Main Article Content
Abstract
This study aims to develop a learning trajectory for ratio material using rice fertilization context. This study employed validation studies within a design research framework involving seventh-grade students at a junior high school in Belitang. Data was collected through student activity sheets, observations, and interviews. Design research consists of three stages: experimental preparation, experimental design, and retrospective analysis. The findings indicate that the learning trajectory for ratio material in the rice fertilization context includes identifying patterns and relationships between two quantities, determining the simplest unit ratios, representing ratios using ratios table, and solving real-world ratio-related problems. These results confirm that the designed learning trajectory serves as a practical reference for teachers in designing ratio learning that is meaningful, applicable, and relevant to the life experiences of students in agricultural areas, while supporting sustainable agriculture.
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Copyright (c) 2025 Filian Yunita Sari, Zulkardi Zulkardi, Ratu Ilma Indra Putri, Ely Susanti

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References
- Akker, J. V., & Gravemeijer, K. (2010). Introducing Educational Design Research. In Educational Design Research. (3rd ed.). Netzodruk, Enschede. https://research.utwente.nl/files/14472302/Introduction_20to_20education_20design_20research.pdf
- Andini, W., & Jupri, A. (2017). Student obstacles in ratio and proportion learning. Journal of Physics: Conference Series, 812(1). https://iopscience.iop.org/article/10.1088/1742-6596/812/1/012048
- Benson-O’Connor, C. D., McDaniel, C., & Carr, J. (2019). Bringing math to life: provide students opportunities to connect their lives to math. Networks: An Online Journal for Teacher Research, 21(2). https://doi.org/10.4148/2470-6353.1299
- Bustang, Zulkardi, Darmawijoyo, Dolk, M., & van Eerde, D. (2013). Developing a local instruction theory for learning the concept of angle through visual field activities and spatial representations. International Education Studies, 6(8), 58–70. https://doi.org/10.5539/ies.v6n8p58
- Chaim, B. D., Keret, Y., & Ilany, B.-S. (2012). Ratio and Proportion Research and Teaching in Mathematics Teachers’ Education (Pre- and In-Service Mathematics Teachers of Elementary and Middle School Classes). In Springer eBooks: Sense Publishers. https://doi.org/10.1007/978-94-6091-784-4
- Civak, A. R., Işıksal Bostan, M., & Yemen Karpuzcu, S. (2022). Development of a hypothetical learning trajectory for enhancing proportional reasoning. Hacettepe University Journal of Education, 37(1), 345–365. https://doi.org/10.16986/HUJE.2020063485
- Diba, D. M. S., & Prabawanto, S. (2019). The analysis of students’ answers in solving ratio and proportion problems. International Conference on Mathematics and Science Education. https://doi.org/10.1088/1742-6596/1157/3/032114
- Dougherty, B., Bryant, D. P., Bryant, B. R., & Shin, M. (2017). Helping students with mathematics difficulties understand ratios and proportions. TEACHING Exceptional Children, 49(2). https://exceptionalchildren.org/journal/helping-students-mathematics-difficulties-understand-ratios-and-proportions
- Fauziah, A., Putri, R. I. I., & Zulkardi. (2022). Collaborative learning through lesson study in PMRI training for primary school pre-service teacher: the simulation of polygon matter. Infinity Journal, 11(1), 1–16. https://doi.org/10.22460/infinity.v11i1.p1-16
- Gravemeijer, K., & Cobb, P. (2006). Design Research from A Learning Design Perspective Dalam Jvd. Akker, K. Gravemeijer, S. Mckenney, & N. Nieveen (Editor), Educational Design Research (17-51). Routledge Taylor and Francis Group 4. https://doi.org/10.4324/9780203088364
- Gunawan, W., & Hadi, S. (2024). The effect of a realistic mathematics education (RME) approach and reasoning ability on students’ conceptual and procedural understanding. Contemporary Educational Researches Journal, 14(2), 90–102. https://doi.org/10.18844/cerj.v14i2.9318
