Designing a learning trajectory for cube nets using the Engklek game: An educational design research study within realistic mathematics education
DOI:
https://doi.org/10.29408/jel.v12i1.32583Keywords:
cube nets, design research, engklek game, learning trajectory, realistic mathematics educationAbstract
The tendency to teach geometry mechanically highlights the need for mathematics instruction grounded in students’ daily contexts. This study designs a Learning Trajectory (LT) for cube nets using the engklek game—a traditional Indonesian hopping game—as a realistic context, framed within Realistic Mathematics Education (RME), and implemented through educational design research (validation study type). Conducted at one school in Kudus, the study involved 21 Grade 5 students across three phases: preparing for the experiment, designing the experiment (pilot and teaching experiment), and retrospective analysis. The resulting LT comprises four sequenced activities aligned with RME’s modeling levels—situational, model-of, model-for, and formal—through which students progressed from informal understandings (e.g., engklek kapal terbang as a cube net) to formal knowledge: recognizing eleven distinct cube nets organized into four pattern categories (1-4-1, 2-3-1, 2-2-2, and 3-3). This LT contributes to the expansion of instructional design in mathematics and provides teachers with a culturally responsive, game-based pedagogical tool that bridges students’ informal experiences with formal geometric concepts.
References
Afghohani, A., Cahyaningrum, F. S., & Astutiningtyas, E. L. (2024). Exploring ethnomathematics in the traditional game named engklek. Ethnomathematics Journal, 5(2), 84–99. https://doi.org/10.21831/ej.v5i2.66609
Alangui, W. V. (2017). Ethnomathematics and culturally relevant mathematics education in the Philippines. In M. Rosa, L. Shirley, M. E. Gavarrete, & W. V Alangui (Eds.), Ethnomathematics and its Diverse Approaches for Mathematics Education (pp. 183–208). Springer. https://doi.org/10.1007/978-3-319-59220-6_8
Alotaibi, M. S. (2024). Game-based learning in early childhood education: a systematic review and meta-analysis. Frontiers in Psychology, 15(1307881), 1–11. https://doi.org/10.3389/fpsyg.2024.1307881
Altakhyneh, B. H. (2020). The effect of mathematics anxiety on the achievement of middle school students in Amman. Journal of Education and Learning (EduLearn), 14(3), 338–344. https://doi.org/10.11591/edulearn.v14i3.15886
Amanda, A. P. D., Zuliana, E., & Riswari, L. A. (2024). Application of kahoot online media to critical thinking in mathematics for grade V students of SDN 1 Tambahrejo. Range: Jurnal Pendidikan Matematika, 6(1), 96–111. https://doi.org/10.32938/jpm.v6i1.7380
Anisa, Y., Siregar, R. F., & Hafiz, M. (2023). Ethnomathematics as an exploration of cultural mathematical concepts in traditional Indonesian engklek games. Asian Research Journal of Mathematics, 19(7), 65–75. https://doi.org/10.9734/arjom/2023/v19i7680
Arisetyawan, A., Ariadi, R. A., & Fauzi, I. (2025). Didactic design of cube nets based on ethnomathematics of engklek game in elementary school : An educational innovation. Jurnal Kependidikan, 11(1), 26–38. https://doi.org/10.33394/jk.v11i1.12838
Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge. https://doi.org/10.4324/9780203701010
Battista, M. T. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-based classroom. Journal for Research in Mathematics Education, 30(4), 417–448. https://doi.org/10.2307/749708
Chen, P. Y., Hwang, G. J., Yeh, S. Y., Chen, Y. T., Chen, T. W., & Chien, C. H. (2022). Three decades of game-based learning in science and mathematics education: an integrated bibliometric analysis and systematic review. Journal of Computers in Education, 9(3), 455–476. https://doi.org/10.1007/s40692-021-00210-y
Chen, W. H., & Ja’Faruddin. (2021). Traditional houses and projective geometry: Building numbers and projective coordinates. Journal of Applied Mathematics, 2021(1), 1–25. https://doi.org/10.1155/2021/9928900
Clements, D. H., Lizcano, R., & Sarama, J. (2023). Research and pedagogies for early math. Education Sciences, 13(8), 1–27. https://doi.org/10.3390/educsci13080839
Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. Routledge.
