Farklı Sınıf Seviyelerindeki Ortaokul Öğrencilerinde Cebirsel Düşünme: Örüntülerde Genelleme Hakkındaki Algıları

Yıl 2016, Cilt: 10 Sayı: 2, 243 - 272, 30.12.2016

Dilek Girit , Didem Akyüz

https://doi.org/10.17522/balikesirnef.277815

Cited By: 2

Öz

Cebir, genel olarak
sembolleri manipüle etmek olarak görülürken, cebirsel düşünmenin genelleme ile
ilgili olduğu kabul edilir. Örüntüler, erken yaşlardaki çocukların cebirsel
düşünmelerini geliştirmek için genelleme ile kullanılabilir. Örüntüleri
genelleme bağlamında, bu çalışmanın amacı cebirsel düşünmenin geliştiği
ortaokul yıllarındaki farklı sınıf seviyelerindeki öğrencilerin akıl yürütme ve
çözüm stratejilerini araştırmaktır. Öncelikle, 154 ortaokul öğrencisine sayı,
şekil ve tablo şeklinde temsil edilen farklı tipte örüntü soruları sorulmuştur.  Sonra, her bir sınıf seviyesinden (6., 7. ve
8.sınıf) iki öğrenci ile, öğrencilerin farklı temsillerle gösterilen
örüntülerdeki ilişkiyi nasıl yorumladıkları ve hangi stratejileri
kullandıklarını incelemek için görüşmeler yapılmıştır. Çalışmanın bulguları, sınıf
seviyeleri arttıkça, öğrencilerin cebirsel sembolleri kullanmaya daha eğilimli
olduğunu göstermektedir. Bununla birlikte, öğrencilerin değişken kavramı ile
ilgili algılarında sıkıntılar olduğu görülmüştür.

