Ortaokul Öğretmenlerinin İspatla İlişkili Etkinliklere Katılımlarının Doğasının İncelenmesi

Yıl 2019, Cilt: 48 Sayı: 1, 100 - 130, 21.04.2019

Muhammed Fatih Dogan

Öz

Akıl yürütme ve ispat matematik öğrenmede ve öğretmede çok önemli olmasına ve okul matematiğinde daha fazla yer edinmiş olmasına rağmen, hem öğrenciler hem öğretmenler, ispatla ilgili etkinliklerde büyük zorluklarla karşılaşmaktadır. Bu tür zorluklara neden olan önemli potansiyel neden, öğretmenlerin ispat kavramı ile ilgili anlayışları olabilir. Buna rağmen ortaokulda görev yapan matematik öğretmenlerinin gerekçelendirme ve ispat kavramlarını nasıl öğrendiklerini araştıran çok az çalışma vardır. Bu nedenle, alan yazındaki boşluğu doldurmak için, bu çalışma ortaokul öğretmenlerinin akıl yürütme ve ispat ile ilgili etkinliklerle etkileşimlerine odaklanan yüksek lisans düzeyindeki bir mesleki gelişim dersinin gözlemsel verilerini incelemektedir. Bu çalışmadan elde edilen bulgular, ö ğretmenlerin ispat etkinliklerini çözmede oldukça başarılı olduklarını, ancak tümdengelimsel argümanlar üretmede zorlandıklarını göstermektedir. Bu başarısızlığın arkasındaki bazı nedenler, öğretmenlerin etkinliklere çözüm üretmeyi ispatı tamamlamak olarak görerek ispat yapma ihtiyacı duymamaları ve cebirsel ifadeleri (simgeleştirmeyi) matematiksel fikirleri ifade etmek için uygun bir araç olarak görmemeleri olabilir.

Anahtar Kelimeler

Matematik öğretmen eğitimi, Mesleki gelişim, Akıl yürütme, gerekçelendirme ve ispat

The Nature of Middle School In-Service Teachers’ Engagements in Proving-Related Activities

Öz

Although reasoning and proof in learning and teaching mathematics is crucial and have gained more presence in school mathematics, both students and their teachers face great difficulties when engaging in proving activities. One potential cause for such difficulties might be due to teachers’ conception of proof. However, to date, there are few, if any, studies that have examined how secondary school in-service mathematics teachers learn justification and proof. Thus, in order to fill this gap, this study examines secondary school in-service teachers’ engagement in proving activities by providing observational data from a master’s level professional development course that focuses on teaching reasoning and proof. The findings from this work show that teachers were very successful at engaging in exploration of the proving tasks, but they fail to produce complete-deductive arguments. Some reasons behind this failure were teachers’ lack of a perceived need for justification and proof after exploring the task, and their lack of seeing algebraic symbolization as a viable means of expressing mathematical ideas.

Anahtar Kelimeler

Mathematics teacher education, Professional Development, Reasoning, Justification and Proof

Kaynakça

• Alibert, D. (1988). Toward new customs in the classrooms. For the Learning of Mathematics, 8(2), 31-43.
• Alibert, D., & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.) Advanced Mathematical Thinking (pp. 215-230). Kluwer: The Netherlands.
• Ball, D., Hoyles, C., Jahnke, H., & Movshovitz-Hadar, N. (2002). The teaching of proof. Paper presented at the International Congress of Mathematicians, Beijing, China.
• Bell, A. (1976). A study of pupils’ proof – explanations in mathematical situations. Educational Studies in Mathematics, 7, 23-40.
• Bieda (2010). Enacting proof-related tasks in middle school mathematics: challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351-382.
• Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359-387.
• Chazan, D., & Lueke, H. M. (2009). Exploring tensions between disciplinary knowledge and school mathematics: Implications for reasoning and proof in school mathematics. Teaching and learning mathematics proof across the grades, 21-39.
• Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Council of Chief State School Officers.
• Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches. London: Sage Publications.
• Dogan, M. F. (2015). The Nature of Middle School In-Service Teachers' Engagements in Proving-Related Activities. Unpublished doctoral dissertation, Doctoral dissertation, University of Wisconsin-Madison, USA.
• Dogan, M.F. (2017). Learning Proof as Collective Mathematical Activity. International Conference on Mathematics and Mathematics Education (ICMME-2017), (pp. 910-912), 11-13 May 2017, Harran University, Şanlıurfa.
• Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: strategies for qualitative theory. New Brunswick: Aldine Transaction.
• Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.
• Hanna, G. (1995). Challenges to the importance of proof. For the learning of mathematics, 15(3), 42-49.
• Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23.
• Hanna, G. (2018). Reflections on proof as explanation. In Advances in Mathematics Education Research on Proof and Proving (pp. 3-18). Springer, Cham.
• Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education, III (pp. 234-283). Washington DC: Mathematical Association of America.
• Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. Second handbook of research on mathematics teaching and learning, 2, 805-842.
• Healy L. & Hoyles C., (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.
• Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students?. Cognition and Instruction, 24(1), 73-122.
• Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389-399.
• Jones, K., & Herbst, P. (2012). Proof, proving, and teacher-student interaction: Theories and contexts. In Proof and proving in mathematics education (pp. 261-277). Springer Netherlands.
• Knuth, E. (2002a). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486-491.
• Knuth, E. (2002b). Teachers conceptions of proof in the context of secondary school mathematics. Journal for Research in Mathematics Education, 5(1), 61-88.
• Knuth, E. (2002c). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405.
• Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). New York, NY: Routledge.
• Martin, W.G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal of Research in Mathematics Education, 20(1), 41-51.
• Martin,T.S., McCrone, S.M.S., Bower, M.L.W., & Dindyal, J. (2005). The interplay of teacher and students actions in the teaching and learning of geometric proof. Educational Studies in Mathematics, 60, 95-124.
• Moore, R.C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249-266.
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• Ozgur, Z., Ellis, A.B., Vinsonhaler, R., Dogan, M.F., & Knuth, E. (2019, In Press). From examples to proof: Purposes, strategies, and affordances of example use. Journal of Mathematical Behavior.
• Polya G. (1954). Mathematics and plausible reasoning, Princeton University Press.
• Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589-598.
• Schoenfeld, A.H. (1994). What do we know about mathematics curricula? Journal of Mathematical Behavior, 13(1), 55-80.
• Selden A. & Selden J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36.
• Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3-31.
• Stylianides, A.J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(1), 289-321.
• Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9-16.
• Stylianides, G. J., & Silver, E. A. (2009). Reasoning-and-proving in school mathematics: The case of pattern identification. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 235–249). New York: Routledge.
• Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40, 314-352.
• Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–266). Reston, VA: National Council of Teachers of Mathematics.
• Tall, D. O., & Mejia-Ramos, J. P. (2006). The long-term cognitive development of different types of reasoning and proof. In Conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives, Essen, Germany.
• Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169.
• Yin, R. K. (2006). Case study methods. In J. Green, G. Camilli, & P. Elmore (Eds.), The handbook of complementary methods in education research (pp. 111- 122). Mahwah, NJ: Lawrence Erlbaum Associates.
• Weber, K. (2005). Problem-solving, proving, and learning: The relationship between problem-solving processes and learning opportunities in the activity of proof construction. The Journal of Mathematical Behavior, 24(3), 351-360.