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

Abstract

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.

Keywords

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

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

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

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