Examining of Secondary School Students’ Transition to Algebra in the context of Making Properties in Natural Number System Visible via Generalization

Abstract

The purpose of this study is to examine secondary school students’ transition from arithmetic to algebra in the context of making properties in natural number system visible via generalization. The cross-sectional study, which is one of the developmental research methods, was used in this study. Open-ended written tests were applied to 285 secondary school students from different grades (5th-8th Grades), and 24 students were interviewed clinically. Three questions and additional other questions were prepared to collect the data for this study in which the transition from arithmetic to algebra was investigated; and the data were assessed according to the characterization table prepared. In addition, the changes and developments in the transition process of the students of different grades from arithmetic to algebra data have been investigated with the clinical interview. As a conclusion, it has been observed that as the educational level of the students increase, the transition from arithmetic to algebra changed and developed in a positive manner in the context of generalization of the properties of the natural number system; however, this change and development has been realized at a little rate. There is not a clear differentiation between the 5th and 6th and between the 6th and 7th graders; and the most distinctive change and development among the educational levels has been observed between the 7th and 8th Graders.

Keywords

• Transition to algebra,Commutative,associative and distributive properties,Natural number system,Generalization,Secondary school students

Doğal Sayı Sistemindeki Özellikleri Genelleme Yoluyla Görünür Kılma Bağlamında Ortaokul Öğrencilerinin Cebire Geçişlerinin İncelenmesi

Abstract

Bu çalışmanın amacı, doğal sayı sistemindeki bazı özellikleri genelleme yoluyla görünür kılma bağlamında farklı öğrenim seviyesindeki ortaokul öğrencilerinin aritmetikten cebire geçişlerini incelemektir. Gelişimci araştırmaların bir türü olan enlemesine çalışmanın kullanıldığı bu çalışmada, farklı öğrenim seviyelerindeki 285 ortaokul (5-8.sınıf) öğrencisine açık-uçlu yazılı sınavlar uygulanmış, 24 öğrenciyle ise klinik mülakatlar yürütülmüştür. Veri toplamak amacıyla aritmetikten cebire geçişin inceleneceği bu iki konuyu içeren 3 soru ile ek sorular hazırlanmış ve elde edilen veriler hazırlanan karakterizasyon tablosuna göre değerlendirilmiştir. Ayrıca elde edilen klinik mülakat verileriyle farklı öğrenim seviyelerindeki ortaokul öğrencilerin aritmetikten cebire geçiş sürecindeki değişim ve gelişimleri incelenmiştir. Sonuç olarak öğrencilerin öğrenim seviyesi arttıkça doğal sayı sistemi ile ilgili özellikleri genelleme bağlamında aritmetikten cebire geçişin olumlu yönde değiştiği ve geliştiği görülmüş, ancak bu değişim ve gelişim çok az olmuştur. Özellikle 5 ile 6 ve 6 ile 7.sınıf öğrencileri arasında çok belirgin bir farklılaşma olmamakla birlikte, öğrenim seviyeleri arasındaki en belirgin değişim ve gelişim 7 ile 8.sınıf öğrencileri arasında gerçekleşmiştir.

Keywords

• Cebire geçiş,Değişme,birleşme ve dağılma özellikleri,Doğal sayı sistemi,Genelleme,Ortaokul öğrencileri

References

1. Akkan, Y. (2009). İlköğretim öğrencilerinin aritmetikten cebire geçiş süreçlerinin incelenmesi. Doktora tezi, Karadeniz Teknik Üniversitesi, Fen Bilimleri Enstitüsü.
2. Akkan, Y. (2016). Cebirsel düşünme. E. Bingölbali, S. Arslan & İ., Ö. Zembat (Ed.), Matematik Eğitim Teorileri, (s.43-64), Ankara, Pegem Akademi.
3. Armstrong, B., E. (1995). Teaching patterns, relationships and multiplication as worthwhile mathematical tasks, Teaching Children Mathematics, 1, 446-450.
4. Beatty, R., & Bruce, C. (2012). From patterns to algebra: Lessons for exploring linear relationships. Toronto, ON: Nelson Education.
5. Bishop, J., W. (2000). Linear geometric number patterns: Middle school students’ strategies, Mathematics Education Research Journal,12, 2,107-126.
6. Carpenter, T., P., Levi, L., Franke, M., L. & Zeringue, J., K. (2005). Algebra in elementary school: Developing relational thinking, International Review on Mathematics Education, 31, 1, 53-59.
7. Carpenter, T., P. & Levi, L. Developing conceptions of algebraic reasoning in the primary grades. Research Report Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science. www.wcer.wisc.edu/ncisla /publications/index.html adresinden Aralık 2008’te ulaşılmıştır.
8. Carpenter, T. P., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.

