions as part of a whole, (b) changing fractions to other forms (mixed fractions, decimals, percents) and vice versa, (c ) found how to compare common fractions, (d) found how to compare decimals, and (e) found how to order fractions

Learning trajectory of set is a learning path to get concept of set. However, several teachers did not combine methods, approaches, and ideas in their practical deliveries. This situation becomes a concern for teachers to handle since it will affect the rule without reason so that the accepted concept will not last long in students’ memory. This study aim to describe the learning trajectory using RME models to construct the concept of set. Hypothetical learning trajectory (HLT) was designed using a qualitative method with the realistic mathematics education (RME) of Gravemeijer model as the activity stage begin from preparing for the experiment, pilot experiment, teaching experiment and retrospective analysis. The designed HLT consisted of an objective, activity, and conjecture. This study achieved an understanding of the set concept with applying RME design. By providing examples of contextual mathematics that take place in the learning environment, these outcomes were achieved. Then using media like set cards to model mathematics so that students can advance their own knowledge to the level of formal mathematics. Therefore, the RME-based HLT design can be a solution to obtain the concept of set, primarily in domain definition and set notation to produce a learning trajectory.

Recent research suggests the benefits of supporting a progressive understanding of concepts of inference prior to the teaching of procedures and formal calculations through the study of informal statistical inference (ISI). To contribute to the growing knowledge about the early learning and teaching of statistics, particularly regarding the development of informal inferential reasoning (IIR), we designed a learning trajectory (LT) that addresses ISI in K–4 students (ages 5 to 9 years). This article describes part of the LT in detail, in which third-grade students (n = 12) were introduced to sampling, frequency distribution, randomness and sampling variation as well as to developing a data sense in online lessons. The results of this type of teaching show that the creation and collection of authentic data in a playful context, together with an exploratory analysis of the data as a precursor to utilizing aspects specific to IIR, promoted the integration progress of IIR components in the oral and written informal inferences of students. [ABSTRACT FROM AUTHOR]

Non-Euclidean geometry is an abstract subject and difficult to learn, but mandatory for students. The ethnomathematics approach as a learning approach to improve students' spatial abilities. The aim of this research is to discover new elements of the spatial abilities of non-Euclidean geometry; determine the relationship between spatial abilities for Euclid, Lobachevsky, and Riemann geometry. This study used the micro genetic method with a 2×2 factorial experimental research design. The sample of this research is 100 students of mathematics education. There are three valid and reliable research instruments through expert trials and field trials. Data collection was carried out in two ways, namely tests and observations. Quantitative data were analyzed through ANCOVA, and observational data were analyzed through the percentage of implementation of the learning trajectory stages. The result is that the spatial ability of students who are given the ethnomathematics learning approach is higher than students who are given the conventional learning approach for Lobachevsky geometry material after controlling for the effect of Euclidean geometry spatial ability. Also, the same thing happened for the spatial abilities of Riemann geometry students. The learning trajectory is conveying learning objectives (learning objective); providing ethnomathematics-based visual problems; students do exploration; students make conclusions and summaries of exploration results; and ends with students sharing conclusions/summaries about concepts and principles in geometric systems. It was concluded that learning non-Euclid geometry through learning paths with an ethnomathematics approach had a positive impact on increasing students' spatial abilities. [ABSTRACT FROM AUTHOR]

Rice farmer activities can be used as a context for understanding mathematical concepts. There are many studies on Hypothetical Learning Trajectory (HLT) on direct proportion material with a cultural context background. However, there is no research that focuses on using the context of Pandanwangi rice farmers to facilitate students in understanding material in direct proportion.This study aims to design Hypothetical Learning Trajectory (HLT) in Scientific Approach on Material Direct Proportion in the Context of Rice Farmers' Activities in Pandanwangi Cianjur, who are familiar with student culture in Cianjur. This research is part of design research. The research design consisted of HLT, namely learning objectives, learning activities, and student learning hypotheses. The result of the research is the set of HLT in the context of Pandanwangi Rice Farmer Activities, which includes learning objectives, learning activities, and the assumptions of each activity. This HLT can be a promising solution to overcome students' difficulties in understanding the concept of direct proportion and values in a cultural context to improve students' character.

