The Practice of Reflection Based on Didactical Design Research: An Analysis of the Geometry Transformation Material

Iik Nurhikmayati; Mohamad Gilar Jatisunda; Nunuy Ratnawulan

doi:10.31764/jtam.v6i3.8441

The Practice of Reflection Based on Didactical Design Research: An Analysis of the Geometry Transformation Material

Iik Nurhikmayati, Mohamad Gilar Jatisunda, Nunuy Ratnawulan

Abstract

This study aims to identify the possibility of learning obstacles in the concept of Geometric Transformation based on the meaning of the Geometric Transformation concept that students have regarding their experience in obtaining the definition of the Geometric Transformation concept. This study uses a qualitative method and Didactical Design Research for methodological framework that contains three stages of analysis: prospective, metapedadidactic, and retrospective. Teachers who will carry out the learning reflection process based on didactical design research were chosen through purposive sampling as research participants. Forty-eight students took the written test, and then six students were selected by purposive sampling to participate in in-depth interviews. Data analysis was carried out descriptively by reducing data, presenting data, and drawing conclusions. The result indicates that the meaning of the concept of Geometry Transformation, according to students, was the mapping of points in a plane to a set of points in the same plane; the existence of inconsistencies and ambiguity of meaning, and the emergence of the findings of other meaning units of Geometric Transformation concepts. The experience of student meaning shows a tendency for students to get a sense from what is taught by teachers and books with more procedurally oriented concept meanings. Based on the purpose and experience of students' definition, there are learning obstacles in the Geometric Transformation concept, including ontogenic obstacles, epistemological obstacles, and didactical obstacles. These learning obstacles can be a valuable consideration for improving and developing learning designs related to the concept of Geometric Transformation.

Keywords

Concept Image; Geometics Learning Obstacles; Transformation;

Full Text:

DOWNLOAD [PDF]

References

Ada, T., & Kurtulucs, A. (2010). Students' misconceptions and errors in transformation geometry. International Journal of Mathematical Education in Science and Technology, 41(7), 901–909. https://doi.org/10.1080/0020739X.2010.486451

Albab, I. U., Hartono, Y., & Darmawijoyo, D. (2014). Kemajuan belajar siswa pada geometri transformasi menggunakan aktivitas refleksi geometri. Jurnal Cakrawala Pendidikan, 33(3). https://doi.org/10.21831/cp.v3i3.2378

Ali, A. A., & Reid, N. (2012). Understanding mathematics: Some key factors. European Journal of Educational Research, 1(3), 283–299. https://doi.org/10.12973/eu-jer.1.3.283

Attorps, I. (2007a). Concept definition and concept image: in the case of equations. https://www.diva-portal.org/smash/get/diva2:241363/FULLTEXT01.pdf

Attorps, I. (2007b). Concept definition and concept image: in the case of equations. https://www.diva-portal.org/smash/get/diva2:241363/FULLTEXT01.pdf

Bansilal, S., & Naidoo, J. (2012). Learners engaging with transformation geometry. South African Journal of Education, 32(1), 26–39. https://doi.org/10.15700/saje.v32n1a452

Bardini, C., Pierce, R., Vincent, J., & King, D. (2014). Undergraduate Mathematics Students' Understanding of the Concept of Function. Indonesian Mathematical Society Journal on Mathematics Education, 5(2), 85–107.

Barnard, T., & Tall, D. (1997). Cognitive units, connections and mathematical proof. PME Conference, 2, 2–41. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.478.3214&rep=rep1&type=pdf

Bechberger, L., & Kühnberger, K.-U. (2019). Formalized Conceptual Spaces with a Geometric Representation of Correlations. In Conceptual Spaces: Elaborations and Applications (pp. 29–58). Springer. https://doi.org/10.1007/978-3-030-12800-5_3

Brousseau, G. (2006a). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970--1990 (Vol. 19). Springer Science & Business Media.

