Pre-Servıce Prımary Teachers? Opınıons on Proof: Formal Proof-Enactıve Proof

Year 2014, Issue: 30, 23 - 55, 11.07.2014

Bekir Doruk , Yasemin Kıymaz Tuğba Horzum , Zekiye Morkoyunlu

Abstract

The idea of incorporating proof into students‟ mathematical experiences from the beginning of their schooling raises importance of proof in the early grades. In this respect, primary school teachers have a significant role. Therefore, it might be valuable to investigate the pre-service primary school teachers‟ opinions on both formal and enactive proofs. To this end, 184 pre-service primary school teachers were given a questionnaire on mathematical proofs. Then an enactive proof activity was carried out. Thirty participants took part in this activity. Semi structured interviews were conducted with eleven of the participating pre-service teachers. Based on the findings, pre-service teachers believe proofs are important and necessary for explaining mathematical concepts. However, the majority of the pre-service teachers thought that it was unnecessary to try to prove the propositions already proved. They do not trust themselves on proving. Based on the findings obtained from interviews, pre-service teachers receiving high or low scores from the formal proof questionnaire believe that if possible, proofs should be supported by enactive proofs that this is enjoyable and contribute to meaningful learning and making students love mathematics. Based on conclusions, some suggestions were put forward to overcome difficulties that students face with proof and to help the pre-service primary teachers develop positive attitude towards proof.

Keywords

Pre-service primary teachers, proving, formal proof, enactive proof

SINIF ÖĞRETMENİ ADAYLARININ İSPATLA İLGİLİ GÖRÜŞLERİ: FORMAL İSPAT- TEMSİLİ İSPAT

Abstract

Okula başlamalarıyla birlikte öğrencilerin matematiksel deneyimleri içerisine ispatı yerleştirme fikri erken dönemde ispat konusunun önemini arttırmaktadır. Çocuklara bu dönemlerinde eğitim verecek olan sınıf öğretmenlerine bu açıdan önemli görevler düşmektedir. Buradan hareketle hem formal ispat, hem de informal bir ispatlama etkinliği olan, fiziksel bir hareketi, görsel ve sözel desteği içeren, temsili ispatlara yönelik sınıf öğretmeni adaylarının görüşlerinin araştırılmasının yararlı olacağı düşünülmüştür. Bu amaçla 184 sınıf öğretmeni adayına matematiksel ispat yapmaya yönelik görüş anketi uygulanmış, katılımcılardan 30 kişilik bir grupla temsili ispat yapma etkinliği düzenlenmiştir.  Ardından etkinliğe katılan 11 öğretmen adayı ile yarı yapılandırılmış görüşmeler yapılmıştır. Anketten elde edilen veriler incelendiğinde sınıf öğretmeni adaylarının matematiksel bir sonucun doğruluğuna inanmada, matematiksel olguları açıklamada ispatın önemli ve gerekli olduğunu düşündükleri görülmüştür. Ancak adayların büyük bölümünün zaten ispatlanmış önermeleri kendilerinin ispatlamaya çalışmasının gereksiz olduğunu düşündüğü, ispat yapmayı sevmediği, sıkıcı bulduğu ve bu konuda kendilerine güvenmediği belirlenmiştir. Görüşmelerin analizinde gerek ispatla ilgili görüş anketinden yüksek puan alan, gerekse düşük puan alan öğretmen adaylarının ispatların mümkün olduğu sürece temsili ispatlarla desteklenmesi gerektiği, bunun oldukça eğlenceli olduğu ve anlamlı öğrenmeye, kalıcılığa, matematiği sevmeye katkı sağlayacağı görüşünde oldukları ortaya çıkmıştır.

