Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri

Yıl: 2010 Cilt: 10 Sayı: 2 Sayfa Aralığı: 729 - 769 Metin Dili: Türkçe İndeks Tarihi: 29-07-2022

Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri

Öz:
Çalışmanın amacı, lise matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili pedagojik alan bilgilerini saptamaktır. Çalışma öncesinde KKTC devlet okullarındaki 639 lise son sınıf öğrencisine “a ≠ 0 koşuluyla, $a^0$ =1” ve “0!=1” denklemleri ile ilgili açıklamalarda bulunmaları ve “a ≠ 0 koşuluyla, a ÷ 0’ın ne olduğu” sorulmuştur. Öğrencilerin açıklamalarının oldukça yetersiz oluşu çalışmanın matematik öğretmenleriyle gerçekleştirilmesini anlamlı kılmıştır. Çalışmanın örneklemini KKTC’deki dört ilçedeki okullarda çalışan 58 lise matematik öğretmeni oluşturmaktadır. Öğretmenlerin $a^0$=1, 0!=1 denklemlerini ve a ÷ 0 işlemini nasıl öğrettikleri ile ilgili sorulara verdikleri yanıtlar, daha önce artaştırmacı tarafından tümevarımcı yaklaşımla gerçekleştirilen içerik analizi sonucu ortaya çıkan kategori ve temalar baz alınarak içerik analizi konusunda bilgi sahibi olan üç, son sınıf, matematik öğretmen adayı tarafından tümdengelimci yaklaşımla içerik analizinden geçirilmiştir. Analizler ($X^2$), öğretmenlerin önerdiği yaklaşımların kavramsal anlamadan uzak, öğrencileri ezbere yöneltebilecek potensiyelde olduğunu göstermiştir. Deneyimli öğretmenler yeni öğretmenlere göre kısmen de olsa daha kavramsal yaklaşımlar önermişlerdir. Bu çalışmadan elde edilen bulgular ışığında, matematiğin öğretildiği her ortamda kavramsal anlamanın ön planda düşünülmesi ve aynı zamanda matematiğin kuruluşu ve matematiksel düşünme üzerinde de durulması ve pedagojik alan bilgisinin geliştirilmesinde genel yaklaşımlardan çok, konu temelli pedagojik alan bilgisine ağırlık verilmesi önerilmektedir.
Anahtar Kelime: lise öğrencileri matematik öğretmenleri kavramsal yaklaşım alan bilgisi kuzey kıbrıs sıfırın öğretimi pedagojik bilgi pedagojik içerik bilgisi konu temelli matematik pedagojik alan bigisi

Konular: Eğitim, Eğitim Araştırmaları Matematik

Mathematics teachers' topic-specific pedagogical content knowledge in the context of teaching $a^0$, 0! and a ÷ 0

Öz:
The aim of this study is to explore high-school school mathematics teachers’ topic-specific pedagogical content knowledge. First, 639 high-school students were asked to give explanations about “$a^0$ = 1, 0! = 1” and “a ÷ 0” where a ≠ 0. Weak explanations by the students led to a detailed research on teachers. Fifty-eight high school mathematics teachers in Northern Cyprus were the participants. Th ey were asked to write how they teach the above topics to high school students. Th e researcher determined some categories and themes from the written explanations of the teachers using inductive content analysis. Then, three pre-service mathematics teachers had a four-hour training on deductive content analysis. The pre-service teachers went over the written explanations independently and used the pre-determined categories and themes to code the explanations in a deductive way. The results indicate ($X^2$) that experienced teachers propose more conceptually based instructional strategies than novice teachers. The results also indicate that strategies proposed by all the participants were mainly procedural, fostering memorization. Hence, giving teachers the opportunity to view and teach mathematics in a more constructivist way and more emphasis on topic-specifi c pedagogical content knowledge in teacher training programs are recommended.
Anahtar Kelime: pedagogical content knowledge topic-specific pedagogical content knowledge high school students mathematics teachers conceptual approach knowledge of field northern cyprus teaching zero pedagogical knowledge

