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Ortaokul Öğrencilerinin Aritmetikten Cebire Geçiş Süreçlerinin İncelenmesi: Sembollerin Kullanımı ve Harflerin Anlamı

Year 2016, Volume: 11 Issue: 2, 0 - 0, 23.12.2016

Yaşar Akkan Adnan Baki

Abstract

Bu çalışmanın amacı 5-8.sınıf öğrencilerinin aritmetikten cebire geçiş süreçlerini sembollerin kullanımı ve harflerin anlamını yorumlama bağlamında araştırmak ve araştırmanın sonuçlarına göre önerilerde bulunmaktır. Gelişimci araştırmaların bir türü olan enlemesine çalışmanın kullanıldığı bu çalışmada, farklı öğrenim seviyelerindeki 285 ortaokul öğrencisine yazılı sınavlar uygulanmış, 24 öğrenciyle ise klinik mülakatlar yürütülmüştür. Veri toplamak amacıyla aritmetikten cebire geçişin inceleneceği bu iki konuyu içeren 2 soru ile ek sorular hazırlanmıştır. Bu çalışmanın veri analizinde bu iki konuya ilişkin hazırlanan karakterizasyon tablolarından yararlanılmıştır. Ayrıca yazıya dökülen klinik mülakat verileriyle farklı öğrenim seviyelerindeki ortaokul öğrencilerin aritmetikten cebire geçiş sürecindeki değişim ve gelişimleri incelenmiştir. Sonuç olarak farklı öğrenim seviyelerindeki öğrencilerin öğrenim seviyeleri arttıkça iki konu açısından aritmetikten cebire geçişin olumlu yönde değiştiği ve geliştiği görülmüş, fakat öğrenim seviyeleri arasındaki bu değişim ve gelişim çok az olmuştur.