Anahtar Kelimeler

cebirsel düşünme, erken cebir, örüntülerde genelleme

Kaynakça

• Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning: An International Journal, 9(3), 249–272.
• Barbosa, A., & Vale, I. (2015). Visualization in pattern generalization: Potential and Challenges. Journal of the European Teacher Education Network, 10, 57-70.
• Becker, J. R., & Rivera, F. (2005). Generalization schemes in algebra of beginning high school students. In H. Chick, & J. Vincent (Eds.), Proceedings of the 29th conference of the international group for psychology of mathematics education (Vol. 4) (pp. 121–128). Melbourne, Australia: University of Melbourne.
• Billings, E. (2008). Exploring generalization through pictorial growth patterns. In C. Greenes, & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics: 70th NCTM Yearbook (pp. 279–293). Reston, VA: National Council of Teachers of Mathematics.
• Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers: Algebra eyes and ears, Teaching children mathematics, 10, 70-77.
• Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester, Jr., (Ed.), Second handbook of research on mathematics teaching and learning (Vol II., pp. 669-705). Charlotte, NC: Information Age Publishing.
• Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches. (2nd ed.). Sage publications: USA.
• Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed method approaches. (3rd ed.). Sage publications: USA.
• Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. (4th ed.). Pearson Education, Inc.: Boston.
• Çayır, M. Y., & Akyüz, G. (2015). 9. Sınıf Öğrencilerinin Örüntü Genelleme Problemlerini Çözme Stratejilerinin Belirlenmesi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 9(2), 205-229.
• Driscoll, M. (2001). The fostering of algebraic thinking toolkit: A guide for staff development (Introduction and analyzing written student work module). Portsmouth, NH: Heinemann.
• El Mouhayar, R. R. & Jurdak, M. E., (2016). Variation of student numerical and figural reasoning approaches by pattern generalization type, strategy use and grade level. International Journal of Mathematical Education in Science and Technology. 47(2), 197-215.
• English, L., & Warren, E. (1998). Introducing the variable through pattern exploration. Mathematics Teacher, 91(2), 166–170.
• Ferrara, F. & Sinclair, N. (2016). An early algebra approach to pattern generalization: Actualising the virtual through words, gestures and toilet paper. Educational Studies in Mathematics, 1-19.
• Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly, & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–546). Mahwah, NJ: Lawrence Erlbaum.
• Healy, L., & Hoyles, C. (1999). Visual and symbolic reasoning in mathematics: Making connections with computers. Mathematical Thinking and Learning, 1, 59–84.
• Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DRN based instruction. In S. Campbell & R. Zaskis (Eds.), Learning and teaching number theory, journal of mathematical behavior (pp. 185–212). New Jersey: Albex.
• Hargreaves, M., Threlfall, J., Frobisher, L. & Shorrocks Taylor, D. (1999). Children's strategies with linear and quadratic sequences. In A. Orton (Eds.), Pattern in the Teaching and Learning of Mathematics. London: Cassell.
• Jurdak, M. E., & El Mouhayar, R. R. (2014). Trends in the development of student level of reasoning in pattern generalization tasks across grade level. Educational Studies in Mathematics, 85(1), 75-92.
• Kaput, J. J. (1999). Teaching and learning a new algebra. In E. L. Fennema, & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp.133–156). Mahwah, NJ: Lawrence Erlbaum.
• Katz, V. J. (1997). Algebra and its teaching: An historical survey. Journal of Mathematical Behavior, 16(1), 25-38.
• Kendal, M. & Stacey, K. (2004). Algebra: A world of difference. In K. Stacey & H. Chick (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 329-346). Dordrecht, The Netherlands: Kluwer.
• Kieran, C. (1989). A perspective eon algebraic thinking, in G. Vergnaud, J. Rogalski and M. Artigue (eds.), Proceedings of the 13th Annual Conference of the International Group for the Psychology of Mathematics Education, July 9-13, Paris, France, pp. 163-171.
• Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, (9), 23–26.
• Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding?, Journal of Mathematical Behavior, 25, 299–317.
• Lee, L. (1996). An initiation into algebraic culture through generalization activities, in N. Bednarz, C . Kieran and Lee, L. (eds.), Approaches to algebra: Perspectives for research and teaching, Kluwer Academic Publishers, Dordrecht, pp. 87-106.
• Lincoln , Y. S. , & Guba , E. G. ( 1985). Naturalistic inquiry. Thousand Oaks, CA : Sage .
• MacGregor, M., & Stacey, K. (1996). Origins of students’ interpretation of algebraic notation. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th International Conference for Psychology of Mathematics Education, vol. 3, pp. 289–296. Valencia.
• MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19.
• Magiera, M. T., van den Kieboom, L. A., & Moyer, J. C. (2013). An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking, Educational Studies in Mathematics, 84, 93–113.
• Merriam, S. B. (2009). Qualitative Research: A Guide to Design and Implementation. Jossey Bass Publishers: San Francisco.
• Moss, J., Beatty, R., McNab, S. L., & Eisenband, J. (2006). The potential of geometric sequences to foster young students’ ability to generalize in mathematics. In Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco.
• Moss, J., Beatty, R., Barkin, S., & Shillolo, G. (2008) “What is your theory? what is your rule?” fourth graders build an understanding of functions through patterns and generalizing problems. In C. Greenes, & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics: 70th NCTM Yearbook (pp. 155–168). Reston, VA: National Council of Teachers of Mathematics.
• National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.
• Piaget, J. (1952). The origins of intelligence in children (Vol. 8, No. 5, pp. 18-1952). New York: International Universities Press.
• Rakes, C. R., Valentine, J. C., McGatha, M. B., & Ronau, R. N. (2010). Methods of instructional improvement in algebra: A systematic review and meta-analysis. Review of Educational Research, 80(3), 372–400.
• RAND Mathematics Study Panel Report (2003). Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education. (No: 083303331X) Santa Monico, CA: RAND.
• Rivera, F. D. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297-328.
• Rivera, F. D., & Becker, J. R. (2005). Teacher to teacher: figural and numerical modes of generalizing in algebra. Mathematics Teaching in the Middle School, 11(4), 198-203.
• Stacey, K., & MacGregor, M. (2001). Curriculum reform and approaches to algebra. In R. Lins (Ed.), Perspectives on school algebra (pp. 141–153). Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Steele, D. F., & Johanning, D. J. (2004). A schematic-theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57(1), 65-90.
• Tall, D. (1991). The psychology of advanced mathematical thinking, in D. Tall (ed.), Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht, pp. 3-21.
• Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford, (Ed.). The ideas of algebra, K-12.1988 Yearbook (pp. 8-19). Reston, VA; National Council of Teachers of Mathematics.
• Walkowiak, T. A. (2014). Elementary and middle school students’ analyses of pictorial growth patterns, Journal of Mathematical Behavior, 33, 56-71.
• Warren, E. (1996). Interaction between instructional approaches, students’ reasoning processes, and their understanding of elementary algebra. Unpublished dissertation, Queensland University of Technology.
• Warren, E., & Cooper, T. (2008a). Generalizing the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67, 171-185.
• Warren, E., & Cooper, T. (2008b). Patterns that support early algebraic thinking in the elementary school. In C. Greenes, & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics: 70th NCTM Yearbook (pp. 113–126). Reston, VA: National Council of Teachers of Mathematics.
• Yeap, B. H., & Kaur, B. (2008). Elementary school students engaging in making generalization: A glimpse from a Singapore Classroom. ZDM, 40, 55–64.