Mistake Handling Activities in the Development of Middle School Mathemathics Teachers’ Subject Matter Knowledge: Addition Operation with Fractions

Abstract

In this study, the effects of mistake handling actvities in the development process of middle school mathematic teachers’ subject matter knowledge for addition operation with fractions was examined. For this purpose, teachers’ subject matter knowledge in the context of operation, problem posing, using model, explain and know important points about addition operation with fractions set before applications supported by mistake handling activities. Case study methods from qualitative research approaches was used as the design of study, where seven middle school mathemathics teachers were sampled. Teachers were selected by using purposive sampling technique. Datas were obtained through subject matter knowledge test, focus group inteviews and diaries. Descriptive analysis was applied to the collected data. According to the results of the study, it was seen that problem posing skills of teachers were lacking in the context of subject matter knowledge. 

Keywords

• Mathemathical knowledge for teaching,subject matter knowledge,mistake handling activities,addition operation with fractions,middle school mathemathics teachers

Ortaokul Matematik Öğretmenlerinin Konu Alan Bilgilerinin Gelişiminde Hata Temelli Aktiviteler: Kesirlerle Toplama İşlemi

Abstract

Bu çalışmada hata temelli aktivitelerle ortaokul matematik öğretmenlerinin kesirlerle toplama işlemine yönelik konu alan bilgilerinin gelişimleri incelenmiştir. Bu amaç doğrultusunda hata temelli aktiviteler ile desteklenen uygulama öncesinde öğretmenlerin kesirlerle toplama işlemine yönelik alan bilgileri işlem yapma, problem kurma, model kullanma (resim veya diyagram temsili) ve önemli noktaları açıklama ve bilme bağlamında belirlenmiştir. Nitel yaklaşımlardan durum çalışması yönteminin kullanıldığı bu çalışma yedi ortaokul matematik öğretmeniyle yürütülmüştür. Öğretmenler amaçlı örnekleme yöntemiyle seçilmiştir. Veriler konu alan bilgisi testi, odak grup görüşmeleri ve günlüklerden oluşan doküman kullanılmıştır. Toplanan verilere betimsel analiz uygulanmıştır. Çalışma sonuçlarına göre uygulama öncesinde konu alan bilgisi bağlamında öğretmenlerin problem kurma becerilerinin eksik olduğu görülmüştür. Ayrıca hata temelli aktivite uygulamaları öğretmenlerin kendi kavramsal düzeylerinin farkında olmalarına katkı sağlamıştır. 

Keywords

• Öğretim için matematiksel bilgi,konu alan bilgi,hata temelli aktiviteler,kesirlerle toplama işlemi,ortaokul matematik öğretmeni

References

1. Akpınar B. ve Akdoğan, S. (2010). Negatif bilgi kavramı: hata ve başarısızlıklardan öğrenme. Batı Anadolu Eğitim Bilimleri Dergisi, 1(1), 14-22.
2. Aksu, Z. (2013). Sınıf Öğretmeni Adaylarının Kesirler Konusundaki Pedagojik Alan Bilgileri Gelişimi. Yayımlanmamış Doktora Tezi, Atatürk Üniversitesi Eğitim Bilimler Enstitüsü, Erzurum. An, S., Kulm, G. AND 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.
3. Ball, D. L. (1991a). Research on teaching mathematics: Making subject-matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching: Teachers’ knowledge of subject matter as it relates to their teaching practice (Vol.2, pp. 1–48). Greenwich, CT: JAI Press.
4. Ball, D. L. (1991b). What’s all this talk about “discourse”? Arithmetic Teacher, 39(3), 44–48.
5. Ball, D. L. and Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83-104). Westport, CT: Ablex.
6. Ball, D. L. and Bass H. (2002). toward a practice-based theory ofmathematical knowledge for teaching. Canadıan Mathematıcs Educatıon Study Group, Proceedıngs / Actes 2002 Annual Meetıng, Queen’s University, 3-14.
7. Ball, D. L., Hill, H. C. and Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide?. American Educator, 29(3), 14-17, 20-22, 43-46.
8. Ball, D. L., Thames, M. H. and Phelps, G. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59(5), 389-407.Baturo, A., Cooper, T., Doyle, K. and Grant, E. (2007). Using three levels in design of effective teacher-education tasks: The case of promoting conflicts with intuitive understandings in probability. Journal of Mathematics Teacher Education, 10, 251-259.

