Generalization of Patterns Drawing of High-Performance Students Based on Action, Process, Object, and Schema Theory
Andi Mulawakkan Firdaus , Wasilatul Murtafiah , Marheny Lukitasari , Nurcholif Diah Sri Lestari , Tias Ernawati , Sri Adi Widodo
This study is qualitative with descriptive and aims to determine the process of generalizing the pattern image of high performance students based on t.
- Pub. date: January 15, 2023
- Pages: 421-433
- 524 Downloads
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This study is qualitative with descriptive and aims to determine the process of generalizing the pattern image of high performance students based on the action, process, object, and schema (APOS) theory. The participants in this study were high performance eighth-grade Indonesian junior high school. Assignments and examinations to gauge mathematical aptitude and interviews were used to collect data for the study. The stages of qualitative analysis include data reduction, data presentation, and generating conclusions. This study showed that when given a sequence using a pattern drawing, the subjects used a number sequence pattern to calculate the value of the next term. Students in the action stage interiorize and coordinate by collecting prints from each sequence of numbers in the process stage. After that, they do a reversal so that at the object stage, students do encapsulation, then decapsulate by evaluating the patterns observed and validating the number series patterns they find. Students explain the generalization quality of number sequence patterns at the schema stage by connecting activities, processes, and objects from one concept to actions, processes, and things from other ideas. In addition, students carry out thematization at the schematic stage by connecting existing pattern drawing concepts with general sequences. From these results, it is recommended to improve the problem-solving skill in mathematical pattern problems based on problem-solving by high performance students', such as worksheets for students.
Keywords: APOS, generalization, high-performance, pattern drawing.
References
Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014a). APOS theory: A framework for research and curriculum development in mathematics education. Springer. https://doi.org/10.1007/978-1-4614-7966-6
Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014b). From Piaget's theory to apos theory: Reflective abstraction in learning mathematics and the historical development of apos theory. In I. Arnon, E. Dubonsky, A. Oktaç, S. R. Fuentes, M. Trigueros & K. Weller (Eds.), APOS theory (pp. 5–15). Springer. https://doi.org/10.1007/978-1-4614-7966-6_2
Arseven, A. (2015). Mathematical modelling approach in mathematics education. Universal Journal of Educational Research, 3(12), 973–980. https://doi.org/10.13189/ujer.2015.031204
Astuti, D. P., & Anwar, S. (2018). How to develop teaching material of buffer solution based on SETS? In A. kadarohman., D. Sukyadi., Y. S. Kusumah., A. Permanasari., D. Disman & S. Fatimah (Eds.), Proceeding International Conference on Mathematics and Science Education of Universitas Pendidikan Indonesia (pp. 331–336). Universitas Pendidikan Indonesia. https://bit.ly/3X2QAoy
Aunio, P., & Räsänen, P. (2016). Core numerical skills for learning mathematics in children aged five to eight years–a working model for educators. European Early Childhood Education Research Journal, 24(5), 684–704. https://doi.org/10.1080/1350293X.2014.996424
Bakar, R. (2018). The influence of professional teachers on Padang vocational school students' achievement. Kasetsart Journal of Social Sciences, 39(1), 67–72. https://doi.org/10.1016/j.kjss.2017.12.017
Biggs, J. B., & Collis, K. F. (2014). Evaluating the quality of learning: The SOLO taxonomy (structure of the observed learning outcome). Academic Press.
Björklund, C., & Pramling, N. (2014). Pattern discernment and pseudo-conceptual development in early childhood mathematics education. International Journal of Early Years Education, 22(1), 89–104. https://doi.org/10.1080/09669760.2013.809657
Blake, B., & Pope, T. (2015). Developmental psychology: Incorporating Piaget's and Vygotsky's theories in classrooms. Journal of Cross-Disciplinary Perspectives in Education, 1(1), 59-67. https://bit.ly/3v6i943
Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J.-S. (2015). The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87. https://doi.org/10.5951/jresematheduc.46.1.0039
Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE theory to improve students' graphical understanding of derivative. Eurasia Journal of Mathematics, Science and Technology Education, 14(7), 2947–2967. https://doi.org/10.29333/ejmste/91451
Borji, V., & Martínez-Planell, R. (2020). On students' understanding of implicit differentiation based on APOS theory. Educational Studies in Mathematics, 105(2), 163–179. https://doi.org/10.1007/s10649-020-09991-y
Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach. Routledge. https://doi.org/10.4324/9780203520574
Cooley, L., Trigueros, M., & Baker, B. (2007). Schema thematization: A framework and an example. Journal for Research in Mathematics Education, 38(4), 370–392. https://doi.org/10.2307/30034879
Creswell, J. W. (2012). Research design qualitative, quantitative, and mixed second edition. Sage.