- Hodgen, J., Foster, C., Brown, M., & Martin, D. (2024). Low-attaining secondary school mathematics students’ perspectives on recommended teaching strategies. International Journal of Science and Mathematics Education, 22(6), 1325–1343. https://doi.org/10.1007/s10763-023-10420-8
- Lamon, S. (2020). Teaching Fractions and Ratios For Understanding. in Teaching Fractions and Ratios for Understanding (4th ed.). In Routledge. https://doi.org/10.4324/9781003008057
- Lestari, P., Nurhasanah, F., Ayryuna, D., Chrisnawati, H. E., Kurniawati, I., Kuswardi, Y., & Wulandari, A. N. (2019). Proportional reasoning and belief of pre-service mathematics teachers: The use of modified Authentic Investigation Activities (AIA) model. AIP Conference Proceedings. https://pubs.aip.org/aip/acp/article/2194/1/020056/819515/Proportional-reasoning-and-belief-of-pre-service
- Muttaqin, H., Putri, R. I. I., & Somakim. (2017). Design research on ratio and proportion learning by using ratio table and graph with OKU Timur context at the 7 th grade. Journal on Mathematics Education, 8(2), 211–222. https://doi.org/10.22342/jme.8.2.3969.211-222
- Nofriati, N. F., Hartono, Y., & Somakim, S. (2020). Learning direct and inverse proportion using musi tour. International Journal on Emerging Mathematics Education, 3(2), 139. https://doi.org/10.12928/ijeme.v3i2.13578
- Petit, M. M., Laird, R. E., Wyneken, M. F., Huntoon, F. R., Abele-Austin, M. D., & Sequeira, J. D. (2020). A Focus on Ratios and Proportions. Routledge. https://doi.org/10.4324/9780429353611
- Plomp, T., & Nieveen, N. (2013). Educational Design Research. Educational Design Research, July, 1–206. https://doi.org/10.1007/978-1-4614-3185-5_11
- Rawani, D., Putri, R. I. I., Zulkardi, & Susanti, E. (2023). RME-based local instructional theory for translation and reflection using of South Sumatra dance context. Journal on Mathematics Education, 14(3), 545–562. https://doi.org/10.22342/jme.v14i3.pp545-562
- Samura, A. O., Im, R., & Ruslan, J. (2024). Local instruction theory in the realistic mathematics education approach to improve students’ mathematical proficiency in linear equation topics. Beta: Jurnal Tadris Matematika, 17(1), 59–76. https://doi.org/10.20414/betajtm.v17i1.624
- Sari, F. Y., Ilma, R., Putri, I., & Susanti, E. (2025). How do junior high school students solve proportional reasoning problems ? Jurnal Elemen, 11(2), 363–375. https://doi.org/10.29408/jel.v11i2.27922
- Sari, F. Y., Zulkardi, Putri, R. I. I., Susanti, E., & Nusantara, D. S. (2024). Ratio material learning design using the context of cooking rice to help elementary students understand concepts. Inomatika, 6(1), 54–66. https://doi.org/10.35438/inomatika.v6i1.443
- Setyaningsih, N., Juniati, D., & Suwarsono. (2018). Student ’ s scheme in solving mathematics problems. International Conference on Mathematics: Pure, Applied and Computation. https://iopscience.iop.org/article/10.1088/1742-6596/974/1/012012
- Sumarto, Zulkardi, Darmawijoyo, & van Galen, F. (2013). Ratio table and money context as means to support the development of student’s proportional reasoning. The First South East Asia Design/ Development Research (SEA-DR) International Conference, 427–435. https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://repository.unsri.ac.id/5980/1/P49_Syl_435.pdf&ved=2ahUKEwi58cSc_NyNAxXGS3ADHQ8qGxYQFnoECBgQAQ&usg=AOvVaw0kjz55aITijRASff7KK_2R
- Tamur, M., Kurnila, V. S., Jehadus, E., Nurjaman, A., Mandur, K., & Ndiung, S. (2021). The effect of the realistic mathematics education approach: meta-analysis of the measured mathematical ability angle. Atlantis Press. https://www.atlantis-press.com/proceedings/icmmed-20/125956432
- UNESCO. (2024). Education for Sustainable Development: Learning to Act for Sustainability. United Nations Educational, Scientific and Cultural Organization. https://unesdoc.unesco.org/ark:/48223/pf0000386934
