Ortaokul Matematik Öğretmenlerinin Aritmetik ve Cebir Problemleri Hakkındaki İnanışları

Yıl 2019, Cilt: 21 Sayı: 1, 156 - 176, 29.04.2019

Yaşar Akkan , Mesut Öztürk , Pınar Akkan Betül Küçük Demir

https://doi.org/10.17556/erziefd.431583

Öz

Öğretmenler, matematiği bilmenin ve anlamının
ne demek olduğunu, öğrencilerinin hangi matematiksel görevleri yapmaları
gerektiğini ve bilişsel olarak öğrencilerini zorlayan becerilerin nasıl
geliştirilmesi ve desteklenmesini de anlamalıdırlar. Çünkü öğrencilerin
nasıl/ne düşündüklerini ve yeteneklerinin ne olduğunu bilmek; öğretmenlerin
ders içeriklerine ve sunum stillerine etki etmektedir. Bu bağlamda çalışmanın amacı, öğrencilerin matematiksel içerik edinme
yetkinlikleri ile ilgili ortaokul matematik öğretmenlerinin inançlarını problem
çözme bağlamında farklı açılardan belirlemektir.  Çalışma, nicel araştırma desenlerinden
betimsel araştırma modeline göre yürütülmüştür. Araştırmanın örenklemini
32 ortaokul matematik öğretmeni oluşturmaktadır. Veri toplama aşamasında, Likert ve problem tipi sorulardan oluşan
literatür destekli anketlerden ve araştırmacı tarafından geliştirilen
açık uçlu anket sorularından yararlanılmıştır. Verilerin çözümlenmesinde
betimsel istatistik ve analiz yöntemleri kullanılmıştır. Öğretmenler öğrenciler
için cebir
problemlerin aritmetik problemlere göre, sözel problemlerin ise sembolik
problemlere göre daha zor olduğunu, problemdeki bilinmeyen niceliğin
pozisyonunun zorluk düzeylerini etkilediğini ve öğrencilerin kullanabilecekleri birden çok strateji olduğunu
belirtmişlerdir. Ayrıca öğretmenlerin çoğu, hikaye problemlerinin yeni anlamlar çıkarmada
sembolik problemlere göre daha uygun olduğunu ve sembol öncelikli eğilimi ile
“cebir en iyidir” görüşünü benimsemektedirler.

