Main Article Content
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.
Keywords
Article Details
Copyright (c) 2023 Nadya Amalia Juana, Junet Kaswoto, Sugiman, Aulia Almas Agustin Hidayat
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References
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- Arsoetar, N., & Sugiman, S. (2019). Development of student worksheets based on Realistic Mathematics Education (RME) oriented to mathematical reasoning. Journal Of Physics: Conference Series, 1397(1), 12091. https://doi.org/10.1088/1742-6596/1397/1/012091
- Astuti, W., & Wijaya, A. (2020). Learning trajectory berbasis proyek pada materi definisi himpunan. Jurnal Riset Pendidikan Matematika, 7(2), 254–266. https://doi.org/https://doi.org/10.21831/jrpm.v7i2.16483
- Chan, C. B., & Wilson, O. (2020). Using chakowa’s digitally enhanced learning model to adapt face- to-face EAP materials for online teaching and learning. International Journal of TESOL Studies, 2(1), 83–89. https://doi.org/https://doi.org/10.46451/ijts.2020.06.06
- Chen, C.Y. (2016). A hypothetical learning trajectory of arguing statemants about geometric figures.
- Clements, D. H., Wilson, D. C., & Sarama, J. (2012). Young children’s composition of geometric figures: A learning trajectory. In Hypothetical Learning Trajectories (1 ed., hal. 163–184). Routledge.
- Da, N. T. (2022). Approach to Realistic Mathematics Education in Teaching Calculus for High School Students: A Case of the Application of Derivatives. International Journal of Professional Development, Learners and Learning, 4(1).
- Dwidarti, U., Mampouw, H. L., & Setyadi, D. (2019). Analysis of students' difficulties in solving word problems on set material [in Bahasa]. Jurnal Cendekia: Jurnal Pendidikan Matematika, 3(2), 315–322. https://doi.org/https://doi.org/10.31004/cendekia.v3i2.110
- Fauzan, A, & Diana, F. (2020). Learning trajectory for teaching number patterns using RME approach in junior high schools. Journal of Physics: Conference Series, 1470(1), 12019. https://doi.org/10.1088/1742-6596/1470/1/012019
- Fauzan, Ahmad, Musdi, E., & Afriadi, J. (2018). Developing learning trajectory for teaching statistics at junior high school using RME approach. Journal of Physics: Conference Series, 1088(1), 12040. https://doi.org/10.1088/1742-6596/1088/1/012040
- Freudenthal, H. (2006). Revisiting mathematics education: China lectures (Vol. 9). Springer Science & Business Media.
- Gazali, R. Y. (2016). Pembelajaran matematika yang bermakna. Math Didactic: Jurnal Pendidikan Matematika, 2(3), 181–190.
- Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In Educational design research (hal. 29–63). Routledge.
- Gravemeijer, K. (2020). A socio-constructivist elaboration of realistic mathematics education. In National reflections on the Netherlands didactics of mathematics (hal. 217–233). Springer, Cham.
- Hedayani, E. (2018). Development of learning trajectory-based geometry learning tools oriented to critical and creative thinking skills for junior high school students [in Bahasa]. PYTHAGORAS, 13(2).
- Hery, R. D., Putri, R. I. I., & Hartono, Y. (2018). PMRI-Based Interactive Multimedia Development of Parallel Materials [in Bahasa]. Kreano, Jurnal Matematika Kreatif-Inovatif, 9(1), 78–83. https://doi.org/https://doi.org/10.15294/kreano.v9i1.14367
- Indriani, N., & Julie, H. (2017). Developing learning trajectory on the circumference of a cycle with realistic mathematics education (RME). AIP conference proceedings, 1868(1), 50022. https://doi.org/https://doi.org/10.1063/1.4995149
- Kim, C., Park, S. W., Cozart, J., & Lee, H. (2015). From motivation to engagement: The role of effort regulation of virtual high school students in mathematics courses. Journal of Educational Technology & Society, 18(4), 261–272.
- Kusrini, E., & Rizkianto, I. (2018). Developing Mathematics Learning Materials based on Multiple Intelligence Theory, Learning Trajectory, and Conceptual Knowledge in the Topic of Probability for Eighth Graders. University of Muhammadiyah Malang’s 1st International Conference of Mathematics Education (INCOMED 2017), 301–308. https://doi.org/https://dx.doi.org/10.2991/incomed-17.2018.65
- Marasabessy, R., & Hasanah, A. (2021). Mathematical Reasoning: What Is Its Central Aspect? [in Bahasa]. Jurnal Cendekia: Jurnal Pendidikan Matematika, 5(1), 562–577. https://doi.org/https://doi.org/10.31004/cendekia.v5i1.404
- Maxwell, S. E., Delaney, H. D., & Kelley, K. (2017). Designing experiments and analyzing data: A model comparison perspective. Routledge.
