Construction of Integer Operations of the Gasing-Mathematics Method from the Perspective of APOS Theory

Evy Lalan Langi; Marilyn Lasarus Kondolayuk; Katriani Dea Domassang

doi:10.37680/qalamuna.v17i2.8110

Authors

Evy Lalan Langi Universitas Kristen Indonesia Toraja
Marilyn Lasarus Kondolayuk Universitas Kristen Indonesia Toraja
Katriani Dea Domassang Universitas Kristen Indonesia Toraja

DOI:

https://doi.org/10.37680/qalamuna.v17i2.8110

Keywords:

Action, Project, Scheme, Process, Gasing

Abstract

A low mastery of integer operations remains a significant cause of students’ weak mathematical literacy, as current teaching methods often emphasize mechanical computation over conceptual understanding. This study aims to describe how students construct their understanding of integer operations through the Gasing Method from the perspective of APOS Theory. This research employed a descriptive qualitative design with a task-based case study approach, involving three mathematics education students who had participated in Gasing training and obtained the highest scores on an integer operation test. Data were collected through written tasks and semi-structured interviews, then analyzed using the stages of APOS thinking: Action, Process, Object, and Schema, through iterative coding and triangulation. The results show that the Gasing Method supports the progression of students’ thinking from procedural to conceptual: the Action stage appears in step-by-step computation, the Process stage in mental prediction and verbal explanation, the Object stage in recognizing relationships among operations, and the Schema stage in integrating these into a coherent conceptual framework. The study concludes that the integration of Gasing and APOS provides a powerful pedagogical model that bridges concrete and abstract reasoning, offering meaningful implications for mathematics teachers in designing culturally grounded yet cognitively informed numeracy learning.

References

Adams, T. L. (2020). Mathematical Literacy. Mathematics Teacher: Learning and Teaching PK-12, 113(4), 262–263. https://doi.org/10.5951/mtlt.2019.0397

Afriansyah, A., Arif, D. B., & Islam, K. R. (2024). Development of Civics Learning through COLAKTRA (Congklak Nusantara) Innovation as a Traditional Game-Based Learning Media. Journal of Insan Mulia Education, 2(1), Article 1. https://doi.org/10.59923/joinme.v2i1.93

Agbo, F. J., Olaleye, S. A., Bower, M., & Oyelere, S. S. (2023). Examining the relationships between students’ perceptions of technology, pedagogy, and cognition: The case of immersive virtual reality mini games to foster computational thinking in higher education. Smart Learning Environments, 10(1), 16. https://doi.org/10.1186/s40561-023-00233-1

Ahmad, H., Nurhidayah, & Nurdin. (2018). Analisis Kemampuan Siswa Dalam Menyelesaikan Soal Cerita Pokok Bahasan Program Linear. Journal of MathEducation Nusantara, 1(1), 20–24.

Ahmad, H., Syamsuddin, F., & Latif, A. (2021). The development of student worksheets, assisted by the GeoGebra application, has improved higher-order thinking abilities in mathematics learning. Journal of Physics: Conference Series, 1882(1), 1–8. https://doi.org/10.1088/1742-6596/1882/1/012048

Alexander, P. A. (2019). Individual differences in college-age learners: The importance of relational reasoning for learning and assessment in higher education. British Journal of Educational Psychology, 1–13. https://doi.org/10.1111/bjep.12264

Allsop, Y. (2019). Assessing the computational thinking process using a multiple evaluation approach. International Journal of Child-Computer Interaction, 19, 30–55. https://doi.org/10.1016/j.ijcci.2018.10.004

Arisetyawan, A., & Supriadi, S. (2020). Ethnomathematics study in the calendar system of the Baduy tribe. Ethnomathematics Journal, 1(1), 25–29. https://doi.org/10.21831/ej.v1i1.28013

As Ari, A. R., Mahmudi, A., & Nuerlaelah, E. (2017), our Prospective Mathematic Teachers Are Not Yet Critical Thinkers. Journal on Mathematics Education, 8(2), 145–156. https://doi.org/10.22342/jme.8.2.3961.145-156

Aziz, L. A. (2021). Analisis Kemampuan Computational Thinking Mahasiswa Dalam Menyelesaikan Masalah Matematika. Jurnal Undikma, 9(1), 34–42.

