Strategi Generalisasi Pola pada Siswa Kelas VII
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Abstract
Pola adalah susunan atau struktur objek yang memiliki keteraturan maupun sifat-sifat yang memungkinkan untuk digeneralisasi. Pola merupakan salah satu konsep utama yang berkontribusi untuk memahami konsep-konsep matematika, mengenali hubungan matematika dan menafsirkannya dengan benar. Oleh karena itu, penting untuk mengetahui strategi yang digunakan siswa dalam menggeneralisasi pola dan bagaimana proses berpikir siswa. Tujuan dari penelitian ini adalah untuk menggambarkan strategi yang dipilih enam siswa kelas 7 dalam menggeneralisasi pola. Subjek penelitian diberikan Tugas Generalisasi Pola (TGP). Hasil pengerjaan siswa dianalisis berkaitan dengan strategi yang dipilih siswa dalam menggeneralisasi pola. Selain itu, wawancara semi-terstruktur juga dilakukan dengan tujuan mengungkap bagaimana siswa berpikir dalam menyelesaikan TGP. Data yang terkumpul diklasifikasikan berdasarkan strategi generalisasi. Hasil penelitian menunjukkan bahwa dalam menyelesaikan tugas generalisasi dalam bentuk pola bilangan empat subjek menggunakan strategi menebak dan mengecek, lima subjek menggunakan strategi eksplisit dan tiga subjek menerapkan strategi kontekstual. Sedangkan dalam menggeneralisasi pola visual, delapan subjek menggunakan strategi eksplisit dan empat subjek menggunakan strategi kontekstual. Dalam melanjutkan pola yang dekat, subjek cenderung menggunakan strategi additif (penambahan) dan mereka lebih suka mencari rumus umum dan menggunakan strategi eksplisit untuk mendapatkan langkah yang jauh
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How to Cite
Fadiana, M. (2017). Strategi Generalisasi Pola pada Siswa Kelas VII. PRISMA, Prosiding Seminar Nasional Matematika, 230-240. Retrieved from https://journal.unnes.ac.id/sju/prisma/article/view/21615
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References
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Burns, M. (2000). About teaching mathematics. A-K 8 research (2nd ed.) Sausaluto, California. CA: Math Solutions Publication.
Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM Mathematics Education, 40, 3-22.
Ebersbach, M. and Wilkening, F. (2007). Children’s intuitive mathematics: The development of knowledge about nonlinear growth. Children Development, 78, 296-308.
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Feifei, Y. (2005). “Diagnostic Assessment of Urban Middle School Student Learning of Pre algebra Patternsâ€. Doctoral Dissertation, Ohio State University, USA.
Fox, J. (2005). Child-initiated mathematical patternning in the pre-compulsory years. In H. L. Chick & J. L. Vincent (Eds.),Proceeding of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 313-320). Melbourne: PME.
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Guerrero, L., & Rivera A. (2002). Exploration of patterns and recursive functions. In D. S. Mewborn, P. Sztajn, D. Y. White, H.
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Lan Ma, H. (2007). The potential of patterning activities to generalization. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceeding of Th e 31th Conference of the international Group for the Psychology of Mathematics Education (Vol. 3, pp. 225- 232). Seoul: PME.
Lannin, J. (2003). Developing algebraic reasoning through generalization. Mathematics Teaching in the Middle School, 8(7), 342-348.
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Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 73(7), 231- 258.
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Mulligan, J. ; Mitchelmore, M.; Kemp, C. & Marston, J. (2008). Encouraging mathematical thinking through patterns and structure: An intervention in the first year of schooling. Australian Primary Mathematics Classroom, 13(8), 10-15.
NCTM. (2000). Principles and standards for school mathematics. Reston, VA:NCTM.
Orton, A. (1999). Pattern in the teaching and learning of mathematics. London: Cassel.
Orton, A. & Orton, J. (1999). Pattern and the Approach to Algebra. In A. Orton (Ed.) Pattern in the Teaching and Learning of Mathematics (pp. 104–120). Cassell, London.
Orton, J., Orton A. ve Roper T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (121- 136). London and New York: Cassell.
Papic, M. (2007). Promoting repeating patterns with young children—more than just alternating colours! Australian Primary Mathematics Classroom, 12(3), 8-13.
Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M.
Reys, R. E., Suydam, M. N., Lindquist M. M., & Smith. N. L. (1998). Helping children learn mathematics (5th ed.). Boston, MA: Allyn and Bacon.
Risnick, L.B. Cauzinille-Marmeche, E. and Mathieu, J. (1987). Understanding algebra. In J. A. Sloboda and D. Rogers (ed), Cognitive processes in mathematics. Oxford: Clarendon Press.
Rivera, F. & Becker, J. (2005). Figural and numerical modes of generalizing in Algebra. In Mathematics Teaching in the Middle School, 11(4),198-203.
Steele, D. & Johanning D. I. (2004). A schematic–theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57, 65–90.
Swafford, J. O. & Langrall, C. W. (2000). Grade 6 students’ pre-instructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112.
Tall, D. (1992). The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof. D. Grouws(Ed.) Handbook of Research on Mathematics Teaching and Learning (pp. 495-514). Macmillan Publishing Company, Newyork.
Tanıslı, D. and Ozdas, A. (2009). The strategies of using the generalizing patterns of primary school 5Th Grade students. Educational Sciences: Theory and Practice, 9 (3), 1485-1497.
Zazkis, R. & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.
Amit, M. and Neria, D. (2008). Rising to the challenge: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40, 111-129.