- Meidiana, S., Nyimas, A., & Pratiwi, W. D. (2021). Investigating Student’s Mathematical Reasoning Ability by Using PMRI Based on Emergent Modeling. 1st International Conference on Mathematics and Mathematics Education (ICMMEd 2020), 339–344. https://doi.org/https://dx.doi.org/10.2991/assehr.k.210508.085
- Najwa, W. A., Susiswo, S., & As’ari, A. R. (2019). Horizontal Mathematical Ability of Elementary School Students in Solving Positive Integer Problems [in Bahasa]. Jurnal Pendidikan: Teori, Penelitian, dan Pengembangan, 3(12), 1606–1611. https://doi.org/http://dx.doi.org/10.17977/jptpp.v3i12.12555
- 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/https://doi.org/10.22460/infinity.v8i2.p247-258
- Posamentier, A. S., & Smith, B. (2020). Teaching secondary school mathematics: Techniques and enrichment. World Scientific.
- Purwati, R. (2020). Application of Realistic Mathematic Education (RME) Approach in learning Mathematic to Improve Student Learning Outcomes. International Conference on Elementary Education, 2(1), 729–736.
- Puspaningtyas, N. D., & Ulfa, M. (2020). Numerical Literacy-Based Mathematical Problem Practice for IT Fitrah Insani High School Students [in Bahasa]. Jurnal Pengabdian Masyarakat MIPA Dan Pendidikan MIPA, 4(2), 137–140. https://doi.org/https://doi.org/10.21831/jpmmp.v4i2.37504
- Risdiyanti, I., & Prahmana, R. C. I. (2021a). Designing learning trajectory of set through the Indonesian shadow puppets and Mahabharata stories. Infinity Journal, 10(2), 331–348. https://doi.org/https://doi.org/10.22460/infinity.v10i2.p331-348
- Sarvita, L., & Syarifuddin, H. (2020). The developed hypothetical learning trajectory for integral topic based on realistic mathematics education. Journal of Physics: Conference Series, 1554(1), 12032. https://doi.org/10.1088/1742-6596/1554/1/012032
- Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of’well- taught’mathematics courses. Educational psychologist, 23(2), 145–166. https://doi.org/https://doi.org/10.1207/s15326985ep2302_5
- Simanulang, J. (2014). Development of teaching materials for the set of the rainbow troop context with the Indonesian Realistic Mathematics Education (PMRI) approach for Class VII Junior High Schools [in Bahasa]. Jurnal pendidikan matematika, 8(1), 43–54. https://doi.org/http://dx.doi.org/10.22342/jpm.8.1.1859.43-54
- Sirait, A. R., & Azis, Z. (2017). The Realistic of Mathematic Educational Approach (RME) toward the Ability of the Mathematic Connection of Junior High School in Bukhari Muslim Medan. American Journal of Educational Research, 5(9), 984–989. https://doi.org/10.12691/education-5-9-10
- Surya, A. (2018). Learning trajectory in elementary school (SD) mathematics learning [in Bahasa]. Jurnal Pendidikan Ilmiah, 4(2), 22–26.
- Towe, M. M., & Julie, H. (2020). Developing learning trajectories with the RME of phytagorean theorem. Journal of Physics: Conference Series, 1470(1), 12027. https://doi.org/10.1088/1742-6596/1470/1/012027
- Tunjungsari, A. R., & Tasyanti, T. (2017). Application of PBL with GeoGebra-assisted RME approach to improve mathematical thinking skills [in Bahasa]. PRISMA, Prosiding Seminar Nasional Matematika, 556–566.
