International Journal of Global Education
Yazarlar: Pınar GÜNER, Esen ERSOY, Tuğba TEMİZ
Konular:-
Anahtar Kelimeler:-
Özet: Pattern is a combination of shapes, sounds, actions or symbols in a specific order (Souviney, 1994). The definition of mathematics which is stated as pattern and system science (Goldenberg, Cuoco and Mark, 1998) and the universal language which is used to understand the relationships among these (Olkun and Toluk-Uçar, 2006) shows the importance of pattern in mathematics. Pattern is one of the main concepts that contribute to comprehend mathematical concepts, recognize mathematical relationships and interpret them correctly (Burns, 2000). Therefore, it is important to know the strategies that students used to reach generalization in patterns and how they think in this process in terms of teaching mathematics. The purpose of this study is to describe 7th and 8th grades students` ways of thinking related to pattern and investigate the generalization strategies. This research was carried out total 8 students attending the teaching program at 7th and 8th grades in a primary school in İstanbul in the term of 2012-2013. Of these participants, 4 were at 7th grade and 4 were 8th grade. Open ended 4 problems related to patterns were used as data collection tool. Besides, semi-structured interviews were made with the aim of exposing how students think while solving these problems. The collected data was classified through the generalization strategies in the related literature. As a result of the study it is seen that most of the students use guess and check or explicit strategies whereas few of them apply contextual and only one use addictive strategy. In addition, addictive strategies are usually used in near generalization whereas explicit strategies are used in far generalization. Keywords: Pattern, generalization strategies, 7th and 8th grade students References Akkan Y. and Çakıroğlu Ü. (2012). Doğrusal ve İkinci Dereceden Örüntüleri Genelleştirme Stratejileri: 6-8. Sınıf Öğrencilerinin Karşılaştırılması. Eğitim ve Bilim, 37 (165), 184-194. 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 Investigación 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. Alatorre, J. L. Cortina, M. Saiz, & A. Mendez (Eds.), Proceeding of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 95-101). Merida, Mexico: Universidad Pedagogica Nacional. Bogdan, R. C., & Biklen, S. K. (1998). Qualitative research for education: An ıntroduction to theory and methods (3rd ed.). Boston, MA: Allyn and Bacon. Borgen, K. L., & Manu, S. S. (2002). What do students really understand? Journal of Mathematical Behavior, 21(2), 151-165. 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. English, L., D. and Warren, E. A. (1998). Introducing the variable through pattern exploration. Mathematics Teacher, 912, 166–170. Dörfler, 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 Patterns”. 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. Goldenberg, E.P., Cuoco, A.A. and Mark, J. (1998). A role for geometry in general education. In R. Lehrer & D. Chazan (Eds.) Designing Learning Environments for Developing Understanding of Geometry and Space. Mahwah, NJ: Lawrence Erlbaum Associates, 3-44. Guerrero, L., & Rivera A. (2002). Exploration of patterns and recursive functions. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Heide, R. L. Bryant & K. Nooney (Eds.), Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (24th, Athens, Georgia, October 26-29) (Vol. 1-4, pp. 262-272). Athens, Georgia: PME-NA. Hargreaves, M., Shorrocks-Taylor, D., & Th relfall, J. (1998). Children’s strategies with number patterns. Educational Studies, 24(3), 315-331. 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). Th e 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 Shoice”. 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). MEB, TTKB. (2006). Ortaöğretim Matematik Dersi Öğretim Programı ve Kılavuzu (ss.7,101) . Ankara: MEB Basımevi. 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. Olkun, S. ve Toluk-Uçar, Z. (2006). İlköğretimde matematik öğretimine çağdaş yaklaşımlar. Ankara: Ekinoks Yayınları. 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. Saiz, & A. Mendez (Eds.). Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 2-21). Merida, Mexico: Universidad Pedagogica Nacional. RAND Mathematics Study Panel. (2003). Mathematical Proficiency for all Students: Toward a Strategic Research and Development Program in Mathematics Education. Sta. Monica, CA: RAND. Rawson, B. (1993). Searching for patterns. Education 3-13, 21(3), 26-33. 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. Smith, E. (2003). Stasis and Change: Integrating Patterns, Functions, and Algebra Throughout the K-12 Curriculum. In J. Kilpatrick, W. G. Martin, & D. Schifter (Ed.) A Research Companion to Principles and Standards for School Mathematics. Reston, VA: NCTM. Souviney, R. J. (1994). Learning to teach mathematics (2nd ed.). New York, NY: Merrill. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164. Stake, R. (1995). The art of case study research. Thousand Oaks, CA: Sage 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. Steen, L. A. (1988). The Science of Patterns, Science, 240, 611–616. 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. Tanışlı, D. and Özdas, A. (2009). The strategies of using the generalizing patterns of primary school 5Th Grade students. Educational Sciences: Theory and Practice, 9 (3), 1485-1497. Tanışlı and Köse (2011). Lineer Şekil Örüntülerine İlişkin Genelleme Stratejileri: Görsel ve Sayısal İpuçlarının Etkisi. Eğitim ve Bilim, 36 (160), 184-194. Waters, J. (2004). Mathematical patterning in early childhood settings. In I. Putt, R. Faragher ve M. McLean (Ed.), Mathematics Education For The Third Millennium, Towards 2010, MERGA (Mathematics Education Research Group of Australasia Conference Proceedings).http://www.merga.net.au/publication/conf-_display.php?year=2005 adresinden 12.11.2006 tarihinde alınmıştır. Yıldırım, A. ve Şimşek, H. (2008). Sosyal bilimlerde nitel araştırma yöntemleri (beşinci baskı). Ankara: Seçkin Yayıncılık. Zaskis, R. & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behaviour, 17 (4), 429-439. Zazkis, R. & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.