Pub Date : 2022-12-30DOI: 10.22342/jpm.17.1.19077.89-102
Nadya Amalia Juana, Junet Kaswoto, S. Sugiman, A. Hidayat
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.
{"title":"The Learning Trajectory of Set Concept Using Realistic Mathematics Education (RME)","authors":"Nadya Amalia Juana, Junet Kaswoto, S. Sugiman, A. Hidayat","doi":"10.22342/jpm.17.1.19077.89-102","DOIUrl":"https://doi.org/10.22342/jpm.17.1.19077.89-102","url":null,"abstract":"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.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"152 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74279580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-30DOI: 10.22342/jpm.17.1.18787.21-36
Febrian Febrian, Puji Astuti, S. Susanti
The continuous exploration of mathematics as a human activity triggers the need to research ethnomathematics. This study aimed to identify ethnomathematics in the manufacture of indigenous fish traps (Bubu) from Bintan Regency. This ethnography study uses direct observation, interviews, and documentation. The researcher acts as the main instrument. The data were analyzed using the Spradley analysis technique, namely domain, taxonomic, componential, and cultural theme analysis. Data reduction, data presentation, and conclusions were carried out for each analysis. The results show that there are mathematical activities in designing, counting, and measuring length dimensions in Bubu's making. In these activities, there are mathematical concepts, including three-dimensional figures, the net of three-dimensional figures, curves, odd numbers, sequences with their attributes, bilateral symmetry, symmetry axes, figurative numbers, the congruence of plane figures, and length measurements with non-standardized units. These results showed that the Bubu maker already had a geometric sense through the symmetrical concept that became the basis for two activities such as counting and measuring, similar to the results of ethnomathematical research on the Yupiaq Eskimo community in Alaska and the Carolina Islanders in Micronesia. This study provides ideas to utilize everyday phenomena in teaching mathematics as a starting point prior to learning mathematics more formally.
{"title":"Ethnomathematical Study on Indigenous Fish Trap: Example from Kijang, Bintan Regency","authors":"Febrian Febrian, Puji Astuti, S. Susanti","doi":"10.22342/jpm.17.1.18787.21-36","DOIUrl":"https://doi.org/10.22342/jpm.17.1.18787.21-36","url":null,"abstract":"The continuous exploration of mathematics as a human activity triggers the need to research ethnomathematics. This study aimed to identify ethnomathematics in the manufacture of indigenous fish traps (Bubu) from Bintan Regency. This ethnography study uses direct observation, interviews, and documentation. The researcher acts as the main instrument. The data were analyzed using the Spradley analysis technique, namely domain, taxonomic, componential, and cultural theme analysis. Data reduction, data presentation, and conclusions were carried out for each analysis. The results show that there are mathematical activities in designing, counting, and measuring length dimensions in Bubu's making. In these activities, there are mathematical concepts, including three-dimensional figures, the net of three-dimensional figures, curves, odd numbers, sequences with their attributes, bilateral symmetry, symmetry axes, figurative numbers, the congruence of plane figures, and length measurements with non-standardized units. These results showed that the Bubu maker already had a geometric sense through the symmetrical concept that became the basis for two activities such as counting and measuring, similar to the results of ethnomathematical research on the Yupiaq Eskimo community in Alaska and the Carolina Islanders in Micronesia. This study provides ideas to utilize everyday phenomena in teaching mathematics as a starting point prior to learning mathematics more formally.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91259710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-30DOI: 10.22342/jpm.17.1.17765.37-54
Sisca Puspita Sepriliani, Z. Zulkardi, S. Somakim
This research aims to develop a PISA-type content quantity that uses the context of religious holidays during the pandemic that is valid, practical, and potentially affects mathematical literacy skills. This research uses development research with two stages: preliminary and formative evaluation. This research also uses Inquiry-Based Learning (IBL) model in the learning process. This study involved eighth-grade students aged 13-15 years of various abilities. Data collection and analysis techniques were documentation, walkthroughs, observation, interviews, and tests. The research was carried out offline and online (Zoom and WhatsApp Group (WAG)). In this study, sharing activities were produced as well as a PISA-type jumping task with content quantity characteristics in the context of religious holiday during the pandemic by the PISA 2022 framework where what needs to be considered are mathematical literacy skills and use of language that is by language standards that can be applied and well interpreted by the students. Based on the students' answers, it can be seen that the questions and activities are included in the practical category because they can be solved well by students. From the results, it can be concluded that the developed PISA-like numeracy and activity has a potential effect on mathematical literacy skills and life in the context of religious days during the pandemic. In addition, IBL model can improve students' communication skills in solving PISA-type math problems and activities.