Keywords

Sınıf öğretmen adayları, ispat yapma, formal ispat, temsili ispat

References

• matematik ve matematik öğretimi gibi derslerde, matematiksel ilişkileri, önermeleri
• açıklarken somut modellerden yararlanarak temsili ispatlar yapma gibi farklı ispatlama
• deneyimleri yaşamaları önemsenmelidir.
• Almeida, D. (2001). Pupils‟ Proof Potential, International Journal of Mathematical Education in Science and Technology, 32(1), 53-60.
• Altıparmak, K. ve Öziş, T. (2005). Matematiksel İspat ve Matematiksel Muhakemenin Gelişimi Üzerine Bir İnceleme. Ege Eğitim Dergisi, 6(1), 25-37.
• Arslan, Ç. (2007). İlköğretim Öğrencilerinde Muhakeme Etme ve İspatlama Düşüncesinin Gelişimi, Yayımlanmamış Doktora Tezi, Uludağ Üniversitesi, Bursa.
• Balacheff, N. (1991). The Benefits and Limits of Social İnteraction: The Case of Mathematical Proof. In A. Bishop, S. Mellin-Olsen ve J. Van Dormolen (Eds.), Mathematical Knowledge: Its Growth Through Teaching (175–192). The Netherlands: Kluwer AcademicPublishers.
• Balacheff, N. (1988). Aspects of Proof in Pupils‟ Practice of School Mathematics. In D. Pimm (ed.), Mathematics, Teachers and Children, Hodder and Stoughton, London, pp. 216– 235.
• Ball, D. L., Hoyles, C., Jahnke, H. N. and Movshovıtz-Hadar, N. (2002). The Teaching of Proof. In L. I. Tatsien (Ed.), Proceedings of the International Congress of Mathematicians (Vol. III). Beijing: Higher Education Press, pp. 907–920.
• Bruner, J. S. (1966). Toward a Theory of Instruction. Cambridge, MA: Belknap Press.
• Chazan, D. (1990). Quasi-Empirical Views of Mathematics and Mathematics Teaching. Interchange, 21(1), 14–23.
• Clements, D. H.(1999). „Concrete‟ Manipulatives, Concrete Ideas. Contemporary Issues in Early Childhood, Vol. 1, No 1, 45-60.
• De Villers, M. (1990). The Role and Function of Proof With Sketchpad. Pythagoras, vol.24, p.17 -24.
• Dienes, Z. P. and Golding, E. W. (1971). Approach to Modern Mathematics. New York: Herder and Herder.
• Francis, G. (1996). Mathematical Visualization: Standing At the Crossroads. Retrieved March 15, 2013, from http://www.oldweb. cecm.sfu.ca/projects/PhilVisMath/vis96panel.html .
• Fuson, K. C. And Briars, D. J. (1990). Using a Base-Ten Blocks Learning / Teaching Approach for First and Second Grade Placevalueand Multidigit Addition and Subtraction. Journal for Research in Mathematics Education, 21, 180–206.
• Gholamazad, S. (2007) . Pre-service Elementary School Teachers‟ Experiences with the Process of Creating Proofs. In Woo, J. H., Lew, H. C., Park, K. S. ve Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 265-272. Seoul: PME.
• Gholamazad, S., Liljedahl, P. and Zazkis, R. (2003). One Line Proof: What Can Go Wrong? In N.A. Pateman, B. J. Dougherty, and J. Zilliox (Eds.), Proceedings 27th Conf. of the Int. Group for the Psychology of Mathematics Education, Vol. 2, pp. 437-444, 13-18 July, 2003, Honolulu, HI.
• Güler, G., Özdemir E. ve Dikici, R., (2012). Öğretmen Adaylarının Matematiksel Tümevarım Yoluyla İspat Becerileri Ve Matematiksel İspat Hakkındaki Görüşleri. Kastamonu Eğitim Dergisi, 20(1), 219-236.
• Hanna, G. (2000). Proof, Explanation and Exploration: An Overview. Educational Studies in Mathematics, 44, 5–23.
• Hanna, G. (1995). Challenges to The Importance of Proof. For the Learning of Mathematics, 15(3), 42–49.
• Hawkins, M. (2007). Teaching Geometric Reasoning: Proof by Pictures, Unpublished Master Thesis, North Carolina State University, Raleigh, North Carolina, USA.
• Healy, L. and Hoyles, C. (2000). A Study of Proof Conceptions in Algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
• İskenderoğlu, T.A., Baki, A. ve Palancı, M. (2011). Matematiksel Kanıt Yapmaya Yönelik Görüş Ölçeği: Geçerlik ve Güvenirlik Çalışması, Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 5(1), 181-203.