Konular: Eğitim, Eğitim Araştırmaları Matematik
Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Bibliyografik
  • Adey, P. & Shayer, M. J. (1994). Really raising standards. London: Routledge.
  • An, S., Kulm, G., & Wu, Z. (2004). Th e pedagogical content knowledge of middle school mathematics teachers in China and the United States. Journal of Mathematics Teachers Education, 7, 145-172.
  • Arsham, H. (2008). Zero in four dimensions: Historical, psychological, cultural and logical. Retrieved December 17, 2008, from http://www.pantaneto.co.uk/issue5/ arsham.html.
  • Ball, D. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132-144.
  • Ball, D. L. & Bass, H. (2000). Interweaving content an 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.
  • Ball, D. L., Hill, H. H., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Mathematical Educator, Fall, 14-46.
  • Baturo, A. & Nason, R. (1996) Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31, 235-268.
  • Bolyard, J. J. & Packenham, P. S. (2008). A review of the literature on mathematics and science teacher quality. Peabody Journal of Education, 83, 509-535.
  • Boyer, C. B. (1968). A history of mathematics. New York: John Wiley & Sons. Brickhouse, N. W. (1990). Teachers’ beliefs about the nature of science and their relationship to classroom practice. Journal of Teacher Education, 41(3), 53-62.
  • Burton, M., Daane, C. T., & Giesen, J. (2008). Infusing math content into a methods course: Impacting content knowledge for teaching. Issues in the Undergraduate Mathematics Preparation of School Teachers: Th e Journal, 1 (Content Knowledge). Retrieved January 20, 2009, from http://www.k-12prep.math.ttu.edu.
  • Chinnappan, M. & Lawson, M. J. (2005). A framework for analysis of teachers’ geometric content knowledge and geometric knowledge for teaching. Journal of Mathematics Teacher Education, 8, 197-221.
  • Clark, C. M. & Peterson, P. L. (1986). Teachers’ thought processes. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd Edt., pp. 255-296). New York: Macmillan.
  • Clarke, B. (1995). Expecting the unexpected: Critical incidents in the mathematics classrooms. Unpublished doctoral dissertation, University of Wisconsin-Madison.
  • Cooney, T. & Wiegel, H. (2003). Examing the mathematics in mathematics teacher education. In A. Bishop et al. (Eds.), Second international handbook of mathematics education (pp. 795-828). Dordrecht: Kluwer Academic Publishers.
  • Çakmak, M. (1999). Novice and Experienced Teachers’ Strategies for Mathematics Teaching in English and Turkish Primary Classrooms. Unpublished doctoral thesis, Leicester University, England.
  • Davis, P. J. & Hersch, R. (1981). Th e mathematical experience. London: Penguin Books.
  • Eisenhart, M., Borko, H., Underhill, R., Brown, A., Jones, D., & Agard, P. (1993). Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding. Journal for Research in Mathematics Education, 24(1), 8-40.
  • Even, R. & Tirosh, D. (1995). Subject-matter knowledge and knowledge about the students as sources of teacher presentations of the subject-matter. Educational Studies in Mathematics, 29, 1-20.
  • Eves, H. (1983). Great moments in mathematics before 1650. Washington, DC: Th e Mathematical Association of America.
  • Fawns, R. & Nance, D. (1993). Teacher knowledge, education studies and advanced skills credentials. Australian Journal of Education, 37, 248-258.
  • Fennema, E., & Franke, M. L. (1992). Teacher’s knowledge and its impact. In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147-164). New York: Simon and Schuster Macmillan.
  • Fenstermacher, G. (1986). Philosophy of research on teaching: three aspects. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd Edt, pp. 37-49). New York: Macmillan.
  • Geddis, A. (1993). Transforming subject-matter knowledge: Th e role of pedagogical content knowledge in learning to refl ect on teaching. International Journal of Science Education, 15(6), 673-683.
  • Halpern, D. F. (1992). Enhancing thinking skills in the sciences and mathematics. London: Lawrence Erlbaum.
  • Hanushek, E. (1971). Teacher characteristics and gains in student achievement: Estimation using micro data. Th e American Economic Review, 61, 280-288.
  • Henry, B. (1969). Zero, the troublemaker. Th e Arithmetic Teacher, 16(5), 365-67.
  • Hiebert, J. (1986). Conceptual and procedural knowledge: Th e case of mathematics. Hillsdale, NJ: Erlbaum.
