Examination of Generalization Process in the Context of “ the Circles and Points” Problem

Year 2020, Autumn, 244 - 258, 18.12.2020

Handan Demircioğlu , Halid Tuncay

https://doi.org/10.21733/ibad.732665

Abstract

The aim of this study is to examine the generalization process of preservice mathematics teacher, mathematics teachers and mathematicians, the special situations they take, and the strategies they use in the context of “the circle and points” problem. It was carried out by case study, one of the qualitative research methods. The participants of the study consist of 5 people, one academician, two mathematics teachers and two preservice mathematics teacher. The data were collected with the problem of circles and dots. The data were obtained by clinical interview method, document review and observation. The findings showed that all participants generalized after four or five situations, did not verify in any particular case as a result of generalization and problems were important in developing generalization skills.

Keywords

Generalization, Generalization Process

Genelleme Sürecinin Çember ve Noktalar Problemi Bağlamında İncelenmesi

Abstract

Bu çalışmanın amacı matematik öğretmen adaylarının, matematik öğretmenlerinin ve matematikçinin genelleme sürecini, aldıkları özel durumları, kullandıkları stratejileri çember ve noktalar problemi bağlamında incelemektir. Nitel araştırma yöntemlerinden durum çalışması ile yürütülmüştür. Çalışmanın katılımcılarını bir akademisyen, iki matematik öğretmeni ve iki matematik öğretmeni adayı olmak üzere toplam 5 kişi oluşturmaktadır. Veriler çember ve noktalar problemi ile toplanmıştır. Veriler klinik görüşme yöntemi, doküman incelemesi ve gözlem ile gerçekleştirilmiştir. Elde edilen bulgular tüm katılımcıların dört veya beş durum sonrasında genelleme yaptıklarını, genelleme sonucunda herhangi bir özel durumda doğrulama yapmadıklarını ve genelleme becerilerini geliştirmede problemlerin önemli olduğunu göstermiştir

Keywords

Genelleme, Genelleme Süreci

References

• Bills, L., Ainley, J., & Wilson, K. (2006). Modes of algebraic communication—moving between natural language, spreadsheet formulae and standard notation. For the Learning of Mathematics 26(1), 41–46.
• Becker, J. R., & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. L. Chick, & 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: PME.
• Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches (2nd ed.). Sage Publications, Inc.
• Davydov, V. V. (1990). Types of generalisation in instruction: logical and phsycological problems in the structuring of school curricula. In: J. Kilpactrick (Ed.). Soviet studies in mathematics education, (2). Reston, VA: National Council of Teachers of Mathematics.
• Dörfler, W. (1991). Forms and means of generalization in mathematics. In Mathematical knowledge: Its growth through teaching (pp. 61-85). Springer Netherlands.
• Ellis, A. B. (2007). A taxonomy for categorizing generalizations: generalizing actions and reflection generalizations. The Journal of The Learning Sciences, 16 (2), 221–262.
• Garcia-Cruz, J. A., & Martinon, A. (1998). Levels of genaralizations in linear patterns. Proceeding of the 22 nd Conference of the International Group for the Psychology of Mathematics Education, 2, 329-336.
• Kaput, J. J. (1999). Teaching and learning a new algebra. In E. L. Fennema, & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp.133–156). Mahwah, NJ: Lawrence Erlbaum
• Krutetskii, V. A. (1976). The Psychology of Mathematical Abilities in School Children. Chicago, IL: Universty of Chicago Press
• Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.) Approaches to algebra: Perspectives for research and teaching (pp. 87–106). Dordrecht, Netherlands: Kluwer.
• Lobato, J. E. (2006). Alternative perspectives on the transfer of learning: History, issues, and challenges for future research. The Journal of the Learning Sciences, 15(4), 431-449.
• Lobato, J. (2003). How design experiments can inform a rethinking of transfer and vice versa. Educational Researcher, 32(1), 17-20.
• Malara, N. A. (2012). Generalization processes in the teaching/learning of algebra: students behaviours and teacher role. In: Maj-Tatsis B., Tatsis K. (Eds.) Generalization in mathematics at all educational levels, Wydawnictwo Uniwersitetu Rzeszowskiego, Rzeszów (Poland), 57-90.
• Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, and L. Lee, (Eds.), Approaches to Algebra: Perspectives for Research and Teaching (pp. 65-86). Kluwer: Dordrecht/Boston/London.
• Mason J., Burton L., & Stacey K., (2010), Thinking Mathematically, Addison Wesley, London.
• Mason J & Pimm D. (1984) Generic Examples: seeing the general in the particular, Journal of Educational Studies 15 (3) p277-289.
• Polya, G. (1957). How to solve it: A new aspect of mathematical method. Princeton, N.J:Princeton University Press.
• Radford, L. (2010). Layers of generality and types of generalization in pattern activities. In P. Brosnan, D. B. Erchickk, & L. Flevares (Eds.), 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 37–62). Columbus: PME-NA
• Sriraman, B. (2004). Reflective abstraction, uniframes and the formulation of generalizations. The Journal of Mathematical Behavior, 23, 205–222.
• Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.
• Stacey, K., & MacGregor, M. (2001). Curriculum reform and approaches to algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 141–154). Dordrecht, Netherlands: Kluwer.
• Strachota, S. (2016). Conceptualizing Generalization. International Mathematical Virtual Institute Open Mathematical Education Notes, 6(1), 41-55.
• Stylianides, A. J. (2009). Breaking the equation "empirical argument = proof." Mathematics Teaching, 213, 9-14.
• Stylianides, G. J., & Stylianides, A. J. (2008). Enhancing undergraduate students’ understanding of proof. Electronic proceedings of the 11th Conference on Research in Undergraduate Mathematics Education (http://mathed.asu.edu/crume2008/Proceedings/Stylianides&Stylianides_LONG(21).pdf). San Diego, California.
• Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40 (3).
• Varhol, A., Gunnar O. G. & Hansen, M. N. (2020) Discovering key interactions. How student interactions relate to progress in mathematical generalization Mathematics Education Research Journal https://doi.org/10.1007/s13394-020-00308-z
• Weinberg, P. (2019) Generalizing and Proving in an Elementary Mathematics Teacher Education Program: Moving Beyond Logic. EURASIA Journal of Mathematics, science and technology education, 15(9).
• Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379- 402.
• Zazkis,R. Liljedahl,P. Chernoff,E. (2007). The role of examples in forming and refuting generalizations. ZDM Mathematics Education. DOI 10.1007/s11858-007-0065-9.