Matematik Öğretmen Adaylarının Örüntüler Bağlamında Genelleme ve Doğrulama Bilgileri

Year 2017, Volume: 5 Issue: 3, 195 - 222, 30.11.2017

Dilek Tanışlı , Nilüfer Yavuzsoy Köse , Faik Camci

Abstract

Bu çalışmanın amacı ortaokul matematik öğretmen adaylarının örüntüler bağlamında genellemelerini incelemek, genellemeler için ortaya koydukları doğrulamalarını keşfetmek ve yapılan genelleme ile doğrulama arasındaki ilişkiyi saptamaktır. Temel nitel araştırma yaklaşımının benimsendiği ve sekiz öğretmen adayı ile gerçekleştirilen bu çalışmada, verilerin toplanması klinik görüşme tekniği aracılığıyla gerçekleştirilmiş ve veri toplama aracı olarak doğrusal ilişki içeren ve içermeyen örüntü görevleri kullanılmıştır. Klinik görüşme verileri tematik analiz yöntemi kullanılarak analiz edilmiştir. Çalışma bulguları öğretmen adaylarının örüntüleri genelleme sürecinde belirgin ve belirgin olmayan muhakemeleri kullandıklarını ve doğrulama sürecinde ise tümdengelim ve tümevarım yöntemlerini benimsediklerini ortaya koymuştur. Belirgin muhakemeyi kullanan adayların doğrulama sürecinde genel olarak tümdengelim ve tümevarım yöntemlerini kullanırken, belirgin olmayan muhakemede bulunan adayların ise sadece tümevarım yöntemini kullandıkları bazı adayların da doğrulamalarının deneysel deliller ve dış otoriteden destek alma düzeyinde kaldıklarını göstermiştir. Çalışmada önemli görülen iki sonucun şekil aracılığıyla verilen örüntü görevlerinin genelleme ve doğrulama süreçlerini desteklemesi ve doğrulama yapmanın genelleme yapmayı katkı sağlaması olduğu söylenebilir. Öğretmen eğitiminde alan bilgisi ve alan öğretimine yönelik derslerde genelleme ve doğrulama süreçleri üzerinde adayların daha derin bir anlayış kazanmaları sağlanması önerilebilir.

