Formal İspatın Mevcut Olmadığı Bir Durumda İspat İmajı Var Olabilir Mi?: Başarısız Bir İspat Girişiminin Analizi

Year 2021, Volume: 15 Issue: 1, 1 - 31, 27.06.2021

Ozan Pala , Esra Aksoy , Serkan Narlı

https://doi.org/10.17522/balikesirnef.843527

Cited By: 1

Abstract

Matematiğin temel bileşenlerinden olan ispat ve ispatlamayı farklı perspektiflerden ele alan pek çok teorik çerçeve sunulmuştur. Bunlardan biri olan ispat imajı, Kidron ve Dreyfus’un (2014) iki profesyonel matematikçinin ispat süreci üzerinde yaptığı analizler sayesinde ortaya çıkmıştır. Yazarlar, ispat imajını bileşenleri bağlamında tanımlamış ve bunun formal ispat ile ilişkisini vurgulamıştır. Diğer yandan ispat imajının, formal ispatın ortaya çıkmadığı durumlarda da ortaya çıkabileceği belirtilmesine rağmen böyle bir örnek sunulmamıştır. Teorik çerçevedeki bu boşluk araştırmanın motivasyon kaynağı olarak benimsenmiştir. Çalışma sürecinde çok aşamalı örnekleme yaklaşımı tercih edilmiş ve öncelikle 120 öğretmen adayından cebir ile ilgili iki teoremi ispatlamaları istenmiştir. Daha sonra her iki teoremi de doğru olarak ispatlayabilen 3 katılımcı ile etkinlik temelli mülakatlar gerçekleştirilmiştir. Toplanan veriler üzerinde mikro-analitik analizler yapılmış ve alt bileşenler arasındaki ilişki tartışılmıştır. Ayrıca “aydınlanma” kavramının rolü yorumlanmış ve hislerin etkisi detaylandırılmıştır. Bu sayede katılımcılardan birinin formal ispata ulaşamamasına rağmen ispat imajına sahip olduğu belirlenmiş ona dair verilere bu çalışmada yer verilmiştir.

Keywords

İspat İmajı, İspat, cebir, matematik eğitimi

Can the Proof Image Exist in the Absence of the Formal Proof?: Analyses of an Unsuccessful Proving Attempt

Abstract

As proof and proving are the key elements of mathematics, several frameworks evaluating this process have been presented. Proof image, being one of them, was introduced by Kidron and Dreyfus (2014) through analyses of two mathematicians' activities. Authors clarified it in the context of components, and emphasized its relation with formal proof. However, despite mentioning its possibility, they didn’t present any case where proof image exists without the formal proof. This led us investigating dynamics of such cases. Multi-stage sampling was preferred, and 120 pre-service teachers were asked to prove two theorems about algebra firstly. Then, task-based interviews were conducted with 3 participants, who proved both theorems. Moment-by-moment analyses were conducted and sub-dimensions were discussed in detail. Additionally, role of “enlightenment” was reinterpreted and feeling dimension was elaborated. Consequently, it was identified that one participant had an image although she couldn’t reach the formal proof, and her story was presented.

Keywords

Proof image, proof, algebra, mathematics education

References

• Antonini, S., & Mariotti, M. A. (2008). Indirect proof: what is specific to this way of proving? ZDM, 40(3), 401-412.
• 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, Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Volume 2, 119-126.
• Clore, G. L. (1992). Cognitive phenomenology: Feelings and the construction of judgment. The construction of social judgments, 10, 133-163.
• Davydov, V. V. (1990). Soviet studies in mathematics education: Vol. 2. Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (J. Kilpatrick, Ed., & J. Teller, Trans.). Reston, VA, USA: National Council of Teachers of Mathematics. (Original work published in 1972.)
• Dictionary, O. E. (1989). Oxford English Dictionary. Simpson, JA & Weiner, ESC.
• Fischbein, E. (1982). Intuition and proof. For the learning of mathematics, 3(2), 9-24.
• Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational studies in mathematics, 38(1-3), 11-50.
• Fischbein, H. (1987). Intuition in science and mathematics: An educational approach (Vol. 5). Springer Science & Business Media.
• Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht, The Netherlands: Kluwer.
• Goldin, G. A. (2000). Affective pathways and representation in mathematical problem solving. Mathematical thinking and learning, 2(3), 209-219.
• Grootenboer, P., & Marshman, M. (Eds.). (2016). The Affective Domain, Mathematics, and Mathematics Education. In Mathematics, Affect and Learning (pp. 13-33). Singapore: Springer
• Hannula, M., Evans, J., Philippou, G., & Zan, R. (2004). Affect in mathematics education–exploring theoretical frameworks. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 107-136). Bergen, Norway: PME.
• Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32 (2), 195-222.
• Kidron, I. & Dreyfus, T. (2014). Proof image. Educational Studies in Mathematics, 87(3), 297-321.
• Kidron, I., & Dreyfus, T. (2010). Justification enlightenment and combining constructions of knowledge. Educational Studies in Mathematics, 74, 75-93.
• Knapp, J. (2005). Learning to prove in order to prove to learn. Retrieved on 6-June-2020 at http://mathpost.asu.edu/~sjgm/issues/2005_spring/SJGM_knapp.pdf
• Liljedahl, P. (2004). the AHA! Experience: mathematical contexts, pedagogical implications (Unpublished Doctoral Dissertation). Simon Fraser University, Burnaby. British Columbia, 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, 219–235.
• Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249-266.
• Rota, G.-C. (1997). Indiscrete thoughts. Boston, MA: Birkhäuser.
• 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 International Group for the Psychology of Mathematics Education: Vol. 4 (pp. 169–176). Bergen, Norway: PME.
• 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.
• Treffers, A., & Goffree, F. (1985). Rational analysis of realistic mathematics education—TheWiskobas program. In L. Streefland (Ed.), Proceedings of the 9th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 97–121). Utrecht, The Netherlands: OW&OC.
• Türnüklü, E., & Özcan, B. N. (2014). the Relationship Between Students' Construction of Geometric Knowledge Process Based on RBC Theory And van Hiele Geometric Thinking Levels: Case Study. Mustafa Kemal University Journal of Graduate School of Social Sciences, 11 (27), p. 295-316.
• van der Waerden, B. L. (1954). Einfall und Überlegung: Drei kleine Beiträge zur Psychologie des mathematischen Denkens [Idea and reflection: Three small contributions to the psychology of mathematical thinking]. Basel, Switzerland: Birkhäuser.
• Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational studies in mathematics, 48(1), 101-119.
• Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational studies in mathematics, 56(2-3), 209-234.