Brousseau, G. (2006b). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990 (Vol. 19). Springer Science & Business Media.

Cheng, K. H.-F., & Kong, S. C. (2017). An Approach to Facilitate Coherent Concept Image Formation via Guided Reinvention. In Emerging Practices in Scholarship of Learning and Teaching in a Digital Era (pp. 233–244). Springer. https://doi.org/10.1007/978-981-10-3344-5_15

Clements, D. H., Swaminathan, S., Hannibal, M. A. Z., & Sarama, J. (1999). Young children's concepts of shape. Journal for Research in Mathematics Education, 192–212. https://doi.org/10.2307/749610

Creswel, J. W. (2009a). Research design: Qualitative, quantitative, and mixed methods approaches. Los Angeles: University of Nebraska–Lincoln.

Creswel, J. W. (2009b). Research design: Qualitative, quantitative, and mixed methods approaches. Los Angeles: University of Nebraska--Lincoln.

Creswell, J. W. (2012). Educational Research: Planning, Conducting and evaluating Quantitative and Qualitative Research (4th ed). Pearson education, Inc.

Creswell, J. W., & Clark, V. L. P. (2017). Designing and conducting mixed methods research. Sage publications.

Dedy, E., & Sumiaty, E. (2017). Desain didaktis bahan ajar matematika SMP berbasis learning obstacle dan learning trajectory. JRPM (Jurnal Review Pembelajaran Matematika), 2(1), 69–80. https://doi.org/10.15642/jrpm.2017.2.1.69-80

Edwards, L. D. (1997). Exploring the territory before proof: Student's generalizations in a computer microworld for transformation geometry. International Journal of Computers for Mathematical Learning, 2(3), 187–215. https://doi.org/10.1023/A:1009711521492

Engelke Infante, N., Murphy, K., Glenn, C., & Sealey, V. (2018a). How concept images affect students' interpretations of Newton's method. International Journal of Mathematical Education in Science and Technology, 49(5), 643–659. https://doi.org/10.1080/0020739X.2017.1410737

Engelke Infante, N., Murphy, K., Glenn, C., & Sealey, V. (2018b). How concept images affect students' interpretations of Newton's method. International Journal of Mathematical Education in Science and Technology, 49(5), 643–659. https://doi.org/10.1080/0020739X.2017.1410737

Fauzi, I., & Suryadi, D. (2020). Didactical Design Research untuk Mengembangkan Kompetensi Pedagogik Guru di Sekolah Dasar. INVENTA: Jurnal Pendidikan Guru Sekolah Dasar, 4(1), 58–68. https://doi.org/10.36456/inventa.4.1.a2207

Fendrik, M. (2019). Penggunaan Alat Peraga Dakon Matematika (Dakota) Sebagai Upaya Peningkatan Hasil Belajar Matematika Bagi Siswa Sekolah Dasar. Jurnal Basicedu, 3(2), 702–708. https://doi.org/10.31004/basicedu.v3i2.58

Gagatsis, A., & Kyriakides, L. (2000). Teachers' attitudes towards their pupils' mathematical errors. Educational Research and Evaluation, 6(1), 24–58. https://doi.org/10.1076/1380-3611(200003)6:1;1-I;FT024

Gan, K. S. (1982). Development of mathematical concepts and skills in primary school children. Teaching and Learning, 3(1), 14–20. https://doi.org/https://repository.nie.edu.sg/bitstream/10497/2764/1/TL-3-1-14.pdf

Gärdenfors, P. (2019). From sensations to concepts: a proposal for two learning processes. Review of Philosophy and Psychology, 10(3), 441–464. https://doi.org/10.1007/s13164-017-0379-7

Giraldo, V., Tall, D., & Carvalho, L. M. (2003). Using theoretical-computational conflicts to enrich the concept image of derivative. Research in Mathematics Education, 5(1), 63–78. https://doi.org/10.1080/14794800008520115

