7. Sınıf Öğrencilerinin Cebir Öğretimi Öncesi Matematiksel Çözüm Stratejileri: Eşitlik ve Denklem Konusu

Öz

Öğrencilerin aritmetik bilgilerini, cebirsel düşünmelerini, denklem çözümlerindeki informel akıl yürütmelerini ve karşılaştıkları zorlukları anlamak, onların cebir öğrenme-öğretme süreçlerinin gelişimini açıklamak için önemlidir. Cebirde eşitlik ve denklem öğrenme-öğretme için önemli bir adım öğrencilerin mevcut durumlarının belirlenmesidir. Bu çalışmanın amacı, eşitlik ve denklemler konusu öğretilmeden önce yedinci sınıf öğrencilerinin informel matematiksel çözüm stratejilerini ortaya çıkarmak ve açıklamaktır. Bu çalışma, 2022-2023 eğitim-öğretim yılında bir ortaokulda öğrenim gören 94 yedinci sınıf öğrencisinin katılımıyla gerçekleştirilen bir durum çalışmasıdır. Veri toplama aracı, pilot uygulaması yapılan ve kapsam geçerliliği sağlanmış, dokuz sorudan oluşan açık uçlu bir testtir. Öğrencilerin testteki çözümlerine ilişkin yazılı ifadelerinden elde edilen veriler kodlanarak analiz edilmiştir. Bulgularda, öğrencilerin denklemler ve cebirsel ifadeler arasındaki farkı henüz ayırt edemedikleri görülmüştür. Öğrenciler informel çözümlerinde çoğunlukla aritmetik ve semantik yöntemlerle “bilinmeyeni” bulmaya çalışmışlardır. Öğrencilerin çözümlerdeki hataları aritmetik temelli eksikliklerinin olduğunu göstermektedir. Sonuç olarak, denklemler konusu öğretilmeden önce sınıflarda informel olarak eşitliğin ilişkisel anlamından bahsedilebilir. Ayrıca, aritmetik dönemde öğrenme aşamasında çeşitli etkinliklerle öğrencilerin ilişkisel becerileri kazanmaları desteklenebilir.

Anahtar Kelimeler

• cebir
• denklem
• ortaokul matematiği
• durum çalışması

7th Grade Students’ Mathematical Solution Strategies before Algebra Instruction: Equality and Equations Topic

Öz

Understanding students’ arithmetic knowledge, algebraic thinking, informal reasoning and difficulties in solving equations is essential to articulate the development of algebra learning-teaching process. In algebra, one of the important steps for learning-teaching equality and equations is to determine the current situation of the students. This study aimed to reveal and explain the informal mathematical solution strategies of seventh-grade students before teaching the subject of equality and equations. This study was a case study with the participants of 94 seventh-grade students in a middle school in the 2022-2023 academic year. The data collection tool was an open-ended test, which was piloted and had content validity, consisting of nine questions. Data obtained from students’ written statements in the test were coded and analyzed. In the findings, we saw that students could not yet distinguish between equations and algebraic expressions. The students attempted to find the “unknown” mostly with arithmetic and semantic methods in their informal solutions. The students’ mistakes in the solutions showed that they had arithmetic-based deficiencies. As a result, before teaching equations, the relational meaning of equality can be mentioned informally in classrooms. Moreover, students can be supported to develop relational skills through particular activities during the arithmetic period.

