Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case

Abstract

Recently, studies have been carried out on alternative proof methods due to the change in the perspective of teaching proof and the difficulties of learners in proof. In this context, proof without words, which are presented as an alternative to proof teaching, defined by diagrams or visual representations and require the student to explain how proof is, are discussed in this study. The aim of this study is to examine pre-service mathematics teachers' explanations of proof without words about the sum of consecutive numbers from 1 to n. The data were collected by the proof of the sum of consecutive integers. 27 pre-service teachers from a university in the Middle Anatolia region participated in this study, which was conducted using a basic qualitative research design. At the end of the study, it was seen that most of the pre-service teachers were unable to explain the proof without words of the sum of integers from 1 to n. One of the reason for this may be related to the spatial thinking skills of pre-service teachers. However, there are pre-service teachers who can interpret the visual given in the proof correctly, use the necessary mathematical knowledge, but cannot generalize using the given visual. The reasons why the pre-service teachers could not express the general situation are considered as the lack of algebraic thinking.

Keywords

• Proof
• Proof teaching
• Proof without words
• Visual proof

Ortaöğretim Matematik Öğretmen Adaylarının Sözsüz İspat Becerilerinin Bağlamsal İncelenmesi: 1’den n’ye Kadar Olan Tamsayıların Toplamı Durumu

Abstract

İspata yönelik algının son dönemlerde değişmesi ve öğrenenlerin ispatta zorlanmalarından dolayı alternatif ispat yolları ile ilgili araştırmalar yapılmaktadır. Bu bağlamda, ispat öğretimine alternatif bir yol olarak sunulan, ispatın doğruluğunu göstermekle beraber öğrencinin ispatı açıklamasını gerektiren diyagram veya resimler olarak tanımlanan sözsüz ispatlardan biri bu çalışmada ele alınmıştır. Çalışmanın amacı öğretmen adaylarının 1'den n'ye kadar olan ardışık tamsayıların toplamının sözsüz ispatına yönelik açıklamalarını incelemektir. Veriler, ardışık tamsayılar toplamının sözsüz ispatı ile toplanmıştır. Temel nitel araştırma deseninde gerçekleştirilen bu araştırmaya İç Anadolu Bölgesi'ndeki bir üniversiteden 27 öğretmen adayı katılmıştır. Araştırmanın sonucunda matematik öğretmen adaylarının çoğunun, ardışık tamsayıların toplamı ile ilgili verilen sözsüz ispatı açıklayamadıkları görülmüştür. Bu durumun nedenlerinden birisi öğretmen adaylarının uzamsal düşünme yetenekleridir. Bununla birlikte öğretmen adaylarından sözsüz ispattaki görseli doğru biçimde yorumlayabilen, gerekli olan matematiksel bilgiyi kullanabilen ancak verilen görseli genelleyemeyenler de bulunmaktadır. Öğretmen adaylarının genel durumu ifade edememelerinin nedenleri cebirsel düşünme eksikliği olarak değerlendirilebilir.

Keywords

• İspat
• İspat öğretimi
• Sözsüz ispatlar
• Görsel ispatlar

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