Batlolona, J. R., Laurens, T., Leasa, M., Batlolona, M., Kempa, R., & Enriquez, J. J. (2019). Comparison of Problem-Based Learning and Realistic Mathematics Education to Improve Students' Academic Performance. Jurnal Pendidikan Progresif, 9(2), Article 2.

Baye, M. G., Ayele, M. A., & Wondimuneh, T. E. (2021). Implementing GeoGebra integrated with multi-teaching approaches guided by the APOS theory to enhance students’ conceptual understanding of limits in Ethiopian Universities. Heliyon, 7(5), e07012. https://doi.org/10.1016/j.heliyon.2021.e07012

Bazelais, P., Lemay, D. J., & Doleck, T. (2018). Grit, Mindset, and Academic Performance: A Study of Pre- Pre-Pre-Pre-Pre-Pre-Pre-University Science Students. EURASIA Journal of Mathematics, Science and Technology Education, 14(12).

Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE Theory to Improve Students’ Graphical Understanding of Derivatives. Eurasia Journal of Mathematics, Science and Technology Education, 14(7), 2947–2967. https://doi.org/10.29333/ejmste/91451

C. da Silva, T., B. de Melo, J., & Tedesco, P. (2020). The Creative Process in the Development of Computational Thinking in Higher Education. Proceedings of the 12th International Conference on Computer Supported Education, 215–226. https://doi.org/10.5220/0009346502150226

Clarke, D., & Roche, A. (2018). Using contextualized tasks to engage students in meaningful and worthwhile mathematics learning. Journal of Mathematical Behavior, 51(November 2016), 95–108. https://doi.org/10.1016/j.jmathb.2017.11.006

Dewi, S. (2025). The Manajemen-Strategi Pemasaran Digital untuk Meningkatkan Penjualan Produk Fashion di Era Media dan E-Commerce. Study of Applied Entrepreneurship, 1(3), 135–144. https://doi.org/10.33830/sae.v1i3.11249

Diah, R., & Siregar, N. (2023). Pengaruh Model Pembelajaran TGT (Teams Games Tournament) Modifikasi Metode Gasing Terhadap Hasil Belajar Matematika Siswa. EDUKASIA: Jurnal Pendidikan dan Pembelajaran, 4(2), 1033–1042. https://doi.org/10.62775/edukasia.v4i2.386

Education, T., Chen, Y., & Chang, C. (2018). The Impact of an Integrated Robotics STEM Course with a Sailboat Topic on High School Students ’ Perceptions of Integrative STEM, Interest, and Career Orientation Robots in the STEM Curriculum. EURASIA Journal of Mathematics, Science & Technology Education, 14(12).

Elo, S., Kääriäinen, M., Kanste, O., Pölkki, T., Utriainen, K., & Kyngäs, H. (2019). Qualitative Content Analysis: A Focus on Trustworthiness. Sage Open, 4(1), 2158244014522633. https://doi.org/10.1177/2158244014522633

Fitrieyah, I. D. (2021). Development of Moodle-Based E-Learning as a Mathematics Learning Media to Improve Student Learning Outcomes in Integral Materials. AMCA Journal of Education and Behavioral Change (AJEB), 1(2), 15–19. https://doi.org/10.29037/ajeb.xxx

Ghofur, A., Masrukan, & Rochmad. (2022). Mathematical Literacy Ability in Experiential Learning with Performance Assessment Based on Self-Efficacy. Unnes Journal of Mathematics Education Research, 11(1), 94–101.