References
Amelia, R., Kadarisma, G., Fitriani, N., & Ahmadi, Y. (2020). The effect of online mathematics learning on junior high school mathematic resilience during covid-19 pandemic. Journal of Physics: Conference Series, 1657(1), 12011. https://doi.org/10.1088/1742-6596/1657/1/012011
Amirante, A., Tortorelli, L., & Veronesi, I. (2022). Mathematics, Literature and Art: Getting Passionate about Mathematics Though The Use of Digital Technologies. ICERI2022 Proceedings, 3877–3884. https://doi.org/https://doi.org/10.21125/iceri.2022.0941
Andriani, L. (2019). Analysis of Student Errors in Solving Set Problems in the Mathematics Education Study Program at UIN SUSKA RIAU [in Bahasa]. Jurnal Cendekia: Jurnal Pendidikan Matematika, 3(2), 550–562. https://doi.org/https://doi.org/10.31004/cendekia.v3i2.146
Anggraeni, R., & Kadarisma, G. (2020). Analysis of Mathematical Problem Solving Ability of Grade VII Junior High School Students on Set Material [in Bahasa]. Jurnal Cendekia: Jurnal Pendidikan Matematika, 4(2), 1072–1082. https://doi.org/https://doi.org/10.31004/cendekia.v4i2.334
Arsoetar, N., & Sugiman, S. (2019). Development of student worksheets based on Realistic Mathematics Education (RME) oriented to mathematical reasoning. Journal Of Physics: Conference Series, 1397(1), 12091. https://doi.org/10.1088/1742-6596/1397/1/012091
Astuti, W., & Wijaya, A. (2020). Learning trajectory berbasis proyek pada materi definisi himpunan. Jurnal Riset Pendidikan Matematika, 7(2), 254–266. https://doi.org/https://doi.org/10.21831/jrpm.v7i2.16483
Chan, C. B., & Wilson, O. (2020). Using chakowa’s digitally enhanced learning model to adapt face- to-face EAP materials for online teaching and learning. International Journal of TESOL Studies, 2(1), 83–89. https://doi.org/https://doi.org/10.46451/ijts.2020.06.06
Chen, C.Y. (2016). A hypothetical learning trajectory of arguing statemants about geometric figures.
Clements, D. H., Wilson, D. C., & Sarama, J. (2012). Young children’s composition of geometric figures: A learning trajectory. In Hypothetical Learning Trajectories (1 ed., hal. 163–184). Routledge.
Da, N. T. (2022). Approach to Realistic Mathematics Education in Teaching Calculus for High School Students: A Case of the Application of Derivatives. International Journal of Professional Development, Learners and Learning, 4(1).
Dwidarti, U., Mampouw, H. L., & Setyadi, D. (2019). Analysis of students' difficulties in solving word problems on set material [in Bahasa]. Jurnal Cendekia: Jurnal Pendidikan Matematika, 3(2), 315–322. https://doi.org/https://doi.org/10.31004/cendekia.v3i2.110
Fauzan, A, & Diana, F. (2020). Learning trajectory for teaching number patterns using RME approach in junior high schools. Journal of Physics: Conference Series, 1470(1), 12019. https://doi.org/10.1088/1742-6596/1470/1/012019
Fauzan, Ahmad, Musdi, E., & Afriadi, J. (2018). Developing learning trajectory for teaching statistics at junior high school using RME approach. Journal of Physics: Conference Series, 1088(1), 12040. https://doi.org/10.1088/1742-6596/1088/1/012040
Freudenthal, H. (2006). Revisiting mathematics education: China lectures (Vol. 9). Springer Science & Business Media.
Gazali, R. Y. (2016). Pembelajaran matematika yang bermakna. Math Didactic: Jurnal Pendidikan Matematika, 2(3), 181–190.
Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In Educational design research (hal. 29–63). Routledge.
Gravemeijer, K. (2020). A socio-constructivist elaboration of realistic mathematics education. In National reflections on the Netherlands didactics of mathematics (hal. 217–233). Springer, Cham.
Hedayani, E. (2018). Development of learning trajectory-based geometry learning tools oriented to critical and creative thinking skills for junior high school students [in Bahasa]. PYTHAGORAS, 13(2).
Hery, R. D., Putri, R. I. I., & Hartono, Y. (2018). PMRI-Based Interactive Multimedia Development of Parallel Materials [in Bahasa]. Kreano, Jurnal Matematika Kreatif-Inovatif, 9(1), 78–83. https://doi.org/https://doi.org/10.15294/kreano.v9i1.14367
Indriani, N., & Julie, H. (2017). Developing learning trajectory on the circumference of a cycle with realistic mathematics education (RME). AIP conference proceedings, 1868(1), 50022. https://doi.org/https://doi.org/10.1063/1.4995149
Kim, C., Park, S. W., Cozart, J., & Lee, H. (2015). From motivation to engagement: The role of effort regulation of virtual high school students in mathematics courses. Journal of Educational Technology & Society, 18(4), 261–272.