Penelitian ini merupakan penelitian design research tipe validation study yang bertujuan untuk mengembangkan learning trajectory pada pembelajaran geometri topik geometri tiga dimensi untuk menunjang kemampuan berpikir kualitatif siswa dengan pendekatan Rigorous Mathematical Thinking (RMT). Hypothetical Learning Trajectory (HLT) akan disempurnakan sepanjang pembelajaran yang dilakukan di dalam kelas. Kegiatan dibuat dan dirancang dalam bentuk lembar kerja siswa (LKPD) untuk menunjang proses pembelajaran. Subjek penelitian ini adalah siswa kelas XII SMA Negeri 1 Palembang, sedangkan kelompok fokus penelitian ini adalah 6 siswa dengan kemampuan yang bervariasi yaitu tinggi, sedang, dan rendah yang dipilih berdasarkan hasil prestasi yang didokumentasikan oleh guru dalam semester sebelumnya serta rekomendasi guru mata pelajaran matematika. Ada tiga tahapan dalam melakukan penelitian ini, yaitu: (1) preparing the experiment; (2) design experiment dibagi menjadi preliminary experiment dan teaching experiment; (3) retrospective analysis. Data penelitian dikumpulkan dan akan dianalisis secara kualitatif. Hasil penelitian menunjukkan bahwa lintasan belajar yang dibuat menggunakan tingkat berpikir kualitatif dengan indikator yang sering terlihat pada siswa kelas XII MIPA 7 SMA Negeri 1 Palembang adalah indikator visualisasi dan pelabelan. Lintasan belajar yang diperoleh dari penelitian ini didasarkan pada LKPD yang telah dilakukan oleh siswa dimana masalah berkontribusi untuk membangun micro teaching dalam pembelajaran dimensi ketiga. (10pt) This research is a research design type validation study that aims to develop a learning trajectory on the study of geometry topics of three-dimensional geometry to support students' qualitative thinking skills with a Rigorous Mathematical Thinking (RMT) approach. Hypothetical Learning Trajectory (HLT) will be refined throughout the lesson carried out in the classroom. The activities were created and designed using student worksheets (LKPD) to support the learning process. The subjects of the study were students of SMA Negeri 1 Palembang class XII, meanwhile, the focus group of this study was 6 students with varied abilities, namely high, medium, and low who were selected based on the achievement result documented by the teacher in the previous semester as well as the recommendations of teachers of mathematics subjects. There are three stages in conducting this research, namely: (1) Preparation Phase; (2) Design experiments divided into pilot experiments and teaching experiments; (3) retrospective analysis. Research data is collected and will be analyzed qualitatively. The results showed that the learning trajectory made using qualitative thinking levels with indicators often seen in students of SMA Negeri 1 Palembang class XII MIPA 7 is an indicator of visualization and labeling. The learning trajectory obtained from this study is based on LKPD that has been done by students where the problem contributes to building micro-teaching in third-dimensional learning.

Kegiatan pembelajaran bangun datar yang hanya menitikberatkan pada prosedur formal, tidak mampu mengembangkan lintasan belajar siswa. Oleh karena itu, pembelajaran bangun datar menggunakan konteks permainan tradisional menjadi salah satu alternatif yang dapat dilakukan. Tujuan penelitian ini adalah untuk mendeskripsikan learning trajectory siswa dalam mengkonstruksikan konsep luas bangun datar dengan menggunakan konteks bentuk papan permainan dam-daman sebagai titik awal pembelajaran. Metode penelitian ini menggunakan penelitian kualitatif berupa design research. Hasil dari penelitian ini adalah serangkaian aktivitas pada tingkat situational, referensial, general, dan formal. Pada tingkat situational, pemahaman awal secara langsung bentuk-bentuk bangun datar menggunakan bentuk papan permainan dam-daman. Tingkat referensial, siswa merepresentasi bentuk bangun apa yang ditemukan pada papan permainan dam-daman melalui representasi gambar. Tingkat general, siswa menghitung luas dari suatu bangun dengan menggunakan satu satuan luas yang telah ditentukan. Terakhir, tingkat formal, siswa merekonstruksi pemahaman mereka untuk menentukan rumus masing-masing luas bangun datar. Simpulan dari penelitian ini menunjukkan bahwa papan permainan dam-daman mampu merangsang siswa dalam mengkonstruksikan konsep luas bangun datar seperti persegi panjang, segitiga, jajar genjang, dan trapesium.