Giraldo, V., Tall, D., & Mariano Carvalho, L. (2003). Using theoretical-computational conflicts to enrich the concept image of derivative. Research in Mathematics Education, 5(1), 63–78. https://doi.org/10.1080/14794800008520115

Habineza, F. (2013). A case study of analyzing student teachers' concept images of the definite integral. Rwandan Journal of Education, 1(2), 38–54. https://www.ajol.info/index.php/rje/article/view/111569/101346

Harel, G. (2008). What is mathematics? A pedagogical answer to a philosophical question. In Proof and other dilemmas: Mathematics and philosophy (pp. 265–290). https://pdfs.semanticscholar.org/5f7d/bccaa85a33ef6a7d4a37e23e23a9ec436738.pdf

Hollebrands, K. F. (2003). High school students' understandings of geometric transformations in the context of a technological environment. The Journal of Mathematical Behavior, 22(1), 55–72. https://doi.org/10.1016/S0732-3123(03)00004-X

Jiang, Z. (2008). Explorations and reasoning in the dynamic geometry environment. Tersedia: Http://Atcm. Mathandtech. Org/.(12 Oktober 2010).

Kementerian Pendidikan dan Kebudayaan. (2016). Kementerian Pendidikan dan Kebudayaan. Pedoman Umum Ejaan Bahasa Indonesia. 4th Ed. Jakarta: Badan Pengembangan Dan Pembinaan Bahasa.

Kusmanto, H. (2013). Upaya Meningkatkan Keterampilan Berfikir Kritis Dan Keaktifan Siswa Pada Pembelajaran Matematika Melalui Pendekatan Konstruktivisme (Penelitian Tindakan Kelas Pada Siswa Kelas VII SMPN 2 Ciwaru Kab. Kuningan). Eduma: Mathematics Education Learning and Teaching, 2(2).

Lasmanah, A. (2016). Peningkatan Hasil Belajar Matematika Siswa Melalui Model Kooperatif Teknik Think Pair Share (TPS)(Penelitian Tindakan Kelas Terhadap Siswa Kelas VII-A SMPN Sukasari Sumedang”. Jurnal Analisa, 2(3), 18–25. https://doi.org/10.15575/ja.v2i3.1221

Maher, C. A., Sigley, R., Sullivan, P., & Wilkinson, L. C. (2018). An international perspective on knowledge in teaching mathematics. The Journal of Mathematical Behavior, 51, 71–79. https://doi.org/10.1016/j.jmathb.2018.05.002

Malatjie, F., & Machaba, F. (2019). Exploring mathematics learners' conceptual understanding of coordinates and transformation geometry through concept mapping. EURASIA Journal of Mathematics, Science and Technology Education, 15(12), em1818. https://doi.org/10.29333/ejmste/110784

Martin, G. E. (2012). Transformation geometry: An introduction to symmetry. Springer Science & Business Media.

Maulida, L. (2018). Kajian Concept Image Pada Materi Sistem Pertidaksamaan Linear Dua Variabel. Universitas Pendidikan Indonesia.

Miftah, R., Kurniawati, L., & Solicha, T. P. (2020a). Mengatasi Learning Obstacle Konsep Transformasi Geometri Dengan Didactical Design Research. ALGORITMA: Journal of Mathematics Education, 1(2). https://doi.org/10.15408/ajme.v1i2.14076

Miftah, R., Kurniawati, L., & Solicha, T. P. (2020b). Mengatasi Learning Obstacle Konsep Transformasi Geometri Dengan Didactical Design Research. ALGORITMA: Journal of Mathematics Education, 1(2), 156–166. https://doi.org/10.15408/ajme.v1i2.14076

Miles, M. B., & Huberman, A. M. (1992). Analisis Data Kualitatif. Terjemahan Tjetjep Rohendi Rohidi. Jakarta: Penerbit Universitas Indonesia.

