İspat İmajının Oluşumunda Formal Bilginin Rolü: Sonsuz Kümeler Bağlamında Bir Durum Çalışması

Yıl 2020, Cilt: 11 Sayı: 3, 584 - 618, 15.12.2020

Ozan Pala , Serkan Narlı

https://doi.org/10.16949/turkbilmat.702540

Öz

Matematik eğitimi çalışmalarında ispatlamanın önemine sıklıkla vurgu yapılmasına rağmen araştırmalar üniversite öğrencilerinin bu konuda güçlük çektiğini göstermektedir. İspat sürecinde yaşanan bu güçlüklerin araştırmacılar tarafından ayrıntılı olarak ele alınması sayesinde ispatı farklı perspektiflerden değerlendiren birçok görüş sunulmuştur. Bunlardan biri olan ve ispat sürecinde hem bilişsel hem duyuşsal faktörleri dikkate alan ispat imajı, Kidron ve Dreyfus (2014) tarafından “bağlamda soyutlama” teorik çerçevesi bağlamında sunulmuştur. Ancak, ispat imajı ile formal bilgi arasındaki bağlantı yazarlar tarafından derinleştirilmediğinden, bu makalede bu boşluğun doldurulması amaçlanmıştır. Daha geniş bir doktora tez çalışmasının parçası olan bu çalışma, betimsel türde nitel bir araştırmadır. Çalışmanın katılımcıları ilköğretim matematik öğretmenliği ikinci sınıf öğrencileri arasından ölçüt örnekleme yöntemi ile seçilen üç öğretmen adayıdır. Bu katılımcıların, Cantor Küme Teorisi bağlamında aldıkları bir derse paralel olarak sonsuz kümelerin denkliğine dair etkinlik temelli bireysel mülakatlar (Uygulama I-II-III-IV) gerçekleştirilmiştir. Sonsuzluk konusu, ispat imajının doğasına uygun olarak sezgiselden formele giden bir çerçeveyi barındırdığından çalışmanın bağlamı olarak tercih edilmiştir. İlk çalışmada (Uygulama I) katılımcıların yeterli ön bilgiye sahip olmadıkları durumda gerçekleştirecekleri eylemlerin ispat imajı açısından incelenmesi sağlanmıştır. Temel bilgilerin sunulduğu bir dersin ardından gerçekleştirilen ikinci çalışmada (Uygulama II) katılımcılara aynı soru tekrar yöneltilmiş ve böylece onların formal bilgiye sahipken oluşturdukları süreçlerin incelenmesi sağlanmıştır. Her iki uygulamanın verileri üzerinde yapılan betimsel analizler sonucunda katılımcılardan Ç’nin ilk uygulamada bir ispat imajına sahip olmamasına rağmen ikinci uygulamada sahip olduğu belirlenmiştir. Bu nedenle bu çalışmada onun ispat süreçlerine dair bulgular paylaşılmıştır. Sonuç olarak formal bilginin, ispat imajının oluşumuna olanak veren bileşenlerin her biri ile doğrudan bağlantılı olduğu belirlenmiş ve temel katkıları başlıklar halinde sıralanmıştır.

Anahtar Kelimeler

İspat İmajı, Sonsuzluk, Cantor Küme Teorisi, İspat, Matematik Eğitimi

Destekleyen Kurum

TUBİTAK BİDEB 2211 ve Dokuz Eylül Üniversitesi

Proje Numarası

2019.KB.EGT.008

Role of the Formal Knowledge in the Formation of the Proof Image: A Case Study in the Context of Infinite Sets

Öz

Although the emphases on the importance of proving in mathematics education literature, many studies show that undergraduates have difficulty in this regard. Having researchers discussed these difficulties in detail; many frameworks have been presented evaluating the proof from different perspectives. Being one of them the proof image, which takes into account both cognitive and affective factors in proving, was presented by Kidron and Dreyfus (2014) in the context of the theoretical framework of “abstraction in context”. However, since the authors have not deepened the relationship between the proof image and formal knowledge, this article was intended to fill this gap. In this study, which is part of a larger doctoral thesis, descriptive method one of the qualitative methods was used. The participants of the study were three pre-service teachers selected via criterion sampling method among sophomore elementary school mathematics teacher candidates. In parallel with a course relating to Cantorian Set Theory, task-based individual interviews (Task I-II-III-IV) were conducted in the context of the equivalence of infinite sets. The subject of "infinity" had been chosen as the context of the study since it contains a process that goes from intuitive to formal. In the first task (Task I), the actions that the participants had performed without enough pre-knowledge was examined in terms of the proof image. In the second task (Task II) carried out after a course, in which basic knowledge was presented, the same question was asked to the participants again. Thus, the processes formed with the presence of formal knowledge were analysed. As a result of the descriptive analysis executed on the data of both tasks, it was determined that Ç, who was one of the participants, reached a proof image in the second task although she failed in the first task. Therefore, in this study, findings of her proving activity were shared. Consequently, formal knowledge has been identified to be directly related to each of the components of the proof image and, its main contributions have been listed as headings.

