The Proof Schemes of Preservice Middle School Mathematics Teachers and Investigating the Expressions Revealing These Schemes

Abstract

The aim of this study is to investigate preservice middle school teachers’ proof schemes and how they presented their proof schemes. Clinical method was used to identify the proof schemes of preservice teachers.  For this purpose, clinical interviews about the nature of proof and task based interviews were conducted with the participants in the field of numbers. The Task Based Interview Questions Form and Interview Questions Form about the Nature of Proof were conducted with three female preservice teachers in a single session. Using the content analysis report, it was found that preservice teachers used external proof schemes more frequently than analytic proof schemes, and they used empirical proof schemes less often. It was determined that showing responses on analytical proof schemes was higher in those preservice teachers when compared to the ones with lower level achievements. It was found that the external based opinions of the preservice teachers were found to be related with their characteristics which revealed external based proof scheme. It was also noticed that there could be a relationship between already acquired opinions which were memorized and superficial and the ones which block transforming ideas while making proofs.      

Keywords

• Preservice middle school mathematics teachers,numbers,task based interviews,proof scheme,nature of proof

Ortaokul Matematik Öğretmeni Adaylarının İspat Şemaları ve Bu Şemaları Ortaya Koyan İfadelerinin İncelenmesi

Öz

Mevcut çalışma ile ortaokul matematik öğretmeni adaylarının ispat şemalarının neler olduğunu ve bu şemaları nasıl ortaya koyduklarını araştırmak amaçlanmıştır. Öğretmen adaylarının ispat şemalarının belirlenebilmesi için klinik yöntem kullanılmıştır. Bu amaçla öğretmen adaylarıyla sayılar alanında görev temelli görüşmeler ve ispatın doğasına ilişkin klinik görüşmeler yapılmıştır. 3 kız öğretmen adayına tek bir oturumda Görev Temelli Görüşme Formu ve İspatın Doğasına İlişkin Görüşme Formu yöneltilmiştir. İçerik analizi yöntemi kullanılarak öğretmen adaylarının en çok dışsal, daha sonra analitik ve en az deneysel ispat şemalarını ortaya koyan tepkiler verdikleri belirlenmiştir. Çalışmada daha yüksek başarı düzeyindeki öğretmen adaylarının daha düşük başarı düzeyindeki öğretmen adayına göre analitik ispat şemasını ortaya koyan tepkileri daha sık gösterdikleri belirlenmiştir. Öğretmen adaylarının dışsal kaynaklı fikirlerinin, çoğunlukla onların dışsal alışkanlık edinilmiş ispat şemalarını ortaya çıkaran özellikleri ile ilişkili olduğu belirlenmiştir. Öğretmen adaylarının ispatın doğasına ilişkin önceden edinilmiş ezbere ve yüzeysel fikirleri ile onların ispatı yapılandırırken dönüşüm yapmalarına engel olan fikirlerinin ilişkili olabileceği belirlenmiştir.

Anahtar Kelimeler

• Ortaokul matematik öğretmeni adayları,sayılar,görev temelli görüşmeler,ispat şeması,ispatın doğası

Kaynakça

1. Arslan, S., ve Yıldız, C. (2010). 11. sınıf öğrencilerinin matematiksel düşünmenin aşamalarındaki yaşantılarından yansımalar, Eğitim ve Bilim, 35(156), 17-31.
2. Aydoğdu İskenderoğlu, T. (2016). Kanıt ve kanıt şemaları. E. Bingölbali, S. Arslan, ve İ.Ö. Zembat (Eds.), Matematik Eğitiminde Teoriler içinde (s. 65-83). Pegem Akademi, Ankara.
3. Aydoğdu İskenderoğlu, T. (2003). Farklı sınıf düzeylerindeki öğrencilerin matematik problemlerini kanıtlama süreçleri. (Yüksek Lisans Tezi, Abant İzzet Baysal Üniversitesi, Sosyal Bilimleri Enstitüsü, Bolu). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
4. Aylar, E. (2014). 7. sınıf öğrencilerinde ispat kavramının öğretilebilirliğinin incelenmesi, (Doktora tezi, Hacettepe Üniversitesi, Eğitim Bilimleri Enstitüsü, Ankara). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
5. Baştürk, S. (2010). First-year secondary school mathematics students' conceptions mathematical proofs and proving. Educational Studies, 36(3), 283-298.
6. Bieda, K.N. (2008). The pedagogy of proving in middle school mathematics. (Doctoral dissertation, University of Wisconsin, Madison, USA).
7. Boyle, J.D. (2012). Study of prospective secondary mathematics teachers’ evolving understanding of reasoning-and-proving. (Doctoral dissertation, University of Pittsburgh, USA). Retrieved from https://search.proquest.com/pqdtglobal/docview/1222084018.
8. CadwalladerOlsker, T. (2007). Proof schemes and proof writing. (Doctoral dissertation, Claremont Graduate University, California, USA).

