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

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.

Anahtar Kelimeler:

İspat İmajı, İspat, cebir, matematik eğitimi

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

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,

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