Anahtar Kelimeler

• algebra
• equation
• middle school mathematics
• case study

Kaynakça

1. Akgün, L. (2006). Cebir ve değişken kavramı üzerine. Journal of Qafqaz University, 17(1), 25–29.
2. Akkan, Y. (2009). İlköğretim öğrencilerinin aritmetikten cebire geçiş süreçlerinin incelenmesi (Yayımlanmamış Doktora Tezi). Karadeniz Teknik Üniversitesi, Fen Bilimleri Enstitüsü, Trabzon.
3. Akkan, Y., Baki, A. & Çakıroğlu, Ü. (2012). 5-8. sınıf öğrencilerinin aritmetikten cebire geçiş süreçlerinin problem çözme bağlamında incelenmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 43, 1–13.
4. Akkan, Y., Öztürk, M., Akkan, P. & Demir, B. K. (2019). Ortaokul matematik öğretmenlerinin aritmetik ve cebir problemleri hakkındaki inanışları. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 21(1), 156–176.
5. Armstrong, B. E. (1995). Implementing the professional standards for teaching mathematics: Teaching patterns, relationships, and multiplication as worthwhile mathematical tasks. Teaching Children Mathematics, 1(7), 446–450.
6. Ayan-Civak, R., Işıksal-Bostan, M. & Yemen-Karpuzcu, S. (2024). From informal to formal understandings: Analysing the development of proportional reasoning and its retention. International Journal of Mathematical Education in Science and Technology, 55(7), 1704–1726. https://doi.org/10.1080/0020739X.2022.2160384
7. Bal, A. P. & Karacaoğlu, A. (2017). Cebirsel sözel problemlerde uygulanan çözüm stratejilerinin ve yapılan hataların analizi: Ortaokul örneklemi. Çukurova Üniversitesi Sosyal Bilimler Enstitüsü Dergisi, 26(3), 313–327.
8. Behr, M., Erlwanger, S. & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92(1), 13–15.