本研究旨在开发一种有效、实用且可能影响数学素养技能的pisa类型的内容量，该内容量在大流行期间使用宗教节日的背景。本研究采用发展性研究，分为初步评价和形成性评价两个阶段。本研究还在学习过程中使用了基于探究的学习(IBL)模型。这项研究涉及13-15岁的八年级学生，他们的能力各不相同。数据收集和分析技术包括文档、演练、观察、访谈和测试。这项研究是在线下和线上(Zoom和WhatsApp Group (WAG))进行的。在本研究中，通过PISA 2022框架，在大流行期间的宗教节日背景下，制作了分享活动以及具有内容数量特征的PISA类型跳跃任务，其中需要考虑的是数学素养技能和语言使用，这是通过语言标准，可以被学生应用和很好地解释。从学生的回答可以看出，这些问题和活动都属于实践范畴，因为学生可以很好地解决。从结果可以得出结论，在大流行期间的宗教日子背景下，发达的类似pisa的计算能力和活动对数学素养技能和生活有潜在影响。此外，IBL模式可以提高学生在解决pisa类型数学问题和活动中的沟通能力。
{"title":"Development of PISA-like Activities using the Inquiry-based Learning Model and the Context of Religious Holidays during the Pandemic","authors":"Sisca Puspita Sepriliani, Z. Zulkardi, S. Somakim","doi":"10.22342/jpm.17.1.17765.37-54","DOIUrl":"https://doi.org/10.22342/jpm.17.1.17765.37-54","url":null,"abstract":"This research aims to develop a PISA-type content quantity that uses the context of religious holidays during the pandemic that is valid, practical, and potentially affects mathematical literacy skills. This research uses development research with two stages: preliminary and formative evaluation. This research also uses Inquiry-Based Learning (IBL) model in the learning process. This study involved eighth-grade students aged 13-15 years of various abilities. Data collection and analysis techniques were documentation, walkthroughs, observation, interviews, and tests. The research was carried out offline and online (Zoom and WhatsApp Group (WAG)). In this study, sharing activities were produced as well as a PISA-type jumping task with content quantity characteristics in the context of religious holiday during the pandemic by the PISA 2022 framework where what needs to be considered are mathematical literacy skills and use of language that is by language standards that can be applied and well interpreted by the students. Based on the students' answers, it can be seen that the questions and activities are included in the practical category because they can be solved well by students. From the results, it can be concluded that the developed PISA-like numeracy and activity has a potential effect on mathematical literacy skills and life in the context of religious days during the pandemic. In addition, IBL model can improve students' communication skills in solving PISA-type math problems and activities.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88549697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-30DOI: 10.22342/jpm.17.1.20097.103-120
P. Pramudiani, T. Herman, T. Turmudi, M. Dolk, Belinda Terlouw
At the end of the sixties of last century, the development of Realistic Mathematics Education (RME) started in the Netherlands. At the beginning of this century, the Indonesian adaptation of RME, Pendidikan Matematika Realistik Indonesia (PMRI), started. The implementation of RME / PMRI has proven to be challenging. In this research, a qualitative case study was used to investigate teachers’ perceptions and experiences in implementing RME/PMRI in their classes. Semi-structured interviews were conducted with several Dutch and Indonesian teachers who have joined the RME/PMRI training. We found similarities and differences between the two groups of teachers. Both groups of teachers understand the use of context as a starting point for students to construct mathematical understanding. The Dutch teachers considered the construction of interesting mathematical problems and the use of the guided-reinvention principle as the difficulties but motivated them to do more practice. Indonesian teachers mentioned that for them, the integration of mathematics with other subject areas for integrated thematic learning in the 2013 curriculum was their constraint but it was also a challenge for them to be more creative. These perspectives can become a reference for the development of a localized implementation of learning trajectory in classroom practices.
上世纪60年代末，荷兰开始发展现实主义数学教育。本世纪初，印尼开始对RME进行改编，即Pendidikan Matematika Realistik Indonesia (PMRI)。RME / PMRI的实施已被证明具有挑战性。本研究采用定性个案研究，探讨教师在课堂上实施RME/PMRI的认知与经验。对参加RME/PMRI培训的几名荷兰和印度尼西亚教师进行了半结构化访谈。我们发现了两组老师之间的异同。两组教师都明白使用语境作为学生构建数学理解的起点。荷兰教师认为有趣的数学问题的构造和引导再造原则的使用是困难的，但激励他们做更多的实践。印尼教师提到，对他们来说，在2013年的课程中，数学与其他学科领域的综合主题学习是他们的约束，但这也是他们更具创造力的挑战。这些观点可以为学习轨迹在课堂实践中本地化实施的发展提供参考。
{"title":"What Do Indonesian and Dutch Teachers Find Challenging When Implementing Realistic Mathematics Education?","authors":"P. Pramudiani, T. Herman, T. Turmudi, M. Dolk, Belinda Terlouw","doi":"10.22342/jpm.17.1.20097.103-120","DOIUrl":"https://doi.org/10.22342/jpm.17.1.20097.103-120","url":null,"abstract":"At the end of the sixties of last century, the development of Realistic Mathematics Education (RME) started in the Netherlands. At the beginning of this century, the Indonesian adaptation of RME, Pendidikan Matematika Realistik Indonesia (PMRI), started. The implementation of RME / PMRI has proven to be challenging. In this research, a qualitative case study was used to investigate teachers’ perceptions and experiences in implementing RME/PMRI in their classes. Semi-structured interviews were conducted with several Dutch and Indonesian teachers who have joined the RME/PMRI training. We found similarities and differences between the two groups of teachers. Both groups of teachers understand the use of context as a starting point for students to construct mathematical understanding. The Dutch teachers considered the construction of interesting mathematical problems and the use of the guided-reinvention principle as the difficulties but motivated them to do more practice. Indonesian teachers mentioned that for them, the integration of mathematics with other subject areas for integrated thematic learning in the 2013 curriculum was their constraint but it was also a challenge for them to be more creative. These perspectives can become a reference for the development of a localized implementation of learning trajectory in classroom practices.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77360636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-30DOI: 10.22342/jpm.17.1.19905.121-134