Hariati, P. N. S., Rohanita, L., & Safitri, I. (2020). PENGARUH PENGGUNAAN MEDIA VIDEO ANIMASI TERHADAP RESPON SISWA DALAM PEMBELAJARAN MATEMATIKA PADA MATERI OPERASI BILANGAN BULAT. JURNAL PEMBELAJARAN DAN MATEMATIKA SIGMA (JPMS), 6(1). https://doi.org/10.36987/jpms.v6i1.1657

Herlina, S., Kusumah, Y. S., & Juandi, D. (2023). DIGITAL LITERACY: STUDENT PERCEPTION IN MATHEMATICS LEARNING. AKSIOMA: Jurnal Program Studi Pendidikan Matematika, 12(3), Article 3. https://doi.org/10.24127/ajpm.v12i3.7561

Hijriani, L., & Simarmata, J. E. (2023). PELATIHAN MATEMATIKA GASING BAGI SISWA. JMM (Jurnal Masyarakat Mandiri), 7(2), 1425. https://doi.org/10.31764/jmm.v7i2.13633

Huang, Q., Zhang, X., Liu, Y., Yang, W., & Song, Z. (2017). The contribution of parent–child numeracy activities to young Chinese children’s mathematical ability. British Journal of Educational Psychology, 87(3), 328–344. https://doi.org/10.1111/bjep.12152

Irianti, N. P., & Wijaya, E. M. S. (2019). Program Belajar Siswa Berbasis Prinsip Progressive Differentiation dan Integrative Reconciliation. JIPM (Jurnal Ilmiah Pendidikan Matematika), 7(2), 74. https://doi.org/10.25273/jipm.v7i2.3280

Jusmaniar, J., Riani, I., Anderson, E. C., Lee, M. C., & Oktavia, S. W. (2024). Gasing Game: Ethnoscientific Exploration of Circular Motion in Physics Learning on the Coast of East Sumatra to Build the Character of Perseverance. Schrödinger: Journal of Physics Education, 5(1), 1–9. https://doi.org/10.37251/sjpe.v5i1.902

Kabael, T. (2011). Generalizing single-variable functions to two-variable functions, the function machine, and APOS. Educational Sciences: Theory and Practice, 11(1), 484–499.

Kahar, N., & Palupi, R. (2020). Implementasi Metode Simple Additive Weighting Dalam Penentuan Sekolah Dasar Negeri Rujukan/Model Kota Jambi. Jurnal Nasional Teknologi Dan Sistem Informasi, 5(3), 138–147. https://doi.org/10.25077/teknosi.v5i3.2019.138-147

Kandemir, C. M., Kalelioğlu, F., & Gülbahar, Y. (2020). Pedagogy of teaching introductory text‐based programming in terms of computational thinking concepts and practices. Computer Applications in Engineering Education, 29(1), 29–45. https://doi.org/10.1002/cae.22374

Kim, D.-J., Choi, S., & Lim, W. (2017). Sfard’s Commognitive Framework as a Method of Discourse Analysis in Mathematics. International Journal of Cognitive and Language Sciences, 11(11), 481–485.

Kusuma, M. W. K., Jampel, I. N., & Bayu, G. W. (2018). Pengaruh Metode Pembelajaran Matematika Gasing Terhadap Hasil Belajar Matematika. Jurnal Pedagogi Dan Pembelajaran, 1(1), 37–46. https://doi.org/10.23887/jp2.v1i1.19330

Langi, E. L., Juniati, & Abadi. (2021). Understanding definite integral concepts of prospective teachers through actions and processes based on gender differences. Journal of Physics: Conference Series, 1747(1), 012026. https://doi.org/10.1088/1742-6596/1747/1/012026

Langi’, E. L., Juniati, D., & Abadi, A. (2019). Reconstruction of Derivatives Concept by Prospective Teachers Based on APOS Theory Reviewed from Gender Differences. MISEIC 2019. https://miseic.conference.unesa.ac.id/index.php/ocs/miseic2019/paper/view/2173

Langi, E. L., Juniati, D., & Abadi, A. (2023). Students as Prospective Teachers’ Understanding of Integral Based on the APOS Theory in Terms of Gender Difference. Journal of Higher Education Theory and Practice, 23(4). https://doi.org/10.33423/jhetp.v23i4.5897

Maharani, S., Kholid, M. N., Pradana, L. N., & Nusantara, T. (2019). Problem Solving in the Context of Computational Thinking. Infinity Journal of Mathematics Education, 8(2), 109–116.

