Pre-service secondary mathematics teachers' (PSMTs) understanding and ability of constructing a proof is not only important for their own learning process, but also important for these PSMTs to help their future students learn how to do proofs. Therefore, this study is focused on and explains PSMTs' behaviors that they revealed throughout the proving process of a proposition. In this qualitative case study, the participants were fifteen volunteer PSMTs from a public university in Turkey. The participants were given a proposition which they were asked to think aloud and prove it on the blackboard. The findings of this study show that PSMTs' behaviors and thoughts regarding the given proposition were limited. In detail, PSMTs had difficulties in application of mathematical language and notations, understanding the meaning of the given proposition, knowing where to get started on a proof, using examples efficiently, using appropriate and efficient methods to construct the proof, and defining logical structures of the proposition to construct the proof.
Öz: Son yıllarda matematiksel ispatlara verilen önemin artmasıyla birlikte, matematik sınıflarındaki muhakeme etme, ispat yapma ve tartışma durumları, matematik eğitimi araştırmaları için odak konular haline gelmiştir. İspat öğretiminde temel noktalardan birisi ispatın kavranmasıdır. İspatın kavranması sağlayan araçlardan birisi (ispat) kavrama testidir. Bu çalışma İspat Kavrama Testi'ne dayalı bir öğretim uygulamasını içeren geniş bir araştırmanın parçasıdır. Ana çalışma bir eylem araştırması olarak tasarlanmış olup 11. sınıfta öğrenim gören 20 öğrenci ile beş hafta süren (Yang ve Lin'in [2008] modeline dayalı) bir öğretim uygulamasının çok yönlü değerlendirilmesini içermektedir. Bu çalışmada ise söz konusu uygulamada öğretmenin karşılaştığı güçlükler ve sürece müdahale yolları ele alınmaktadır. Söz konusu deneyimleri betimlemek için informal gözlemler ve öğretmen günlükleri kullanılmıştır. Yapılan analiz sonucunda öğretmen deneyimleri üç kategori altında gruplanmıştır. Bunlar; 1-karşılaşılan güçlükler, 2-müdahale yolları ve 3-müdahalelerin sürece yansımasıdır.
In our country, one of the important concepts proposed by the recent curricula has been 'activities' since 2005. From this point of view, both mathematics educators and teachers should focus on and discuss about the importance of activities. This study is designed to examine pre-service secondary mathematics teachers' (PSMTs) perceptions about mathematics learning activities (MLA). Participants are twenty-seven PSMTs. The data include 3 different groups. The first group consists of PSMTs' free writings that describe their definitions of MLA. The second group consists of observation notes of the class discussions which are related to the structure of MLA and MLA's required properties. The last one consists of MLA samples that were designed by PSMTs. In consequence of data analysis, results show that PSMTs did not reach consensus on a unique definition or particular properties of MLA; the definitions spreaded out a broad perspective. PSMTs defined the activity as the classroom application that is connected with real life and other disciplines, compatible with the objectives in teaching programs, is designed to improve students' cognitive skills, student-centered, interesting, easy to learn, and appropriate for groupwork. But, it is found out that PSMTs did not reflect the properties of MLA while they were designing the activities. The results of the study indicated that PSMTs' perceptions of activities and their skills of designing an activity must be improved.
This study is a part of a large scale project in which an action research design is used to teach proof to 11 th grade students. This part of the project aims to identify students' comprehension level through five proof comprehension tests developed by the researchers based on the National Geometry Curriculum. Data were analyzed by considering the framework of Yang and Lin's ( 2008) multilevel model. Results showed none of the students were successful at the most sophisticated level of the proof comprehension tests which requires conducting a proof in various ways or proving different theorems by using the same proof methods. Moreover, the highest proof comprehension was obtained from the level containing knowledge about definition, properties, and meanings of symbols. Achievement and comprehension decreased for components of a proof needing higher level mathematical skills. Based on the study's results, suggestions about teaching proof are provided.
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