References

  • Akkan, Y. (2009). İlköğretim öğrencilerinin aritmetikten cebire geçiş süreçlerinin incelenmesi. Yayımlanmamış doktora tezi. Karadeniz Teknik Üniversitesi, Trabzon.
  • Akkan, Y., Baki, A. ve Çakıroğlu, Ü. (2011). Aritmetik ile cebir arasındaki farklılıklar: Cebir öncesinin önemi. İlköğretim Online, 10(3), 812-823.
  • Akkan, Y., Öztürk, M. ve Akkan, P. (2014). Aritmetik ile cebir arasındaki farklılıklar üzerine bir çalışma. 11. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi Bildiri Özetleri Kitapçığı içinde (s. 1167-1168). Adana: Çukurova Üniversitesi
  • Alexandrou-Leonidou, V. & Philippou, G. (2007). Elementary school students’ understanding and use of the equal sign. Paper presented at CERME 5 Working Group 6, 825–834, Lanarka, Cyprus.
  • Arzarello, F., Bazzini, L. & Chiappini, G. (1993). Cognitive processes in algebraic thinking: towards a theoretical framework. Proceedings PME 17(1), 138-145.
  • Behr, M., Erlwanger, S. & Nichols, E. (1980). How children view the equal sign. Mathematics Teaching, 92, 13-15.
  • Booth, L. (1984). Algebra: Children’s strategies and errors. Windsor, UK: NFER-Nelson.
  • Booth, L. R. (1988). Children's difficulties in beginning algebra. In A. F. Coxford (Eds.). The Ideas of Algebra, K-12, 20–32. Reston, VA: NCTM.
  • Carpenter, T. P. & Levi, L.(2000). Developing Conceptions of Algebraic Reasoning in the Primary Grades. Research Report Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science. www.wcer.wisc.edu/ncisla /publications/index.html, 11 Aralık 2008
  • Chalouh, L. & Hersovics, N. (1988). Teaching algebra expressions in a meaningfulway, in the ıdeas of algebra, K-12: Yearbook. 33-42, A.F. Coxford (Ed.), NCTM.
  • ClementJ. (1982).Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16-30.
  • Cooper, T. J., Boulton-Lewis, G., Athew, B., Willss, L. & Mutch, S. (1997). The transition arithmetic to algebra: Initial understandings of equals, operations and variable. International Group for the Psychology of Math. Education, 21, 2, 89-96.
  • Cooper, T. J., Williams, A. M. & Baturo, A. R. (1999a). Equals, expressions, equations, and the meaning of variable: A teaching experiment. In Making the difference:Proceedings of the 22nd Annual Conference of the Mathematics Education Research Group of Australasia. Adelaide, South Australia: MERGA.
  • Cooper, T. J., Baturo, A. R. & Williams, A. M. (1999b). Unknowns, patterns, relationships, concrete materials and teaching the meaning of the algebraic expressions, 3 x. In E. B. Ogena & E. F. Golla (Eds.), Mathematics for the 21st century, Proceedings of the 8th South East Asian Conference on Mathematics Education, 127-136. Manila, Philippines.
  • Çepni, S. (2007). Arastırma ve proje çalısmalarına giriş, Trabzon: Celepler Matbaacılık.
  • Dede, Y. (2003). ARCS motivasyon modeli ve Öğe Gösterim Teorisi'ne (Component Display Theory) dayalı yaklaşımın öğrencilerin değişken kavramını öğrenme düzeylerine ve motivasyonlarına etkisi. Yayımlanmamış doktora tezi. Gazi Üniversitesi, Ankara.
  • Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10, Portsmouth, NH: Heinemann.
  • English, L. D. & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Erlbaum.
  • Filloy, E. & Rojana, T. (1989). Solving equations: The transition from arithmetic to algebra, For The Learning of Mathematics, 9(2), 19 - 25.
  • Goldin, G. A. (1998). Observing mathematical problem solving through task-based ınterviews, In A.R. Teppo (Ed.), Qualitative research methods in mathematics mathematics education, NCTM.
  • Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In S.Wagner & C. Kieran (Eds.), Research ıssues in the learning and teaching of algebra, 60-92. Reston, VA: NCTM, Hillsdale, NJ: Lawrence Erlbaum.
  • Hersovics, N. & Linchevski, L., 1994. A cognative gap between arithmetic and algebra, Educational Studies in Mathematics, 27(1), 59-78
  • Hunting, R.P. (1997). Clinical interview methods in mathematics education research and practice, Journal of Mathematical Behaviour, 16(2), 145-165.
  • Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner & C. Kieran (Eds.). Resaearch ıssues in the learning and teaching of algebra, 33-56. Reston, VA: NCTM.
  • Kieran, C. (1990). Cognitive processes involved in learning school algebra. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and Cognition, 96-112. Cambridge: Cambridge University Press.
  • Kieran, C. (1992). The learning and teaching of school algebra. In D.A. Grouws (Eds.). Handbook of research on mathematics teaching and learning, 390-419. New York: Macmillan.