Maharani, S., Nusantara, T., As’ari, A. R., & Qohar, A. (2019). Analyticity and systematicity in students of mathematics education in solving non-routine problems. Mathematics and Statistics, 7(2). https://doi.org/10.13189/ms.2019.070204

Martínez-Planell, R., & Trigueros, M. (2019). Using cycles of research in APOS: The case of functions of two variables. The Journal of Mathematical Behavior, 55, 100687. https://doi.org/10.1016/j.jmathb.2019.01.003

Melati, E., Fayola, A. D., Hita, I. P. A. D., Saputra, A. M. A., Zamzami, Z., & Ninasari, A. (2023). Pemanfaatan Animasi sebagai Media Pembelajaran Berbasis Teknologi untuk Meningkatkan Motivasi Belajar. Journal on Education, 6(1), Article 1. https://doi.org/10.31004/joe.v6i1.2988

Mulyawati, I., & Sarwinda, W. (2020). IbM Workshop Metode Matematika Gasing Bagi Guru SD Muhammadiyah Se Jakarta Timur. Jurnal Pengabdian Masyarakat MIPA Dan Pendidikan MIPA, 4(2), 79–85. https://doi.org/10.21831/jpmmp.v4i2.37495

Nga, N. T., Dung, T. M., Trung, L. T. B. T., Nguyen, T.-T., Tong, D. H., Van, T. Q., & Uyen, B. P. (2023). The Effectiveness of Teaching Derivatives in Vietnamese High Schools Using APOS Theory and ACE Learning Cycle. European Journal of Educational Research, 12(1), 507. https://doi.org/10.12973/eu-jer.12.1.507

Novita, D. (2021). ANALISIS KEMAMPUAN LITERASI SISWA DALAM MENYELESAIKAN MASALAH MATEMATIKA DITINJAU DARI BRAIN DOMINANCE. Skripsi, 53(9), 6.

Oktaç, A., Trigueros, M., & Romo, A. (2019). Apos Theory: Connecting Research and Teaching. For the Learning of Mathematics, 39(1), 33–37.

Pertiwi, H. D. P., Maharani, S., & Darmadi, D. (2025). The effectiveness of problem-based learning models assisted with ethno-fun on learning outcomes, as reviewed by students’ computational thinking. Al-Jabar : Jurnal Pendidikan Matematika, 16(01), 117–128. https://doi.org/10.24042/ajpm.v16i1.22421

Prananda, G., Friska, S. Y., & Susilawati, W. O. (2021). Pengaruh Media Konkret Terhadap Hasil Belajar Materi Operasi Hitung Campuran Bilangan Bulat Siswa Kelas IV Sekolah Dasar. JEMS: Jurnal Edukasi Matematika Dan Sains, 9(1), Article 1. https://doi.org/10.25273/jems.v9i1.8421

Prasetyo, E., Jatmiko, B., & Jufriansah, A. (2020). Literature Study of Understanding the Physical Concepts of Straight Motion Materials Using the Gasing Method. Indonesian Review of Physics, 3(2), 40. https://doi.org/10.12928/irip.v3i2.2236

Psycharis, S., Kalovrektis, K., Xenakis, A., Paliokas, I., Patrinopoulos, M., Georgiakakis, P., Iatrou, P., Theodorou, P., Papageorgiou, T., & Ntourou, V. (2021). The Impact of Physical Computing and Computational Pedagogy on Girls’ Self–Efficacy and Computational Thinking Practice. 2021 IEEE Global Engineering Education Conference (EDUCON), 308–315. https://doi.org/10.1109/educon46332.2021.9454003

Putra, E. S., Putri, R. I. I., & Susanti, E. (2018). PISA-Like Problems With Swimming Context. 5th ICRIEMS Proceedings, 371–378.