Ahmad Rivai, A. Lestari, Nilam Permatasari Munir, Aswar Anas
Mathematical literacy helps an individual recognize the role or use of mathematics in everyday life. One of the factors supporting mathematical literacy is the learning style of the student. This study aimed to describe the mathematical literacy of students in the context of SMP Negeri 1 Palopo based on their answers to PISA test questions by observing their learning styles. The subjects in this study were three eighth graders each representing visual, auditory, and kinesthetic learning styles. The data instruments used were a learning style questionnaire, a mathematical literacy test based on the 2012 PISA test draft, and an interview guide. The results of the learning style questionnaire were analyzed by referring to the indicators of the three learning styles under study, while the results of the PISA test were analyzed by referring to the indicators for each PISA level. The results of this study indicated that the visual student and the auditory student had mathematical literacy at level 3 as shown by their ability to connect and reflect things involved in interpretation and basic reasoning, while the kinesthetic student had mathematical literacy at level 4 as shown by their ability to build and communicate explanations and argumentation based on interpretations, arguments, and actions. Based on the research results, it is suggested that students be accustomed to working on PISA-like problems to improve their mathematical literacy.
数学素养帮助个人认识到数学在日常生活中的作用或使用。支持数学素养的因素之一是学生的学习方式。本研究旨在通过观察学生的学习风格，根据学生对PISA测试问题的回答，描述学生在SMP Negeri 1 Palopo背景下的数学素养。本研究的受试者为三名八年级学生，分别代表视觉、听觉和动觉学习风格。使用的数据工具是一份学习风格问卷，一份基于2012年PISA测试草稿的数学素养测试，以及一份面试指南。学习风格问卷的结果参照所研究的三种学习风格的指标进行分析，而PISA测试的结果参照每个PISA水平的指标进行分析。本研究结果表明，视觉型学生和听觉型学生的数学素养在3级，表现为他们在解释和基本推理中联系和反映事物的能力;而动觉型学生的数学素养在4级，表现为他们在解释、论点和行动的基础上建立和交流解释和论证的能力。根据研究结果，建议学生习惯于解决类似pisa的问题，以提高他们的数学素养。
{"title":"Students' Mathematical Literacy in Solving PISA Problems Observed by Learning Styles","authors":"Ahmad Rivai, A. Lestari, Nilam Permatasari Munir, Aswar Anas","doi":"10.22342/jpm.17.1.19905.121-134","DOIUrl":"https://doi.org/10.22342/jpm.17.1.19905.121-134","url":null,"abstract":"Mathematical literacy helps an individual recognize the role or use of mathematics in everyday life. One of the factors supporting mathematical literacy is the learning style of the student. This study aimed to describe the mathematical literacy of students in the context of SMP Negeri 1 Palopo based on their answers to PISA test questions by observing their learning styles. The subjects in this study were three eighth graders each representing visual, auditory, and kinesthetic learning styles. The data instruments used were a learning style questionnaire, a mathematical literacy test based on the 2012 PISA test draft, and an interview guide. The results of the learning style questionnaire were analyzed by referring to the indicators of the three learning styles under study, while the results of the PISA test were analyzed by referring to the indicators for each PISA level. The results of this study indicated that the visual student and the auditory student had mathematical literacy at level 3 as shown by their ability to connect and reflect things involved in interpretation and basic reasoning, while the kinesthetic student had mathematical literacy at level 4 as shown by their ability to build and communicate explanations and argumentation based on interpretations, arguments, and actions. Based on the research results, it is suggested that students be accustomed to working on PISA-like problems to improve their mathematical literacy.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80776314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-30DOI: 10.22342/jpm.17.1.18866.55-68
A. S. Wahyuningrum, D. Suryadi, T. Turmudi
This study aims to investigate students’ prior knowledge as an obstacle when learning ratio and proportion concept. The study uses an interpretive paradigm which is part of Didactical Design Research. Eighth graders who had learned about ratio and proportion participated in this study. The analysis was carried out qualitatively based on the data from the students’ answers and interviews on their answers when solving ratio and proportion problems to identify learning obstacle, especially ontogenic obstacle regarding the students’ prior knowledge. The result of this study indicates that prior knowledge is one of the ontogenic obstacle in teaching and learning of ratio and proportion. It can be found from their learning experience in understanding the concept. In conclusion, investigating students' prior knowledge is essential for the effectiveness of teaching and learning of ratio and proportion. It is important to overcome ontogenic obstacles and to understand how to activate students’ prior knowledge using the right or appropriate methods when learning ratio and proportion.