  • Kieran, C. & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. In P. S. Wilson (Ed.), Research ideas for the classroom: Middle grades mathematics, 119-139. New York: Macmillan.
  • Kinzel, M. T. (2000). Charecterizing ways of thinking that underlie college students ınterpretation and use of algebraic notation. Unpublished doctoral dissertation, The Pennslyvania State University, USA.
  • Knuth, E. J., Alibali, M. W., McNeil, N.M., Weinberg, A. & Stephens, A.C. (2005). Middle school students’ understanding of core algebraic concepts: Equality & variable. Zentralblatt für Didaktik der Mathematik, 37(1), 68-76.
  • Knuth, E. J., Stephens, A. C., McNeil, N. M. & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations, Journal for Research in Mathematics Education, 37, 297-312.
  • Kuchemann, D.: 1981, ‘Algebra’, in K. Hart (ed.), Children's understanding of mathematics: 11–16, (pp. 102–119) London: John Murray.
  • Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra, The Journal of Mathematical Behaviour,14, 113-120.
  • 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, 38–65.
  • Linchevski, L. & Livneh, D. (1999). Sctructure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173-196.
  • Lodholz, R. D. (1993). The transition from arithmetic to algebra. E.L. Edwards (Ed.) Algebra for everyone, 24-33. Reston, VA: NCTM.
  • Macgregor, M. & Stacey, K. (1997a). Students’ understanding of algebraic notation: 11-15, Educational Studies in Mathematics, 33, 1-19.
  • Macgregor, M. & Stacey, K. (1997b). Ideas about symbolism that students bring to algebra, The Mathematics Teacher, 90(2), 110-113.
  • MEB (2009). İlköğretim matematik programı, www.meb.gov.tr. 7 Mart 2009.
  • National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, Va: NCTM
  • Perso, T. (1992). Using diagnostic teaching to overcome misconceptions in algebra, The Mathematical Association of Western Australia.
  • Philipp, R. A. (1999). The many use of algebraic variables, In B. Moses (Ed.), Algebraic thinking, grades 9-12: Readings from NCTM’s school based journals and other publications,150-156, Reston, VA: NCTM.
  • Saenz-Ludlow, A. & Walgamuth, C. (1998). Third grader’s ınterpretation of equality and the equals sign, Educational Studies in Mathematics, 35, 153-187.
  • 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, 21, 1-36.
  • Sfard, A. & Linchevski, L. (1994). the gains and the pitfalls of reification-the case of algebra, Educational Studies in Mathematics, 26(2), 191-228.
  • Skemp, R. (1987). The psychology of learning mathematics, Lawrence Erlbaum Associates, Hillsdale, NJ.
  • Stacey, K. & MacGregor, M. (1997). Building foundations for algebra, Mathematics in the Middle School, 2, 253 – 260.
  • Stacey, K. & MacGregor, M. (2000). Learning the Algebraic Method of Solving Problems, Journal of Mathematical Behaviour, 18(2), 149-167.
  • Swadener, M. & Soedjadi, R. (1988). Values, mathematics education and the task of developing pupils’ personalities: An Indonesian perspective, Educational StudiesIn Mathematics, 19(2), 193-208.
  • Tall, D., Gray, E., Bin Ali, M. B., Crowley, L., DeMarois, P., McGowen, M., Pitta, D., Pinto, M., Thomas, M. & Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking, Canadian journal of science, 1, 80-104, Mathematics and Technology Education.
  • Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In B. Moses (Eds.). Algebraic Thinking Grades K-12, 7-14. Reston, VA: NCTM.
  • Van Amerom, B. (2002). Reinvention of early algebra: Developmental research on the transition from arithmetic to algebra. Unpublished doctoral dissertation. University of Utrecht, The Netherlands.
  • Wagner, S. (1983). What are these things called variables? Mathematics Teacher, October, 474-478.
  • Wagner, S. & Kieran, C. (1989). Research issues in the learning and teaching of algebra. Reston, VA: NCTM.
  • Wagner, S. & Parker, S. (1993). Advancing algebra. In P. S. Wilson, (Ed.), Research ideas for the classroom: High school mathematics, 117-139, New York: Macmillan Publishing Company.
  • Williams, A. M. & Cooper, T. J. (2001). Moving from arithmetic to algebra under the time pressures of real classrooms. In H. Chick, K. Stacey, Jill Vincent, & John Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra, 665-662. Melbourne: University of Melbourne.
  • Yaman, H., Toluk, Z. ve Olkun, S. (2003). İlköğretim öğrencileri eşittir işaretini nasıl algılamaktadırlar? Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24,142-151.
  • Yıldırım, C. (2000). Matematiksel düşünme, Remzi Kitapevi, İstanbul. 373.
  • Zaskıs, R. & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions, Journal of Mathematical Behaviour, 17(4), 429-439.