Anahtar Kelimeler

Proof Image, Infinity, Cantorian Set Theory, Proof, Mathematics Education

Proje Numarası

2019.KB.EGT.008

Kaynakça

• Almeida, D. (2000). A survey of mathematics undergraduates’ interaction with proof: Some implications for mathematics education. International Journal of Mathematical Education in Science and Technology, 31(6), 896-890.
• Altun, M. (2005). Matematik öğretimi. Bursa: Aktüel Yayınevi.
• Antonini, S. & Mariotti, M. A. (2007). Indirect proof: An interpreting model. In D. Pitta-Pantazi, & G. Philippou (Eds.), Proceedings of the 5th Congress of the European Society for Research in Mathematics Education (pp. 541–550). Cyprus: Larnaca.
• Atwood, P. R. (2001). Learning to construct proofs in a first course on mathematical proof (Unpublished doctoral dissertation). Western Michigan University, USA.
• Baker, D., & Campbell, C. (2004). Fostering the development of mathematical thinking: Observations from a proofs course. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 14(4), 345-353.
• Barnard, A. D. & Tall, D. O. (1997). Cognitive Units, Connections, and Mathematical Proof. In E. Pehkonen, (Ed.), Proceedings of the 21st Annual Conference for the Psychology of Mathematics Education, Vol. 2 (pp. 41–48). Lahti, Finland.
• Bikner-Ahsbahs, A. (2004). Towards The Emergence of Constructing Mathematical Meanings. In M. J. Hoines and A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Pychology of Mathematics Education, Vol. 2 (pp. 119-126). Bergen, Norway.
• Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö. E., Karadeniz, Ş. ve Demirel, F. (2013). Bilimsel araştırma yöntemleri. Ankara: Pegem Akademi Yayınevi.
• Clore, G. L. (1992). Cognitive phenomenology: Feelings and the construction of judgment. In L. L. Martin, & A. Tesser (Eds.), The construction of social judgments (pp. 133-163). Hillsdale, NJ: Erlbaum.
• Davydov, V. V. (1990). Types of Generalization in Instruction: Logical and Psychological Problems in the Structuring of School Curricula. In J. Kilpatrick (Ed.) and J. Teller (Trans.), Soviet Studies in Mathematics Education: Volume 2. Reston, VA: NCTM. (Original work published 1972).
• Doruk, M. ve Kaplan, A. (2017). İlköğretim matematik öğretmeni adaylarının analiz alanında yaptıkları ispatların özellikleri. Mehmet Akif Ersoy Üniversitesi Eğitim Fakültesi Dergisi, 44, 467-498.
• Dreyfus, T. (1999). Why Johnny can’t prove. Educational Studies in Mathematics, 38(1), 85-109.
• Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9-24.
• Fischbein, E. (1994). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. W. Scholz, R. Sträßer & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 231 245). Dordrecht, The Netherlands: Kluwer.
• Cantor, G. (1891). "Ueber eine elementare frage der mannigfaltigkeitslehre". Jahresbericht der Deutschen Mathematiker-Vereinigung, 1, 75-78.
• Güler, G. & Dikici, R. (2014). Examining prospective mathematics teachers' proof processes for algebraic concepts. International Journal of Mathematical Education in Science and Technology, 45(4), 475-497.
• Güler, G., Özdemir, E. ve Dikici, R. (2012). Öğretmen adaylarının matematiksel tümevarım yoluyla ispat becerileri ve matematiksel ispat hakkındaki görüşleri. Kastamonu Eğitim Dergisi, 20(1), 219-236.
• Güney, Z. ve Özkoç, M. (2015). Soyut Matematik. İzmir: Dinozor Kitabevi.
• 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. Research in Collegiate Mathematics Education, 3(7), 234-282.
• 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). Greenwich, CT: Information Age.
• Hart, E. W. (1994). A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory. In J. J. Kaput & E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results (pp.49–62.), Washington, DC: Mathematical Association of America.
• Herbst, P. (2002). Engaging students in proving: A double bind on the teacher. Journal for Research in Mathematics Education, 33(3), 176-203.
• Hershkowitz, R., Schwarz, B. B. & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32(2), 195-222.
• Jones, K. (2000). The student experience of mathematical proof at university level. International Journal of Mathematical Education in Science and Technology, 31(1), 53-60.
• Kaptan, S. (1998). Bilimsel araştırma ve istatistik teknikleri. Ankara: Tekışık Web Ofset.
• Kidron, I. & Dreyfus, T. (2009). Justification, enlightenment and the explanatory nature of proof. In F.-L. Lin, F.-J. Hsieh, G. Hanna & M. de Villiers (Eds.), Proceedings of the ICMI
• Study 19 Conference: Proof and Proving in Mathematics Education, Vol. 1 (pp. 244–249). Taipei, Taiwan.
