International Association of Educators   |  ISSN: 2834-7919   |  e-ISSN: 1554-5210

Original article | International Journal of Progressive Education 2021, Vol. 17(2) 29-54

Instructional Explanations of Class Teachers and Primary School Mathematics Teachers about Division

Ebru Korkmaz

pp. 29 - 54   |  DOI: https://doi.org/10.29329/ijpe.2021.332.3   |  Manu. Number: MANU-2003-19-0001.R1

Published online: April 07, 2021  |   Number of Views: 116  |  Number of Download: 611


Abstract

This study is a qualitative research which was conducted in order to reveal the instructional explanations of class teachers and primary school mathematics teachers working in state schools about division. A semi-structured interview form with three open-ended questions about division, prepared for this purpose, was examined by the experts. The semi-structured interview form inluded three questions asking the teachers to solve the long division operations of 3385: 13 = ?, 1238: 12 =? and 102102: 12 =? using the mathematical table of digits with a descriptive language as if they were telling the primary school students the solutions. While the first two questions were suitable with the 5th grade learning outcomes, the third question was suitable with a high level learning outcome. The main purpose of asking the 3rd question was to evaluate the instructional explanation of the teachers in a problem of different difficulty. The study group consisted of 34 teachers, 16 of whom were primary school mathematics teachers and 18 of whom were class teachers, working at central primary schools in a province located in Eastern Anatolia region of Turkey. The content analysis of the data showed that not all of the teachers could interpret the operation of division regarding the concept of digit accurately, and their division was result and reasoning oriented. However, it was found that few teachers made generalizations in a similar way. It was also seen that teachers who were at problem-solving level according to Kinach’s (2002b) comprehension level framework could not make sense of the logic underlying the division. In addition, the reason why zero (0) was moved to the quotient and when the divisor sought in remaining number should be completed by the teachers could not be clarified because they did not know the logic of the division.

Keywords: Mathematical Knowledge, Instructional Explanations, Operation of Division, Maths Teachers, Class Teachers


How to Cite this Article?

APA 6th edition
Korkmaz, E. (2021). Instructional Explanations of Class Teachers and Primary School Mathematics Teachers about Division . International Journal of Progressive Education, 17(2), 29-54. doi: 10.29329/ijpe.2021.332.3

Harvard
Korkmaz, E. (2021). Instructional Explanations of Class Teachers and Primary School Mathematics Teachers about Division . International Journal of Progressive Education, 17(2), pp. 29-54.

Chicago 16th edition
Korkmaz, Ebru (2021). "Instructional Explanations of Class Teachers and Primary School Mathematics Teachers about Division ". International Journal of Progressive Education 17 (2):29-54. doi:10.29329/ijpe.2021.332.3.