Muah, T. (2016). Penggunaan Model Pembelajaran Problem Based Instruction (PBI) untuk Meningkatkan Keaktifan dan Hasil Belajar Matematika Siswa Kelas 9B Semester Gasal Tahun Pelajaran 2014/2015 SMP Negeri 2 Tuntang-Semarang. Scholaria: Jurnal Pendidikan Dan Kebudayaan, 6(1), 41–53. https://doi.org/10.24246/j.scholaria.2016.v6.i1.p41-53

NCTM. (2000). Principles and standarts for school mathematics NCTM.

Nurul’Azizah, A. (2019). Upaya Peningkatan Hasil Belajar Matematika Melalui Model Project Based Learning Siswa Kelas V SD. Jartika, 2(1), 194–204. http://journal.rekarta.co.id/index.php/jartika/article/download/280/273/730

Ouvrier-Buffet, C. (2004). Construction of Mathematical Definitions: An Epistemological and Didactical Study. International Group for the Psychology of Mathematics Education.

Portnoy, N., Grundmeier, T. A., & Graham, K. J. (2006). Students' understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs. The Journal of Mathematical Behavior, 25(3), 196–207. https://doi.org/10.1016/j.jmathb.2006.09.002

Priatina, Y. (2018). Upaya Peningkatan Hasil Belajar Matematika Menggunakan Model Pembelajaran Kooperatif Tipe STAD pada Materi Bangun Ruang Sisi Datar. JKPM (Jurnal Kajian Pendidikan Matematika), 4(1), 67–78. https://doi.org/10.30998/jkpm.v4i1.3062

Rosita, C. D., Maharani, A., Tonah, T., & Munfi, M. (2020a). Learning Obstacle Siswa SMP Pada Materi Lingkaran. AKSIOMA: Jurnal Program Studi Pendidikan Matematika, 9(2), 467–479. https://doi.org/10.24127/ajpm.v9i2.2735

Rösken, B., & Rolka, K. (2007). Integrating intuition: The role of concept image and concept definition for students' learning of integral calculus. In The Montana Mathematics Enthusiast (Vol. 3, pp. 181–204). Citeseer. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.537.4751&rep=rep1&type=pdf

Sadi, Ö., & Daugyar, M. (2015). High School Students' Epistemological Beliefs, Conceptions of Learning, and Self-efficacy for Learning Biology: A Study of Their Structural Models. Eurasia Journal of Mathematics, Science & Technology Education, 11(5). https://doi.org/10.12973/eurasia.2015.1375a

Septyawan, S. R. (2018). Learning obstacles pada konsep fungsi: sebuah studi fenomenologi hermeneutik.

Sullivan, P. (2011). Teaching mathematics: Using research-informed strategies.

Suryadi, D. (2010a). Menciptakan proses belajar aktif: Kajian dari sudut pandang teori belajar dan teori didaktik. Bandung: Tidak Diterbitkan.

Suryadi, D. (2010b). Menciptakan proses belajar aktif: Kajian dari sudut pandang teori belajar dan teori didaktik. Bandung: Tidak Diterbitkan.

Suryadi, D. (2010c). Menciptakan proses belajar aktif: Kajian dari sudut pandang teori belajar dan teori didaktik. Bandung: Tidak Diterbitkan.

Suryadi, D. (2014). Penelitian Desain Didaktis (DDR) dan Kemandirian Berpikir. Seminar Nasional Pendidikan Matematika UAD.

Suryadi, D. (2015). Refleksi Kritis Tradisi Pendidikan Matematika dan Sebuah Gagasan Alternatif. In Pendidikan Disiplin Ilmu Abad 21: Sebuah Kajian Prospektif. (pp. 122–147).

Suryadi, D. (2019a). Landasan Filosofis Penelitian Desain Didaktis (DDR). In Landasan Filosofis Penelitian Desain Didaktis (DDR) (pp. 5–17). Gapura Press.