  • Hill, H. C., Ball, D. L., & Schilling, S. C. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specifi c knowledge of students. Journal for Research in Mathematics Education, 39(7), 372-400.
  • Hinton, P. R. (1996). Statistics explained. New York: Routledge.
  • Hitchinson, E. (1996). Pre-service teachers knowledge: A contrast of beliefs and knowledge of ratio and proportion. Unpublished doctoral thesis, University of Wisconsin-Madison.
  • Kaplan, R. (1999). Th e nothing that is: A natural history of zero. Oxford: Oxford University Press,
  • Lampert, M. (1988). Th e teacher’s role in reinventing the meaning of mathematical knowing in the classroom. In M. J. Behr et al. (Eds.), Proceedings of the 10th Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 433-480). DeKalb, IL: Northern Illinois University
  • Lantz, C. A. & Nebenzahl, E. (1996). Behavior and interpretation of the kappa statistics: resolution of the two paradoxes. Journal of Clinical Epidemiology, 49, 431.
  • Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.
  • Magnusson, S., Krajcik, J., & Borko, H. (1999). Nature, sources, and development of pedagogical content knowledge for science teaching. In J. Gess-Newsome & N. Lederman (Eds.), Examining pedagogical content knowledge (pp. 95-132). Dordrecht: Kluwer.
  • Mason, J. & Spence, M. (1999). Beyond mere knowledge of mathematics: Th e importance of knowing-to act in the moment. Educational Studies in Mathematics, 38, 135-161.
  • Munby, H., Russel, T., & Martin, A. K. (2001). Teacher knowledge and how it develops. In V. Richardson (Ed.), Handbook of research on teaching (4th Edt., pp. 877– 904). Washington, DC: American Educational Research Association.
  • Nakiboğlu, C. & Karakoç, Ö. (2005). Th e forth knowledge domain a teacher should have: Th e pedagogical content knowledge. Kuram ve Uygulamada Eğitim Bilimleri/ Educational Sciences: Th eory & Practice, 5(1), 201-206.
  • Nespor, J. (1987). Th e role of beliefs in the practice of teaching. Journal of Curriculum Studies, 19(4), 317-328.
  • Neubrand, J. (2006). Th e TIMSS 1995 and 1999 video studies: In search for appropriate units of analysis. In F. K. S. Leung, K. D. Graf, & F. J. Lopez-Real (Eds.), Mathematics education in diff erent cultural traditions: A comparative study of East Asia and the West. - Th e 13th ICMI Study (Vol. 9., pp. 291-318). Berlin, Heidelberg, New York: Springer.
  • Neubrand, M. (2008, March). Knowledge of teachers – knowledge of students: Conceptualizations and outcomes of a mathematics teacher education study in Germany. Paper presented at the Symposium on the Occasion of the 100th Anniversary of ICMI (Rome, March 5-8, 2008) Workıng Group 2: Th e professional formation of teachers, Carl-von-Ossietzky-University, Oldenburg, Germany.
  • Nickerson, R., Perkins, D., & Smith, E. (1985). Th e teaching of thinking. Hillsdale, NJ: Erlbaum.
  • Ore, O. (1988). Number theory and its history. New York: Dover Publications.
  • Pogliani, L., Randić, M., & Trinajstić, N. (1998). Much ado about nothing-an introductive inquiry about zero. International Journal of Mathematics Education, Science and Technology, 29(5), 729-744.
  • Potari, D., Zachariades, T., Christou, K., Kyriazis, G., & Pitta-Padazi, D. (2007). Teachers’ mathematical and pedagogical awarness in calculus teaching. Proccedings of CERME 5, Larnaca, Cyprus.
  • Quinn, J. R., Lamberg, T. D., & Perrin, J. R. (2008). Teacher perceptions of division by zero. Th e Clearing House, 81(3), 101-104.
  • Reys, R. E. (1974). Division and zero-An area of needed research. Th e Arithmetic Teacher, 21(2), 153-56.
  • Rowland, T., Huckstep, P., & Th waites, A. (2004). Refl ecting on prospective elementary teachers’ mathematics content knowledge. In M. J. Høines and A. B. Fugelstad, (Eds.), Proceedings of the 28th International Group for the Psychology of Mathematics Education (Vol. 4, pp. 121-128). Bergen, Norway: Bergen University College.
  • Rowland, T., Huckstep, P., & Th waites, A. (2005). Elementary teachers’ mathematical subject knowledge: Th e knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255-281.
  • Sanders S. E. & Morris H. J. (2000). Exposing student teachers’ content knowledge: Empowerment or debilitation? Educational Studies, 26(4), 397-408.
  • Schoenfeld, A. (1992). Learning to think mathematically. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning. New York: Macmillan. Seife, C. (2000). Zero: Th e biography of a dangerous ıdea. Penguin: Harmondsworth.