Clements, D. H., & Sarama, J. (2025). Systematic review of learning trajectories in early mathematics. ZDM - Mathematics Education, 57(4), 637–650. https://doi.org/10.1007/s11858-024-01644-1
DePascale, M., & Ramani, G. B. (2025). The role of math games for children’s early math learning: A systematic review. Journal of Numerical Cognition, 11(Article e14897), 1–24. https://doi.org/10.5964/jnc.14897
Ellis, A. B., Ozgur, Z., Kulow, T., Dogan, M. F., & Amidon, J. (2016). An exponential growth learning trajectory: Students’ emerging understanding of exponential growth through covariation. Mathematical Thinking and Learning, 18(3), 151–181. https://doi.org/10.1080/10986065.2016.1183090
Erşen, Z. B., & Ergül, E. (2022). Trends of game-based learning in mathematics education: A systematic review. International Journal of Contemporary Educational Research, 9(3), 603–623. https://doi.org/10.33200/ijcer.1109501
Facciaroni, L., Gambini, A., & Mazza, L. (2023). The difficulties in geometry: A quantitative analysis based on results of mathematics competitions in Italy. European Journal of Science and Mathematics Education, 11(2), 259–270. https://doi.org/10.30935/scimath/12590
Fauzi, I., Arisetyawan, A., & Chano, J. (2023). Integration of Indonesian culture in the didactic design of the concept of fractions in elementary schools. International Online Journal of Primary Education, 12(4), 263–284. https://doi.org/10.55020/iojpe.1336481
Fauzi, L. M., Hanum, F., Jailani, J., & Jatmiko, J. (2022). Ethnomathematics: Mathematical ideas and educational values on the architecture of Sasak traditional residence. International Journal of Evaluation and Research in Education, 11(1), 250–259. https://doi.org/10.11591/ijere.v11i1.21775
Fauzi, L. M., Hayati, N., & Satriawan, R. (2025). Ethnomathematics study and educational values in fundamental activities at the Langgar Pusaka of Sapit Village. Jurnal Math Educator Nusantara, 11(1), 21–32. https://doi.org/10.29407/jmen.v11i1.25097
Fauzi, L. M., Hayati, N., Satriawan, R., & Fahrurrozi. (2023). Perceptions of geometry and cultural values on traditional woven fabric motifs of the Sasak people. Jurnal Elemen, 9(1), 153–167. https://doi.org/10.29408/jel.v9i1.6873
Fauzi, L. M., Hayati, N., Wardi, Z., & Yazid, M. (2024). The influence of a realistic mathematics learning approach, interaction between students in terms of intrinsic motivation on mathematics learning outcomes. Jurnal Math Educator Nusantara, 10(1), 179–191. https://doi.org/10.29407/jmen.v10i1.22431
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3(3), 413–435. https://doi.org/10.1007/BF00302305
Freudenthal, H. (2002). Revisiting mathematics education: China lectures. Springer Dordrecht. https://doi.org/10.1007/0-306-47202-3
Ginting, S. M. B., & Rakhmawati, F. (2024). The influence of realistic mathematics education (rme) model based on ethnomathematics of engklek game on students’ mathematical problem solving abilities. Prisma Sains : Jurnal Pengkajian Ilmu Dan Pembelajaran Matematika Dan IPA IKIP Mataram, 12(3), 428–438. https://doi.org/10.33394/j-ps.v12i3.11905
Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Aldine Transaction.
Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht University.
Gravemeijer, K. (2006). Design research from the learning design perspective. In J. Van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational Design Research (pp. 45–85). Routledge. https://doi.org/10.4324/9780203088364
Gravemeijer, K. (2007). Emergent modelling as a precursor to mathematical modelling. In W. Blum, P. L. Galbraith, H. W. Henn, & M. Niss (Eds.), Modelling and Applications in Mathematics Education (pp. 137–144). Springer. https://doi.org/10.1007/978-0-387-29822-1_12
Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1), 111–129. https://doi.org/10.1023/A:1003749919816
Gravemeijer, K., & Terwel, J. (2000). Hans Freudenthal: A mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777–796. https://doi.org/10.1080/00220270050167170
Grigaliūnienė, M., Lehtinen, E., Verschaffel, L., & Depaepe, F. (2025). Systematic review of research on pedagogical content knowledge in mathematics: Insights from a topic-specific approach. ZDM – Mathematics Education, 57(4), 777–794. https://doi.org/10.1007/s11858-025-01684-1