Recent advances in pedagogical research have called attention to the dynamic nature of the teaching and learning process in which the actors mutually influence one another. The understanding of how this works in the brain—the specialized neural networks related to this process—is often limited to neuroscientists but are slowly becoming available to other learning scientists, including teachers. A transdisciplinary approach combining the best information about observable teaching-learning processes from education with newer information from the neurosciences may aid in resolving fundamental questions in the learning process. Teachers’ professional formation and development is often structured in segmented topical ways (e.g., pedagogy, evaluation, planning, classroom management, social–emotional learning), to identify important content knowledge (e.g., art, reading, mathematics, STEM), or to appreciate life skills (e.g., collaboration, critical thinking, social–emotional learning). While important, knowledge about the brain, the organ responsible for learning, is typically absent from teacher education. This paper reexamines the evidence from neuroconstructivism and the hierarchy of learning trajectories and combines it with evidence from psychology and the ways humans interact during the teaching-learning process to suggest radical neuroconstructivism as a framework within which to organize teachers’ professional development. The radical neuroconstructivism framework may contribute to making the content knowledge of teachers’ continual professional development more visible.

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 research aimed to generate a learning trajectory in an introduction to early mathematics, precisely to measure learning using educational games and Realistic Mathematics Education (RME) and to describe young children’s curiosity in learning early mathematics. Children need to have an understanding to take a measurement and use a learning trajectory to learn how to differentiate measurable and countable quantities and to tell the longer and the greater in number. The design of this research involves three phases, namely, preliminary design, experimentation, and retrospective analysis. A Hypothetical Learning Trajectory (HLT) using educational games was developed by collecting data from documentation, observations, and interviews. The results obtained from the implementation in the classroom showed that the educational games were able to help young children get to know early mathematics, particularly on measurements of lengths and volumes. There is an association with several factors in the selection of realistic problems. Communications during the game could stimulate an introduction to measurement. The use of objects in students’ vicinity also contributed to their experiments relating to measurement. The results of this research inform policymaking for teachers in designing learning in the classroom in introducing early mathematics based on educational games and Realistic Mathematics Education (RME).

This systematic review examined evidence of the utility of learning progression (LP)–based assessments to inform teaching and student learning in classroom contexts. Fifty-nine studies met inclusion criteria and were analyzed against four research questions. Evidence highlighted their potential for supporting judgments about learning, informing instructional and learning decisions, and improving teacher learning and development. Although 23 studies measured student achievement, reporting positive overall effects, only 6 adopted the experimental designs necessary for causal claims. Using LP-based assessment for formative purposes was well supported. Limited evidence was found regarding summative and accountability uses. Findings show that LP-based assessment design and use requires trade-offs relating to standardization and scale. Teachers need opportunities for negotiation when making judgments and integrating LP-based assessments into existing curriculum and policy contexts. Future research should examine student use of LP assessments and find a balance between standardization and customization to meet the needs of diverse learners and local contexts. [ABSTRACT FROM AUTHOR]

Mathematics learning trajectories (LTs) for students above elementary school level are widely investigated. Recently, LTs for kindergarteners have also attracted attention, but in those studies the LTs were based on Western samples, and it is unclear whether they also involved culture and gender differences. Therefore, the purposes of this study were twofold: (1) construct a counting and arithmetic LT based on an Eastern sample and (2) show its similarities and differences by gender. The constructed LT contains 13 hypothesized levels of mathematical concepts according to previous research, and 59 kindergarteners (26 boys and 33 girls) participated in this study and completed a counting and arithmetic test to examine empirically the theoretical LT. The results showed that empirically, there were eight and nine conceptual levels for boys and girls, respectively, and boys and girls mastered concepts in a similar order (basic arithmetic→basic counting→advanced counting→mediocre arithmetic→advanced arithmetic), with the first part differing from the hypothesized LT. Within this developmental progression, girls showed a different path from advanced counting to mediocre arithmetic. The findings show gender and culture differences for the LTs for kindergarteners, which contradicts most previous research based on Western samples. [ABSTRACT FROM AUTHOR]