9. Chazan, D. (1993). High school geometry students’ justification for their views of empirial evidence and mathematical proof, Educational Studies in Mathematics, 24(4), 359-387.
10. Ceylan, T. (2002). Geogebra yazılımı ortamında ilköğretim matematik öğretmen adaylarının geometrik ispat biçimlerinin incelenmesi. (Yüksek Lisans Tezi, Ankara Üniversitesi, Eğitim Bilimleri Enstitüsü, Ankara). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
11. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In R. Lesh, & A. Kelly (Eds) Handbook of Research Design in Mathematics and Science Education (pp. 547-589). Lawrence Erlbaum, Hillsdale, NJ.
12. Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students, British Educational Research Journal, 20(1), 41–54.
13. Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/Math/ internet adresinden 5.03.2017 tarihinde elde edildi.
14. Demiray, E. (2013). An investigation of pre-service middle school mathematics teachers’ achievement levels in mathematical proof and the reasons of their wrong interpretations. (Doctoral Dissertation, Orta Doğu Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, Ankara). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
15. Ellis, A.B. (2007). Connections between generalizing and justifiying; Students’ reasoning with linear relationships, Journal for Research in Mathematics Education, 38(3), 194-229.
16. Flores, A. (2006). How do students know what they learn in middle school mathematics is true?, School Science and Mathematics, 106(3), 124-132.
17. Gholamazad, S., Liljedahl, P., & Zazkis, R. (2004, October). What Counts as Proof? Investigation of Preservice Elementary Teachers' Evaluation of Presented 'Proofs’, Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Toronto, Canada.
18. Goetting, M. M. (1995). The college students' understanding of mathematical proof. (Unpublished doctoral dissertation). The University of Maryland, USA.
19. Goldin, G. A. (2000). A scientific perspective on structured, task based interviews in mathematics education research. İçinde A. E. Kelly , & R.A. Lesh (Ed.), Handbook of research design in mathematics and science education (517-545). Mahwah: Lawrence Erlbaum Associates Publishers.
20. Grigoriadou, O. (2012). Reasoning in geometry. How first learning to appreciate the generality of arguments helps students come to grips with the notion of proof. (Unpublished master’s thesis). University of Amsterdam, Holland.
21. Güler, G. (2013). Matematik öğretmeni adaylarının cebir öğrenme alanındaki ispat süreçlerinin incelenmesi, (Doktora Tezi, Atatürk Üniversitesi, Eğitim Bilimleri Enstitüsü, Erzurum). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
22. Güler, G., ve Dikici, R. (2012). Ortaöğretim matematik öğretmeni adaylarının matematiksel ispat hakkındaki görüşleri, Kastamonu Eğitim Dergisi, 20(2), 571-590.
23. Güler, G., Özdemir, E., ve Dikici, R (2012). Öğretmen adaylarının matematiksel tümevarım yoluyla ispat becerileri ve matematiksel ispat hakkındaki görüşleri, Kastamonu Eğitim Dergisi, 20(1), 219-236.
24. Güner, P. (2012). Matematik öğretmen adaylarının ispat yapma süreçlerinde DNR tabanlı öğretime göre anlama ve düşünme yollarının incelenmesi. (Yüksek Lisans Tezi, Marmara Üniversitesi, , Eğitim Bilimleri Enstütüsü, İstanbul). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
25. Harel, G. (2014). Deductive reasoning in mathematics education. In S.Lerman (Eds.), Encyclopedia of Mathematics Education (pp. 143-147). Springer, London.
26. Harel, G. (2007). Students’ proof schemes revisited. In P. Boero (Eds.), Theorems in school. From history, epistemology and cognition to classroom practice (pp. 65-78). Sense Publishers, Rotterdam.
27. Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R. Zaskis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 185–212). Ablex Publishing Corporation, New Jersey.
28. Harel G., & Rabin, J. M. (2010) Teaching practices that can promote the authoritative proof scheme, Canadian Journal of Science, Mathematics and Technology Education, 10(2), 139-159.
29. Harel, G., & Sowder, L. (1998). Students proof schemes: Results from exploratory studies, CBMS Issues in Mathematics education, 7, 234-283.
30. Haverhals, N.J. (2011). Student’s development in proof: A longitudinal study. (Doctoral disseration, The University of Montana, Missoula). https://scholarworks.umt.edu/etd/923/ adresinden edinilmiştir.
31. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra, Journal for Research in Mathematics Education, 31(4), 396-428.
32. Heinze, A., & Reiss, K. (2003, February). Reasoning and proof: Methodological knowledge as a component of proof competence., CERME 3 Third Conference of the European Society for Research in Mathematics Education, Bellaria, Italy.