Diana, P., Marethi, I., & Pamungkas, A. S. (2020). Kemampuan pemahaman konsep matematis siswa: Ditinjau dari kategori kecemasan matematik [Understanding students' mathematical concepts skill: In terms of the category of mathematical anxiety]. Supremum Journal of Mathematics Education, 4(1), 24–32. https://doi.org/10.35706/sjme.v4i1.2033
Dubinsky, E. (2000). Using a theory of learning in college mathematics course. Teaching and Learning Undergraduate Mathematics, 12, 10–15. https://bit.ly/3YGZNnN
Dubinsky, E. D., & Mcdonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education. George State University. https://doi.org/10.1007/0-306-47231-7_25
Fernández Cueli, M. S., Areces Martínez, D., García Fernández, T., Alves, R. A. T., & González Castro, P. (2020). Attention, inhibitory control and early mathematical skills in preschool students. Psicothema, 32(2), 237-244. https://doi.org/10.7334/psicothema2019.225
Firdaus, A. M., Juniati, D., & Wijayanti, P. (2019). The characteristics of junior high school students in pattern generalization. Journal of Physics: Conference Series, 1157, Article 80. https://doi.org/10.1088/1742-6596/1157/4/042080
Font Moll, V., Trigueros, M., Badillo, E., & Rubio, N. (2016). Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA. Educational Studies in Mathematics, 91(1), 107–122. https://doi.org/10.1007/s10649-015-9639-6
Fox, J. (2005). Child-Initiated mathematical patterning in the pre-compulsory years. In H. L. Chick & J. L. Vincent (Eds.), Proceedings The 29th Conference of the International Group for the Psychology of Mathematics Education (pp. 313-320). Psychology of Mathematics Education (PME).
Fox, J. (2006). Connecting algebraic development to mathematical patterning in early childhood. In J. Novotna., H. Moraova., M. Kratka., & N. Stehlokova (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (pp. 89-96). Charles University, Prague.
Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2012). How to design and evaluate research in education. McGraw-Hill Companies.
Fuentealba, C., Badillo, E., Sánchez-Matamoros, G., & Cárcamo, A. (2019). The understanding of the derivative concept in higher education. Eurasia Journal of Mathematics, Science and Technology Education, 15(2), Article em1662. https://doi.org/10.29333/ejmste/100640
Glaesser, J. (2019). Competence in educational theory and practice: A critical discussion. Oxford Review of Education, 45(1), 70–85. https://doi.org/10.1080/03054985.2018.1493987
Hawes, Z., Moss, J., Caswell, B., Seo, J., & Ansari, D. (2019). Relations between numerical, spatial, and executive function skills and mathematics achievement: A latent-variable approach. Cognitive Psychology, 109, 68–90. https://doi.org/10.1016/j.cogpsych.2018.12.002
Herbert, K., & Brown, R. H. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics, 3(6), 340–345. https://doi.org/10.5951/TCM.3.6.0340
Hodnik Čadež, T., & Manfreda Kolar, V. (2015). Comparison of types of generalizations and problem-solving schemas used to solve a mathematical problem. Educational Studies in Mathematics, 89(2), 283–306. https://doi.org/10.1007/s10649-015-9598-y
Iuculano, T., Rosenberg-Lee, M., Supekar, K., Lynch, C. J., Khouzam, A., Phillips, J., Uddin, L. Q., & Menon, V. (2014). Brain organization underlying superior mathematical abilities in children with autism. Biological Psychiatry, 75(3), 223–230. https://doi.org/10.1016/j.biopsych.2013.06.018
Johnston-Wilder, S., & Mason, J. (2005). Developing thinking in geometry. Sage.
Juniati, D., & Budayasa, K. (2017). Construction of learning strategies to combine culture elements and technology in teaching group theory. World Transactions on Engineering and Technology Education, 15(3), 206–211.
Kusumadewi, R. F., Kusmaryono, I., Lail, I. J., & Saputro, B. A. (2019). Analisis struktur kognitif siswa kelas Iv sekolah dasar dalam menyelesaikan masalah pembagian bilangan bulat [Analysis of cognitive structure of grade IV elementary school students in solving integer division problems]. Journal of Medives: Journal of Mathematics Education IKIP Veteran Semarang, 3(2), 251-259. https://doi.org/10.31331/medivesveteran.v3i2.875
Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258. https://doi.org/10.1207/s15327833mtl0703_3
Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding? The Journal of Mathematical Behavior, 25(4), 299–317. https://doi.org/10.1016/j.jmathb.2006.11.004
Lave, J. (1990). The culture of acquisition and the practice of understanding. In J. Stigler, R. Schweder & G. Herdt (Eds.), Cultural psychology: Essays on comparative human development (pp. 309-327). Cambridge University Press. https://doi.org/10.1017/CBO9781139173728.010
Lee, L., & Freiman, V. (2004). Tracking primary students' understanding of patterns. In A. B. Fuglestad., & M. J. Hoines (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 415-171). Bergen University College.