Kusrini, E., & Rizkianto, I. (2018). Developing Mathematics Learning Materials based on Multiple Intelligence Theory, Learning Trajectory, and Conceptual Knowledge in the Topic of Probability for Eighth Graders. University of Muhammadiyah Malang’s 1st International Conference of Mathematics Education (INCOMED 2017), 301–308. https://doi.org/https://dx.doi.org/10.2991/incomed-17.2018.65
Marasabessy, R., & Hasanah, A. (2021). Mathematical Reasoning: What Is Its Central Aspect? [in Bahasa]. Jurnal Cendekia: Jurnal Pendidikan Matematika, 5(1), 562–577. https://doi.org/https://doi.org/10.31004/cendekia.v5i1.404
Maxwell, S. E., Delaney, H. D., & Kelley, K. (2017). Designing experiments and analyzing data: A model comparison perspective. Routledge.
Meidiana, S., Nyimas, A., & Pratiwi, W. D. (2021). Investigating Student’s Mathematical Reasoning Ability by Using PMRI Based on Emergent Modeling. 1st International Conference on Mathematics and Mathematics Education (ICMMEd 2020), 339–344. https://doi.org/https://dx.doi.org/10.2991/assehr.k.210508.085
Najwa, W. A., Susiswo, S., & As’ari, A. R. (2019). Horizontal Mathematical Ability of Elementary School Students in Solving Positive Integer Problems [in Bahasa]. Jurnal Pendidikan: Teori, Penelitian, dan Pengembangan, 3(12), 1606–1611. https://doi.org/http://dx.doi.org/10.17977/jptpp.v3i12.12555
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/https://doi.org/10.22460/infinity.v8i2.p247-258
Posamentier, A. S., & Smith, B. (2020). Teaching secondary school mathematics: Techniques and enrichment. World Scientific.
Purwati, R. (2020). Application of Realistic Mathematic Education (RME) Approach in learning Mathematic to Improve Student Learning Outcomes. International Conference on Elementary Education, 2(1), 729–736.
Puspaningtyas, N. D., & Ulfa, M. (2020). Numerical Literacy-Based Mathematical Problem Practice for IT Fitrah Insani High School Students [in Bahasa]. Jurnal Pengabdian Masyarakat MIPA Dan Pendidikan MIPA, 4(2), 137–140. https://doi.org/https://doi.org/10.21831/jpmmp.v4i2.37504
Risdiyanti, I., & Prahmana, R. C. I. (2021a). Designing learning trajectory of set through the Indonesian shadow puppets and Mahabharata stories. Infinity Journal, 10(2), 331–348. https://doi.org/https://doi.org/10.22460/infinity.v10i2.p331-348
Sarvita, L., & Syarifuddin, H. (2020). The developed hypothetical learning trajectory for integral topic based on realistic mathematics education. Journal of Physics: Conference Series, 1554(1), 12032. https://doi.org/10.1088/1742-6596/1554/1/012032
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of’well- taught’mathematics courses. Educational psychologist, 23(2), 145–166. https://doi.org/https://doi.org/10.1207/s15326985ep2302_5
Simanulang, J. (2014). Development of teaching materials for the set of the rainbow troop context with the Indonesian Realistic Mathematics Education (PMRI) approach for Class VII Junior High Schools [in Bahasa]. Jurnal pendidikan matematika, 8(1), 43–54. https://doi.org/http://dx.doi.org/10.22342/jpm.8.1.1859.43-54
Sirait, A. R., & Azis, Z. (2017). The Realistic of Mathematic Educational Approach (RME) toward the Ability of the Mathematic Connection of Junior High School in Bukhari Muslim Medan. American Journal of Educational Research, 5(9), 984–989. https://doi.org/10.12691/education-5-9-10
Surya, A. (2018). Learning trajectory in elementary school (SD) mathematics learning [in Bahasa]. Jurnal Pendidikan Ilmiah, 4(2), 22–26.
Towe, M. M., & Julie, H. (2020). Developing learning trajectories with the RME of phytagorean theorem. Journal of Physics: Conference Series, 1470(1), 12027. https://doi.org/10.1088/1742-6596/1470/1/012027
Tunjungsari, A. R., & Tasyanti, T. (2017). Application of PBL with GeoGebra-assisted RME approach to improve mathematical thinking skills [in Bahasa]. PRISMA, Prosiding Seminar Nasional Matematika, 556–566.