33. Housman, D., & Porter, M. (2003). Proof schemes and learning strategies of above-avarage mathematics students, Educational Studies in Mathematics, 53(2), 139-158.
34. Houssart, J., & Evens, H. (2011). Conducting task‐based interviews with pairs of children: consensus, conflict, knowledge construction and turn taking. International Journal of Research & Method in Education, 34(1), 63-79.
35. İmamoğlu, Y. (2010). An investigation of fresmen and senior mathematics and teaching matematics students’ conceptionsand practices regarding proof. (Doctoral dissertation, Bogazici University, Fen Bilimleri Enstitüsü, İstanbul). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
36. İskenderoğlu, T. (2010). İlköğretim matematik öğretmeni adaylarının kanıtlamayla ilgili görüşleri ve kullandıkları kanıt şemaları (Doktora tezi, Karadeniz Teknik Üniversitesi, Fen Bilimleri Enstitüsü, Trabzon). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
37. İskenderoğlu, T., Baki, A., & İskenderoğlu, M.(2010). Proof schemes used by first grade of preservice mathematics teachers about function topic, Procedia Social and Behavioral Sciences, 9, 531-536.
38. Knuth, E.J. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405.
39. Knuth, E.J. (2002b). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5, 61-88.
40. Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ productions of mathematical justification. In M. Blanton, D. Stylianou, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (153–212). Routledge, NY.
41. Knuth, E., Slaughter, M., Choppin, J., & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, & K. Noony (Ed.), Proceedings of the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol 4), (1693–1700). OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education, Columbus.
42. Koichu, B. (2009, May). What can pre-service teachers learn from interviewing high school students on proof and proving?. ICMI Study 19 Conference: Proof and Proving in Mathematics Education, Volume 2, Taipei, Taiwan.
43. Koichu, B., & Harel, G. (2007). Triadic interaction in clinical task-based interviews with mathematics teachers. Educational Studies in Mathematics 65(3), 349-365.
44. Liu, Y., & Manouchehri, A. (2013). Middle school children’s mathematical reasoning and proving schemes, Investigations in Mathematics Learning, 6(1), 18-40.
45. Maher, C.A., & Sigley, R. (2014). Task based interview in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (579-582). London: Springer.
46. Martin, T.S., Soucy McCrone, S.M., Wallece Bower M.L., & Dindyal, J. (2005). The interplay of teacher and student actions in the teaching and learning of geometric proof, Educational Studies in Mathematics, 60(1), 95-124.
47. Martin, G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51.
48. MEB (2017). T.C. Milli Eğitim Bakanlığı Matematik Dersi Öğretim Programı (1,2,3,4,5,6,7 ve 8. Sınıflar). Ankara: Milli Eğitim Bakanlığı.
49. Mejia-Ramos, J. P. & Tall, D. (2005), ‘Personal and public aspects of formal proof: a theory and a single-case study’, In D. Hewitt and A. Noyes (Eds), Proceedings of the sixth British Congress of Mathematics Education held at the University of Warwick, (pp. 97-104). Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=8EF1334F83CAA307CC4E0B550BA53E0C?doi=10.1.1.377.5416&rep=rep1&type=pdf.
50. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in mathematics, 27(3), 249-266.
51. Moralı, S. , Uğurel, I., Türnüklü, E., ve Yeşildere, S. (2006). Matematik öğretmen adaylarının ispat yapmaya yönelik görüşleri, Kastamonu Eğitim Dergisi, 14(1), 147-160.
52. National Council of Teachers of Mathematics (NCTM) (2000). Principles and Standards for School Mathematics, Reston.VA: NCTM.
53. Norby, K. (2013). Investigating viable arguments: pre-service mathematics teachers' construction and evaluation of arguments (Doctoral Dissertation, Montana State University, Bozeman, Montana) .Retrieved from https://scholarworks.montana.edu/xmlui/handle/1/2903.
54. Oehrtman, M., & Lawson, A. E. (2008). Connecting science and mathematics: The nature of proof and disproof in science and mathematics. International Journal of Science and Mathematics Education, 6(2), 377-403.
55. Oflaz, G., Bulut, N., & Akcakin, V. (2016). Pre-service classroom teachers’ proof schemes in geometry: a case study of three pre-service teachers. Eurasian Journal of Educational Research, 63, 133-152.
56. Opper, S. (1977). Piaget’s Clinical Method, The Journal of Children’s Mathematical Behavior, 1(1), 90-107.