This paper comprises the results of a design study that aims at developing a theoretically and empirically based learning trajectory on statistical inference for 9th-grade students. Based on theories of informal statistical inference, an 8-step learning trajectory was designed. The trajectory consisted of two similar four step sequences: (1) experimenting with a physical black box, (2) visualizing distributions, (3) examining sampling distributions using simulation software, and (4) interpreting sampling distributions to make inferences in real -life contexts. Sequence I included only categorical data and Sequence II regarded numerical data. The learning trajectory was implemented in an intervention among 267 students. To examine the effects of the trajectory on students' understanding of statistical inference, we analyzed their posttest results after the intervention. To investigate how the stepwise trajectory fostered the learning process, students' worksheets during each learning step were analyzed. The posttest results showed that students who followed the learning trajectory scored significantly higher on statistical inference and on concepts related to each step than students of a comparison group (n = 217) who followed the regular curriculum. Worksheet analysis demonstrated that the 8-step trajectory was beneficial to students' learning processes. We conclude that ideas of repeated sampling with a black box and statistical modeling seem fruitful for introducing statistical inference. Both ideas invite more advanced follow-up activities, such as hypothesis testing and comparing groups. This suggests that statistics curricula with a descriptive focus can be transformed to a more inferential focus, to anticipate on subsequent steps in students' statistics education. [ABSTRACT FROM AUTHOR]

Students' mastery of geometry topics affects their ability to understand other mathematical topics. In addition, students are required to have higher-order thinking skills. Previous research shows that ICT media played an important role in improving students’ higher-order thinking skills. This study was carried out to produce a cuboid volume learning trajectory in ICT-assisted learning that can support students' higher-order thinking skills. This research employed validation design which consisted of three main stages, namely preliminary, experiment, and retrospective analysis. It was held during two main cycles. Sixty-four students of the eighth-grade in Palembang, Ogan Ilir, Manado, and West Papua were involved as research subjects. Data were collected through validation sheets, observations, interviews, and documents in the form of student worksheets. Then, the data were analyzed qualitatively and described narratively. The learning design developed was able to help students use higher-order thinking skills where students analyzed, evaluated, and used their creativity in answering the problems given. The results of this study can inform the policy making for teachers in designing mathematics learning and for lecturers to carry out mentoring for teachers in designing mathematics learning based on Realistic Mathematics Education (RME) and ICT media.

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.

Initiated by the Program for International Student Assessment (PISA), Mathematical Key Competencies (MKCs), which integrate into the process of solving situational problems, becomes a typical example of Mathematical Key Competencies assessment. Cognitive Diagnostic Assessment, as a new generation of measurement theory, integrates the measurement objectives into the cognitive process model via cognitive analysis, and helps increase understanding of students' mastery over fine-grained knowledge points. This paper analyzed 12 PISA test items and calibrated their attributes, thus forming a cognitive model based on six PISA's MKCs, which were namely Mathematical Abstraction, Logical Reasoning, Intuitive Imagination, Mathematical Modeling, Mathematical Operation and Data Analysis. Through the comparison of models fit for DINA, DINO, RRUM, ACDM, GDM, LCDM, LLM, G-DINA and Mixed Model, the LCDM with a good model fit was selected to analyze the data of 19, 454 students in eight countries, and comparisons of the six MKCs among these countries were obtained. Through analyzing the knowledge states, combing the prerequisite relationships between the attributes, and exploring the learning trajectories of students' MKCs in different countries, we found that the students' performance in China was the best among all countries for each of the six MKCs. In all other countries, the students' performance in Logical Reasoning and Intuitive Imagination were weaker than the other attributes. An analysis of learning trajectories found that Russia, Singapore, Australia and Finland had very similar main learning trajectories (highlighted in red), and the main learning trajectories of Russia and Singapore are the same; however, there exist distinct differences in learning trajectories between China and the United States, where the learning trajectories in China is complicated with varied branches, while the ones in the United States is relatively simple with fewer branches. [ABSTRACT FROM AUTHOR]

Learning trajectories are built upon progressions of mathematical understandings that are typical of the general population of students. As such, they are useful frameworks for exploring how understandings of diverse learners may be similar or different from their peers, which has implications for tailoring instruction. The purpose of this teaching experiment was to explore a diverse learner's understandings about equipartitioning and relational reasoning. Across eleven 45-min individualized sessions, the equipartitioning learning trajectory (EPLT) served as the framework for investigating the student's thinking and learning. Findings illustrate how a student's actual trajectory can be focused on developing a many-to-one meaning for fractions and relational reasoning, rather than strictly adhering to the sequence of proficiencies hypothesized by the EPLT. Further, the student's engagement with cognitive elements which characterize mental activity reveals ways instruction might be tailored to support a student's relational reasoning. Implications include a more nuanced perspective on individual learning of equipartitioning and important considerations for educators who support relational reasoning in diverse learners. [ABSTRACT FROM AUTHOR]

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.