{"title":"Students’ Prior Knowledge as an Ontogenic Obstacle on the Topic of Ratio and Proportion","authors":"A. S. Wahyuningrum, D. Suryadi, T. Turmudi","doi":"10.22342/jpm.17.1.18866.55-68","DOIUrl":"https://doi.org/10.22342/jpm.17.1.18866.55-68","url":null,"abstract":"This study aims to investigate students’ prior knowledge as an obstacle when learning ratio and proportion concept. The study uses an interpretive paradigm which is part of Didactical Design Research. Eighth graders who had learned about ratio and proportion participated in this study. The analysis was carried out qualitatively based on the data from the students’ answers and interviews on their answers when solving ratio and proportion problems to identify learning obstacle, especially ontogenic obstacle regarding the students’ prior knowledge. The result of this study indicates that prior knowledge is one of the ontogenic obstacle in teaching and learning of ratio and proportion. It can be found from their learning experience in understanding the concept. In conclusion, investigating students' prior knowledge is essential for the effectiveness of teaching and learning of ratio and proportion. It is important to overcome ontogenic obstacles and to understand how to activate students’ prior knowledge using the right or appropriate methods when learning ratio and proportion.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84090443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-14DOI: 10.21831/pythagoras.v17i2.51400
Emut Emut
Kajian literatur tentang teori graf, khususnya planaritas suatu graf, yaitu menentukan apakah suatu graf itu termasuk graf planar atau bukan sudah banyak dibahas. Pada artikel ini pembahasan sedikit berbeda yaitu melakukan generalisasi atau perumuman teorema planaritas untuk dan teorema planaritas untuk Eksistensi perumuman ini penting karena dapat membuktikan planaritas dan sekaligus serta planaritas beberapa graf yang terkait. Pembuktian ketidakplanaran graf lengkap dan memberikan manfaat besar terhadap pengembangan teori planaritas graf dan memantapkan jaminan kebenaran pada terapannya. Urgensi pembuktiannya memiliki peran yang besar dalam menentukan planaritas graf-graf yang terkait, baik isomorfik atau subdivisi. Salah satu produk yang dihasilkan adalah teorema Kuratovski yang memberikan syarat perlu dan cukup suatu graf merupakan graf planar. Proses generalisasi dilakukan melalui kajian terhadap sifat-sifat khusus pada dan juga sifat-sifat khusus yang dimiliki . Sifat-sifat khusus tersebut diperumum sehingga diperoleh suatu sifat yang berlaku baik untuk maupun . Berdasarkan hasil generalisasi dari sifat tersebut, kemudian dikombinasikan dengan teorema pertidaksamaan Euler menghasilkan suatu teorema yaitu jika suatu graf planar terhubung, dan panjang sikel terpendeknya adalah , dengan maka berlaku £ Manfaat dari generalisasi ini dapat juga digunakan pada pembuktian dan secara langsung dan beberapa graf terkait secara mudah. Generalization of Euler's Inequality to Prove Planarity of Graphs K_5 and K_(3,3) AbstractThe study of literature on graph theory, especially the planarity of a graph, which is to determine whether a graph is a planar graph or a non-planar graph, has been widely discussed. This article's discussion is slightly different, namely generalizing the planarity theorem for and the planarity theorem for This generalization is important because it can prove the planarity of and and the planarity of several related graphs. Proving the unplanarity of complete graphs and provide significant benefits to developing graph planarity theory and strengthens the guarantee of truth in its application. The urgency of the proof has a significant role in determining the planarity of the related graphs, either isomorphic or subdivision. One of the products of its role is the birth of Kuratovski's theorem, which provides the necessary and sufficient conditions for a planar graph. The generalization process is carried out by studying the special properties of and . These unique properties are generalized to obtain a valid property for and . Based on the results of the generalization of these properties, then combined with the Euler inequality theorem and the resulting theorem is if is a planar graph, connected, and the length of the shortest cycle is k, with then applies £ (n-2). The benefits of this generalization can be used to prove and directly and some related graphs quickly. Penulisan artikel ini bertujuan untuk m