Barbosa, A., Vale, I and Palhares, P. (2012). Pattern Tasks: Thinking Processes Used by 6th Grade Students. Revista Latino americana de Investigacion en Matematica Educativa, 15 (3) : 73- 293
Becker, J. R., & Rivera, F. (2006). Sixth graders’ fi gural and numerical strategies for generalizing patterns in algebra (1). In S.
Burns, M. (2000). About teaching mathematics. A-K 8 research (2nd ed.) Sausaluto, California. CA: Math Solutions Publication.
Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM Mathematics Education, 40, 3-22.
Ebersbach, M. and Wilkening, F. (2007). Children’s intuitive mathematics: The development of knowledge about nonlinear growth. Children Development, 78, 296-308.
Dorfler, W.: 1991, ‘Forms and means of generalization in mathematics’, in A.J. Bishop (ed.), Mathematical Knowledge: Its Growth through Teaching, Kluwer Academic Publishers, Dordrecht, pp. 63–85.
Feifei, Y. (2005). “Diagnostic Assessment of Urban Middle School Student Learning of Pre algebra Patternsâ€. Doctoral Dissertation, Ohio State University, USA.
Fox, J. (2005). Child-initiated mathematical patternning in the pre-compulsory years. In H. L. Chick & J. L. Vincent (Eds.),Proceeding of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 313-320). Melbourne: PME.
Garcia-Cruz, J. A. and Martinon, A. (1997). Actions and Invariant Schemata in Linear Generalizing Problems. In E. Pehkonen (Ed.) Proceedings of the 21th Conference of the International Group for the Psychology of Mathematics Education. (pp. 289-296). University of Helsinki.
Guerrero, L., & Rivera A. (2002). Exploration of patterns and recursive functions. In D. S. Mewborn, P. Sztajn, D. Y. White, H.
Hargreaves, M., Shorrocks-Taylor, D., & Th relfall, J. (1999). Children’s strategies with number patterns. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 67-83). London and New York, NY: Cassell.
Herbert, K., & Brown, R. H. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics, 3, 123-128. Krebs, A. S. (2003). Middle grade students’ algebraic understanding in a reform curriculum. School Science and Mathematics, 103, 233-243.
Krebs, A. S. (2005). Studying students’ rea. Mathematics Teaching in Th e Middle School, 10(6), 284-287.
Lan Ma, H. (2007). The potential of patterning activities to generalization. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceeding of Th e 31th Conference of the international Group for the Psychology of Mathematics Education (Vol. 3, pp. 225- 232). Seoul: PME.
Lannin, J. (2003). Developing algebraic reasoning through generalization. Mathematics Teaching in the Middle School, 8(7), 342-348.
Lannin, J., Barker, D. and Townsend, B. (2006). Algebraic generalization strategies: factors influencing student strategy selection. Mathematics Education Research Journal, 18 (3), 3-28.
Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 73(7), 231- 258.
Ley, A., F. (2005). “A Cross- Sectional Investigation of Elementary School Students’ Ability to Work with Linear Generalizing Patterns: The Impact of Format and Age on Accuracy and Atrategy Shoiceâ€. Master Dissertation,Toronto University,Canada.
Lesley, L., & Freiman, V. (2004). Tracking primary students’ understanding of patterns. In M. J. Hoines, & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 415- 422). Bergen, Norway: PME.
Mason, J. (1996), Expressing Generality and Roots of Algebra. In N. Bednarz, C.
Kieran and L. Lee (eds.), Approaches to Algebra, Perspectives for Research and Teaching (pp. 65–86). Dordrecht: Kluwer Academic Publishers.
Mason, J., Johnston-Wilder, S. and Graham, A. (2005). Developing Thinking in Algebra. London: Sage (Paul Chapman).
Mulligan, J. ; Mitchelmore, M.; Kemp, C. & Marston, J. (2008). Encouraging mathematical thinking through patterns and structure: An intervention in the first year of schooling. Australian Primary Mathematics Classroom, 13(8), 10-15.
NCTM. (2000). Principles and standards for school mathematics. Reston, VA:NCTM.
Orton, A. (1999). Pattern in the teaching and learning of mathematics. London: Cassel.
Orton, A. & Orton, J. (1999). Pattern and the Approach to Algebra. In A. Orton (Ed.) Pattern in the Teaching and Learning of Mathematics (pp. 104–120). Cassell, London.
Orton, J., Orton A. ve Roper T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (121- 136). London and New York: Cassell.
Papic, M. (2007). Promoting repeating patterns with young children—more than just alternating colours! Australian Primary Mathematics Classroom, 12(3), 8-13.
Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M.
Reys, R. E., Suydam, M. N., Lindquist M. M., & Smith. N. L. (1998). Helping children learn mathematics (5th ed.). Boston, MA: Allyn and Bacon.
Risnick, L.B. Cauzinille-Marmeche, E. and Mathieu, J. (1987). Understanding algebra. In J. A. Sloboda and D. Rogers (ed), Cognitive processes in mathematics. Oxford: Clarendon Press.
Rivera, F. & Becker, J. (2005). Figural and numerical modes of generalizing in Algebra. In Mathematics Teaching in the Middle School, 11(4),198-203.
Steele, D. & Johanning D. I. (2004). A schematic–theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57, 65–90.
Swafford, J. O. & Langrall, C. W. (2000). Grade 6 students’ pre-instructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112.
Tall, D. (1992). The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof. D. Grouws(Ed.) Handbook of Research on Mathematics Teaching and Learning (pp. 495-514). Macmillan Publishing Company, Newyork.
Tanıslı, D. and Ozdas, A. (2009). The strategies of using the generalizing patterns of primary school 5Th Grade students. Educational Sciences: Theory and Practice, 9 (3), 1485-1497.
Zazkis, R. & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.