(1) Background: This research shows how the identification of conceptual advances (CA) that determine the transition between the stages of a progression model in a learning trajectory will help prospective primary teachers (PPT) to develop the teaching competence "professional noticing of students' mathematical thinking". Conceptual advances are key moments in the construction of mathematical structures and involve a change in the way students understand mathematical relationships. (2) Methods: A teaching module has been designed in which students of the Teaching Degree will analyze the responses of primary education students to tasks of pattern generalization from the identification and use of conceptual advances. (3) Results: The results of the teaching module show that professional noticing can be developed in suitable teaching environments. (4) Conclusions: The recognition of conceptual advances helps to interpret students' thinking and learning trajectories which are effective tools to structure and develop professional noticing. [ABSTRACT FROM AUTHOR]

Abstract: This study aims at determining the role of the number line media in helping students understand and obtain local instructional theory in learning integer operations in the 4th Grade of Madrasah Ibtidaiyah in Tarakan. This research is design with three stages, namely preliminary design, experimental design, and retrospective analysis. In the preliminary design stage, a number line media along with the hypothetical learning trajectory is made. Then the teaching experiment is conducted to compare the expected learning trajectory with the actual learning trajectory that takes place. In the final stage, a retrospective analysis is carried out by viewing the video recording of learning and evaluation results. From the analysis, it was found that the use of number lines can help students to understand the operations of addition and subtraction of integers. This study also developed local instructional theory related to learning integers using number lines. Abstrak: Penelitian ini bertujuan untuk mengetahui peranan media garis bilangan dalam membantu pemahaman siswa serta mengembangkan teori instruksional lokal (local intructional theory) pada pembelajaran operasi bilangan bulat di kelas IV di salah satu Madrasah Ibtidaiyah di Kota Tarakan. Desain penelitian ini terdiri dari tiga tahap, yaitu desain pendahuluan, desain percobaan, dan analisis retrospektif. Pada tahap desain pendahuluan dilakukan pembuatan media garis bilangan beserta dugaan lintasan belajar, kemudian dilakukan percobaan pengajaran untuk membandingkan lintasan belajar yang diharapkan dengan lintasan belajar yang sesungguhnya terjadi. Pada tahap akhir dilakukan analisis retrospektif dengan melihat rekaman video pembelajaran dan hasil evaluasi. Dari analisis tersebut ditemui bahwa penggunaan garis bilangan dapat membantu siswa untuk memahami operasi penjumlahan dan pengurangan bilangan bulat. Penelitian ini juga mengembangkan teori instruksional lokal terkait pembelajaran bilangan bulat menggunakan garis bilangan.

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.

This research is aimed to describe a learning trajectory for probability through game-based learning. The research employed design research consisting of three stages: preparing for the experiment, design experiment, and retrospective analysis. A hypothetical learning trajectory (HLT) using Sudoku and Snake-and-ladder games was developed by collecting data through documentation, interviews, and classroom observations. The HLT was implemented in the classroom to investigate students’ actual learning trajectory. The results of this research indicate that the games helped students understand the concept of probability. The learning trajectory for probability based on game-based learning is seen from the perspective of four levels of emergent modeling. In the first level – ‘situational level’ – Sudoku and Ladder-and-Snake games were played by students. The second level is the ‘referential level’ where the rules of the games were used as a starting point to learn the concept of probability. Communication during game playing stimulated students' knowledge about random events, sample spaces, sample points, and events. At the third level – ‘general level’ – students used tree and table diagrams to generalize possible outcomes of an experiment and develop an understanding of sample spaces and sample points. Lastly, at the ‘formal level’ students developed their informal knowledge into formal concepts of probabilities.

This research-based on the findings that elementary school students already have mathematical critical thinking skills but have not reached the whole stage of the skills. Critical thinking skills of mathematics is essential for students because critical mathematical thinking can solve the problems and as a consideration to make decisions. This research aims to: (1) explore the critical thinking skills of mathematics of the subject of the volume of cubes and beams; (2) to determine student's difficulties in solving mathematical critical thinking skills test questions on the subject of the volume of cubes and beams. The research used a case study method of 28 students in one of the Islamic elementary schools in Bandung Regency. The data collected in the form of qualitative data obtained through tests of mathematical critical thinking skills, interviews, and document analysis. Data analysis techniques used thematic analysis by analyzing students' learning trajectory. The results showed that most students were still in the low category in the ability of critical thinking. As for the difficulties experienced by students, namely in providing simple explanations and working on problems that are not systematic. For this reason, the improving of students' mathematical critical thinking skills need further research.