关于格拉比理论的文献研究，尤其是关于平面平面的研究，即确定这种感觉是否包括平面平面。在这篇文章中，对实行这种提法的行星定理和行星定理的根本进行一点不同的讨论，因为它可以证明植物的存在，同时也可以证明植物的存在。证明平面平面是完全的，这对行星理论的发展有很大的好处，并巩固了对其治疗的真实性的保证。证词的紧迫性在决定与昆虫相关的行星格拉夫方面发挥了重要作用，无论是孤立的还是分裂的。由此产生的产品之一是库拉托夫斯基定理，它提供了必要的条件和足够的条件。归纳是通过对特定属性及其属性的研究来实现的。这些特殊的品质得到了适用于或适用于的品质。根据这种特质的概括,然后加上pertidaksamaan欧拉定理产生一个定理,即如果有格拉芙planar连接,长sikel terpendeknya是概括的£好处就适用,也可以用于证明和容易地直接和一些相关的格拉芙。欧拉对图形的可预测性性(3.3)，特别是图的文学研究，特别是图的行星，以确定一个图形是否一个平面或一个非平面图形，已被广泛否认。这篇文章的讨论则相对较小，namely概括了这个世代的行星和行星的原理是重要的，因为它可以证明几个相关的图形的行星和行星。揭示完整的景观和提供必要的支持来开发图图的行星理论和加强真理在其应用中的保证。证据的紧迫性是关系石刻的行星的决心，无论是孤立的还是分裂的。它的角色之一是库拉托夫斯基的诞生，这证明了对平面的需求和适应。由研究丹的特殊财产所引起的一般诉讼。这些独特的属性被用来指定一个有效的属性。generalization》改编自the results of这些财产,然后一起组合《欧拉不平等定理与无resulting定理是如果是a planar graph,连通和《shortest周期长度正好是k,然后一起applies£(已经开始)。这一代人的好处可以用来证明和恢复，一些相关的图形迅速发展。撰写本文的目的是对K_5进行普遍性植物定理，对k_3.3进行planatic定理。这个定理的存在之所以重要，是因为它可以证明K_5、同时证明k_3.3和多位行星相关的存在。这篇文章是关于格拉夫理论的文献研究的结果，尤其是平面理论的平面研究，即确定该学科是平面还是平面研究。K_5和k_3.3的彻底剥夺平面平面效果对planaritas graf理论的发展产生了巨大的好处，并巩固了对其治疗的真实性的保证。证词的紧迫性在决定与昆虫相关的行星格拉夫方面发挥了重要作用，无论是孤立的还是分裂的。其作用的产品之一是库拉托夫斯基定理的诞生，它提供了必要的条件和足够的平面平面。黑塔夫斯基定理解释说，格拉夫G是一个图形，如果而且只有G不包含K5或k3.3或k3.3的等等变量子。定理过程是通过对K_5的特殊属性和k_3.3的特殊属性的研究来实现的。这些特殊品质得到了有利于K_5和k_3.3的品质。根据结果概括这种特质,然后结合在pertidaksamaan欧拉定理和定理,即如果G有格拉芙planar产生连接,| V (G) | = n, | E (G) | = m、长度sikel terpendeknya是k和k3,那么适用。这个概括定理可以直接证明K_5和k_3.3，也可以很容易地相互关联。
{"title":"Generalisasi Pertidaksamaan Euler untuk Membuktikan Planaritas Graf K_5 dan K_(3,3)","authors":"Emut Emut","doi":"10.21831/pythagoras.v17i2.51400","DOIUrl":"https://doi.org/10.21831/pythagoras.v17i2.51400","url":null,"abstract":"Kajian literatur tentang teori graf, khususnya planaritas suatu graf, yaitu menentukan apakah suatu graf itu termasuk graf planar atau bukan sudah banyak dibahas. Pada artikel ini pembahasan sedikit berbeda yaitu melakukan generalisasi atau perumuman teorema planaritas untuk dan teorema planaritas untuk Eksistensi perumuman ini penting karena dapat membuktikan planaritas dan sekaligus serta planaritas beberapa graf yang terkait. Pembuktian ketidakplanaran graf lengkap dan memberikan manfaat besar terhadap pengembangan teori planaritas graf dan memantapkan jaminan kebenaran pada terapannya. Urgensi pembuktiannya memiliki peran yang besar dalam menentukan planaritas graf-graf yang terkait, baik isomorfik atau subdivisi. Salah satu produk yang dihasilkan adalah teorema Kuratovski yang memberikan syarat perlu dan cukup suatu graf merupakan graf planar. Proses generalisasi dilakukan melalui kajian terhadap sifat-sifat khusus pada dan juga sifat-sifat khusus yang dimiliki . Sifat-sifat khusus tersebut diperumum sehingga diperoleh suatu sifat yang berlaku baik untuk maupun . Berdasarkan hasil generalisasi dari sifat tersebut, kemudian dikombinasikan dengan teorema pertidaksamaan Euler menghasilkan suatu teorema yaitu jika suatu graf planar terhubung, dan panjang sikel terpendeknya adalah , dengan maka berlaku £ Manfaat dari generalisasi ini dapat juga digunakan pada pembuktian dan secara langsung dan beberapa graf terkait secara mudah. Generalization of Euler's Inequality to Prove Planarity of Graphs K_5 and K_(3,3) AbstractThe study of literature on graph theory, especially the planarity of a graph, which is to determine whether a graph is a planar graph or a non-planar graph, has been widely discussed. This article's discussion is slightly different, namely generalizing the planarity theorem for and the planarity theorem for This generalization is important because it can prove the planarity of and and the planarity of several related graphs. Proving the unplanarity of complete graphs and provide significant benefits to developing graph planarity theory and strengthens the guarantee of truth in its application. The urgency of the proof has a significant role in determining the planarity of the related graphs, either isomorphic or subdivision. One of the products of its role is the birth of Kuratovski's theorem, which provides the necessary and sufficient conditions for a planar graph. The generalization process is carried out by studying the special properties of and . These unique properties are generalized to obtain a valid property for and . Based on the results of the generalization of these properties, then combined with the Euler inequality theorem and the resulting theorem is if is a planar graph, connected, and the length of the shortest cycle is k, with then applies £ (n-2). The benefits of this generalization can be used to prove and directly and some related graphs quickly. Penulisan artikel ini bertujuan untuk m","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"241 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83231666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-14DOI: 10.21831/pythagoras.v17i2.26986