9. Birgin, O. & Demirören, K. (2020). Ortaokul Yedinci ve Sekizinci Sınıf Öğrencilerinin Cebirsel İfadeler Konusundaki
10. Başarı Performanslarının İncelenmesi. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 50, 99–117. https://doi.org/10.9779/pauefd.567616
11. Booth, L. R. (1988). Children’s difficulties in beginning algebra. A. F. Coxford (Edt.), The Ideas of Algebra, K-12 (1988 Yearbook) içinde (s. 20–32). Reston, VA: National Council of Teachers of Mathematics.
12. Büyüköztürk, Ş., Kılıç-Çakmak, E., Akgün, Ö. E., Karadeniz, Ş. & Demirel, F. (2008). Bilimsel Araştırma Yöntemleri (Geliştirilmiş 2. Baskı). Ankara: Pegem Akademi.
13. Cai, J. & Knuth, E. (Eds.). (2011). Early algebraization: A global dialogue from multiple perspectives. Springer Science & Business Media.
14. Cai, J. & Knuth, E. J. (2005). The development of students’ algebraic thinking in earlier grades from curricular, instructional, and learning perspectives. Zentralblatt für didaktik der mathematik, 37(1), 1–4.
15. Carpenter, T. P. & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Research Report: National Center for Improving Student Learning and Achievement in Mathematics and Science, Wisconsin University, Madison.
16. Clement, J. (1982). Students’ preconceptions in introductory mechanics. American Journal of physics, 50(1), 66–71.
17. Clement, J., Lochhead, J. & Monk, G. S. (1981). Translation difficulties in learning mathematics. The American Mathematical Monthly, 88(4), 286–290.
18. Cooper, T., Boulton-Lewis, G., Atweh, W., Pillay, H., Wilss, L. & Mutch, S. (1997). The transition from arithmetic to algebra: Initial understandings of equals, operations and variable. Proceedings of Psychology of Maths Education 21. University of Helsinki, Jyvaskyla, Finland.
19. CCSSO [Council of Chief State School Officers] (2010). Common core state standards for mathematics. Washington, DC: Council of Chief State School Officers.
20. Çakmak Gürel, Z. & Okur, M. (2018). 7. ve 8. sınıf öğrencilerinin eşitlik ve denklem konusundaki kavram yanılgıları. Cumhuriyet Uluslararası Eğitim Dergisi, 6(4), 479–507. https://doi.org/10.30703/cije.342074
21. Dede Y. & Argün, Z. (2003). Cebir, öğrencilere niçin zor gelmektedir? Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24, 180–185.
22. English, L. & Halford, S. (1995). Mathematics Education. New Jersey:Lawrence Erlbaum Associates.
23. Eren, E. & Obay, M. (2023). Ortaokul matematik öğretmenlerinin öğrencilerin sembolleştirme becerisinin matematik öğrenme ve başarılarına etkisine ilişkin görüşleri. Iğdır Üniversitesi Sosyal Bilimler Dergisi, 32, 46–67.
24. Eriksson, H. (2022). Teaching algebraic thinking within early algebra–a literature review. Twelfth Congress of the European Society for Research in Mathematics Education (CERME12), Feb 2022, Bozen-Bolzano, Italy. Hal-03744603
25. French, D. (2002). Teaching and learning algebra. A&C Black.
26. Fujii, T. (2003). Probing students’ understanding of variables through cognitive conflict: ıs the concept of a variable so difficult for students to understand. In PME CONFERENCE (Vol. 1) içinde (s. 47–66).
27. Gray, E. M. & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A "proceptual" view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140.
28. Gülpek, P. (2020). İlköğretim 7. ve 8. sınıf öğrencilerinin cebirsel düşünme düzeylerinin gelişimi (Yayınlanmamış doktora tezi). Bursa Uludağ Üniversitesi, Eğitim Bilimleri Enstitüsü, Bursa.
29. Gürbüz, R. & Akkan, Y. (2008). A comparison of different grade students' transition levels from arithmetic to algebra: A Case for 'Equation' Subject. Eğitim ve Bilim, 33(148), 64–76.
30. Harvey, J. G. (1995). The influence of technology on the teaching and learning of algebra. Journal of Mathematical Behavior, 14(1), 75–109.
31. Herscovics, N. & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78.
32. Kabael, T. & Akın, A. (2016). Yedinci sınıf öğrencilerinin cebirsel sözel problemlerini çözerken kullandıkları stratejiler ve niceliksel muhakeme becerileri. Kastamonu Eğitim Dergisi, 24(2), 875–894.
33. Kaput, J. (2008). Algebra from a symbolization point of view. J Kaput, D. W. Carraher. & M. L. Blanton (Eds.), Algebra in the early grades içinde (s. 19–56). New York: Routledge.
34. Kaya, D. (2017). Yedinci sınıf öğrencilerinin cebirsel düşünme düzeyleri ile becerilerinin incelenmesi. Bartın Üniversitesi Eğitim Fakültesi Dergisi, 6(2), 657-675.
35. Kaya, D., Keşan, C., İzgiol, D. & Erkuş, Y. (2016). Yedinci sınıf öğrencilerinin cebirsel muhakeme becerilerine yönelik başarı düzeyi. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 7(1), 142-163. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317–326.
36. Kieran, C. (1990). Cognitive processes involved in learning school algebra. P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education içinde (s. 96–112). Cambridge University.
37. Kieran, C., Booker, G., Filloy, E., Vergnaud, G. & Wheeler, D. (1990). Cognitive processes involved in learning school algebra. Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education, 97–136.
38. Kieran, C. (1992). The learning and teaching of school algebra. D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics içinde (s. 390–419). Macmillan Publishing Co., Inc.
39. Kieran, C. (2004). The core of algebra: Reflections on its main activities. K. Stacey, H. Chick, M. Kendal (Edt.) The future of the teaching and learning of algebra the 12th ICMI study içinde (s. 21–33). Dordrecht: Springer.
40. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. F. K. Lester (Edt.), Second handbook of research on mathematics teaching and learning içinde (s. 707–762). Charlotte, NC: New Age Publishing, Reston, VA: National Council of Teachers of Mathematics.
41. Kieran, C. & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. P. S. Wilson (Edt.) Research ideas for the classroom: Middle grades mathematics içinde (s. 119–139). New York: Macmillan.
42. Kieran, C., Pang, J., Schifter, D. & Fong Ng, S. (2016). Early algebra. Research into its nature, its learning, its teaching. Switzerland: Springer International Publishing AG. https://doi.org/10.1007/978-3-319-32258-2
43. Lee, H. Y. & Chang, K. Y. (2012). Algebraic reasoning abilities of elementary school students and early algebra instruction (1). School Mathematics, 14(4), 445-468.
44. Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra. The Journal of Mathematical Behavior, 14(1), 113–120.
45. Linchevski, L. & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30(1), 39–65.
46. MEB [Milli Eğitim Bakanlığı] (2018). Matematik dersi (5-8.Sınıflar) öğretim programı, http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=329
47. NCTM [National Council of Teachers of Mathematics] (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
48. Palabıyık, U. & Akkuş-İspir, O. (2011). Örüntü temelli cebir öğretiminin öğrencilerin cebirsel düşünme becerileri matematiğe karşı tutumlarına etkisi. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 30(30), 111–123.
49. Perso, T. (1992). Making the most of errors. Australian Mathematics Teacher, 48(2), 12-14.
50. Radford, L. (2022). Introducing equations in early algebra. ZDM Mathematics Education, 54(6), 1151–1167. https://doi.org/10.1007/s11858-022-01422-x
51. Rosnick, P. (1981). Some misconceptions concerning the concept of variable. The Mathematics Teacher, 74(6), 418–420.
52. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.
53. Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14(1), 15–39.
54. Stacey, K. & MacGregor, M. (1997). Building foundations for algebra. Mathematics Teaching in the Middle School, 2(4), 252–260.
55. Stacey, K. & MacGregor, M. (1999). Learning the algebraic method of solving problems. The Journal of Mathematical Behavior, 18(2), 149–167.
56. Stephens, M. & Wang, X. (2008). Investigating some junctures in relational thinking: a study of year 6 and year 7 students from Australia and China. Journal of Mathematics Education, 1(1), 28–39.
57. Strauss, A. & Corbin, J. (1990). Basics of qualitative research. Sage publications.
58. Sutherland, R. & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. The Journal of Mathematical Behavior, 12(4), 353–383.
59. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. D. A. Grouws (Edt.) Handbook of research on mathematics teaching and learning içinde (s. 495–511). Macmillan, New York.
60. Tondorf, A. & Prediger, S. (2022). Connecting characterizations of equivalence of expressions: design research in Grade 5 by bridging graphical and symbolic representations. Educational Studies in Mathematics, 111(3), 399–422.
61. Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. B. Moses (Edt.), Algebraic Thinking, Grades K–12: Readings from NCTM’s School-Based Journals and Other Publications içinde (s. 7–13). Reston, Va: National Council of Teachers of Mathematics.
62. Van Amerom, B. A. (2002). Reinvention of early algebra: Developmental research on the transition from arithmetic to algebra. (Unpublished doctoral dissertation). University of Utrecht, The Netherlands.
63. Van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra. Educational Studies in Mathematics, 54(1), 63–75.
64. Van de Walle, J. A., Karp, K. S. & Bay-Williams, J. W. (2013). Elementary and middle school mathematics: Teaching developmentally (10th ed.) Boston, MA: Pearson Education.
65. Van Dooren, W., Verschaffel, L. & Onghena, P. (2003). Pre-service teachers' preferred strategies for solving arithmetic and algebra word problems. Journal of mathematics teacher education, 6(1), 27–52. https://doi.org/10.1023/A:1022109006658
66. Vance, J. H. (1998). Number operations from an algebraic perspective. Teaching children mathematics, 4(5), 282–285.
67. van Oers, B. (2001). Educational forms of initiation in mathematical culture. Educational Studies in Mathematics, 46 (1-3), 59–85. https://doi.org/10.1023/A:1014031507535
68. Warren, E. (2005). Patterns supporting the development of early algebraic thinking. P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Edt.), Bildiriler kitabı içinde (s. 759–766). Building connections: research, theory and practice: Proceedings of the 28th Annual Conference of the Mathematics Education Group of Australasia. Melbourne: MERGA.
69. Xie, S. & Cai, J. (2022). Fifth graders’ learning to solve equations: the impact of early arithmetic strategies. ZDM Mathematics Education, 54, 1169–1179. https://doi.org/10.1007/s11858-022-01417-8
70. Yaman, H., Toluk, Z. & Olkun, S. (2003). İlköğretim öğrencileri eşit işaretini nasıl algılamaktadırlar?. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24, 142–151.
71. Yıldırım, A. & Şimşek, H. (2005). Sosyal bilimlerde nitel araştırma yöntemleri (5. Baskı). Ankara: Seçkin Yayıncılık.
72. Zwanch, K. (2022). Examining middle grades students' solutions to word problems that can be modeled by systems of equations using the number sequences lens. Journal of Mathematical Behavior, 66. https://doi.org/10.1016/j.jmathb.2022.100960