9. Beckmann, S. (2008). Mathematics for elemantary teachers (Second Edition). USA: Addison-Wesley (Pearson Education).
10. Bell. A. (1983). Diagnostic teaching of additive and multiplicative problems. In R. Hershkowits (Ed.). Proceedings of the Seventh International Conference for the Psychology of Mathematics Education (pp. 205-210). Rehovot. Israel: Weizmann Institute of Science.
11. Borasi, R. (1986). On the Educational Roles of Mathematical Errors: Beyond Diagnosis and Remediation. Ph.D. Dissertation, State University of New York at Bufalo.
12. Borasi, R. (1988). Towards a Reconceptualization of the Role of Errors in Education: The Need for New Metaphors. Annual Meeting of the American Educational Research Association, New Orleans, LA.
13. Borasi, R. (1989). Students’ Constructive Uses of Mathematical Errors: A Taxonomy. Annual Meeting of the American Educational Research Association, San Francisco.
14. Borasi R. (1994). Capitalizing on Errors as "Springboards for Inquiry": A Teaching Experiment. Journal for Research in Mathematics Education, 25(21), 66- 208.
15. Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Norwood, NJ: Ablex Publishing Corporation.
16. Buckreis, W. F. (1999). Elementary mathematics teacher subject matter knowledge and its relationship to teaching and learning. Submittedfor the degree Doctor of Philosophy, Oregon State University, USA.
17. Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö. E., Karadeniz, Ş. ve Demirel, F. (2011). Bilimsel araştırma yöntemleri (8. Baskı). Ankara: Pegem Akademi.
18. Carpenter, T. P., Franke, M. L. and Levi, L. (2003), Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School, New Hampshire, USA.Creswell, J. W. (2015). Nitel araştırma yöntemleri: Beş yaklaşıma göre nitel araştırma ve araştırma deseni (3. Baskıdan Çeviri). (Çeviri Editörleri: M. Bütün ve S. B. Demir). Ankara: Siyasal Yayın Dağıtım.
19. Dorgan, K. (1994). What textbooks offer for instruction in fraction concepts. Teaching Mathematics 1(3), 150-155.
20. Ekiz, D. (2009). Bilimsel araştırma yöntemleri (2. baskı). Ankara: Anı Yayıncılık.
21. Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24(2), 94-116.
22. Grossman P L. (1990). The making of a teacher: teacher knowledge and teacher education. New York: Teachers College Press.
23. Güler, A., Halıcıoğlu, M. B., ve Taşğın, S. (2013). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin Yayıncılık.
24. Heinze, A. (2005). Mistake-handling activities in german mathematics classroom. In H.L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education(Vol. 3, pp. 105-112). Melbourne (Australien): Melbourne University.
25. Heinze, A. and Reiss, K. (2007). Mistake-Handling Activities in the Mathematics Classroom: Effects of an In-Service Teacher Training on Students’ Performance in Geometry. In J.-H. Woo, H.-C. Lew, K.-S. Park & D.-Y. Seo (Eds.), Proceedings of the 31stConference of the International Group for the Psychology of Mathematics Education (Vol. 3, 9-16). Seoul: PME.
26. Hill, H. C., Ball, D. L. and Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400. Işık, C. ve Kar, T. (2012). 7.sınıf öğrencilerinin kesirlerde toplama işlemine yönelik kurdukları problemlerin analizi. İlköğretim Online, 11(4), 1021-1035.
27. Işıksal, M. (2006). A study on pre-service elementary mathematicsteachers’ subject matter knowledge and pedagogical content knowledge regarding the multiplicatıon and division of fractions. Unpublished doctoral dissertation, Middle East Technical University, Turkey.
28. Jaworski, B. (2001). Developing mathematics teaching: Teachers, teacher-educators and researchers as co-learners’. In F.L. Lin and T.J. Cooney (Eds.), Making sense of mathematics teacher education, Kluwer, Dordrecht.
29. Kar, T. (2014). Ortaokul matematik öğretmenlerinin öğretim için matematiksel bilgisinin problem kurma bağlamında incelenmesi: Kesirlerle toplama işlemi örneği. Yayımlanmamış doktora tezi, Atatürk Üniversitesi Eğitim Bilimleri Enstitüsü: Erzurum.