A. Mahmudi, S. Sugiman, K. Hernawati, Himmawati Puji Lestari
Penelitian pengembangan ini bertujuan untuk mengembangkan bahan ajar matematika berbasis kontekstual yang valid, praktis, dan efektif dengan model pengembangan ADDIE yang terdiri atas lima langkah pengembangan, yaitu Analysis, Design, Development, Implementation, dan Evaluation. Pengembangan bahan ajar ini penting untuk memfasilitas siswa dalam membangun pemahaman dan kebermaknaan dalam belajar matematika yang terjadi ketika siswa memahami keterkaitan antara suatu pengetahuan dengan pengetahuan lain atau dengan konteks kehidupan sehari-hari. Pembelajaran kontekstual dilaksanakan dengan strategi REACT yang terdiri atas lima aktivitas produktif, yaitu relating (mengaitkan materi pembelajaran dengan konteks), experiencing (melakukan eksplorasi untuk menemukan konsep atau pengetahuan), applying (menerapkan pengetahuan yang telah dikonstruksi), cooperating (bekerjasama untuk menyelesaikan masalah), dan transferring (menerapkan pengetahuan pada situasi atau masalah baru). Struktur penyajian bahan ajar ini diawali dengan penyajian konteks atau masalah yang sesuai dengan suatu konsep. Pemahaman dan penyelesaian terhadap masalah tersebut dijadikan dasar untuk membahas konsep-konsep matematis. Bahan ajar juga dilengkapi dengan berbagai soal-soal latihan yang berupa masalah kontekstual untuk memfasilitasi siswa untuk mengaplikasikan konsep. Insrumen penelitian ini adalah lembar kevalidan bahan ajar, angket kepraktisan bahan ajar, angket respon siswa terhadap bahan ajar, dan tes hasil belajar matematika. Hasil penelitian ini adalah bahan ajar matematika berbasis kontekstual yang valid, efektif, dan praktis.Development of contextual mathematics teaching material AbstractThis development research aims to develop mathematics contextual teaching materials that are valid, practical, and effective with the ADDIE development model consisting of five development steps, namely Analysis, Design, Development, Implementation, and Evaluation. The development of the teaching materials is important to facilitate students in building meaningfulness in learning mathematics. Meaningfulness can be obtained when students understand the relationship between concepts with other concepts and with the context in everyday life. Contextual learning is implemented with a REACT strategy consisting of five productive activities: relating, experiencing, applying, cooperating and transferring. The structure of contextual-based mathematical teaching materials begins with an exploration of the context or problem as a basis for constructing a mathematical concept. Teaching materials are also equipped with various problems of contextual problem solving as a concept application that has been constructed. The instrument of this research is the validation sheet, the instrument of the practicality of the teaching materials, the questionnaire of the students’ response to the teaching materials and the test of achievement. The results of this study are mathematics contextual materials that are vali
{"title":"Pengembangan Bahan Ajar Matematika Berbasis Kontekstual","authors":"A. Mahmudi, S. Sugiman, K. Hernawati, Himmawati Puji Lestari","doi":"10.21831/pythagoras.v17i2.26986","DOIUrl":"https://doi.org/10.21831/pythagoras.v17i2.26986","url":null,"abstract":"Penelitian pengembangan ini bertujuan untuk mengembangkan bahan ajar matematika berbasis kontekstual yang valid, praktis, dan efektif dengan model pengembangan ADDIE yang terdiri atas lima langkah pengembangan, yaitu Analysis, Design, Development, Implementation, dan Evaluation. Pengembangan bahan ajar ini penting untuk memfasilitas siswa dalam membangun pemahaman dan kebermaknaan dalam belajar matematika yang terjadi ketika siswa memahami keterkaitan antara suatu pengetahuan dengan pengetahuan lain atau dengan konteks kehidupan sehari-hari. Pembelajaran kontekstual dilaksanakan dengan strategi REACT yang terdiri atas lima aktivitas produktif, yaitu relating (mengaitkan materi pembelajaran dengan konteks), experiencing (melakukan eksplorasi untuk menemukan konsep atau pengetahuan), applying (menerapkan pengetahuan yang telah dikonstruksi), cooperating (bekerjasama untuk menyelesaikan masalah), dan transferring (menerapkan pengetahuan pada situasi atau masalah baru). Struktur penyajian bahan ajar ini diawali dengan penyajian konteks