Cebir Öncesi: 3, 4 ve 5. Sınıf Öğrencilerinin Fonksiyonel İlişkileri Genelleme Düzeyleri

Öz

Bu çalışmanın amacı, cebir öncesi dönemde olan üçüncü, dördüncü ve beşinci sınıf öğrencilerinin fonksiyonel ilişkileri genelleme düzeylerini belirlemektir. Araştırma yöntemi olarak temel nitel araştırma yaklaşımı benimsenmiştir. Araştırmanın katılımcıları cebir öncesi dönemde olan orta sosyo-ekonomik düzeyde yer alan bir okulda öğrenim gören üçüncü sınıflardan 45, dördüncü sınıflardan 36 ve beşinci sınıflardan 35 kişi olmak üzere toplam 116 öğrenciden oluşmaktadır. Veriler araştırma amacı doğrultusunda açık uçlu sorular yardımıyla toplanmıştır. Verilerin analizinde tematik analiz yöntemi kullanılmıştır. Bu bağlamda alanyazında daha önce geliştirilmiş olan fonksiyonel düşünme düzeyleri dikkate alınmış ve bu düzeylerden bazıları bu araştırma kapsamında kullanılmış aynı zamanda bazı düzeylere ilişkin alt düzeyler oluşturulmuştur. Tüm sınıf düzeylerindeki öğrencilerin genel olarak fonksiyonel düşünmenin birçok göstergesine sahip olduğu görülmüştür. Ancak genel kuralı y=mx+n formunda olan ilişkileri genelleyebilme ve temsil etmede bazı öğrencilerin daha çok zorlandığı araştırmanın önemli görülen sonucundan birisi olmuştur. Bu sonuç cebir öncesi dönemde yer alan öğrencilerin fonksiyonel düşünmelerinin geliştirilebilir olduğu göstermektedir. Bu nedenle daha erken yaşlarda fonksiyonel düşünmeyi geliştirici etkinliklerin programlarda ve ders kitaplarında artırılması gerekmektedir.

Anahtar Kelimeler

• Cebir öncesi,fonksiyonel düşünme,genelleme

Early Algebra: The Levels of 3th, 4th and 5th Grade Students’ Generalisations of Functional Relationship

Abstract

The purpose of this study is to determine the students’ levels of generalisation of functional relationships in third, fourth and fifth grade. The design of the study was a basic qualitative research study. Participants were 116 students; 45 from the third grade, 36 from the fourth grade and 35 from the fifth grade from a middle school. The data were collected with the help of open-ended questions. Thematic analysis was used to analyze the data. In this context, the levels of functional thinking previously developed in the literature have been taken into consideration and some of these levels have been used in this research and at the same time some levels of lower levels have been formed. It has been seen that most students at all grade levels have many indications of functional thinking in general but some students have more difficulty in generalizing and representing the general rule y=mx+n. This result shows that the functional thinking of the students in the early algebra period can be improved. Therefore, it is necessary to increase the activities that develop functional thinking in curriculum and textbooks at an earlier age.

Keywords

• Early algebra,functional thinking,functional relationship,generalization

Kaynakça

1. Akkan, Y., Baki, A., & Çakıroğlu, Ü. (2011). Aritmetik ile cebir arasındaki farklılıklar: Cebir öncesinin önemi. İlköğretim Online, 10(3), 812-823.
2. Amit, M., & Neria, D. (2008). “Rising to the challenge”: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. The International Journal on Mathematics Education, 40, 111-129.
3. Blanton, M.L. (2008). Algebra and the elementary classroom. Transforming thinking, transforming practice. Portsmouth, NH: Heinemann.
4. Blanton, M. L., & Kaput, J. J. (2004). Elementary grade students’ capacity for functional thinking. In M. J. Høines, & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 135–142). Bergen, Norway: PME.
5. Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 5–23). Heidelberg, Germany: Springer.
6. Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Gardiner, A. (2015). Children’s use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17(1), 34–63. doi:10.1080/ 10986065.2015.981939
7. Blanton, M., Brizuela, B. M., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in six-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511-558.
8. Carraher, D.W., Martinez, M.V., & Schliemann, A.D. (2008). Early algebra and mathematical generalization. The International Journal on Mathematics Education, 40, 3-22.

9. Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37, 87-115.
10. Confrey, J. Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66-86.
11. Garcia-Cruz, J.A., & Martinon, A. (1997). Actions and invariant schemata in linear generalizing problems. In E. Pehkonen (Ed.) Proceeding of the 21th Conference of the International Group for the Psychology of Mathematics Education, 2, 289-296. Universty of Helsinki.
12. Herbert, K. & R. Brown. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics, 3, 340-344.
13. Hoffkamp, A. (2011). The use of interactive visualizations to foster the understanding of concepts of calculus: design principles and empirical results. The International Journal on Mathematics Education, 43, 359–372.
14. Kabael, U. T. & Tanışlı, D. (2010). Cebirsel düşünme sürecinde örüntüden fonksiyona öğretim. İlköğretim Online, 9(1), 213-228.
15. Kaput, J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. In S. Fennel (Ed.), The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium (pp. 25–26). Washington, DC: National Research Council, National Academy Press.
16. Kaput, J., Carraher, D., & Blanton, M. (2008). Algebra in the early grades. New York, NY: Erlbaum.
17. Lannin, J. K. (2005). Generalization and justification: The chalenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258.
18. 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.
19. Ley, A. F. (2005). A cross-sectional investigation of elementary school student’s ability to work with linear generalizing patterns: The impact of format and age on accuracy and strategy choice. Masters Abstract International, 44(2), 124. Liamputtong, P. (2009). Qualitative data analysis: conceptual and practical considerations. Health Promotion Journal of Australia, 20(2), 133-139.
20. Martinez, M., & Brizuela, B. M. (2006). A third grader’s way of thinking about linear function tables. Journal of Mathematical Behavior, 25, 285–298.
21. Merriam, S. B. & Tisdell, E.J. (2016). Qualitative research: A guide to design and implementation (3rd ed). San Francisco, CA: Jossey-Bass.
22. Miller, J. (2016, July). Young indigenous students en route to generalising growing patterns. Paper presented of the 39th Annual Meeting of the Mathematics Education Research Group of Australasia, Adelaide, South Australia.
23. National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Reston, VA: NCTM Publications. Orton, A. & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (104-120). London and New York: Cassell.
24. Rivera, F. (2007). Visualizing as a mathematical way of knowing: understanding figural generalization. Mathematics Teacher, 101(1), 69-75.
25. Sasman, M. C., Linchevski, L., & Olivier, A. (1999). Th e infl uence of diff erent representations on children’s generalisation thinking processes. In J. Kuiper (Eds.), Proceedings of the Seventh Annual Conference of the Southern African Association for Research in Mathematics and Science Education (pp. 406-415). Harare, Zimbabwe.
26. Smith, E. (2003). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. Kaput, D. Carraher, & M. Blanton (Eds.), The Development of Algebraic Reasoning in the Context of Elementary Mathematics. ss. 95-132.
27. Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematic,. 20, 147-164.
28. Stephens, A., Fonger, N. L., Blanton, M., & Knuth, E. (2016, April). Elementary Students’ Generalization and Representation of Functional Relationships: A Learning Progressions Approach. Poster to be presented at the Annual Meeting of the American Education Research Association, Washington, DC.95-132 Stephens, A. C.,
29. Fonger, N. L., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Gardiner, A. (2017). A learning progression for elementary students’ functional thinking. Mathematical Thinking and Learning, 19(3), 143-166.
30. Tanışlı, D. (2011). Functional thinking ways in relation to linear function tables of elementary school students. Journal of Mathematical Behavior, 30(3), 206-223.
31. Vollrath, H. J. (1986). Search strategies as indicators of functional thinking. Educational Studies in Mathematics. 17, 387-400.
32. Warren, E. & Cooper, T. (2005). Introducing functional thinking in year 2: A case study of early algebra teaching. Contemporary Issues in Early Childhood, 6 (2), 150-162.
33. Warren, E., & Cooper, T. J. (2008). Patterns that support early algebraic thinking in the elementary school. In C. E. Greenes, & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics. 70th yearbook of the national council of teachers of mathematics (pp. 113–126). Reston, VA: NCTM.
34. Warren, E. A., Cooper, T. J., & Lamb, J. T. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning. Journal of Mathematical Behavior, 25, 208–223.
35. Wilkie, K. J. (2016). Students’ use of variables and multiple representations in generalizing functional relationships prior to secondary school. Educational Studies in Mathematics, 93, 333–361.
36. Yeşildere-İmre, S., Akkoç, H. ve Baştürk-Şahin, B. N. (2017). Ortaokul öğrencilerinin farklı temsil biçimlerini kullanarak matematiksel genelleme yapma becerileri. Turkish Journal of Computer and Mathematics Education, 8(1), 103-129.