Jeon, K. (2009). Mathematics hiding in the nets for a cube. Teaching Children Mathematics, 15(7), 394–399. https://doi.org/10.5951/TCM.15.7.0394
Kamalodeen, V. J., Ramsawak-Jodha, N., Figaro-Henry, S., Jaggernauth, S. J., & Dedovets, Z. (2021). Designing gamification for geometry in elementary schools: insights from the designers. Smart Learning Environments, 8(1), 1–24. https://doi.org/10.1186/s40561-021-00181-8
Kamid, K., Rohati, R., Rahmalisa, Y., Anggo, M., Septi, S. E., Azzahra, M. Z., & Nawahdani, A. M. (2021). Engklek Game” in mathematics: How difference and relationship student attitude towards science process skills? Cypriot Journal of Educational Sciences, 16(6), 3109–3123. https://doi.org/10.18844/cjes.v16i6.6500
Karakoç, B., Eryılmaz, K., Turan Özpolat, E., & Yıldırım, İ. (2022). The effect of game-based learning on student achievement: A meta-analysis study. Technology, Knowledge and Learning, 27(1), 207–222. https://doi.org/10.1007/s10758-020-09471-5
Kolb, S. M. (2012). Grounded theory and the constant comparative method: Valid research strategies for educators. Journal of Emerging Trends in Educational Research and Policy Studies, 3(1), 83–86. https://journals.co.za/doi/abs/10.10520/EJC135409
Kolovou, M. (2022). Embracing culturally relevant education in mathematics and science: A literature review. The Urban Review, 55(1), 133–172. https://doi.org/10.1007/s11256-022-00643-4
Kristanto, W., & Wibowo, H. (2023). Use of engklek in character education: Early childhoon education. The International Journal of Early Childhood Learning, 30(2), 53–72. https://doi.org/10.18848/2327-7939/CGP/v30i02/53-72
Listiawati, N., Sabon, S. S., Siswantari, Subijanto, Wibowo, S., Zulkardi, & Riyanto, B. (2023). Analysis of implementing Realistic Mathematics Education principles to enhance mathematics competence of slow learner students. Journal on Mathematics Education, 14(4), 683–700. https://doi.org/10.22342/jme.v14i4.pp683-700
Mark, S. L., & Id-Deen, L. (2022). Examining pre-service mathematics and science teachers’ plans to implement culturally relevant pedagogy. Educational Action Research, 30(5), 725–746. https://doi.org/10.1080/09650792.2020.1775670
Moradzadeh, S., & Ebrahimi, A. N. (2020). Islamic geometric patterns in higher dimensions. Nexus Network Journal, 22(3), 777–798. https://doi.org/10.1007/s00004-020-00486-0
Murniati, Sugiman, Rohman, A., & Zuliana, E. (2025). Hermeneutics of mathematics learning in Yogyakarta state middle schools. Multidisciplinary Science Journal, 7(10), 1–13. https://doi.org/10.31893/multiscience.2025500
Nugroho, F. A. W., Zuliana, E., & Amaliyah, F. (2025). Effectiveness of RME assisted by story calendar media for elementary school students’ mathematical literacy ability. Indonesian Journal of Educational Development (IJED), 6(1), 214–227. https://doi.org/10.59672/ijed.v6i1.4668
Nuraida, I., & Amam, A. (2019). Hypothetical learning trajectory in realistic mathematics education to improve the mathematical communication of junior high school students. Infinity Journal, 8(2), 247–258. https://doi.org/10.22460/infinity.v8i2.p247-258
Palinussa, A. L., Molle, J. S., & Gaspersz, M. (2021). Realistic mathematics education: Mathematical reasoning and communication skills in rural contexts. International Journal of Evaluation and Research in Education (IJERE), 10(2), 522–534. https://doi.org/10.11591/ijere.v10i2.20640
Pavlovičová, G., Bočková, V., & Laššová, K. (2022). Spatial ability and geometric thinking of students of teacher training for primary education. TEM Journal, 11(1), 388–395. https://doi.org/10.18421/TEM111-49
Pavlovičová, G., & Švecová, V. (2015). The development of spatial skills through discovering in the geometrical education at primary school. Procedia - Social and Behavioral Sciences, 186(1), 990–997. https://doi.org/10.1016/j.sbspro.2015.04.189
Pawartani, T., Suyono, S., Rufiana, I. S., Kusumaningrum, S. R., & Dewi, R. S. I. (2024). Knitting tradition and mathematics: Systematizing the literature on Indonesian traditional games and mathematical dimensions. Al-Islah: Jurnal Pendidikan, 16(2), 2082–2093. https://doi.org/10.35445/alishlah.v16i2.5166