- Utari, R. S., Putri, R. I. I., & Hartono, Y. (2015). Supporting 7 th students’ proportional reasoning using Palembang culture as context and ratio table as model. The Third South East Asia Design/Development Research International Conference, 344–352. https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://repository.unsri.ac.id/id/eprint/6841/contents&ved=2ahUKEwio0Oeu-9yNAxUKUGcHHRAHPVMQFnoECBkQAQ&usg=AOvVaw3Xzw1YAMyPvKMAMu_3ei16
- Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic Mathematics Education. In Encyclopedia of Mathematics Education (In S. Lerm). Springer. https://webspace.science.uu.nl/~heuve108/BEWAAR/download-velle/rme/VdHeuvel-Drijvers_2014_ENCYCLOPEDIA_RME-ref_OCR.pdf
- Wahyuni, I. (2022). Proportional Reasoning. Bantul: Lembaga Ladang Kata. https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://digilib.uinkhas.ac.id/20495/1/PENALARAN%2520PROPORSIONAL%2520%252B%2520SAMPUL.pdf&ved=2ahUKEwi6o_CHp9uNAxW2RWcHHeHnM-UQFnoECBcQAQ&usg=AOvVaw1G8HOO2RZTHuOZSeJk58hH
- Wahyuningrum, A. S., Suryadi, D., & Turmudi, T. (2022). Students’ prior knowledge as an ontogenic obstacle on the topic of ratio and proportion. Jurnal Pendidikan Matematika, 17(1), 55–68. https://doi.org/10.22342/jpm.17.1.18866.55-68
- Walle, V. de. (2008). Elementary and Secondary School Mathematics: Teaching Development. Jakarta: Erlangga. https://opac.umuslim.ac.id/index.php?subject=%22MATEMATIKA+SEKOLAH+DASAR+DAN+MENENGAH%22&search=Search
- Wijaya, A. P., Yunarti, T., & Coesamin, M. (2019). The analyzing of students’ learning obstacles in understanding proportion. Journal of Physics: Conference Series, 1280(4). https://doi.org/10.1088/1742-6596/1280/4/042022
- Zulfah, E., Fauzan, A., & Made Arnawa, I. (2024). Development of local instructional theory on parallelogram topics based on rme to improve mathematical problem-solving skills. Journal of World Science, 3(3), 391–397. https://doi.org/10.58344/jws.v3i3.582
- Zulkardi, & Putri, R. I. I. (2019). New School Mathematics Curricula, PISA and PMRI in Indonesia. In School Mathematics Curricula: Asian Perspectives and Glimpses of Reform (In C. P. V, pp. 39–49). Springer Singapore. https://doi.org/10.1007/978-981-13-6312-2_3
- Zulkardi, & Putri, R. I. I. (2020). Supporting mathematics teachers to develop jumping task using PISA framework (JUMPISA). Mathematics Education Journal, 14(2), 199–210. https://doi.org/10.22342/jpm.14.2.12115.199-210
- Zulkardi, Z., Putri, R. I. I., & Wijaya, A. (2020). Two Decades of Realistic Mathematics Education in Indonesia. In International Reflections on the Netherlands Didactics of Mathematics. ICME-13 Monographs (In M. van, pp. 325–340). https://doi.org/10.1007/978-3-030-20223-1_18
References
Akker, J. V., & Gravemeijer, K. (2010). Introducing Educational Design Research. In Educational Design Research. (3rd ed.). Netzodruk, Enschede. https://research.utwente.nl/files/14472302/Introduction_20to_20education_20design_20research.pdf
Andini, W., & Jupri, A. (2017). Student obstacles in ratio and proportion learning. Journal of Physics: Conference Series, 812(1). https://iopscience.iop.org/article/10.1088/1742-6596/812/1/012048
Benson-O’Connor, C. D., McDaniel, C., & Carr, J. (2019). Bringing math to life: provide students opportunities to connect their lives to math. Networks: An Online Journal for Teacher Research, 21(2). https://doi.org/10.4148/2470-6353.1299
Bustang, Zulkardi, Darmawijoyo, Dolk, M., & van Eerde, D. (2013). Developing a local instruction theory for learning the concept of angle through visual field activities and spatial representations. International Education Studies, 6(8), 58–70. https://doi.org/10.5539/ies.v6n8p58
Chaim, B. D., Keret, Y., & Ilany, B.-S. (2012). Ratio and Proportion Research and Teaching in Mathematics Teachers’ Education (Pre- and In-Service Mathematics Teachers of Elementary and Middle School Classes). In Springer eBooks: Sense Publishers. https://doi.org/10.1007/978-94-6091-784-4
