Square Root Expressions and Irrational Numbers in Middle School Mathematics Textbooks Throughout the History of the Republic

Abstract

Irrational numbers, a fundamental set that extends rational numbers to real numbers, have been a subject of limited study in the context of textbook analyses. This study, therefore, examines how square root expressions and irrational numbers have been addressed in middle school mathematics textbooks throughout the history of the Republic of Türkiye. The research was conducted using a comprehensive approach, with qualitative research methods and content analysis was applied to 37 mathematics textbooks (seven 5th grade, twelve 6th grade, nine 7th grade, and nine 8th grade) from 1932 to 2022. Data were collected from mathematics textbooks available at the Ferit Ragıp Tuncor Archive and Documentation Library affiliated with the Ministry of National Education. The results of this study indicate that changes in mathematics curricula are reflected in textbooks, and this reflection is directly influenced by the learning theory adopted in the curriculum. The study also provides insights into how irrational numbers are introduced in textbooks based on the sequence of topics, and recommendations are made based on the findings.

Keywords

• Irrational numbers
• Square root expressions
• Mathematics textbooks
• Mathematics curriculum
• Document analysis

Cumhuriyet Tarihi Boyunca Ortaokul Matematik Ders Kitaplarında Kareköklü İfadeler ve İrrasyonel Sayılar

Abstract

İrrasyonel sayılar, rasyonel sayıların gerçel sayılara genişletilmesini sağlayan temel bir kümedir ancak bu kavram ile ilgili ders kitabı analizlerine yönelik çok az araştırma yapılmıştır. Bu çalışmada Cumhuriyet tarihi boyunca ortaokul matematik ders kitaplarında kareköklü ifadelerin ve irrasyonel sayıların öğretimsel bağlamda nasıl ele alındığı incelenmiştir. Nitel araştırma türlerinden belge analizi kullanılarak yürütülen çalışmada 1932’den 2022 yılına kadar toplam 37 adet matematik ders kitabı (yedi 5. sınıf, on iki 6. sınıf, dokuz 7. sınıf ve dokuz 8. sınıf) üzerinde içerik analizi yapılmıştır. Veriler Millî Eğitim Bakanlığına bağlı Ferit Ragıp Tuncor Arşiv ve Dokümantasyon Kütüphanesindeki matematik ders kitapları ile sınırlıdır. Bu çalışmanın sonuçları matematik öğretim programlarındaki değişikliklerin ders kitaplarına da yansıtıldığını ve bu yansımanın öğretim programında benimsenen öğrenme kuramından doğrudan etkilendiğini göstermektedir. Diğer yandan irrasyonel sayıların ders kitaplarında hangi konu sıralamasını takip ederek tanıtıldığına ilişkin sonuçlar elde edilmiş ve sonuçlara bağlı olarak öneriler sunulmuştur.

Keywords

• İrrasyonel sayılar
• Kareköklü ifadeler
• Matematik ders kitapları
• Matematik öğretim programları
• Belge analizi

References

1. Adıgüzel, N. (2013). İlköğretim matematik öğretmen adayları ve 8. sınıf öğrencilerinin irrasyonel sayılarla ilgili bilgileri ve bu konudaki kavram yanılgıları [Yayımlanmamış yüksek lisans tezi]. Necmettin Erbakan Üniversitesi.
2. Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2022). Australian Curriculum. https://v9.australiancurriculum.edu.au/
3. Agarwal, R. P., & Agarwal, H. (2021). Origin of irrational numbers and their approximations. Computation, 9(3), 1–49. https://doi.org/10.3390/computation9030029
4. Arbour, D. (2012). Students' understanding of real, rational, and irrational numbers [Unpublished master's thesis]. Concordia University.
5. Arcavi, A., Bruckheimer, M., & Ben-Zvi, R. (1987). History of mathematics for teachers: The case of irrational numbers. For the Learning of Mathematics, 7(2), 18–23. https://www.jstor.org/stable/40247891
6. Argün, Z., Arıkan, A., Bulut, S., & Halıcıoğlu, S. (2014). Temel matematik kavramların künyesi. Gazi Kitabevi.
7. Argün, Z., Arıkan, A., Bulut, S., & Sriraman, B. (2010). A brief history of mathematics education in Turkey: K-12 mathematics curricula. ZDM Mathematics Education, 42, 429–441. https://doi.org/10.1007/s11858-010-0250-0
8. Bakır, N. Ş. (2011). 10. sınıf öğrencilerinin matematik dersi sayılar alt öğrenme alanındaki başarı düzeyleri ve düşünme süreçlerinin incelenmesi [Yayımlanmamış yüksek lisans tezi]. Gazi Üniversitesi.

9. Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351–382. https://doi.org/10.5951/jresematheduc.41.4.0351
10. Bowen, G. A. (2009). Document analysis as a qualitative research method. Qualitative Research Journal, 9(2), 27–40. https://doi.org/10.3316/QRJ0902027
11. Budak, Ş. (2003). Atatürk’ün eğitim felsefesi ve geliştirdiği eğitim sisteminin değiştirilmesi. Millî Eğitim Dergisi, 160, 16–17.
12. Chang, C. C., & Silalahi, S. M. (2017). A review and content analysis of mathematics textbooks in educational research. Problems of Education in the 21st Century, 75(3), 235. https://www.proquest.com/scholarly-journals/review-content-analysis-mathematics-textbooks/docview/2343792753/se-2?accountid=11248
13. Crisan, C. (2014, April). The case of the square root: Ambiguous treatment and pedagogical implications for prospective mathematics teachers. Proceedings of the 8the British Congress of Mathematics Education Nottingham, UK.
14. Common Core State Standarts Initiative [CCSSI]. (2010). Common core state standards for mathematics. https://learning.ccsso.org/wp-content/uploads/2022/11/ADA-Compliant-Math-Standards.pdf
15. Çakıroğlu, E. (2013). Matematiksel kavramların tanımlanması. İ. Ö. Zembat, M. F. Özmantar, E. Bingölbali, H. Şandır, & A. Delice (Ed.), Tanımları ve tarihsel gelişimleriyle matematiksel kavramlar kitabı içinde (ss. 1–14). Pegem Akademi.
16. Çevikbaş, M., & Argün, Z. (2017). Geleceğin matematik öğretmenlerinin rasyonel ve irrasyonel sayi kavramlari konusundaki bilgileri. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 30(2), 551–581. https://doi.org/10.19171/uefad.368968
17. Erdem Uzun, Ö., & Dost, Ş. (2023). Content analysis of qualitative studies on irrational numbers in Turkey: A meta-synthesis study. Mehmet Akif Ersoy Üniversitesi Eğitim Fakültesi Dergisi, (65), 1-11. https://doi.org/10.21764/maeuefd.1078863
18. Fan, L. (2013). Textbook research as scientific research: towards a common ground on issues and methods of research on mathematics textbooks. ZDM Mathematics Education, 45, 765-777. https://doi.org/10.1007/s11858-013-0530-6
19. Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: development status and directions. ZDM Mathematics Education, 45, 633–646. https://doi.org/10.1007/s11858-013-0539-x
20. Fereday, J., & Muir-Cochrane, E. (2006). Demonstrating rigor using thematic analysis: A hybrid approach of inductive and deductive coding and theme development. International Journal Of Qualitative Methods, 5(1), 80–92. https://doi.org/10.1177/160940690600500107
21. Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational numbers in high-school students and prospective teachers. Educational Studies in Mathematics, 29(1), 29–44. https://doi.org/10.1007/BF01273899
22. Glasnović Gracin, D. (2014). Mathematics textbook as an object of research. Croatian Journal of Education: Hrvatski časopis za odgoj i obrazovanje, 16 (Sp. Ed. 3), 211–237.
23. González-Martín, A. S., Giraldo, V., & Souto, A. M. (2013). The introduction of real numbers in secondary education: an institutional analysis of textbooks. Research in Mathematics Education, 15(3), 230–248. https://doi.org/10.1080/14794802.2013.803778
24. Guven, B., Cekmez, E., & Karatas, I. (2011). Examining preservice elementary mathematics teachers' understandings about irrational numbers. PRIMUS, 21(5), 401–416. https://doi.org/10.1080/10511970903256928
25. Hong, D. S., & Runnalls, C. (2020). Understanding length× width× height with modified tasks. International Journal of Mathematical Education in Science and Technology, 51(4), 614–625. https://doi.org/10.1080/0020739X.2019.1583383
26. Kidron, I. (2018). Students’ conceptions of irrational numbers. International Journal of Research in Undergraduate Mathematics Education, 4(1), 94–118. https://doi.org/10.1007/s40753-018-0071-z
27. Kuchemann, D., & Hoyles, C. (2009). From empirical to structural reasoning in mathematics: Tracking changes over time. In Despina A. Stylianou, Maria L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades (171-190). Routledge.
28. Leylek, R. (2020). Türkiye, Finlandiya ve Kanada’da matematik ders kitaplarındaki bazı ortak konuların göstergebilimsel analizi [Yayımlanmamış doktora tezi]. Hacettepe Üniversitesi.