Reflina. (2020). Kesulitan mahasiswa calon guru matematika dalam menyelesaikan soal pembuktian matematis pada mata kuliah geometri. Jurnal Analisa, 6(1), 80–90.

Rikayanti. (2017). Pengembangan Bahan Ajar Berbasis Software Matlab pada Mata Kuliah Metode Numerik. Pasundan Journal of Research in Mathematics Learning and Education, 2(2), 95–108.

Rizaldi, D. F., Atiqoh, F., Dwikoranto, D., Prahani, B. K., Wibowo, F. C., & Astutik, S. (2023). Analysis of Physics Concepts in Gaming. International Journal of Emerging Research and Review, 1(1), 000005–000005. https://doi.org/10.56707/ijoerar.v1i1.5

Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS Theory. The Journal of Mathematical Behavior, 39, 100–120. https://doi.org/10.1016/j.jmathb.2015.06.005

Sari, L. P., Handika, M., Rosita, E., Sari, M., Anggoro, B. S., & Putra, F. G. (2019). The Flipped Classroom Strategy using Learning Video: Applied toward the Ability to Understand Mathematical Concepts. Journal of Physics: Conference Series, 1155(1), 1–5. https://doi.org/10.1088/1742-6596/1155/1/012088

Sary, R. M., & Ristiana, R. (2019). PEMBELAJARAN KELILING DAN LUAS BANGUN DATAR MENGGUNAKAN METODE MATEMATIKA GASING. Journal of Honai Math, 2(2), 143–150. https://doi.org/10.30862/jhm.v2i2.66

Srintin, A. S., Setyadi, D., & Mampouw, H. L. (2019). Pengembangan Media Permainan Kartu Umino Pada Pembelajaran Matematika Operasi Bilangan Bulat. Jurnal Cendekia : Jurnal Pendidikan Matematika, 3(1), 126–138. https://doi.org/10.31004/cendekia.v3i1.89

Suarsana, I. M., Mahayukti, G. A., Sudarma, I. K., & Pujawan, A. A. G. S. (2019). The Effect of Interactive Mathematics Learning Media on Mathematical Conceptual Understanding of Probability among Hearing-impaired Students. Journal of Physics: Conference Series, 1165(1), 1–9. https://doi.org/10.1088/1742-6596/1165/1/012021

Syafarina, G. A., & Zaenuddin, Z. (2025). Prediksi Klik Iklan Online Menggunakan Regresi Logistik: Studi Empiris Tentang Pendekatan Berbasis Data. Technologia : Jurnal Ilmiah, 16(2), 226–231. https://doi.org/10.31602/tji.v16i2.15943

Twarog, K. (2023). Preparing Teachers for Inclusion in a Homogeneous Society: A Case Study of a Czech Univesity's Approach to Inclusive Education. England. https://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-221605

Wijaya, A. P. (2020). Gaya Kognitif Field Dependent Dan Tingkat Pemahaman Konsep Matematis Antara Pembelajaran Langsung Dan STAD. Jurnal Derivat: Jurnal Matematika Dan Pendidikan Matematika, 3(2), 1–16. https://doi.org/10.31316/j.derivat.v3i2.713

Wu, W. R., & Yang, K. L. (2022). The relationships between computational and mathematical thinking: A review study on tasks. Cogent Education, 9(1), 1–19. https://doi.org/10.1080/2331186X.2022.2098929

Zaleha. (2018). Hasil Belajar Operasi Hitung Bilangan Bulat Melalui Implementasi Metode Jarimatika Pada Siswa Kelas V Sekolah Dasar Negeri Gambah, Barabai, Hulu Sungai Tengah. Jurnal Penelitian Tindakan dan Pendidikan, 4(1).