30. Kar, T. ve Işık, C. (2014). Ortaokul yedinci sınıf öğrencilerinin kesirlerle çıkarma işlemine kurdukları problemlerin analizi. İlköğretim Online, 13(4), 1223-1239.
31. Klymchuk, S. and Kachapova, F. (2012) Paradoxes and counterexamples in teaching and learning of probability at university. International Journal of MathematicalEducation in Science and Technology, 43(6), 803-811.
32. Kuntze, S. and Reiss, K. (2006). Evaluational research on a video-based in-service mathematics teacher training project -reported instructional practice and judgementson instructional quality. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of theInternational Group for the Psychology of Mathematics Education, (Vol. 4, pp.1-8.). Prague: PME.
33. Leinhardt, G. and Smith, D. A. (1985). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 77, 247-271.
34. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the Unite States. Mahwah, NJ: Lawrence Erlbaum Associates.
35. Margolinas, C., Coulange, L. and Bessot, A. (2005). What can the teacher learn in the classroom?. Educational Studies in Mathematics, 59, 205-234.
36. Marton, R. L. (1955). İlkokulda aritmetik öğretimi. (Çev. Yakalıoğlu, A.). İstanbul: Maarif Basımevi.
37. MEB (2018). Matematik Dersi Öğretim Programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar). http://ttkb.meb.gov.tr/.
38. Mewborn, D. (2001). Teachers content knowledge, teacher education, and their effects on the preparation of elementary teachers in the United States. Mathematics Teacher Education and Development, 3, 28-36.
39. Misquitta, R. (2011). A review of the literature: Fraction instruction for struggling learners in mathematics. Learning Disabilities Research & Practice, 26(2), 109– 119.
40. Movshovitz-Hadar N. and Hadas R. (1990). Perspective education of math teachers using paradoxes, Educational Studies in Mathematics, 21, 265-287.
41. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. NCTM.
42. Park, S. and Oliver, J. S. (2008). Evisiting the conceptualisation of pedagogical content knowledge (PCK): PCK as a conceptual tool to understand teachrs as professionals. Research in Science Education 38(3), 261-284.
43. Rach, S., Ufer, S. and Heinze, A. (2013). Learning from errors: effects of teachers training on students' attitudes towards and their individual use of errors, PNA, 8(1), 21-30.
44. Ryan, J. and McCrae, B. (2005). Subject matter knowledge: Mathematical errors and misconceptions of beginning pre-service teachers. In P. C. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce & A. Roche (Eds.), 28th Annual Conferenceof the Mathematics Education Research Group of Australasia, (Vol. 2,pp. 641-648). Deakin University, Melbourne.
45. Ryan, J. and Williams, J. (2007). Children’s mathematics 4–15 learning from errors andmisconceptions.(First published). New York: McGraw-Hill Companies. Schoenfeld, A. E. (1992). Learning to think mathematically: Problem solving, metacognition and sense making inmathematics. In Grouws D. A. (ed.), Handbook of Researchon Mathematics Teaching and Learning, 334-370.
46. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Rersearcher, 15(2), 4–14.
47. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–22.
48. Tirosh, D. (2000). Enhancing pre-service teachers' knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.
49. Van De Walle, J. A., Karp, K. S. and Bay-Williams, J. M. (2012). İlkokul ve Ortaokul Matematiği Gelişimsel Yaklaşımla Öğretim.(Çeviri Editörü: Soner Durmuş). Ankara: Nobel Akademik Yayıncılık.
50. Van den Kieboom, L. A. (2008). Developing and using mathematical knowledge for teaching fractions: A case study of preservice teachers. Submitted for the degree Doctor of Philosophy, Marquette University, ABD.
51. Yıldırım, A. ve Şimşek, H (2008). Sosyal bilimlerde nitel araştırma yöntemleri (6. baskı). Ankara: Seçkin Yayıncılık.