• Kidron, I. & Dreyfus, T. (2010). Justification enlightenment and combining constructions of knowledge. Educational Studies in Mathematics, 74(1), 75-93.
• Kidron, I. & Dreyfus, T. (2014). Proof image. Educational Studies in Mathematics, 87(3), 297-321.
• Knapp, J. (2005). Learning to prove in order to prove to learn. Retrieved February 29, 2020 from https://mathpost.asu.edu/~sjgm/issues/2005_spring/SJGM_knapp.pdf.
• Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teachers Education, 5, 61-88.
• Ko, Y. Y. & Knuth, E. (2009). Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions. The Journal of Mathematical Behavior, 28(1), 68-77.
• Kolar, V. M. & Čadež, T. H. (2012). Analysis of factors influencing the understanding of the concept of infinity. Educational Studies in Mathematics, 80(3), 389-412.
• Liljedahl, P. G. (2004). the AHA! Experience: Mathematical contexts, pedagogical implications (Unpublished doctoral dissertation). Simon Fraser University, Canada.
• Liljedahl, P. G. (2005). Mathematical discovery and affect: The effect of AHA! experiences on undergraduate mathematics students. International Journal of Mathematical Education in Science and Technology, 36(3), 219-234.
• Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249-266.
• Pala, O., & Narlı, S. (2018b). Examining proof schemes of prospective mathematics teachers towards countability concept. Necatibey Faculty of Education Electronic Journal of Science & Mathematics Education, 12(2), 136-166.
• Pala, O. ve Narlı, S. (2018a). Matematik öğretmeni adaylarının sonsuz kümelerin denkliği ile ilgili ispatlama yaklaşımları ve yaşadıkları güçlükler. Turkish Journal of Computer and Mathematics Education, 9(3), 449-475.
• Rota, G. C. (1997). Indiscrete thoughts. Boston, MA: Birkhäuser.
• Sandefur, J., Mason, J., Stylianides, G. J. & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83(3), 323-340.
• Sarı, M., Altun, A. ve Aşkar, P. (2007). Üniversite öğrencilerinin analiz dersi kapsamında matematiksel kanıtlama süreçleri: Örnek olay çalışması. Ankara Üniversitesi Eğitim Bilimleri Fakültesi Dergisi, 40(2), 295-319.
• Schoenfeld, A. H. (1994). What do we know about mathematics curricula?. The Journal of Mathematical Behavior, 13(1), 55-80.
• Schwarz, B., Dreyfus, T., Hadas, N., & Hershkowitz, R. (2004). Teacher guidance of knowledge construction. In M. J. Hoines, & A.B. Fuglesad (Eds.), Proceedings of the 28th conference of the ınternational group for the psychology of mathematics education (pp. 169-176). Bergen, Norway.
• Selden, A., McKee, K., & Selden, J. (2010). Affect, behavioural schemas and the proving process. International Journal of Mathematical Education in Science and Technology, 41(2), 199-215.
• Selden, J. & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151.
• Selden, A., & Selden, J. (2007). Overcoming students’difficulties in learning to understand and construct proofs (Technical Report No: 2007-1). Cookeville, TN: Tennesse Technological University, Department of Mathematics. Retrieved February 21, 2020 from http://www.math.tntech.edu/techreports/TR_2007_1.pdf
• Sönmez, V. ve Alacapınar, F. G. (2011). Örneklendirilmiş bilimsel araştırma yöntemleri. Ankara: Anı Yayıncılık.
• Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(1), 289-321.
• Tall, D. (1998, August). The cognitive development of proof: Is mathematical proof for all or for some. Paper presented at Conference of the University of Chicago School Mathematics Project, Chicago.
• 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.
• Thompson, D. R., Senk, S. L. & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43(3), 253-295.
• Tsamir, P. (1999). The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38, 209– 234.
• Türnüklü, E. ve Özcan, B. (2014). Öğrencilerin geometride RBC teorisine göre bilgiyi oluşturma süreçleri ile Van Hiele geometrik düşünme düzeyleri arasındaki ilişki: Örnek olay çalışması. Mustafa Kemal Üniversitesi Sosyal Bilimler Enstitüsü Dergisi, 11(27), 295-316.
• Weber, K. & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(3), 209-234.
• Weber, K. & Alcock, L. (2009). Proof in advanced mathematics classes. In D. A. Stylianou, M. L. Blanton & E. J. Knuth (Eds.), Teaching and learning across the grades: A K-16 perspective (pp. 323–338). New York, NY: Routledge.
• Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101-119.
• Weber, K. (2006). Investigating and teaching the processes used to construct proofs. In F. Hitt, G. Harel & S. Hauk (Eds.), Research in college mathematics education, VI (pp. 197-232). RI: American Mathematical Society.
• Yıldırım A. ve Şimşek H. (2013). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin Yayıncılık.