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Exploration on students’ understanding layers in solving arithmagon problems
Sri Rahayuningsih, Universitas Negeri Malang, Indonesia
Sukoriyanto Sukoriyanto, Universitas Negeri Malang, Indonesia
Abd. Qohar, Universitas Negeri Malang, Indonesia
Abstract
The research aims at exploring the characteristics of ‘don’t need’ boundaries of students in solving arithmagon problems using qualitative descriptive approach. The participants win this research are 23 eight-grader junior high school students. The research used the instruments including test questions and unstructured interviews. The data collection procedure began with giving test questions to 23 participants. All of them could make mathematical diagrams/drawings/model precisely according to the information presented from the problem. The student who used effective and clear sentences in solving problems though could not solve the problem in a structured manner is called as Subject 1 (S1). The student who did not use effective and clear structured sentences is called as Subject 2 (S2). The student who used effective and clear sentences as well as structured in solving problems is called Subject 3 (S3). This research used triangulation method by exploring data obtained from the test and interview results. According to the research result, it is concluded that S1 crossed the first ‘don’t need’ boundaries when solving the addition and multiplication arithmagon problems. S2 crossed the first and second “don’t need” boundaries when solving addition arithmagon problem and only crossed the first ‘don’t need’ boundaries in solving multiplication arithmagon problems. However, S3 crossed the second ‘don’t need’ boundaries in solving arithmagon problem of addition and multiplication. Through this research, it is hoped that further research will be able to find and explore the third ‘don’t need’ boundaries in solving mathematical problems.
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REFERENCES
Afriyani, D., Sa’dijah, C., Subanji, & Muksar, M. (2018). Characteristics of Students’ Mathematical Understanding in Solving Multiple Representation Task based on Solo Taxonomy. International Electronic Journal of Mathematics Education, 13(3), 281–287. https://doi.org/10.12973/iejme/3920
Borgen, K. L. (2006). From mathematics learner to mathematics teacher: Preservice teachers’ Growth of Understanding of Teaching and Learning Mathematics. In Unpublished doctoral dissertation.
Burke, M. J., Kehle, P. E., Kennedy, P. A., & John, D. St. (2006). Extending Number and Operation Activities: Number Triangles. In Navigating through Number and Operations in Grades 9-12 (Illustrate, p. 63). NCTM.
Dehaene, S. (2009). Origins of Mathematical Intuitions The Case of Arithmetic. 259, 232–259. https://doi.org/10.1111/j.1749-6632.2009.04469.x
Droujkova, M. A., Berenson, S. B., Slaten, K., & Tombes, S. (n.d.). A Conceptual Framework for Studying Teacher Preparation : The Pirie-Kieren Model, Collective Understanding, and Metaphor. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 289–296). PME.
Fauziyyah, F. A., & Kriswandani. (2018). Description Profile of Understanding Layer Concept of Conic Section of Mathematics Education Students 2016 of FKIP UKSW. International Conference on Science, Mathematics, and Education, 218(ICoMSE 2017), 16–25.
Franco, J. A., Champion, J., & Lyons, J. W. (2016). Arithmagons and Geometrically Invariant Multiplicative Integer Partitions. 85, 87–95.
Franco, J. A., Champion, J. O. E., & Lyons, J. W. (1991). Arithmagons and Geometrically Invariant Multiplicative Integer Partitions. Mathematics Subject Classification, 1–9.
George, L. G. (2017). Children’s learning of the partitive quotient fraction sub-construct and the elaboration of the don’t need boundary feature of the Pirie-Kieren theory. In Doctoral Dissertation. University of Southampton.
Guajardo, M. G. D. H. (2004). “Solutioning”: A Model of Students’ Problem-Solving Processes. In Disertation. The University of Warwick.
Gülkılık, H., Uğurlu, H. H., & Yürük, N. (2015). Examining Students’ Mathematical Understanding of Geometric Transformations Using the Pirie-Kieren Model. Educational Sciences: Theory & Practice, 15(6), 1531–1548. https://doi.org/10.12738/estp.2015.6.0056
Güner, P., & Uygun, T. (2019). Examining Students ’ Mathematical Understanding of Patterns by Pirie-Kieren Model. Hacettepe University Journal of Education, 1–23. https://doi.org/10.16986/HUJE.2019056035
Karunia, E. P., & Mulyono. (2016). Analisis Kemampuan Pemahaman Konsep Siswa Kelas VII Berdasarkan Gaya Belajar dalam Model Knisley. Seminar Nasional Matematika X Universitas Negeri Semarang, 337–346.
L. Junsay, M. (2016). Reflective Learning and Prospective Teachers’ Conceptual Understanding, Critical Thinking, Problem Solving, and Mathematical Communication Skills. Research in Pedagogy, 6(2), 43–58. https://doi.org/10.17810/2015.34
Laine, A., Ahtee, M., Näveri, L., Pehkonen, E., & Hannula, M. (2018). Teachers ’ influence on the quality of pupils ’ written explanations – Third-graders solving a simplified arithmagon task during a mathematics lesson. LUMAT: International Journal on Math, Science and Technology Education, 6(1), 87–104. https://doi.org/10.31129/LUMAT.6.1.255
Lambertus. (2016). Developing skills understanding of mathematical. International Journal of Education and Research, 4(7), 315–326. https://www.ijern.com/journal/2016/July-2016/25.pdf
Lawan, A. (2011). Growth of students’ understanding of part-whole sub-construct of rational number on the layers of Pirie-Kierien theory. In H. Venkat & A. A. Essien (Eds.), Proceedings of the 17th Annual AMESA National Congress (Vol. 1, pp. 69–80). AMESA.