atau masalah yang sesuai dengan suatu konsep. Pemahaman dan penyelesaian terhadap masalah tersebut dijadikan dasar untuk membahas konsep-konsep matematis. Bahan ajar juga dilengkapi dengan berbagai soal-soal latihan yang berupa masalah kontekstual untuk memfasilitasi siswa untuk mengaplikasikan konsep. Insrumen penelitian ini adalah lembar kevalidan bahan ajar, angket kepraktisan bahan ajar, angket respon siswa terhadap bahan ajar, dan tes hasil belajar matematika. Hasil penelitian ini adalah bahan ajar matematika berbasis kontekstual yang valid, efektif, dan praktis.Development of contextual mathematics teaching material AbstractThis development research aims to develop mathematics contextual teaching materials that are valid, practical, and effective with the ADDIE development model consisting of five development steps, namely Analysis, Design, Development, Implementation, and Evaluation. The development of the teaching materials is important to facilitate students in building meaningfulness in learning mathematics. Meaningfulness can be obtained when students understand the relationship between concepts with other concepts and with the context in everyday life. Contextual learning is implemented with a REACT strategy consisting of five productive activities: relating, experiencing, applying, cooperating and transferring. The structure of contextual-based mathematical teaching materials begins with an exploration of the context or problem as a basis for constructing a mathematical concept. Teaching materials are also equipped with various problems of contextual problem solving as a concept application that has been constructed. The instrument of this research is the validation sheet, the instrument of the practicality of the teaching materials, the questionnaire of the students’ response to the teaching materials and the test of achievement. The results of this study are mathematics contextual materials that are vali","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83975196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-14DOI: 10.21831/pythagoras.v17i2.52985
Afifah Hayati, L. Harini, Ambar Winarni, Nur’aini Muhassanah
Penelitian ini bertujuan untuk memberikan definisi pemetaan (psi,varphi)_omega-kontraksi dalam ruang p-metrik modular, memberikan teorema titik tetap untuk pemetaan (psi,varphi)_omega-kontraksi pada ruang p-metrik modular, dan memberikan aplikasi dari teorema titik tetap tersebut. Penelitian ini menggunakan metode studi literatur. Hasil penelitian menunjukkan bahwa pemetaan (psi,varphi)_omega-kontraksi didefinisikan dalam ruang -metrik modular dengan memperumum pemetaan (psi,varphi)_omega-kontraksi dalam ruang p-metrik dan teorema titik tetap untuk pemetaan tersebut pada ruang p-metrik modular yang juga merupakan perumuman dari teorema titik tetap tersebut pada ruang p-metrik dengan penambahan beberapa sifat yang diasumsikan. Selain itu, hasil penilitian lainnya adalah aplikasi teorema titik tetap tersebut yang menjamin eksistensi solusi suatu persamaan integral yang juga merupakan perumuman dari aplikasi teorema titik tetap tersebut dalam ruang p-metrik. Dari hasil tersebut, dapat disimpulkan bahwa pemetaan (psi,varphi)_omega-kontraksi dapat didefinisikan dalam ruang p-metrik modular dan dapat dibuktikan teorema titik tetap untuk pemetaan (psi,varphi)_omega-kontraksi pada ruang p-metrik modular beserta aplikasi dari teorema titik tetap tersebut yang menjamin eksistensi solusi suatu persamaan integral.
{"title":"Teori Titik Tetap untuk Pemetaan (ψ,φ)_Ω-Kontraksi pada Ruang p-Metrik Modular Berorder","authors":"Afifah Hayati, L. Harini, Ambar Winarni, Nur’aini Muhassanah","doi":"10.21831/pythagoras.v17i2.52985","DOIUrl":"https://doi.org/10.21831/pythagoras.v17i2.52985","url":null,"abstract":"Penelitian ini bertujuan untuk memberikan definisi pemetaan (psi,varphi)_omega-kontraksi dalam ruang p-metrik modular, memberikan teorema titik tetap untuk pemetaan (psi,varphi)_omega-kontraksi pada ruang p-metrik modular, dan memberikan aplikasi dari teorema titik tetap tersebut. Penelitian ini menggunakan metode studi literatur. Hasil penelitian menunjukkan bahwa pemetaan (psi,varphi)_omega-kontraksi didefinisikan dalam ruang -metrik modular dengan memperumum pemetaan (psi,varphi)_omega-kontraksi dalam ruang p-metrik dan teorema titik tetap untuk pemetaan tersebut pada ruang p-metrik modular yang juga merupakan perumuman dari teorema titik tetap tersebut pada ruang p-metrik dengan penambahan beberapa sifat yang diasumsikan. Selain itu, hasil penilitian lainnya adalah aplikasi teorema titik tetap tersebut yang menjamin eksistensi solusi suatu persamaan integral yang juga merupakan perumuman dari aplikasi teorema titik tetap tersebut dalam ruang p-metrik. Dari hasil tersebut, dapat disimpulkan bahwa pemetaan (psi,varphi)_omega-kontraksi dapat didefinisikan dalam ruang p-metrik modular dan dapat dibuktikan teorema titik tetap untuk pemetaan (psi,varphi)_omega-kontraksi pada ruang p-metrik modular beserta aplikasi dari teorema titik tetap tersebut yang menjamin eksistensi solusi suatu persamaan integral.