Suryadi, D. (2019b). Landasan Perancangan Penelitian Desain Didaktis (DDR). In Landasan Filosofis Penelitian Desain Didaktis (DDR) (pp. 43–58). Gapura Press.

Syahmani, S., Rusmansyah, R., Winarti, A., & Almubarak, A. (2020). Penulisan artikel ilmiah berbasis Penelitian Tindakan Kelas (PTK) untuk meningkatkan kualitas pengajaran dan pembelajaran di SMA Banjarmasin Kalimantan Selatan. Jurnal Terapan Abdimas, 5(2), 163–172. https://doi.org/10.25273/jta.v5i2.5615

Tall, D., & Vinner, S. (1981a). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619

Tall, D., & Vinner, S. (1981b). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619

Tekin-Sitrava, R. (2017). Middle Grade Students' Concept Images of Algebraic Concepts. Journal of Education and Learning, 6(3), 299–304. https://doi.org/https://files.eric.ed.gov/fulltext/EJ1141677.pdf

Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In Symbolizing, modeling and tool use in mathematics education (pp. 197–220). Springer. https://doi.org/10.1007/978-94-017-3194-2_12

Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2015a). Early-years teachers' concept images and concept definitions: triangles, circles, and cylinders. ZDM, 47(3), 497–509. https://doi.org/10.1007/s11858-014-0641-8

Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2015b). Early-years teachers' concept images and concept definitions: triangles, circles, and cylinders. ZDM, 47(3), 497–509. https://doi.org/10.1007/s11858-014-0641-8

Uriarte, F. A. (2008a). Introduction to knowledge management: A brief introduction to the basic elements of knowledge management for non-practitioners interested in understanding the subject. Asean Foundation.

Uriarte, F. A. (2008b). Introduction to knowledge management: A brief introduction to the basic elements of knowledge management for non-practitioners interested in understanding the subject. Asean Foundation.

Uygun, T. (2020). An inquiry-based design research for teaching geometric transformations by developing mathematical practices in dynamic geometry environment. Mathematics Education Research Journal, 1–27. https://doi.org/10.1007/s13394-020-00314-1

Uygun, T., & Akyüz, D. (2019). Developing Subject Matter Knowledge through Argumentation. International Journal of Research in Education and Science, 5(2), 532–547.

Viholainen, A. (2008). Incoherence of a concept image and erroneous conclusions in the case of differentiability. The Mathematics Enthusiast, 5(2), 231–248. https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1104&context=tme

Vinner, S. (1983a). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305. https://doi.org/10.1080/0020739830140305

Vinner, S. (1983b). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305. https://doi.org/10.1080/0020739830140305

Vinner, S. (2002). The role of definitions in the teaching and learning of mathematics. In Advanced mathematical thinking (pp. 65–81). Springer. https://doi.org/10.1007/0-306-47203-1_5

Yanik, H. B. (2011). Prospective middle school mathematics teachers' preconceptions of geometric translations. Educational Studies in Mathematics, 78(2), 231–260. https://doi.org/10.1007/s10649-011-9324-3

Yanik, H. B., & Flores, A. (2009). Understanding rigid geometric transformations: Jeff's learning path for translation. The Journal of Mathematical Behavior, 28(1), 41–57. https://doi.org/10.1016/j.jmathb.2009.04.003

Yusuf, Y., Titat, N., & Yuliawati, T. (2017). Analisis hambatan belajar (learning obstacle) siswa SMP pada materi statistika. AKSIOMA: Jurnal Matematika Dan Pendidikan Matematika, 8(1), 76–86. https://doi.org/10.26877/aks.v8i1.1509

Zhang, X., Clements, M. A., & Ellerton, N. F. (2015). Enriching student concept images: Teaching and learning fractions through a multiple-embodiment approach. Mathematics Education Research Journal, 27(2), 201–231. https://doi.org/10.1007/s13394-014-0137-4

DOI: https://doi.org/10.31764/jtam.v6i3.8441