Anahtar Kelimeler

Aritmetik, Cebir, Problem Çözme

Kaynakça

• Akgün, L. (2006). Cebir ve değişken kavramı üzerine. Journal of Qafqaz University, 17(1). 25- 29.
• Akkan, Y. (2009). İlköğretim öğrencilerinin aritmetikten cebire geçiş süreçlerinin incelenmesi, Yayımlanmamış Doktora Tezi, Karadeniz Teknik Üniversitesi, Trabzon, Türkiye.
• Akkan, Y., Baki, A. ve Çakıroğlu, Ü. (2011). Aritmetik ile cebir arasındaki farklılıklar: Cebir öncesinin önemi. İlköğretim Online, 10(3), 812-823.Akkan, Y., Baki, A., Çakıroğlu, Ü. (2012). 5-8. sınıf öğrencilerinin aritmetikten cebire geçiş süreçlerinin problem çözme bağlamında incelenmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 43,01-13.
• Alexandrou-Leonidou, V., & Philippou, N. G. (2005). Teachers’ beliefs about students’ development of the pre-algebraic concepts of equation. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 41- 48). Melbourne, AU: PME.
• Baek, J. M. (2008). Developing algebraic thinking through explorations in multiplication. In C. E. Greenes (Ed.), Algebra and algebraic thinking in school mathematics (Vol. 70, pp. 141- 154). Reston, VA: NCTM.
• Ball, D. L. (1992). Teaching mathematics for understanding: What do teachers need to know about subject matter? In M. Kennedy (Ed.), Teaching academic subjects to diverse learners (pp. 63-83). New York, NY: Teachers College.
• Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.Baranes, R., Perry, M., & Stigler, J. W. (1989). Activation of real-world knowledge in the solution of word problems. Cognition and Instruction, 6, 287–318.Başol, G. (2015). Eğitimde ölçme ve değerlendirme, Ankara: Pegem Akademi
• Benckert, S. (1997). Context and conversation in physics education. 07.03.2018 tarihinde http://www.nshu.se/ download/3018/benckert_sylvia_97.pdf adresinden alınmıştır.
• Borko, H., & Livingston, C. (1989). Cognition and improvisation: Differences in mathematics instruction by expert and novice teachers. American Educational Research Journal, 26, 473–498.
• Borko, H., & Shavelson, R. (1990). Teacher decision making. In B. F. Jones & L. Idol (Eds.), Dimensions of thinking and cognitive instruction (pp. 311–346). Elmhurst, IL: North Central Regional Educational Laboratory and Hillsdale, NJ: Erlbaum.
• Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers’ classrooms. Journal for Research in Mathematics Education, 40(2), 119–156.
• Büyüköztürk, Ş., Çakmak, E.K., Akgün, Ö.E., Karadeniz, Ş., & Demirel,F. (2013). Bilimsel araştırma yöntemleri. (17. baskı). Ankara: Pegem Akademi.
• Cai, J. & Moyer, J. C. (2008). Developing algebraic thinking in earlier grades: Some insights from international comparative studies. In C. E. Greene & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 169- 182). National Council of Teachers of Mathematics 2008 Yearbook. Reston, VA: NCTM.
• Cai, J., & Moyer, J. C. (2007). Developing algebraic thinking in earlier grades: Some insight from international comparative studies. 07.03.2018 tarihinde http://citeseerx.ist.psu. edu/viewdoc /download?doi=10.1.1. 496.8095&rep=rep1&type=pdf adresinden alınmıştır.
• Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97, 3-20.
• Carpenter, T.P. & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. 11.08.2008 tarihinde www.wcer.wisc.edu/ncisla /publications/ index.html. adresinden alınmıştır.
• Carraher, D, W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37 (2), 87- 115.
• Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 21–29.
• Case, R. & Okamoto, Y. (1996). The role of central conceptual structures in the development of children’s thought, Monographs of the Society for Research in Child Development, 61(1–2, Serial No. 246).
• Case, R. (1991a). General and specific views of the mind, its structure and its development. In R. Case (Ed.), The mind's staircase (pp. 3-15). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Cotton, T. (1993). Children’s impressions of mathematics. Mathematics Teacher, 143, 14-17.
• Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ beliefs structures and their influence on instructional practices. Journal Math Teacher Education, 12, 325 – 346.
• De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics teaching and learning. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 491–549). New York: Simon & Schuster Macmillan.
• Fenstermacher, G. (1979). A philosophical consideration of recent research on teacher effectiveness. In L. S. Shulman (Ed.), Review of educational research (Vol. 6, pp. 157–185). Itasca, IL: Peacock.
• Garner, R., & Alexander, P. A. (1994). Beliefs about text and instruction with text. Hillsdale, NJ: Lawrence Erlbaum Associates.
• Gomez-Chacon, I. (2000). Affective influences in the knowledge of mathematics. Educational Studies in Mathematics 43, 149–168.
• Greeno, J. G., Collins, A. M., & Resnick, L. (1996). Cognition and learning. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 15–46). New York: Simon & Schuster Macmillan.
• Hall, R., Kibler, D., Wenger, E., & Truxaw, C. (1989). Exploring the episodic structure of algebra story problem solving. Cognition and Instruction, 6, 223–283.Hersovics, N., & Linchevski, L. (1994). A cognative gap between arithmetic and algebra. Educational Studies in Mathematics, 27 (1), 59-78.