Liang, C. B. (2003). An exploratory study of Singapore prospective mathematics teachers’ problem-solving abilities. Educational Research Association of Singapore (ERAS) Conference, 438–449.
Lin, C.-Y., Kuo, Y.-C., & Yang, D.-C. (2014). Comparing U.S. and Taiwanese Preservice Teachers’ Solving Triangular Arithmagon Problems. In S. Oesterle, C. Nicol, P. Liljedahl, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 6, p. 347). PME.
Martin, L. C. (2008). Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie-Kieren Theory. Journal of Mathematical Behavior, 27(1), 64–85. https://doi.org/10.1016/j.jmathb.2008.04.001
Martin, L. C., & Towers, J. (2014). Growing mathematical understanding through Collective Image Making, Collective Image Having, and Collective Property Noticing. Educational Studies in Mathematics, 88(1), 3–18. https://doi.org/10.1007/s10649-014-9552-4
Martin, L. C., & Towers, J. (2016). Folding back , thickening and mathematical met-befores. The Journal of Mathematical Behavior, 43, 89–97. https://doi.org/10.1016/j.jmathb.2016.07.002
Mason, J., Burton, L., & Stacey, K. (2010). Something to think about. In Thinking Mathematically (2nd ed., pp. 146–180). Pearson Education Limited.
National Council of Teacher of Mathematics. (2000). Executive Summary: Principles and Standards for School Mathematics. In National Council of Teachers of Mathematics (pp. 1–6). http://www.nctm.org/uploadedFiles/Math_Standards/12752_exec_pssm.pdf
Parameswaran, R. (2010). Expert Mathematicians’ Approach to Understanding Definitions. The Mathematics Educator, 20(1), 43–51.
Permendiknas RI 2006 No. 22. (n.d.). Standar Isi Untuk Satuan Pendidikan Dasar dan Menengah (No. 22).
Pirie, S., & Kieren, T. (1989). A Recursive Theory of Mathematical Understanding. For the Learning of Mathematics, 9(3), 7–11. https://doi.org/10.2307/40248156
Pirie, S., & Kieren, T. (1994a). Growth in Mathematical Understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 165–190.
Pirie, S., & Kieren, T. (1994b). Growth in Mathematical Understanding: How Can We Characterise It and How Can We Represent It? Educational Studies in Mathematics, 26(2), 165–190. https://doi.org/10.1007/BF01273662
Pirie, S., Martin, L., & Kieren, T. (1996). Folding Back to Collect: Knowing You Know What You Need to Know. Proceedings from the 20th Conference of the International Group for the Psychology of Mathematics Education, 4, 147–154.
Pramudiani, P., Zulkardi, Yusuf Hartono, & Amerom, B. van. (2011). A Concrete Situation For Learning Decimals. IndoMS. J.M.E, 2(2), 215–230.
Rahayuningsih, S., Sa’dijah, C., Sukoriyanto, & Abadyo. (2019). Layers of Conceptual Understanding of. 12(3), 29–39.
Sagala, V. (2016). Profil Lapisan Pemahaman Konsep Turunan Fungsi Dan Bentuk Folding Back Mahasiswa Calon Guru Berkemampuan Matematika Tinggi Berdasarkan Gender. Journal of Mathematics Education, Science and Technology, 1(2), 47–62.
Sierpinska, A. (1994). Understanding in mathematics. The Falmer Press.
Slaten, K. M. (2006). Effective Teaching and Uses of Instructional Representations in Secondary Geometry: A Comparison of a Novice and an Experienced Mathematics Teacher. Dissertation. https://repository.lib.ncsu.edu/bitstream/handle/1840.16/5481/etd.pdf?sequence=1
Susiswo. (2014). Folding Back Mahasiswa Dalam Menyelesaikan Masalah Limit (Issue November). Universitas Negeri Malang.
Syafri, F. S., & Isran, D. (2016). Pembelajaran Matematika Dengan Model Teori Pirie Dan Kieren. Edudikara, 1(1), 42-50 (ISSN 2541-0261).
Thom, J. S., & Pirie, S. E. B. (2006). Looking at the complexity of two young children ’ s understanding of number. Journal of Mathematical Behavior, 25, 185–195. https://doi.org/10.1016/j.jmathb.2006.09.004
Utami, A. D. (2018). Six Levels of Indonesian Primary School Students ’ Mental Model in Comprehending the Concept of Integer. 11(4), 29–44. https://doi.org/10.12973/iji.2018.1143a
Utami, A. D., Sa’dijah, C., Subanji, & Irawati, S. (2019). Students’ pre-initial mental model: The case of Indonesian first-year of college students. International Journal of Instruction, 12(1), 1173–1188. https://doi.org/10.29333/iji.2019.12175a
Utami, I. W., & Rosyidi, A. H. (2016). Profil Lapisan Pemahaman Property Noticing Siswa Pada Materi Logaritma Ditinjau Dari Perbedaan Jenis Kelamin. Jurnal Ilmiah Pendidikan Matematika, 1(5), 21-29 ISSN : 2301-9085.
Zulkardi. (2002). Developing A Learning Environment on Realistic Mathematics Education for Indonesian Student Teachers. University of Twente.
DOI: https://doi.org/10.21831/cp.v41i1.33837
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