Year 2016, Volume: 11 Issue: 2, 0 - 0, 23.12.2016

Yaşar Akkan Adnan Baki

Abstract

References

  • Akkan, Y. (2009). İlköğretim öğrencilerinin aritmetikten cebire geçiş süreçlerinin incelenmesi. Yayımlanmamış doktora tezi. Karadeniz Teknik Üniversitesi, Trabzon.
  • Akkan, Y., Baki, A. ve Çakıroğlu, Ü. (2011). Aritmetik ile cebir arasındaki farklılıklar: Cebir öncesinin önemi. İlköğretim Online, 10(3), 812-823.
  • Akkan, Y., Öztürk, M. ve Akkan, P. (2014). Aritmetik ile cebir arasındaki farklılıklar üzerine bir çalışma. 11. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi Bildiri Özetleri Kitapçığı içinde (s. 1167-1168). Adana: Çukurova Üniversitesi
  • Alexandrou-Leonidou, V. & Philippou, G. (2007). Elementary school students’ understanding and use of the equal sign. Paper presented at CERME 5 Working Group 6, 825–834, Lanarka, Cyprus.
  • Arzarello, F., Bazzini, L. & Chiappini, G. (1993). Cognitive processes in algebraic thinking: towards a theoretical framework. Proceedings PME 17(1), 138-145.
  • Behr, M., Erlwanger, S. & Nichols, E. (1980). How children view the equal sign. Mathematics Teaching, 92, 13-15.
  • Booth, L. (1984). Algebra: Children’s strategies and errors. Windsor, UK: NFER-Nelson.
  • Booth, L. R. (1988). Children's difficulties in beginning algebra. In A. F. Coxford (Eds.). The Ideas of Algebra, K-12, 20–32. Reston, VA: NCTM.
  • Carpenter, T. P. & Levi, L.(2000). Developing Conceptions of Algebraic Reasoning in the Primary Grades. Research Report Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science. www.wcer.wisc.edu/ncisla /publications/index.html, 11 Aralık 2008
  • Chalouh, L. & Hersovics, N. (1988). Teaching algebra expressions in a meaningfulway, in the ıdeas of algebra, K-12: Yearbook. 33-42, A.F. Coxford (Ed.), NCTM.
  • ClementJ. (1982).Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16-30.
  • Cooper, T. J., Boulton-Lewis, G., Athew, B., Willss, L. & Mutch, S. (1997). The transition arithmetic to algebra: Initial understandings of equals, operations and variable. International Group for the Psychology of Math. Education, 21, 2, 89-96.
  • Cooper, T. J., Williams, A. M. & Baturo, A. R. (1999a). Equals, expressions, equations, and the meaning of variable: A teaching experiment. In Making the difference:Proceedings of the 22nd Annual Conference of the Mathematics Education Research Group of Australasia. Adelaide, South Australia: MERGA.
  • Cooper, T. J., Baturo, A. R. & Williams, A. M. (1999b). Unknowns, patterns, relationships, concrete materials and teaching the meaning of the algebraic expressions, 3 x. In E. B. Ogena & E. F. Golla (Eds.), Mathematics for the 21st century, Proceedings of the 8th South East Asian Conference on Mathematics Education, 127-136. Manila, Philippines.
  • Çepni, S. (2007). Arastırma ve proje çalısmalarına giriş, Trabzon: Celepler Matbaacılık.
  • Dede, Y. (2003). ARCS motivasyon modeli ve Öğe Gösterim Teorisi'ne (Component Display Theory) dayalı yaklaşımın öğrencilerin değişken kavramını öğrenme düzeylerine ve motivasyonlarına etkisi. Yayımlanmamış doktora tezi. Gazi Üniversitesi, Ankara.
  • Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10, Portsmouth, NH: Heinemann.
  • English, L. D. & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Erlbaum.
  • Filloy, E. & Rojana, T. (1989). Solving equations: The transition from arithmetic to algebra, For The Learning of Mathematics, 9(2), 19 - 25.
  • Goldin, G. A. (1998). Observing mathematical problem solving through task-based ınterviews, In A.R. Teppo (Ed.), Qualitative research methods in mathematics mathematics education, NCTM.
  • Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In S.Wagner & C. Kieran (Eds.), Research ıssues in the learning and teaching of algebra, 60-92. Reston, VA: NCTM, Hillsdale, NJ: Lawrence Erlbaum.
  • Hersovics, N. & Linchevski, L., 1994. A cognative gap between arithmetic and algebra, Educational Studies in Mathematics, 27(1), 59-78
  • Hunting, R.P. (1997). Clinical interview methods in mathematics education research and practice, Journal of Mathematical Behaviour, 16(2), 145-165.
  • Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner & C. Kieran (Eds.). Resaearch ıssues in the learning and teaching of algebra, 33-56. Reston, VA: NCTM.
  • Kieran, C. (1990). Cognitive processes involved in learning school algebra. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and Cognition, 96-112. Cambridge: Cambridge University Press.
  • Kieran, C. (1992). The learning and teaching of school algebra. In D.A. Grouws (Eds.). Handbook of research on mathematics teaching and learning, 390-419. New York: Macmillan.
  • Kieran, C. & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. In P. S. Wilson (Ed.), Research ideas for the classroom: Middle grades mathematics, 119-139. New York: Macmillan.