57. Ören, D. (2007). An investigation of 10th grade students’ proof schemes in geometry with respect to their cognitive styles and gender. (Master’s Dissertation, Middle East Technical University, Fen Bilimleri Enstitüsü, Ankara). Retrieved from http://tez2.yok.gov.tr/
58. Özer, Ö., ve Arıkan, A. (2002, Eylül). Lise matematik derslerinde öğrencilerin ispat yapabilme düzeyleri. V.Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi, Ankara, Bildiriler Kitabı, II, 1083-1089. http://old.fedu.metu.edu.tr/ufbmek-5/netscape/b_kitabi/PDF/Matematik/Bildiri/t245d.pdf adresinden edinilmiştir.
59. Pekşen Sağır, P. (2013). Matematik öğretmen adaylarının ispat yapma süreçlerinin incelenmesi. (Yüksek lisans tezi, Marmara Üniversitesi, Eğitim Bilimleri Enstitüsü, İstanbul). http://tez2.yok.gov.tr/ adresinden edinilmiştir.
60. Plaxco, D.B. (2011). Relationship between students’ proof schemes and justifications. (Master’s Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, USA).
61. Recio, A.M., & Godino, J.D. (2001). Institutional and personal meanings of mathematical proof, Educational Studies in Mathematics, 48(1), 83-99.
62. Riley, K.J. (2003). An investigation of prospective secondary mathematics teachers’ conceptions of proof and refutations. (Doctoral Dissertation) Montana State Univesity, Bozeman, Montana.
63. Sarı, M., Altun, A., & Aşkar, P. (2007). Undergraduate students’ mathematical proof processes in a calculus course: case study, Journal of Faculty of Educational Sciences, 40(2), 295-319.
64. Schabel, C.J. (2001). An instructional model to improve proof writing in college number theory. (Doctoral dissertation, Portland State University, USA). Retrieved from https://elibrary.ru/item.asp?id=5231587
65. Schoenfeld, A. (2002). Research methods in (mathematics) education. İçinde L.D. English (Ed.), Handbook of international research in mathematics education (435-488). Mahwah: Lawrance Erlbaum Associates Publishers.
66. Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36.
67. Soto, O.D. (2010). Teacher change in the context of a prof-centered professional development. (Doctoral Dissertation, San Diego State University, San Diego, USA).Retrieved from http://sdsu-dspace.calstate.edu/handle/10211.10/384.
68. Soucy Mccrone, S. M., & Martin, T. S. (2004, October). The impact of teacher actions on student proof schemes in geometry, North American Chapter of the International Group for the Psychology of Mathematics Education, Totonto, Ontario, Canada.
69. Sowder, L., & Harel, G. (2003). Case studies of mathematics majors’ proof understanding, production, and appreciation, Canadian Journal of Science, Mathematics and Technology Education, 3(2), 251-267. Sowder, L., & Harel, G. (1998). Types of students’ justifications, The Mathematics Teacher, 91(8), 670-675.
70. Stylianides, A., & Stylianides, G. (2009). Proof construction and evaluation. Educational Studies in Mathematics, 72, 237–253.
71. Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10, 145-166.
72. Stylinou, D., Chae, N., & Blanton, M. (2006, November). Students’ proof schemes: A closer look at what characterizes students’ proof conceptions, Proceedings of the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico.
73. Şen, C., & Güler, G. (2015). Examination of secondary school seventh graders’ proof skills and proof schemes. Universal Journal of Educational Research, 3(9), 617-631.
74. Şengül, S., ve Güner, P. (2013). DNR tabanlı öğretime göre matematik öğretmen adaylarının ispat şemalarının incelenmesi, International Journal of Social Science, 6(2), 869-878.
75. Uygan, C., Tanışlı, D., ve Köse, N.Y. (2014). İlköğretim matematik öğretmeni adaylarının kanıt bağlamındaki inançlarının, kanıtlama süreçlerinin ve örnek kanıtları değerlendirme süreçlerinin incelenmesi, Turkish Journal of Computer and Mathematics Education, 5(2), 137-157.
76. Varghese, T. (2007). Student teachers' conceptions of mathematical proof, Faculty of Graduate Studies and Research. (Master’s Thesis, University of Alberta, Admonton).
77. Weber, K. (2010). Mathematics majors' perceptions of conviction, validity, and proof, Mathematical Thinking and Learning, 12(4), 306-336.
78. Yin, R. K. (2003). Case study research: Design and methods. Thousand Oaks, CA: Sage.
79. Yoo, S. (2008). Effects of Traditional and Problem-Based Instruction on Conceptions of Proof and Pedagogy in Undergraduates and Prospective Mathematics Teachers (Doktora Tezi, The University of Texas, Austin, USA).
80. Zaimoğlu, Ş. (2012). 8. sınıf öğrencilerinin geometrik ispat süreci ve eğilimleri. (Yüksek lisans tezi, Kastamonu Üniversitesi, Fen Bilimleri Enstitüsü, Kastamonu). http://tez2.yok.gov.tr/ adresinden edinilmiştir.