References
  1. Albayrak, M. & Şimşek M. (2017). Yetişkinlerin davranışlarını değiştirmenin güçlüğü:  bölme örneği. Erzincan Üniversitesi Eğitim Fakültesi Dergisi.19(1) 184-198. [Google Scholar]
  2. Ambrose, R., Baek, J.-M. & Carpenter, T.P. (2003) Children’s invention of multiplication and division algorithms. In A. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Recent research and theory. Mahwah, NJ: Erlbaum. [Google Scholar]
  3. Anghileri, J. (1989). An investigation of young children's understanding of multiplication. Educational Studies in Mathematics, 20(4), 367-385. [Google Scholar]
  4. An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school, mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 7, 145–172. [Google Scholar]
  5. Arslan Kılcan, S. (2006). İlköğretim matematik öğretmenlerinin kesirlerle bölmeye ilişkin kavramsal bilgi düzeyleri. (Yayınlanmamış Yüksek Lisans Tezi). Sosyal Bilimler Enstitüsü, Abant İzzet Baysal Üniversitesi, Bolu. [Google Scholar]
  6. Baki, M. (2013). Pre-service classroom teachers' mathematical knowledge and instructional explanations associated with division. Education and Science, 38(167), 300-311. [Google Scholar]
  7. Ball, D. L. (1990a). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466.  [Google Scholar]
  8. Ball, D. L. (1990b). Prospective elementary and secondary teachers‟ understanding of division. Journal for Research in Mathematics Education, 21(2), 132–144. [Google Scholar]
  9. Borko, H. & Putnam, R. (1996). Learning to teach. In D. Berliner & R. Calfee (eds.), Handbook of Educational Psychology (pp, 673-708). New York: Mcmillan. [Google Scholar]
  10. Berg, B. L. & Lune, H. (2019). Sosyal bilimlerde nitel araştırma yöntemleri. (Çev. Ed. Asım Arı). Konya: Eğitim Yayınevi. [Google Scholar]
  11. Bryant, P. (1997). Mathematical understanding in the nursery school years. In T. Nunes & P. Bryant, (Eds.), Learning and teaching mathematics: An international perspective (pp. 53-67). East Sussex, UK: Psychology Press. [Google Scholar]
  12. Bütün, M. & Baki, A. (2019). İlköğretim matematik öğretmeni adaylarının matematiği öğretme bilgilerinin gelişimi. Cumhuriyet International Journal of Education, 8(1), 300-322. [Google Scholar]
  13. http://dx.doi.org/10.30703/cije.493676 [Google Scholar]
  14. Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided ınstruction: A knowledge base for reform in mathematics ınstruction. The Elementary School Journal, 97(1), 3–20. [Google Scholar]
  15. Charalambous, C. Y., Hill, C. H. & Ball, D. L. (2011). Prospective teachers’learning to provide instructional explanations: How does it look and what might it take? Journal of Mathematics Teacher Education, 14, 441-463. DOI 10.1007/s10857-011-9182-z [Google Scholar]
  16. Creswell, J. W. (1998). Qualitative inquiry and research design: Choosing among five traditions. California: Sage Publications. [Google Scholar]
  17. Çakmak, M. (2004). İlköğretimde matematik öğretimi ve öğretmenin rolü. Retrieved 10.02.2020, from http://www.matder.org.tr/ilkogretimde-matematik-ogretimi-ve-ogretmenin-rolu/ [Google Scholar]
  18. Çepni, S. (2012). Araştırma ve proje çalışmalarına giriş (6. baskı). Trabzon: Celepler Matbaacılık. [Google Scholar]
  19. Çiltaş, A. & Işık, A. (2012). Matematiksel modelleme yönteminin akademik başarıya etkisi. Çağdaş Eğitim Dergisi Akademik, 2, 57-67. [Google Scholar]
  20. Çimen Ev, E. & Tat, T. (2018). Sınıf öğrencilerinin bölme işleminde kalanın yorumlanması konusunda problem kurma becerilerinin incelenmesi.  Eğitim ve Öğretim Araştırmaları Dergisi. 7(4). 1-11. [Google Scholar]
  21. Daehler, K. & Shinohara, M. (2001). A complete circuit is a complete circle: exploring the potential of case materials and methods to develop teachers‟ content knowledge and pedagogical content knowledge of science. Research in Science Education, 31, 267-288. [Google Scholar]