Civak, A. R., Işıksal Bostan, M., & Yemen Karpuzcu, S. (2022). Development of a hypothetical learning trajectory for enhancing proportional reasoning. Hacettepe University Journal of Education, 37(1), 345–365. https://doi.org/10.16986/HUJE.2020063485
Diba, D. M. S., & Prabawanto, S. (2019). The analysis of students’ answers in solving ratio and proportion problems. International Conference on Mathematics and Science Education. https://doi.org/10.1088/1742-6596/1157/3/032114
Dougherty, B., Bryant, D. P., Bryant, B. R., & Shin, M. (2017). Helping students with mathematics difficulties understand ratios and proportions. TEACHING Exceptional Children, 49(2). https://exceptionalchildren.org/journal/helping-students-mathematics-difficulties-understand-ratios-and-proportions
Fauziah, A., Putri, R. I. I., & Zulkardi. (2022). Collaborative learning through lesson study in PMRI training for primary school pre-service teacher: the simulation of polygon matter. Infinity Journal, 11(1), 1–16. https://doi.org/10.22460/infinity.v11i1.p1-16
Gravemeijer, K., & Cobb, P. (2006). Design Research from A Learning Design Perspective Dalam Jvd. Akker, K. Gravemeijer, S. Mckenney, & N. Nieveen (Editor), Educational Design Research (17-51). Routledge Taylor and Francis Group 4. https://doi.org/10.4324/9780203088364
Gunawan, W., & Hadi, S. (2024). The effect of a realistic mathematics education (RME) approach and reasoning ability on students’ conceptual and procedural understanding. Contemporary Educational Researches Journal, 14(2), 90–102. https://doi.org/10.18844/cerj.v14i2.9318
Hodgen, J., Foster, C., Brown, M., & Martin, D. (2024). Low-attaining secondary school mathematics students’ perspectives on recommended teaching strategies. International Journal of Science and Mathematics Education, 22(6), 1325–1343. https://doi.org/10.1007/s10763-023-10420-8
Lamon, S. (2020). Teaching Fractions and Ratios For Understanding. in Teaching Fractions and Ratios for Understanding (4th ed.). In Routledge. https://doi.org/10.4324/9781003008057
Lestari, P., Nurhasanah, F., Ayryuna, D., Chrisnawati, H. E., Kurniawati, I., Kuswardi, Y., & Wulandari, A. N. (2019). Proportional reasoning and belief of pre-service mathematics teachers: The use of modified Authentic Investigation Activities (AIA) model. AIP Conference Proceedings. https://pubs.aip.org/aip/acp/article/2194/1/020056/819515/Proportional-reasoning-and-belief-of-pre-service
Muttaqin, H., Putri, R. I. I., & Somakim. (2017). Design research on ratio and proportion learning by using ratio table and graph with OKU Timur context at the 7 th grade. Journal on Mathematics Education, 8(2), 211–222. https://doi.org/10.22342/jme.8.2.3969.211-222
Nofriati, N. F., Hartono, Y., & Somakim, S. (2020). Learning direct and inverse proportion using musi tour. International Journal on Emerging Mathematics Education, 3(2), 139. https://doi.org/10.12928/ijeme.v3i2.13578
Petit, M. M., Laird, R. E., Wyneken, M. F., Huntoon, F. R., Abele-Austin, M. D., & Sequeira, J. D. (2020). A Focus on Ratios and Proportions. Routledge. https://doi.org/10.4324/9780429353611
Plomp, T., & Nieveen, N. (2013). Educational Design Research. Educational Design Research, July, 1–206. https://doi.org/10.1007/978-1-4614-3185-5_11
Rawani, D., Putri, R. I. I., Zulkardi, & Susanti, E. (2023). RME-based local instructional theory for translation and reflection using of South Sumatra dance context. Journal on Mathematics Education, 14(3), 545–562. https://doi.org/10.22342/jme.v14i3.pp545-562
Samura, A. O., Im, R., & Ruslan, J. (2024). Local instruction theory in the realistic mathematics education approach to improve students’ mathematical proficiency in linear equation topics. Beta: Jurnal Tadris Matematika, 17(1), 59–76. https://doi.org/10.20414/betajtm.v17i1.624
Sari, F. Y., Ilma, R., Putri, I., & Susanti, E. (2025). How do junior high school students solve proportional reasoning problems ? Jurnal Elemen, 11(2), 363–375. https://doi.org/10.29408/jel.v11i2.27922