29. Millî Eğitim Bakanlığı [MEB]. (2018). İlköğretim matematik dersi öğretim programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar). https://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=329
30. Millî Eğitim Bakanlığı [MEB]. (2021, 14 Ekim). Millî Eğitim Bakanlığı Ders Kitapları ve Eğitim Araçları Yönetmeliği. Resmi Gazete (Sayı: 31628). https://www.resmigazete.gov.tr/eskiler/2021/10/20211014-1.htm
31. Millî Eğitim Bakanlığı [MEB]. (2023). Ferit Ragıp Tuncor arşiv ve dokümantasyon kütüphanesi. https://arsivkutuphanesi.meb.gov.tr/Home/Hakkimizda
32. Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In Reconsidering conceptual change: Issues in theory and practice (pp. 232–257). Springer. https://doi.org/10.1007/0-306-47637-1
33. Merriam, S. B., & Tisdell, E. J. (2016). Qualitative research: A guide to design and implementation. John Wiley & Sons.
34. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Sage Publications.
35. Morgan, H. (2022). Conducting a qualitative document analysis. Qualitative Report, 27(1), 64–77. https://doi.org/10.46743/2160-3715/2022.5044
36. National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
37. Oral, B. (Ed.) (2019). Öğrenme öğretme kuram ve yaklaşımları (5. Bs.). Pegem Akademi.
38. Ontario Ministry of Education [OME]. (2020). The Ontario curriculum, grades 1-8: Mathematics. https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics/grades/g8-math/home
39. Özmantar, M., Akkoç, H., Kuşdemir Kayıran, B., & Özyurt, M. (Ed.) (2020). Ortaokul matematik öğretim programları tarihsel bir inceleme (3. Baskı). Pegem Akademi.
40. Patel, P., & Varma, S. (2018). How the abstract becomes concrete: Irrational numbers are understood relative to natural numbers and perfect squares. Cognitive Science, 42(5), 1642–1676. https://doi.org/10.1111/cogs.12619
41. Patton, M. Q. (2015). Qualitative research and methods: Integrating theory and practice. SAGE Publications. Rezat, S., Fan, L., & Pepin, B. (2021). Mathematics textbooks and curriculum resources as instruments for change. ZDM Mathematics Education, 53, 1189–1206. https://doi.org/10.1007/s11858-021-01309-3
42. Rizos, I., & Adam, M. (2022). Mathematics students' conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), 1–15. https://doi.org/10.29333/iejme/11977
43. Shiver, J., & Klosterman, P. (2022). Making irrational numbers real. Middle School Journal, 53(1), 36–42. https://doi.org/10.1080/00940771.2021.1997534
44. Sirotic, N., & Zazkis, R. (2007a). Irrational numbers on the number line–where are they? International Journal of Mathematical Education in Science and Technology, 38(4), 477–488. https://doi.org/10.1080/00207390601151828
45. Sirotic, N., & Zazkis, R. (2007b). Irrational numbers: The gap between formal and intuitive knowledge. Educational Studies in Mathematics, 65(1), 49–76. https://doi.org/10.1007/s10649-006-9041-5
46. Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16. https://www.jstor.org/stable/40248592
47. Tavşan, S., & Pusmaz, A. (2020). İlköğretim matematik öğretmen adaylarının pi sayısı bağlamındaki kavram tanımlarının incelenmesi. Ondokuz Mayıs Üniversitesi Eğitim Fakültesi Dergisi, 39(3 100. Yıl Eğitim Sempozyumu Özel Sayı), 260–274. https://doi.org/10.7822/omuefd.681540
48. Wiesman, J. L. (2015). Enhancing students' understanding of square roots. Mathematics Teaching in the Middle School, 20(9), 556–558. https://doi.org/10.5951/mathteacmiddscho.20.9.0556
49. Winicki-Landman, G., & Leikin, R. (2000). On equivalent and non-equivalent definitions: Part 1. For the Learning of Mathematics, 20(1), 17–21. https://www.jstor.org/stable/40248314
50. Voskoglou, M., & Kosyvas, G. (2012). Analyzing students' difficulties in understanding real numbers. Redimat-Revista De Investigacion En Didactica De Las Matematicas, 1(3), 301–226. https://doi.org/10.4471/redimat.2012.16
51. Zazkis, R. (2005). Representing numbers: prime and irrational. International Journal of Mathematical Education in Science and Technology, 36(2-3), 207–217. https://doi.org/10.1080/00207390412331316951
52. Zazkis, R., & Sirotic, N. (2010). Representing and defining irrational numbers: Exposing the missing link. Research in Collegiate Mathematics Education, 7, 1–27.