  22. Eroğlu, D. & Tanışlı, D. (2015). Ortaokul matematik öğretmenlerinin temsil kullanımına ilişkin öğrenci ve öğretim stratejileri bilgileri. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 9(1), 275-307. [Google Scholar]
  23. Fennema, E. & Franke, M. L.(1992). Teachers’ knowledge and its impact. In D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning (pp 147-164). New York: Macmillan. [Google Scholar]
  24. Gökkurt,B., Şahin, Ö. & Soylu, Y. (2012). Matematik öğretmenlerinin matematiksel alan bilgileri ile pedagojik alan bilgileri arasındaki ilişkinin incelenmesi.  The Journal of Academic Social Science Studies (JASSS), 5(8), 997-1012. [Google Scholar]
  25. Grossman, P.L. (1990). The making of a teacher: teacher knowledge and teacher education. New York: Teachers College Press. [Google Scholar]
  26. Hacıömeroğlu, G. (2013). Sınıf öğretmeni adaylarının öğretim için matematiksel bilgisi: öğrencilerin toplama ve çıkarma işlemlerine ilişkin çözümlerinin analizi. Eğitim ve Bilim, 38(168), 332-346. [Google Scholar]
  27. Hill, H.C., Ball, D.B. & Schilling, S.G. (2008). Unpacking pedagogical content knowledge: conceptualizing and measuring teachers‟ topic-specific knowledge of students. Journal for Research in Mathematics Teacher Education, 39 (4), 372-400. [Google Scholar]
  28. Kahan, J., Cooper, D. & Bethea, K. (2003). The role of mathematics teachers‟ content knowledge in their teaching: a framework for research applied to a study of student teachers. Journal of Mathematics Teacher Education, 6, 223-252. [Google Scholar]
  29. Kılcan, S. A. (2006). İlköğretim matematik öğretmenlerinin kesirlerle bölmeye ilişkin kavramsal bilgi düzeyleri. (Yayımlanmamış Yüksek Lisans Tezi). Sosyal Bilimler Enstitüsü. BİBÜ, Bolu. [Google Scholar]
  30. Kinach, B. M. (2002a). Understanding and learning-to-explain by representing mathematics: Epistemological dilemmas facing teacher educators in the Secondary mathematics “methods” course. Journal of Mathematics Teacher Education, 5, 153–186. [Google Scholar]
  31. Kinach, B. M. (2002b). A cognitive strategy for developing prospective teachers. “Pedagogical content knowledge in the secondary mathematics methods course”: Toward a model of effective practice.” Teaching and Teacher Education, 18(1), 51–71. [Google Scholar]
  32. Kouba, V. L. (1989). Children's solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 147-158. [Google Scholar]
  33. Kaasila, R., Pehkonen, E. & Hellinen, A. (2010). Finnish pre-service teachers’ and upper secondary students’ understanding of division and reasoning strategies used. Educational Studies in Mathematics, 73(3), 247-261. [Google Scholar]
  34. Leinhardt, G. & Smith, D. (1985). Expertise in Mathematics Instruction: subject matter knowledge. Journal of Educational Psychology, 77(3), 247-271. [Google Scholar]
  35. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates Press. [Google Scholar]
  36. Mcduffy, A. (2004). Mathematics teaching as a deliberate practice: an investigation of elementary pre-services teachers‟ reflective thinking during student student teaching. Journal of Mathematics Teacher Education, 7, 33-61. [Google Scholar]
  37. McMillan, J.H. & Schumacher, S. (2010). Research in education: Evidence-basedinquiry (7th Edition). London: Pearson. [Google Scholar]
  38. Milli Eğitim Bakanlığı (MEB) (2005). İlköğretim (1-5) matematik dersi öğretim programı. Ankara: Milli Eğitim Yayınları. [Google Scholar]
  39. Milli Eğitim Bakanlığı [MEB] (2018). Matematik dersi öğretim programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar). Ankara, Türkiye: MEB. [Google Scholar]
  40. Monte-Sano, C. (2011). Learning to open up history for students: Pre-service teachers' emerging pedagogical content knowledge. Journal of Teacher Education, 62(3), 260-272. doi: Doi 10.1177/0022487110397842. [Google Scholar] [Crossref] 