9. Cooper, T., J., Baturo, A., R. & Williams, A., M. (1999). Unknowns, patterns, relationships, concrete materials and teaching the meaning of the algebraic expressions, 3 x. In E. B. Ogena & E. F. Golla (Eds.), Mathematics for the 21st century, Proceedings of the 8th South East Asian Conference on Mathematics Education, (pp. 127-136). Manila, Philippines: SEACME.
10. Çepni, S. (2007). Arastırma ve proje çalısmalarına giriş, Celepler Matbaacılık, Genişletilmiş 3. Baskı, Trabzon.
11. Demana, F. & Leitzel, J. (1988). Establishing fundamental concepts through numerical problem solving, In A.F. Coxford (Ed.), The ideas of algebra, K-12, (pp. 61-68), Reston, VA: NCTM.
12. Goldin, G., A. (1998). Observing mathematical problem solving through task-based interviews, In A.R. Teppo (Ed.), Qualitative Research Methods in Mathematics Mathematics Education, NCTM.
13. Hunting, R., P. (1997). Clinical interview methods in mathematics education research and practice, Journal of Mathematical Behaviour, 16, 2, 145-165.
14. Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235–272). New York, NY: Lawrence Erlbaum Associates.
15. Karasar, N. (1995). Bilimsel araştırma yöntemi. Ankara: 3A Araştırma Eğitim Danışmanlık.
16. Kieran, C. (1992). The learning and teaching of school algebra. In D.A. Grouws (Ed.). Handbook of Research on Mathematics Teaching and Learning, (pp.390-419). New York: Macmillan.
17. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
18. Lannin, J., K. (2003). Developing algebraic reasoning through generalization, Mathematics Teaching in the Middle School, 8, 7, 342-348.
19. Linchevski, L. & Herscovics, N. (1994). Cognitive obstacles in pre-algebra. In J.P da Ponte & J. F. Matos (Eds.). Proceedings of the 18th conference of the International Group for the Psychology of Mathematics Education, 3, (pp.176-183).
20. Linchevski, L. & Livneh, D. (1999). Sctructure sense: The relationship between algebraic and numerical contexts, Educational Studies in Mathematics, 40, 173-196.
21. Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra, The Journal of Mathematical Behaviour, 14, 113-120.
22. MacGregor, M. (1998). How students interpret equations: Intuition versus taught procedures. In H. Steinberg, M.G. Bartolini Bussi, & A. Sierpinska (Eds.), Language communication in the mathematics classroom, (pp. 262-270). Reston, VA: NCTM.
23. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, &L. Lee (Eds.). Approaches to Algebra, (pp.65-111). London: Kluwer Academic Publishers.
24. Milgram, R., J. (2005). The mathematics preservice teachers need to know. Stanford, CA: Stanford University.
25. Okazaki, M. (2006). Semiotic chaining in an expression constructing activity aimed at the transition from arithmetic to algebra, In Novotna, J., Moraova, H., Kratka, M. & Stehlıkova, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 4, (pp. 257-264). Prague: PME. 4- 257.
26. Orton, A.& Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Eds.), Pattern in the Teaching and Learning of Mathematics, (pp. 104–120). Cassell, London.
27. Pillay, H., Wilss, L.& Boulton-Lewis, G. (1998). Sequential development of algebra knowledge: A cognitive analysis, Mathematics Education Research Journal,10, 87–102.
28. Smith, E. (2003). Stasis and change: Integrating patterns, functions, and algebra throughout the K-12 curriculum. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics, Reston, VA: NCTM.
29. Stacey, K. The transition from arithmetic thinking to algebraic thinking, University of Melbourne, Australia, staff.edfac.unimelb.edu.au/~kayecs/IMECstacey ALGEB RA.doc adresinden Mart 2008’te ulaşılmıştır.
30. Tabach, M. & Friedlander, A. (2003). The role of context in learning beginnig algebra, Proceedings of the Third Conference of theEuropean Society for Research in Mathematics Education, Bellaria, Italia.
31. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning, (pp. 495-514). Macmillan Publishing Company, Newyork.
32. Vance, J. (1998). Number operations from an algebraic perspective, Teaching Children Mathematics, 4, 282-285.
33. Warren, E. (2003). Young children’s understanding of equals: A longitudinal study. In N. Pateman, G. Dougherty, J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education and the 25th Conference of Psychology of Mathematics Education North America, 4, (pp.379-387). College of Education: University of Hawaii.
34. Warren, E. & Cooper, T., J. (2008). Generalising the pattern role of visual growht patterns: Actions that support 8 year olds’ thinking, Educational Studies Mathematics, 67, 2, 171- 185.
35. Zaskıs, R. & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions, Journal of Mathematical Behaviour, 17, 4, 429-439.