• Johnson, B. & Christensen, L. (2004). Educational research: Quantitative, qualitative and mixed approaches. Pearson Education, Inc., Second Edition, 562 p, Boston.
• Kalchman, M., & Case, R. (1998). Teaching mathematical functions in primary and middle school: An approach based on neo-Piagetian theory. Scientia Paedagogica Experimentalis, 35, 7–54.
• 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, (pp. 119-139). New York: Macmillan.
• Kieran, C. (1991). A procedural-structural perspective on algebra research. In Furinghetti, F. (Ed.), Proceedings of the Fifteenth International Conference for the Psychology of Mathematics Education, Genoa, Italy, 2, 245–253.
• Kieran, C. (1992). The learning and teaching of school algebra. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York: Macmillan.
• Koedinger, K. R., Nathan, M. J., & Tabachneck, H. J. M. (1996). Early algebra problem solving: A difficulty factors analysis (Tech. Rep.). Pittsburgh, PA: Carnegie Mellon University.
• Koedinger, K.R., & Nathan, M.J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of The Learning Sciences, 13(2), 129-164.
• Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra. The Journal of Mathematical Behaviour, 14, 113-120.
• Lodholz, R. D. (1993). The transition from arithmetic to algebra. E.L. Edwards (Ed.), Algebra for everyone, 24-33. Reston, VA: NCTM.
• McMillan, J.W. & Schumacher, S. (2014). Research in education: Evidence-based inquiry. Boston: Pearson
• Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30, 122–147.
• Nathan, M. J. (2003). Confronting teachers’ beliefs about algebra development: Investigating an approach for professional development (Technical Report No. 03- 04). Boulder, CO: University of Colorado, Institute of Cognitive Science.
• Nathan, M. J., & Koedinger, K. R. (2000a). An investigation of teachers’ beliefs of students’ algebraic development. Cognition and Instruction, 18, 209- 237.
• Nathan, M. J., & Koedinger, K. R. (2000b). Teachers’ and researcher’s beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31(2), 168 -190.
• Nathan, M. J., & Koellner, K. (2007). A framework for understanding cultivating the transition from arithmetic to algebraic reasoning. Mathematical Thinking and Learning, 9(3), 179- 192.
• Nathan, M. J., & Petrosino, A. J. (2003). Expert blind spot among preservice teachers. American Educational Research Jounal, 40, 905- 928.
• Nathan, M. J., Kintsch, W., & Young, E. (1992). A theory of algebra-word-problem comprehension and its implications for the design of learning environments. Cognition and Instruction, 9, 329–389.
• Nathan, M. J., Knuth, E., & Elliott, R. (1998, April). Analytic and social scaffolding in the mathematics classroom: One teacher’s changing practices. Presentation at the annual meeting of the American Educational Research Association, San Diego, CA.
• Nathan, M., & Knuth, E. (2003). The study of whole classroom mathematical discourse and teacher change. Cognition & Instruction, 21(2), 175–207.
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• Philipp, R. A. (2008). Motivating prospective elementary school teachers to learn mathematics by focusing upon children’s mathematical thinking. Issues in Teacher Education, 17 (2), 7- 24.
• Raymond, A. M. (1997). Inconsistency between a beginning elementary school teacher's mathematics beliefs and teaching practices. Journal for Research in Mathematics Education, 28(6), 552-575.
• Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children’s problem-solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153–196). New York: Academic Press.
• Sakellis, F.M. (2011). Teachers’ beliefs about students’ cognition from arithmetic to algebraic concepts. Unpublished doctoral dissertation, Wilmington University, US.
• Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4, 1–94.
• Sherin, M. G. (2004). New perspectives on the role of video in teacher education. In J. Brophy (Ed.), Using video in teacher education (pp. 1–28). Amsterdam: Elsevier.
• Sherin, M. G. (2007). The development of teachers' professional vision in video clubs. In R. Goldman, R. Pea, B. Barron, & S. Derry (Eds.), Video research in the learning sciences (pp. 383-395). Hillsdale, NJ: Lawrence Erlbaum.
• Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15 (2), 4-14.
• Stacey, K., & MacGregor, M. (2000). Learning the algebraic method of solving problems. Journal of Mathematical Behaviour, 18 (2), 149-167.
• Sutherland, R., & Rojano, T.A. (1993). Spreadsheet approach to solving algebraic problems. The Journal of Mathematics Behavior, 12(4), 353-383.
• Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 127–146). New York: Macmillan.
• Usiskin, Z. (1999). Why is algebra important to learn. B. Moses (Ed.), Algebraic thinking: Grades 9-12, Readings from NCTM’s School Based Journals and Other Publications (pp. 22-30). 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.
• Van De Walle, J. A., Karp, K. S. & Bay-Williams J. M. (2013). İlkokul ve ortaokul matematiği. S. Durmuş (Çev.). Ankara: Nobel Yayın Dağıtım.
• Williams, A. & Cooper, T. (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, (pp.665-662). Melbourne: University of Melbourne.