  41. Newsome, J. G. (1999). Pedagogıcal content knowledge: An introduction and orientation. In GessNewsome, J. & Ledermen, N.G. (eds.), Examining Pedagogical Content Knowledge. (pp.3-17). Dordrecht: Kluwer Academic Publishers. [Google Scholar]
  42. Olkun, S. & Toluk Uçar, Z. (2009). İlköğretimde etkinlik temelli matematik öğretimi (4. Baskı). Ankara: Maya Akademi. [Google Scholar]
  43. Oral, N. (2020).  5. Sınıf öğrencilerinin doğal sayılarla bölme işleminde yaşadığı zorluklar ve bu zorlukların nedenleri.  (Yayınlanmamış Yüksek Lisans Tezi).  Eğitim Bilimleri Enstitüsü,   Matematik ve Fen  Bilimleri Eğitimi Anabilim Dalı, Pamukkale Üniversitesi, Denizli. [Google Scholar]
  44. Penso, S. (2002). Pedagogical content knowledge: How do student teachers identify and describe the causes of their pupils learning difficulties? Asia-Pacific Journal of Teacher Education, 30 (1), 25-37. [Google Scholar]
  45. Shulman, L. S. (1986). Those who understand: Knowledge Growth in Teaching. Educational Researcher. 15 (2), 4-14. [Google Scholar]
  46. Simon, M. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 24(3), 233-254. [Google Scholar]
  47. Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 181-189). Hillsdale, NJ: Lawrence Elrbaum. [Google Scholar]
  48. Southwell, B. & Penglase, M, (2005). Mathematical knowledge of preservice primary teachers. In. H. L. Chick and J. L. Vincent (Eds.), International Group for the Psychology of Mathematics Education, l4, 209-216. [Google Scholar]
  49. Staley, K. N. (2004). Tracing the development of understanding rate of change: a case study of changes in a pre-service teacher’s pedagogical content knowledge. (Unpublished Phd Thesis). Graduate Faculty of North Carolina State University, Raleigh. http://www.lib.ncsu.edu/resolver/1840.16/3872  [Google Scholar]
  50. Steffe, L. P. (1988). Children’s construction of number sequences and multiplying schemes. Number Concepts and Operations in the Middle Grades, 2, 119-140. [Google Scholar]
  51. Tekin Sitrava, R., Özel, Z. & Işık, A (2020). Sınıf Öğretmeni adaylarının bölme işleminin anlamına dair alan bilgilerinin incelenmesi. Kastamonu Eğitim Dergisi, 28(2).  [Google Scholar]
  52. Tekin Sitrava, R. (2018). An investigation of prospective mathematics teachers' knowledge of basic algorithms with whole numbers: A case of Turkey. European Journal of Educational Research, 7(3), 513-528. [Google Scholar]
  53. Thompson, P.W., Carson, M. & Silverman, J. (2007). The desing of task in support of teachers’development of coherent masthematical meanings. Jornal for Mathematics Teacher Education, 10(4-6), 415- 432. [Google Scholar]
  54. Tirosh, D., Fischbein, E., Graeber, A., & Wilson, J.W. (1999). Prospective elementary teachers’ conceptions of rational numbers. Access: http:// jwilson. coe. uga. edu/Texts. Folder/ Tirosh/ Pros.El.Tchrs.html. [Google Scholar]
  55. Türnüklü, E. B. (2005). Matematik öğretmen adaylarının pedagojik alan bilgileri ile matematiksel alan bilgileri arasındaki ilişki. Eğitim Araştırmaları Dergisi, 21, 234–247. [Google Scholar]
  56. Thanhieser, E. (2009). Preservice elementaray school teachers’ conceptions of multidiğit whole numbers. Journal for Research in mathematics Education, 40, 252-281. [Google Scholar]
  57. Toluk Uçar, Z. (2011). Öğretmen Adaylarının Pedagojik İçerik Bilgisi: Öğretimsel Açıklamalar. Turkish Journal of Computer and Mathematics Education, 2 (2), 87-102. [Google Scholar]
  58. Uşak, M. (2005). Fen bilgisi öğretmen adaylarının çiçekli bitkiler konusundaki pedagojik alan bilgileri. (Yayınlanmamış Doktora Tezi). Gazi Üniversitesi Eğitim Bilimleri Enstitüsü, Ankara. [Google Scholar]
  59. Yeşildere, S. & Akkoç, H. (2010). Matematik Öğretmen Adaylarının Sayı Örüntülerine İlişkin Pedagojik Alan Bilgilerinin Konuya Özel Stratejiler Bağlamında İncelenmesi. Ondokuz Mayıs Üniversitesi Eğitim Fakültesi Dergisi, 29 (1), 125-149. [Google Scholar]
  60. Yıldırım, A. & Şimşek, H. (2011). Sosyal bilimlerde nitel araştırma yöntemleri (8. baskı). Ankara: Seçkin Yayıncılık. [Google Scholar]