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75378037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-14DOI: 10.21831/pythagoras.v17i2.53625
Raihan Abimanyu Suharman, H. Hartono
Manggis merupakan salah satu komoditas buah asli Indonesia yang memiliki prospek pasar yang menjanjikan, terlebih dalam pasar ekspor. Namun masih ada permasalahan dalam hal penyortiran buah hasil panen. Buah manggis hasil panen disortir berdasarkan kematangan buahnya untuk tujuan pasar yaitu pasar ekspor dan pasar domestik. Faktor penentu kematangan buah manggis adalah warna dan tekstur dari kulit buahnya. Penelitian ini bertujuan untuk mengklasifikasi kematangan buah manggis berdasarkan warna dan tekstur menggunakan algoritma Naive Bayes. Fitur warna dan tekstur yang diekstraksi adalah kontras, korelasi, energi, homogenitas, entropi, standar deviasi, mean, varians, skewness, dan kurtosis. Fitur diekstraksi dari citra RGB, citra grayscale, citra HSV, dan citra CIELAB. Hasil ekstraksi fitur melewati tahap seleksi fitur menggunakan algoritma Minimum Redundancy Maximum Relevance (MRMR). Metode klasifikasi yang digunakan adalah metode Naive Bayes. Model klasifikasi Naive Bayes menggunakan parameter sebanyak 13 fitur dalam pembangunan modelnya yaitu mean R, mean G, standar deviasi G, mean Saturation, mean Hue, standar deviasi Hue, standar deviasi Value, mean a*, mean b*, standar deviasi a*, standar deviasi b*, varians a*, dan kontras. Hasil klasifikasi kematangan buah manggis menggunakan algoritma Naive Bayes memperoleh tingkat akurasi sebesar 95,7% dengan sensitivitas, spesifisitas, dan presisi untuk kelas matang sebesar 93,3%, 96,8%, dan 93,3%. Sensitivitas, spesifisitas, dan presisi untuk kelas mentah masing-masing sebesar 100%. Sensitivitas, spesifisitas, dan presisi untuk kelas setengah matang sebesar 93,3%, 96,9%, dan 93,3%.
{"title":"Klasifikasi Kematangan Manggis Berdasarkan Fitur Warna dan Tekstur Menggunakan Algoritma Naive Bayes","authors":"Raihan Abimanyu Suharman, H. Hartono","doi":"10.21831/pythagoras.v17i2.53625","DOIUrl":"https://doi.org/10.21831/pythagoras.v17i2.53625","url":null,"abstract":"Manggis merupakan salah satu komoditas buah asli Indonesia yang memiliki prospek pasar yang menjanjikan, terlebih dalam pasar ekspor. Namun masih ada permasalahan dalam hal penyortiran buah hasil panen. Buah manggis hasil panen disortir berdasarkan kematangan buahnya untuk tujuan pasar yaitu pasar ekspor dan pasar domestik. Faktor penentu kematangan buah manggis adalah warna dan tekstur dari kulit buahnya. Penelitian ini bertujuan untuk mengklasifikasi kematangan buah manggis berdasarkan warna dan tekstur menggunakan algoritma Naive Bayes. Fitur warna dan tekstur yang diekstraksi adalah kontras, korelasi, energi, homogenitas, entropi, standar deviasi, mean, varians, skewness, dan kurtosis. Fitur diekstraksi dari citra RGB, citra grayscale, citra HSV, dan citra CIELAB. Hasil ekstraksi fitur melewati tahap seleksi fitur menggunakan algoritma Minimum Redundancy Maximum Relevance (MRMR). Metode klasifikasi yang digunakan adalah metode Naive Bayes. Model klasifikasi Naive Bayes menggunakan parameter sebanyak 13 fitur dalam pembangunan modelnya yaitu mean R, mean G, standar deviasi G, mean Saturation, mean Hue, standar deviasi Hue, standar deviasi Value, mean a*, mean b*, standar deviasi a*, standar deviasi b*, varians a*, dan kontras. Hasil klasifikasi kematangan buah manggis menggunakan algoritma Naive Bayes memperoleh tingkat akurasi sebesar 95,7% dengan sensitivitas, spesifisitas, dan presisi untuk kelas matang sebesar 93,3%, 96,8%, dan 93,3%. Sensitivitas, spesifisitas, dan presisi untuk kelas mentah masing-masing sebesar 100%. Sensitivitas, spesifisitas, dan presisi untuk kelas setengah matang sebesar 93,3%, 96,9%, dan 93,3%.","PeriodicalId":31653,"journal":{"name":"Pythagoras Jurnal pendidikan Matematika","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77553379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：