  • Shavelson, R. J. & Webb, N. M. (1991). Generalizability theory: A primer. Th ousand Oaks, CA: Sage.
  • Shulman, L. S. (1986). Th ose who understand: Knowledge growth in teaching. Educational Researcher, 15, 4-14.
  • Shulman, L. S. (1987). Knowledge and teaching: Foundation of the new reform. Harvard Educational Review, 57(1), 1-22.
  • Spence, M. (1996). Psychologizing algebra: Case studies of knowing in the moment. Unpublished doctoral dissertation, University of Wisconsin-Madison.
  • Sternberg, R. J. (1994). Answering questions and questioning answers: Guiding children to intellectual excellence. Phi Delta Kappan, 76(2), 136-138.
  • Strauss, A. & Corbin, J. (1990). Basics of qualitative research. London, UK: Sage Publications.
  • Strauss, R. P. & Sawyer, E. A. (1986). Some new evidence on teacher and student competencies. Economics of Education Review, 5, 41-48.
  • Suryanarayan, E. R. (1996). Th e brahmagupta polynomials. Fibonacci Quart, 34, 30- 39.
  • Swartz, R. J. & Parks, S. (1994). Infusing the teaching of critical and creative thinking into elementary instruction. Pacifi c Grove, CA: Critical Th inking Press & Software.
  • Tamir, P. (1988). Subject matter and related pedagogical knowledge in teacher education. Teaching and Teacher Education, 4(2), 99-110.
  • Tishman, S., Perkins, D., & Jay, E. (1995). Th e thinking classroom. Boston, MA: Allyn & Bacon.
  • Tobin, K. & Fraser, B. J. (1989). Barriers to higher-level cognitive learning in high school science. Science Education, 73(6), 659-682.
  • Tsamir, P., Sheffer, R. & Tirosh, D. (2000). Intuitions and undefi ned operations: Th e cases of division by zero. Focus on Learning in Mathematics, 22(1), 1-16.
  • Türnüklü, E. B. (2005). Th e relationship between pedagogical and mathematical content knowledge of pre-service mathematics teachers. Eurasian Journal of Educational Research, 21, 234-247.
  • Veal, W. R. & MaKinster, J. (1999). Pedagogical content knowledge taxonomies. Electronic Journal of Science Education, 3(4). Retrieved April 13, 2008, from http:// www.unr.edu/homepage/crowther/ejse/vealmak.html.
  • Vorob’Ev, N. N. (1961). Fibonacci numbers. New York: Blaisdell.
  • Wells, D. (1997). Th e penguin dictionary of curious and ınteresting numbers. (Rev. ed.). London: Penguin Books.
  • Wilson, S. & Floden, R. E. (2003). Creating effective teachers: Concise answers for hard questions. An addendum to the report “Teacher Preparation Research: Current Knowledge, Gaps, and Recommendations.” Denver, CO: Education Commission of the States. (ERIC Document Reproduction Service No. ED 476366)
  • Yıldırım, A. & Şimsek, H. (2008). Sosyal bilimlerde nitel araştırma yöntemleri (7. basım). Ankara: Seçkin Yayınları.
  • Zohar, A. (2004). Higher order thinking in science classrooms: Students’ learning and teachers’ professional development. Dordrecht: Kluwer Academic Press.
APA CANKOY O (2010). Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. , 729 - 769.
Chicago CANKOY Osman Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. (2010): 729 - 769.
MLA CANKOY Osman Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. , 2010, ss.729 - 769.
AMA CANKOY O Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. . 2010; 729 - 769.
Vancouver CANKOY O Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. . 2010; 729 - 769.
IEEE CANKOY O "Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri." , ss.729 - 769, 2010.
ISNAD CANKOY, Osman. "Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri". (2010), 729-769.
APA CANKOY O (2010). Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. Kuram ve Uygulamada Eğitim Bilimleri, 10(2), 729 - 769.
Chicago CANKOY Osman Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. Kuram ve Uygulamada Eğitim Bilimleri 10, no.2 (2010): 729 - 769.
MLA CANKOY Osman Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. Kuram ve Uygulamada Eğitim Bilimleri, vol.10, no.2, 2010, ss.729 - 769.
AMA CANKOY O Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. Kuram ve Uygulamada Eğitim Bilimleri. 2010; 10(2): 729 - 769.
Vancouver CANKOY O Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri. Kuram ve Uygulamada Eğitim Bilimleri. 2010; 10(2): 729 - 769.
IEEE CANKOY O "Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri." Kuram ve Uygulamada Eğitim Bilimleri, 10, ss.729 - 769, 2010.
ISNAD CANKOY, Osman. "Matematik öğretmenlerinin $a^0$, 0! ve a ÷ 0 ile ilgili konu temelli pedagojik alan bilgileri". Kuram ve Uygulamada Eğitim Bilimleri 10/2 (2010), 729-769.