Prahmana, R. C. I., Arnal-Palacián, M., Risdiyanti, I., & Ramadhani, R. (2023). Trivium curriculum in Ethno-RME approach: An impactful insight from ethnomathematics and realistic mathematics education. Jurnal Elemen, 9(1), 298–316. https://doi.org/10.29408/jel.v9i1.7262
Prahmana, R. C. I., & Kusumah, Y. S. (2016). The hypothetical learning trajectory on research in mathematics education using research-based learning. Pedagogika, 123(3), 42–54. https://doi.org/10.15823/p.2016.32
Prahmana, R. C. I., Risdiyanti, I., Peni, N. R. N., Ristiana, N., & Ramadhani, R. (2025). Javanese folklore with moral values: An impactful context in learning relations and functions. Journal on Mathematics Education, 16(1), 197–224. https://doi.org/10.22342/jme.v16i1.pp197-224
Putra, H. D., & Ramdhani, S. (2025). How to make “Sumedang” tofu: An ethnomathematics context from West Java. Journal of Honai Math, 8(1), 57–70. https://doi.org/10.30862/jhm.v8i1.690
Putri, D. A., Zulkardi, Susanti, E., & Meryansumayeka. (2025). A local instructional theory for learning the concept of arithmetic sequence through pocah piring game. Jurnal Elemen, 11(3), 533–551. https://doi.org/10.29408/jel.v11i3.29672
Qirom, M. S., & Juandi, D. (2023). Learning from Indonesia: A systematic literature review on the implementation of traditional Indonesian games for mathematics education. Beta: Jurnal Tadris Matematika, 16(2), 136–150. https://doi.org/10.20414/betajtm.v16i2.570
Rahayu, C., Putri, R. I. I., Zulkardi, & Hartono, Y. (2022). Curiosity: A game-based early mathematics case. Journal on Mathematics Education, 13(2), 275–288. https://doi.org/10.22342/jme.v13i2.pp275-288
Rahmadi, J., Wahyu, Y., & Oktari, V. (2024). Implementation of creative problem-solving model with RME approach on mathematics problem-solving ability. Jurnal Elemen, 10(1), 43–54. https://doi.org/10.29408/jel.v10i1.19788
Rahmi, F., Yonalta, S., & Rozana, I. (2025). Ethnomathematics on the Saluang: Analysis of mathematical concepts in Minangkabau musical instruments. Jurnal Elemen, 11(2), 328–345. https://doi.org/10.29408/jel.v11i2.27874
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
Riswari, L. A., Nugroho, F. A. W., & Susanti, O. I. (2023). Analisis kemampuan pemecahan masalah matematis siswa madrasah ibtidaiyah berdasarkan gender pada materi bangun datar [Analysis of elementary madrasah students’ mathematical problem-solving abilities based on gender in flat shape material]. Jurnal Penelitian Pembelajaran Matematika Sekolah (JP2MS), 7(2), 181–191. https://doi.org/10.33369/jp2ms.7.2.181-191
Rizki, M. D., Nugroho, F. A. W., Hapsari, C. R., Paramitha, A. P. A., Haliza, D. M. N., & Lutfiana, N. (2023). Pengaruh kemampuan berhitung terhadap nilai pelajaran matematika siswa kelas VI SD [The effect of numeracy skills on the mathematics scores of sixth-grade elementary school students]. Peshum: Jurnal Pendidikan, Sosial Dan Humaniora, 2(5), 913–920. https://doi.org/10.56799/peshum.v2i5.2239
Sahrudin, A., Budiarto, M. T., & Manuharawati. (2022). Epistemic action of junior high school students with low spatial ability in constructing cube nets. International Journal of Educational Methodology, 8(2), 221–230. https://doi.org/10.12973/ijem.8.2.221
Salma, R., Fevironika, D. O., & Zuliana, E. (2022). Ethnomathematical study of Jepara Troso Ikat weaving motifs in two dimensional geometry mathematics. Kontinu: Jurnal Penelitian Didaktik Matematika, 6(2), 102–115. https://doi.org/10.30659/kontinu.6.2.102-115
Sari, D. N., Husna, A., Amalia, R., Syafitri, L., Rangkuti, P. M., Shobirin, A., & Rambe, M. R. R. (2023). Analysis of learning difficulties in mathematics in the material build a class VI SDN 060820 Ball Room. Edumaspul-Journal of Education, 7(1), 953–957. https://doi.org/10.33487/edumaspul.v7i1.5731
Sholihah, M., Widadah, S., & Sukriyah, D. (2022). Development of student worksheet based on ethnomathematics traditional engklek games on cube and rectangular pyramid. Proceedings of the Eighth Southeast Asia Design Research (SEA-DR) & the Second Science, Technology, Education, Arts, Culture, and Humanity (STEACH) International Conference (SEADR-STEACH 2021), 627(1), 144–152. https://doi.org/10.2991/assehr.k.211229.023
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. https://doi.org/10.2307/749205