Sari, F. Y., Zulkardi, Putri, R. I. I., Susanti, E., & Nusantara, D. S. (2024). Ratio material learning design using the context of cooking rice to help elementary students understand concepts. Inomatika, 6(1), 54–66. https://doi.org/10.35438/inomatika.v6i1.443
Setyaningsih, N., Juniati, D., & Suwarsono. (2018). Student ’ s scheme in solving mathematics problems. International Conference on Mathematics: Pure, Applied and Computation. https://iopscience.iop.org/article/10.1088/1742-6596/974/1/012012
Sumarto, Zulkardi, Darmawijoyo, & van Galen, F. (2013). Ratio table and money context as means to support the development of student’s proportional reasoning. The First South East Asia Design/ Development Research (SEA-DR) International Conference, 427–435. https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://repository.unsri.ac.id/5980/1/P49_Syl_435.pdf&ved=2ahUKEwi58cSc_NyNAxXGS3ADHQ8qGxYQFnoECBgQAQ&usg=AOvVaw0kjz55aITijRASff7KK_2R
Tamur, M., Kurnila, V. S., Jehadus, E., Nurjaman, A., Mandur, K., & Ndiung, S. (2021). The effect of the realistic mathematics education approach: meta-analysis of the measured mathematical ability angle. Atlantis Press. https://www.atlantis-press.com/proceedings/icmmed-20/125956432
UNESCO. (2024). Education for Sustainable Development: Learning to Act for Sustainability. United Nations Educational, Scientific and Cultural Organization. https://unesdoc.unesco.org/ark:/48223/pf0000386934
Utari, R. S., Putri, R. I. I., & Hartono, Y. (2015). Supporting 7 th students’ proportional reasoning using Palembang culture as context and ratio table as model. The Third South East Asia Design/Development Research International Conference, 344–352. https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://repository.unsri.ac.id/id/eprint/6841/contents&ved=2ahUKEwio0Oeu-9yNAxUKUGcHHRAHPVMQFnoECBkQAQ&usg=AOvVaw3Xzw1YAMyPvKMAMu_3ei16
Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic Mathematics Education. In Encyclopedia of Mathematics Education (In S. Lerm). Springer. https://webspace.science.uu.nl/~heuve108/BEWAAR/download-velle/rme/VdHeuvel-Drijvers_2014_ENCYCLOPEDIA_RME-ref_OCR.pdf
Wahyuni, I. (2022). Proportional Reasoning. Bantul: Lembaga Ladang Kata. https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://digilib.uinkhas.ac.id/20495/1/PENALARAN%2520PROPORSIONAL%2520%252B%2520SAMPUL.pdf&ved=2ahUKEwi6o_CHp9uNAxW2RWcHHeHnM-UQFnoECBcQAQ&usg=AOvVaw1G8HOO2RZTHuOZSeJk58hH
Wahyuningrum, A. S., Suryadi, D., & Turmudi, T. (2022). Students’ prior knowledge as an ontogenic obstacle on the topic of ratio and proportion. Jurnal Pendidikan Matematika, 17(1), 55–68. https://doi.org/10.22342/jpm.17.1.18866.55-68
Walle, V. de. (2008). Elementary and Secondary School Mathematics: Teaching Development. Jakarta: Erlangga. https://opac.umuslim.ac.id/index.php?subject=%22MATEMATIKA+SEKOLAH+DASAR+DAN+MENENGAH%22&search=Search
Wijaya, A. P., Yunarti, T., & Coesamin, M. (2019). The analyzing of students’ learning obstacles in understanding proportion. Journal of Physics: Conference Series, 1280(4). https://doi.org/10.1088/1742-6596/1280/4/042022
Zulfah, E., Fauzan, A., & Made Arnawa, I. (2024). Development of local instructional theory on parallelogram topics based on rme to improve mathematical problem-solving skills. Journal of World Science, 3(3), 391–397. https://doi.org/10.58344/jws.v3i3.582
Zulkardi, & Putri, R. I. I. (2019). New School Mathematics Curricula, PISA and PMRI in Indonesia. In School Mathematics Curricula: Asian Perspectives and Glimpses of Reform (In C. P. V, pp. 39–49). Springer Singapore. https://doi.org/10.1007/978-981-13-6312-2_3
Zulkardi, & Putri, R. I. I. (2020). Supporting mathematics teachers to develop jumping task using PISA framework (JUMPISA). Mathematics Education Journal, 14(2), 199–210. https://doi.org/10.22342/jpm.14.2.12115.199-210
Zulkardi, Z., Putri, R. I. I., & Wijaya, A. (2020). Two Decades of Realistic Mathematics Education in Indonesia. In International Reflections on the Netherlands Didactics of Mathematics. ICME-13 Monographs (In M. van, pp. 325–340). https://doi.org/10.1007/978-3-030-20223-1_18