Lee, L., & Freiman, V. (2006). Developing algebraic thinking through pattern exploration. Mathematics Teaching in the Middle School, 11(9), 428–433. https://doi.org/10.5951/MTMS.11.9.0428
Madani, R. A. (2019). Analysis of educational quality, a goal of education for all policy. Higher Education Studies, 9(1), 100–109. https://doi.org/10.5539/hes.v9n1p100
Maharaj, A. (2008). Some insights from research literature for teaching and learning mathematics. South African Journal of Education, 28(3), 401–414. https://doi.org/10.15700/saje.v28n3a182
Maharaj, A. (2013). An APOS analysis of natural science students' understanding of derivatives. South African Journal of Education, 33(1), 1–19. https://doi.org/10.15700/saje.v33n1a458
Marion, M., Zulkardi, Z., & Somakim, S. (2015). Desain pembelajaran pola bilangan menggunakan model jaring laba-laba di SMP [Number pattern learning design using the spider web model in junior high school]. Jurnal Kependidikan: Penelitian Inovasi Pembelajaran, 45(1), 44-61.
Mazana, Y. M., Montero, C. S., & Casmir, R. O. (2019). Investigating students' attitude towards learning mathematics. International Electronic Journal of Mathematics Education, 14(1), 207-231. https://doi.org/10.29333/iejme/3997
McClelland, J. L., & Rumelhart, D. E. (1985). Distributed memory and the representation of general and specific information. Jurnal of Experimental Psychology: General, 114(2), 159-197. https://doi.org/10.1037//0096-3445.114.2.159
Miles, M. B., Huberman, A. M., & Saldaña, J. (2018). Qualitative data analysis: A methods sourcebook. Sage.
Montague-Smith, A., Cotton, T., Hansen, A., & Price, A. J. (2017). Mathematics in early years education. Routledge. https://doi.org/10.4324/9781315189109
Morrison, B. B., Margulieux, L. E., & Guzdial, M. (2015). Subgoals, context, and worked examples in learning computing problem solving. In B. Dorn., J. Sheard., & Q. Cutts (Eds.), Proceedings of the Eleventh Annual International Conference on International Computing Education Research (pp. 21–29). Association for Computing Machinery. https://doi.org/10.1145/2787622.2787733
Muhtarom, M., Juniati, D., & Siswono, T. Y. E. (2019). Examining Prospective Teachers' Belief and Pedagogical Content Knowledge Towards Teaching Practice in Mathematics Class: A Case Study. Journal on Mathematics Education, 10(2), 185–202. https://doi.org/10.22342/jme.10.2.7326.185-202
Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21, 33–49. https://doi.org/10.1007/BF03217544
Mutaqin, E. J. (2017). Analisis learning trajectory matematis dalam konsep perkalian bilangan cacah di kelas rendah sekolah dasar [Mathematical learning trajectory analysis in the concept of multiplication of whole numbers in low grade elementary school]. DWIJA CENDEKIA: Jurnal Riset Pedagogik, 1(1), 19–33. https://doi.org/10.20961/jdc.v1i1.13054
Ndlovu, Z., & Brijlall, D. (2015). Pre-service teachers' mental constructions of concepts in matrix algebra. African Journal of Research in Mathematics, Science and Technology Education, 19(2), 156–171. https://doi.org/10.1080/10288457.2015.1028717
Nern, A., Pfeiffer, B. D., & Rubin, G. M. (2015). Optimized tools for multicolor stochastic labeling reveal diverse stereotyped cell arrangements in the fly visual system. Proceedings of the National Academy of Sciences, 112(22), E2967–E2976. https://doi.org/10.1073/pnas.1506763112
Radford, L., & Peirce, C. S. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre., J. L. Cortina., M. Sáiz., & A. Méndez (Eds.), Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 2–21). Universidad Pedagógica Nacional.
