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By taking the role as a mentor and a facilitator, a teacher in the 4 th grade of elementary school needs to look at the condition of the students in the concrete thinking stage. Learning process needs to be adjusted such that the abstract objects in mathematics can be represented through concrete objects as a bridge to enter the knowledge that the students already had, especially for the material of fraction. This research aims to analyze the achievement and the improvement of students' mathematics reasoning ability through the implementation of Indonesian realistic mathematics education (PMRI) approach. The research subject consisted of 51 students in the experiment group and 45 students in the control group which categorized into three levels (low, intermediate, and high). The result suggests that the achievement and the improvement of students' reasoning ability in the mathematics learning using PMRI approach are better than the conventional learning. AbstrakSebagai pembimbing sekaligus fasilitator, seorang guru di kelas 4 sekolah dasar perlu memperhatikan kondisi para siswanya yang berada pada taraf berfikir konkret. Proses belajar perlu disesuaikan sehingga objek abstrak pada matematika dapat "direpresentasikan" melalui benda konkret sebagai jembatan memasuki pengetahuan yang telah dimiliki siswa, khususnya terhadap materi pecahan. Penelitian ini bertujuan untuk mengkaji pencapaian dan peningkatan kemampuan penalaran matematika (KPM) siswa melalui penerapan pendekatan pembelajaran pendidikan matematika realistik Indonesia (PMRI). Subjek penelitian terdiri dari 51orang kelas eksperimen dan 45 orang kelas kontrol yang terbagi menjadi tiga level (rendah, sedang dan tinggi). Hasil penelitian menunjukkan pencapaian dan peningkatan KPM siswa pada pembelajaran matematika dengan pendekatan PMRI lebih baik dari pembelajaran konvensional.

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In general, many people still view mathematics as a subject that is far from reality and culture in everyday life. Historically, in fact, mathematics is very close to daily life and was developed by humans in response to the surrounding phenomena. Indonesia has diverse cultures, including in Yogyakarta. This culture can be used to explore mathematical concepts as a transformational effort to bring mathematics closer to the reality and perception of its people. Besides, we can use culture as the basis of learning mathematics in schools. Therefore, this study seeks to explore a mathematical concept of geometry transformation in the Yogyakarta batik pattern. This is an ethnography study. The research data were collected through observations, literature studies, and interviews with the batik culture practitioner and artist to understand the batik techniques and moral, historical, and philosophical values in each batik motif. This study's results indicate that in Yogyakarta batik, it uses the concept of geometry transformation in the making of Yogyakarta's unique Batik motif. Besides that, each motif or pattern also contains local values. These, namely moral, historical, and philosophical values, can be felt, reflected, and applied in daily life, such as values that teach leadership, good deeds, and so on.

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Mathematics exists as rooted in human ideas, ways, and techniques to respond to their environments. Along with its development, mathematics has been taught through formal education or schooling in the flat- not flexible ways and far from students' social and cultural lives. Indonesia is rich with culture, including the Special Region of Yogyakarta (DIY). It is called a city of culture. Educators are not yet aware of this richness in the learning process to integrate it as the starting point. It is hard to find in mathematics textbooks in Indonesia, which put cultural context as starting points. Therefore, this study aims to explore Yogyakarta's culture in terms of contexts used in mathematics learning.  It is an ethnography study. Furthermore, the data was collected through literature artifacts, field observation, and interviews with resource persons who understand the seasons, system, and calculation of birth and death days. It is to clarify the researcher's understanding of the literature. This study showed that Yogyakarta's people use mathematical modeling to determine the seasons' system and funerary dates. These models have the potential to be used as a starting point in learning mathematics.

Several studies on learning mathematics for rural area's student indicate that students have difficulty in understanding the concept of division operation. Students are more likely to be introduced by the use of the formula without involving the concept itself and learning division separate the concrete situation of learning process. This underlies the researcher to design division operation learning in the Mathematics of GASING (Math GASING), which always starts from concrete to abstract level. The research method used is a design research which describes how the Math GASING make a real contribution of students understanding in the concept of division operation.