• Jones, K. (2000). The Student Experience of Mathematical Proof at University Level, International Journal of Mathematical Education in Science and Technology, vol. 31, no.1, p. 53 -60.
• Knuth, E. J. (2002). Teachers‟ Conceptions of Proof in the Context of Secondary School Mathematics, Journal of Mathematics Teacher Education, 5, 1, 61-88.
• Martin, W. G. and Harel, G. (1989). Proof Frames of Preservice Elementary Teachers. Journal for Research in Mathematics Education, 20(1), 41–5
• Mejıa-Ramos, J. P. (2005). Aspects of Proof In Mathematics Research. Hewitt, D. (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 25(2), 61- 66.
• Moralı, S., Uğurel, I., Türnüklü, S. ve Yeşildere, S. (2006). Matematik Öğretmen Adaylarının İspat Yapamaya Yönelik Görüşleri. Kastamonu Eğitim Dergisi, 14(1), 147-160.
• Natıonal Councıl of Teachers of Mathematıcs [NCTM] (2000). Principles and Standards for School Mathematics. Reston, VA: Author.
• Palais, R.S. (1999). The Visualization of Mathematics: Toward a Mathematical Exploratorium. Notices of the AMS 46(6), 647-658
• Piaget, J. (1971). Biology and knowledge. Chicago: The University of Chicago Press.
• Raphael, D. and Wahlstrom, M. (1989). The Influence of Instructional Aids on Mathematics Achievement. Journal for Researchin Mathematics Education, 20, 173-190.
• Sarı, M., Altun, A.ve Aşkar, P. (2007). Undergraduate Students‟ Mathematical Proof Processes in a Calculus Course: A Case Study. Ankara University Journal of Faculty of Educational Sciences, 40(2), 295-319.
• Schoenfeld, A. (1994). What Do We Know About Mathematics Curricula? Journal of Mathematical Behavior, 13(1), 55–80.
• Selden, J., and Selden, A. (2009). Understanding The Proof Construction Process. In F.-L Lin, F.-J. Hsieh, G. Hanna, ve M. de Villiers (Eds.), Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education, Vol. 2. (pp. 196-201). Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
• Skemp, R. R. (1987). The Psychology of Learning Mathematics, Hillsdale, NJ: Lawrence Erlbaum.
• Sowder, L. and Harel, G. (1998). Types of Students‟ Justifications. Mathematics Teacher, 91(8), 670–675.
• Stylianides, G. J. (2007a). Investigating the Guidance Offered to Teachers in Curriculum Materials: The Case of Proof in Mathematics, International Journal of Science and Mathematics Education, 6, 191-215.
• Stylianides, A. J. (2007b). The Notion of Proof in the Context of Elementary School Mathematics. Educational Studies in Mathematics. 65: 1–20.
• Stylianides, G. J., Stylianides, A. J. and Philippou, G. N. (2007). Preservice Teachers‟ Knowledge of Proof by Mathematical Induction. Journal of Mathematics Teacher Education, 10, 145–166.
• Tall, D. and Mejia-Ramos, J. P. (2006). The Long-Term Cognitive Development of Different
• Types of Reasoning and Proof. Conference on Explanation and Proof in Mathematics:
• Philosophical and Educational Perspectives, Universitat Duisburg-Essen, Kasım 1–4, 2006.
• Tall, D. (1999). The Cognitive Development of Proof: Is Mathematical Proof for All or for
• Some? In Z. Usiskin (Ed.), Developments in School Mathematics Education Around the
• World, vol, 4, pp.117-136. Reston, Virginia: NCTM.
• Thompson, A. G. (1992). Teachers‟ Beliefs and Conceptions: A Synthesis of the Research. In D. A. Grows (ed.), Handbook of Research on Mathematics Teaching and Learning, pp. 127-146, New York: Macmillan.
• Turker, B., Alkas, Ç., Aylar, E., Gurel, R. ve Akkuş İspir, O. (2010). The Views of Elementary Mathematics Education Preservice Teachers on Proving. International Journal of Human and Social Sciences(423-427) 5:7.
• Weber, K. (2001). Student Difficulty in Constructing Proof: The Need for Strategic Knowledge. Educational Studies in Mathematics, 48, 101–119
• Wu, H. (1996). The role of Euclidean geometry in high school. Journal of Mathematical Behavior, 15, 221–237.
• Yıldırım, A. ve Şimşek, H. (2005). Sosyal Bilimlerde Nitel Araştırma Yöntemleri (Göz. Geç. 5. Bs.). Ankara: Seçkin Yayıncılık.