Radford, L. (2008a). Diagrammatic thinking: Notes on Peirce's semiotics and epistemology. PNA, 3(1), 1–18. https://doi.org/10.30827/pna.v3i1.6192
Radford, L. (2008b). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM - International Journal on Mathematics Education, 40(1), 83–96. https://doi.org/10.1007/s11858-007-0061-0
Rafiola, R., Setyosari, P., Radjah, C., & Ramli, M. (2020). The effect of learning motivation, self-efficacy, and blended learning on students' achievement in the industrial revolution 4.0. International Journal of Emerging Technologies in Learning, 15(8),. 71–82. https://doi.org/10.3991/ijet.v15i08.12525
Reys, R., Lindquist, M. M., Lambdin, D. V., & Smith, N. L. (2020). Children learning mathematics. The Arithmetic Teacher, 10(4), 179–182. https://doi.org/10.5951/at.10.4.0179
Richland, L. E., & Simms, N. (2015). Analogy, higher order thinking, and education. Wiley Interdisciplinary Reviews: Cognitive Science, 6(2), 177-192. https://doi.org/10.1002/wcs.1336
Rosa, M., D’Ambrosio, U., Orey, D. C., Shirley, L., Alangui, W. V., Palhares, P., & Gavarrete, M. E. (2016). Current and future perspectives of ethnomathematics as a program. Springer. https://doi.org/10.1007/978-3-319-30120-4
Rosa, M., & Orey, D. C. (2016). State of the art in Ethnomathematics. In M. Rosa, U D’Ambrosio, D. C. Orey, L. Shirley, W. V. Alangui, P. Palhares, & M. E. Gavarrete (Eds.). Current and future perspectives of ethnomathematics as a program (pp. 11–37). Springer. https://doi.org/10.1007/978-3-319-30120-4_3
Samson, D. A. (2011). The heuristic significance of enacted visualisation [Unpublished doctoral dissertation]. Rhodes University
Singer, F. M., Sheffield, L. J., Freiman, V., & Brandl, M. (2016). Research on and activities for mathematically gifted students. Springer. https://doi.org/10.1007/978-3-319-39450-3
Situmorang, M., Sitorus, M., Hutabarat, W., & Situmorang, Z. (2015). The development of innovative chemistry learning material for bilingual senior high school students in Indonesia. International Education Studies, 8(10), 72–85. https://doi.org/10.5539/ies.v8n10p72
Smith, E. (2017). 5 representational thinking as a framework for introducing functions in the elementary curriculum. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 133–160). Routledge. https://doi.org/10.4324/9781315097435-6
Suciati, S., Munadi, S., Sugiman, S., & Febriyanti, W. D. R. (2020). Design and validation of mathematical literacy instruments for assessment for learning in Indonesia. European Journal of Educational Research, 9(2), 865–875. https://doi.org/10.12973/eu-jer.9.2.865
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), Article em1744. https://doi.org/10.29333/ejmste/108457
Syarifuddin, S. (2018). Pengaruh pembelajaran kooperatif tipe jigsaw dan tipe group investigation (GI) terhadap ketercapaian kompetensi dan kemampuan komunikasi matematika siswa di SMA [The effect of jigsaw cooperative learning and group investigation (GI) types on the achievement of students' mathematics competence and communication skills in high school]. Jurnal Ilmiah Mandala Education, 4(1), 163–172. https://doi.org/10.36312/jime.v4i1.338
Tikekar, V. G. (2009). Deceptive patterns in mathematics. Sutra: International Journal of Mathematical Science Education, 2(1), 13–21. https://bit.ly/3VAWeO8
Wang, L., Uhrig, L., Jarraya, B., & Dehaene, S. (2015). Representation of numerical and sequential patterns in macaque and human brains. Current Biology, 25(15), 1966–1974. https://doi.org/10.1016/j.cub.2015.06.035
Warren, E., & Cooper, T. (2007). Repeating patterns and multiplicative thinking: Analysis of classroom interactions with 9-year-old students that support the transition from the known to the novel. The Journal of Classroom Interaction, 2(1), 7–17. http://www.jstor.org/stable/23869442
Xu, C., Lafay, A., Douglas, H., Di Lonardo Burr, S., LeFevre, J.-A., Osana, H. P., Skwarchuk, S.-L., Wylie, J., Simms, V., & Maloney, E. A. (2022). The role of mathematical language skills in arithmetic fluency and word-problem solving for first-and second-language learners. Journal of Educational Psychology, 114(3), 513-539. https://doi.org/10.1037/edu0000673
Zazkis, R., & Liljedahl, P. (2002a). Arithmetic sequence as a bridge between conceptual fields. Canadian Journal of Science, Mathematics and Technology Education, 2(1), 91–118. https://doi.org/10.1080/14926150209556501
Zazkis, R., & Liljedahl, P. (2002b). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379–402. https://doi.org/10.1023/A:1020291317178