Sulistyawati, E., Nirmala, B. I., & Hamidah, D. (2025). Ethnomathematical exploration of the Remo dance as a source for mathematics learning. Jurnal Elemen, 11(1), 34–49. https://doi.org/10.29408/jel.v11i1.26997
Sunzuma, G., & Maharaj, A. (2019). Teacher-related challenges affecting the integration of ethnomathematics approaches into the teaching of geometry. Eurasia Journal of Mathematics, Science and Technology Education, 15(9), 1–15. https://doi.org/10.29333/ejmste/108457
Sunzuma, G., & Maharaj, A. (2022). Zimbabwean in-service teachers’ views of geometry: an ethnomathematics perspective. International Journal of Mathematical Education in Science and Technology, 53(9), 2504–2515. https://doi.org/10.1080/0020739X.2021.1919770
Supriadi, S. (2019). Didactic design of sundanese ethnomathematics learning for primary school students. International Journal of Learning, Teaching and Educational Research, 18(11), 154–175. https://doi.org/10.26803/ijlter.18.11.9
Tarkar, A., Matalka, B., Cartwright, M., & Kloos, H. (2022). Student-guided math practice in elementary school: Relation among math anxiety, emotional self-efficacy, and children’s choices when practicing math. Education Sciences, 12(9), 1–12. https://doi.org/10.3390/educsci12090611
Treffers, A. (1987a). Integrated column arithmetic according to progressive schematisation. Educational Studies in Mathematics, 18(1), 125–145. https://doi.org/10.1007/BF00314723
Treffers, A. (1987b). Three dimensions: A model of goal and theory description in mathematics instruction—the Wiskobas project. Springer. https://doi.org/10.1007/978-94-009-3707-9
Treffers, A. (1993). Wiskobas and Freudenthal realistic mathematics education. Educational Studies in Mathematics, 25(1), 89–108. https://doi.org/10.1007/BF01274104
Van den Akker, J. (1999). Principles and methods of development research. In J. Van den Akker, R. M. Branch, K. Gustafson, N. Nieveen, & T. Plomp (Eds.), Design Approaches and Tools in Education and Training (pp. 1–14). Springer. https://doi.org/10.1007/978-94-011-4255-7_1
Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35. https://doi.org/10.1023/B:EDUC.0000005212.03219.dc
Van den Heuvel-Panhuizen, M., & Drijvers, P. (2020). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 713–717). Springer. https://doi.org/10.1007/978-3-030-15789-0_170
Wijaya, A., Elmaini, & Doorman, M. (2021). A learning trajectory for probability: A case of game-based learning. Journal on Mathematics Education, 12(1), 1–16. https://doi.org/10.22342/jme.12.1.12836.1-16
Wilson, P. H., Sztajn, P., Edgington, C., & Myers, M. (2015). Teachers’ uses of a learning trajectory in student-centered instructional practices. Journal of Teacher Education, 66(3), 227–244. https://doi.org/10.1177/0022487115574104
Windari, R. A., & Amir, M. F. (2024). Realistic mathematics education model for logical reasoning of primary students. Jurnal Math Educator Nusantara, 10(2), 221–230. https://doi.org/10.29407/jmen.v10i2.22275
Zakaria, & Dewantara, A. H. (2024). Understanding multiplication concept: Exploring PMRI-enhanced thematic learning using traditional games for primary teachers. Jurnal Elemen, 10(1), 55–69. https://doi.org/10.29408/jel.v10i1.19838
Zayyadi, M., Misriyana, S., & Ratnadi, D. (2023). Ethnomathematics exploration through traditional Madurese games in elementary school mathematics learning. Mathema: Jurnal Pendidikan Matematika, 5(2), 325–335. https://doi.org/10.33365/jm.v5i2.3716
Zuliana, E., Dwiningrum, S. I. A., Wijaya, A., & Hukom, J. (2025). The effect of culture-based mathematics learning instruction on mathematical skills: a meta-analytic study. Journal of Education and Learning (EduLearn), 19(1), 191–201. https://doi.org/10.11591/edulearn.v19i1.21172
Zuliana, E., Dwiningrum, S. I. A., Wijaya, A., & Purnomo, Y. W. (2023). The geometrical patterns and philosophical value of Javanese traditional mosque architecture for mathematics learning in primary school: An ethnomathematic study. Journal of Education Culture and Society, 14(2), 512–532. https://doi.org/10.15503/jecs2023.2.512.532
Zuliana, E., & Nugroho, F. A. W. (2025). Ethnomathematical exploration: Mathematical concepts in the Kolong pigeon game. Journal of Honai Math, 8(3), 437–454. https://doi.org/10.30862/jhm.v8i3.976
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