  • Kinzel, M. T. (2000). Charecterizing ways of thinking that underlie college students ınterpretation and use of algebraic notation. Unpublished doctoral dissertation, The Pennslyvania State University, USA.
  • Knuth, E. J., Alibali, M. W., McNeil, N.M., Weinberg, A. & Stephens, A.C. (2005). Middle school students’ understanding of core algebraic concepts: Equality & variable. Zentralblatt für Didaktik der Mathematik, 37(1), 68-76.
  • Knuth, E. J., Stephens, A. C., McNeil, N. M. & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations, Journal for Research in Mathematics Education, 37, 297-312.
  • Kuchemann, D.: 1981, ‘Algebra’, in K. Hart (ed.), Children's understanding of mathematics: 11–16, (pp. 102–119) London: John Murray.
  • Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra, The Journal of Mathematical Behaviour,14, 113-120.
  • 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, 38–65.
  • Linchevski, L. & Livneh, D. (1999). Sctructure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173-196.
  • Lodholz, R. D. (1993). The transition from arithmetic to algebra. E.L. Edwards (Ed.) Algebra for everyone, 24-33. Reston, VA: NCTM.
  • Macgregor, M. & Stacey, K. (1997a). Students’ understanding of algebraic notation: 11-15, Educational Studies in Mathematics, 33, 1-19.
  • Macgregor, M. & Stacey, K. (1997b). Ideas about symbolism that students bring to algebra, The Mathematics Teacher, 90(2), 110-113.
  • MEB (2009). İlköğretim matematik programı, www.meb.gov.tr. 7 Mart 2009.
  • National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, Va: NCTM
  • Perso, T. (1992). Using diagnostic teaching to overcome misconceptions in algebra, The Mathematical Association of Western Australia.
  • Philipp, R. A. (1999). The many use of algebraic variables, In B. Moses (Ed.), Algebraic thinking, grades 9-12: Readings from NCTM’s school based journals and other publications,150-156, Reston, VA: NCTM.
  • Saenz-Ludlow, A. & Walgamuth, C. (1998). Third grader’s ınterpretation of equality and the equals sign, Educational Studies in Mathematics, 35, 153-187.
  • 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, 21, 1-36.
  • Sfard, A. & Linchevski, L. (1994). the gains and the pitfalls of reification-the case of algebra, Educational Studies in Mathematics, 26(2), 191-228.
  • Skemp, R. (1987). The psychology of learning mathematics, Lawrence Erlbaum Associates, Hillsdale, NJ.
  • Stacey, K. & MacGregor, M. (1997). Building foundations for algebra, Mathematics in the Middle School, 2, 253 – 260.
  • Stacey, K. & MacGregor, M. (2000). Learning the Algebraic Method of Solving Problems, Journal of Mathematical Behaviour, 18(2), 149-167.
  • Swadener, M. & Soedjadi, R. (1988). Values, mathematics education and the task of developing pupils’ personalities: An Indonesian perspective, Educational StudiesIn Mathematics, 19(2), 193-208.
  • Tall, D., Gray, E., Bin Ali, M. B., Crowley, L., DeMarois, P., McGowen, M., Pitta, D., Pinto, M., Thomas, M. & Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking, Canadian journal of science, 1, 80-104, Mathematics and Technology Education.
  • Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In B. Moses (Eds.). Algebraic Thinking Grades K-12, 7-14. Reston, VA: NCTM.
  • Van Amerom, B. (2002). Reinvention of early algebra: Developmental research on the transition from arithmetic to algebra. Unpublished doctoral dissertation. University of Utrecht, The Netherlands.
  • Wagner, S. (1983). What are these things called variables? Mathematics Teacher, October, 474-478.
  • Wagner, S. & Kieran, C. (1989). Research issues in the learning and teaching of algebra. Reston, VA: NCTM.
  • Wagner, S. & Parker, S. (1993). Advancing algebra. In P. S. Wilson, (Ed.), Research ideas for the classroom: High school mathematics, 117-139, New York: Macmillan Publishing Company.
  • Williams, A. M. & Cooper, T. J. (2001). Moving from arithmetic to algebra under the time pressures of real classrooms. In H. Chick, K. Stacey, Jill Vincent, & John Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra, 665-662. Melbourne: University of Melbourne.
  • Yaman, H., Toluk, Z. ve Olkun, S. (2003). İlköğretim öğrencileri eşittir işaretini nasıl algılamaktadırlar? Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24,142-151.
  • Yıldırım, C. (2000). Matematiksel düşünme, Remzi Kitapevi, İstanbul. 373.
  • Zaskıs, R. & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions, Journal of Mathematical Behaviour, 17(4), 429-439.
There are 58 citations in total.

Details

Journal Section Research Article
Authors

Yaşar Akkan This is me

Adnan Baki This is me

Publication Date December 23, 2016
Submission Date January 2, 2017
Published in Issue Year 2016 Volume: 11 Issue: 2

Cite

APA Akkan, Y., & Baki, A. (2016). Ortaokul Öğrencilerinin Aritmetikten Cebire Geçiş Süreçlerinin İncelenmesi: Sembollerin Kullanımı ve Harflerin Anlamı. Bayburt Eğitim Fakültesi Dergisi, 11(2).