Keywords

Genelleme, doğrulama, örüntüler, matematik öğretmen adayları

References

• Akkan, Y. (2016). Cebirsel düşünme. E. Bingölbali, S. Arslan ve İ. Ö. Zembat (Edt.), Matematik Eğitiminde Teoriler içinde (ss. 43-64). Ankara: Pegem Akademi.
• Aslan-Tutak, F. ve Köklü, O. (2016). Öğretmek için matematik bilgisi. E. Bingölbali, S. Arslan ve İ. Ö. Zembat (Edt.), Matematik Eğitiminde Teoriler içinde (ss. 701-719). Ankara: Pegem Akademi.
• Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics Teachers and Children (pp. 216–235). London: Holdder ve Stoughton.
• 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.
• Becker, J. R., & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. L. Chick, ve J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 4, pp. 121–128). Melbourne, Australia: University of Melbourne.
• Becker, J. R., & Rivera, F. (2006). Establishing and justifying algebraic generalization at the sixth grade level. In J.Novotna, H. Moraova, M. Kratka, ve N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 4, pp. 465–472). Prague:
• Charles University. Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. New York: Pearson Education.
• Blanton, M., & Kaput, J. (2002). Developing elementary teachers’ algebra “eyes and ears”: Understanding characteristics of professional development that promote generative and self-sustaining change in teacher practice. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA.
• Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai ve E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 5–23). Heidelberg, Germany: Springer.
• Carpenter, T.P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically: Integrating algebra and arithmetic in elementary school. Portsmouth, NH: Heinemann.
• 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. http://files.eric.ed.gov/fulltext/ED470471.pdf adresinden Aralık 2010’da ulaşılmıştır.
• 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.
• Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359–387.
• Chua, B. L., & Hoyles, C. (2012). The effect of different pattern formats on secondary two students’ ability to generalize. In T. Y. Tso (Ed.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 155–162). Taipei, Taiwan: PME.
• Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly ve R. A. Lesh (Eds), Handbook of research design in mathematics and science education (pp. 547–589). London: Lawrence Erlbaum Associates, Publishers.
• Coe, R., & Ruthven, K. (1994). Proof Practices and Constructs of Advanced Mathematics Students. British Educational Research Journal, 20(1), 41-53.
• de Castro, B. (2004). Pre-service teachers’ mathematical reasoning as an imperative for codified conceptual pedagogy in algebra: A case study in teacher education. Asia Pacific Education Review, 5(2), 157–166.
• Ellis, A. B. (2007a). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
• Ellis, A. B. (2007b). A taxonomy for categorizing generalizations: Generalizing actions and reflective generalizations. The Journal of the Learning Sciences, 16(2), 221–262.
• English, L., & Warren, E. (1995). General reasoning processes and elementary algebraic understanding: Implications for instruction. Focus on Learning Problems in Mathematics, 17(4), 1–19.
• Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23.
• Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, ve E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234–283). Providence, RI: American Mathematical Society.
• Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Charlotte, NC: Information Age Publishing.
• Harel, G., & Tall, D. (1989). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38–42.
• Kaput, J. (1999). Teaching and learning a new algebra with understanding. In E. Fennema, ve T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Erlbaum.
• Kieran, C. (1989). A perspective on algebraic thinking. In G. Vernand, J. Rogalski, ve M. Artigue (Eds.), Procedings of the 13th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 163 171), Paris. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.),The Handbook of Research on Mathematics Teaching and Learning (pp. 390–419). New York: Macmillan. Kinach, B.M. (2014). Generalizing: The core of algebraic thinking. The Mathematics Teacher, 107(6), 432 439.
• Kirwan, J. V. (2015). Preservice secondary mathematics teachers’ knowledge of generalization and justification on geometric numerical patterning tasks (Doctoral dissertation). Illinois State University, USA.
• Knuth, E. J., Slaughter, M., Choppin, J., & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In S. Mewborn, P. Sztajn, D.Y. White, H.G. Wiegel, R.L., Bryant,veK. Nooney (Eds.), Proceedings of the 24th Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (Vol. 4, pp. 1693-1700). Athens, GA.
• Koedinger, K. (1998). Intelligent cognitive tutors as modeling tools and ınstructional model. Invited paper for the National Council of Teachers of Mathematics Standarts 2000 Technology Conference.
• Lannin, J. K. (2003). Developing algebraic reasoning through generalization. Mathematics Teaching in the Middle School, 8(7), 342–348.
• Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258.
• Lannin, J., Barker, D., & Townsend, B. (2006). Algebraic generalization strategies: Factors influencing student strategy selection. Mathematics Education Research Journal, 18(3), 3–28.
• Li, X. (2011). Mathematical knowledge for teaching algebraic routines: A case study of solving quadratic equations. Journal of Mathematics Education, 4(2), 1–16.
• Liamputtong, P. (2009). Qualitative data analysis: Conceptual and practical considerations. Health Promotion Journal of Australia, 20(2), 133-139.
• Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, ve L. Lee (Eds.), Approaches to Algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Lee, L. & D. Wheeler: 1987, Algebraic thinking in high school students: their conceptions of generalisation and justification, Research report, Concordia University, Montreal. Mathematical Association: 1945, The Teaching of Algebra in Schools, London, G. Bell and Sons. (The report was first written in 1929.)
• Lo, J.-J., Grant, T. J., & Flowers, J. (2008). Challenges in deepening prospective teachers’ understanding of multiplication through justification. Journal of Mathematics Teacher Education, 11, 5–22.
• Maher, C. A., & Davis, R. B. (1990). Building representations of children’s meanings. In R. B. Davis, C. A. Maher, ve N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics. Reston, VA: NCTM.
• Martin, G.,& Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51.
• Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, ve L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Milli Eğitim Bakanlığı. (2013). Ortaokul matematik dersi (5-6-7-8. Sınıflar) öğretim programı. Ankara: MEB Yayınları.
• Merriam, S. B. (2009). Qualitative research and case study applications in education. San Francisso: Jossey-Bass. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school Mathematics. USA.
• Orton, A., & Orton, J. (1994). Student’s perception and use of pattern and generalization. In J. P. da Ponte, ve J. F. Matos (Eds.), Proceedings of the 18th Conference of the Psychology of Mathematics Education (Vol. 3, pp. 407-414). Lisbon, Portugal: PME.
• Pegg, J., & Redden, E. (1990). Procedures for, and experiences in, introducing algebra in New South Wales. Mathematics Teacher, 83, 386-391.
• Radford, L. (1996). Some reflections on teaching algebra through generalization. In N. Bednarz, C. Kieran, ve L. Lee (Eds.), Approaches to algebra (pp. 107–111). Dordrecht, The Netherlands: Kluwer.
• Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.
• Richardson, K. Berenson, S. & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28, 188–199.
• Rivera, F., & Becker, J. R. (2003). The effects of numerical and figural cues on the induction processes of preservice elementary teachers. In N. Pateman, B. Dougherty, ve J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Vol. 4, pp. 63 – 70). Honolulu, HI: University of Hawaii.
• Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2001). When tables become function tables. In M. v. d. Heuvel-Panhuizen (Ed.), Proceedings of the Twenty-fifth Conference of the International Group for the Psychology of Mathematics Education, (Vol. 4, pp. 145-152). Utrecht, The Netherlands: Freudenthal Institute.
• Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3–31.
• Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147–164.
• Stacey, K., & Macgregor, M. (1997, February). Building foundations for algebra. Mathematics Teaching in the Middle School, 2 (4), 252–260.
• Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31, 447–462.
• Tanışlı, D. (2011). Functional thinking ways in relation to linear function tables of elementary School students. Journal of Mathematical Behavior, 30(3), 206-223.
• Tanisli, D. (2016). How do students prove their learning and teachers their teaching? Do teachers make a difference? Eurasian Journal of Educational Research, 66, 47-70.
• Tanışlı, D. & Yavuzsoy Köse, N. (2011). Lineer şekil örüntülerine İlişkin genelleme stratejileri: Görsel ve sayısal ipuçlarının etkisi. Eğitim ve Bilim, 36(160), 184-198.
• Tanışlı, D. & Özdaş, A. (2009). İlköğretim beşinci sınıf öğrencilerinin örüntüleri genellemede kullandıkları stratejiler. Educational Sciences: Theory & Practice, 9(3), 1453-1497.
• Townsend, B.E., Lannin, J.K., & Barker, D.D. (2009). Promoting Efficient Strategy Use. Mathematics Teaching in the Middle School, 14(9), 542-547.
• Yıldırım, A. ve Şimşek, H. (2005). Sosyal bilimlerde nitel araştırma yöntemleri (Beşinci baskı). Ankara: Seçkin Yayıncılık.
• Yeşildere, S. ve Akkoç, H. (2011). Matematik öğretmen adaylarının şekil örüntülerini genelleme süreçleri. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 30, 141-153.
• Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.
• Waring, S. (2000). Can you prove it? Developing concepts of proof in primary and secondary schools. Leicester, UK: The Mathematical Association. Waring, S.,
• Orton, A., & Roper, T. (1999). Pattern and proof. In A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics (pp. 104–120). London, Cassell.