The purpose of the present study was to investigate preservice secondary mathematics teachers' metacognitive behaviour in the mathematical problemsolving process. The case study methodology was employed with six preservice mathematics teachers, enrolled at one university in Ankara, Turkey. We collected data by using the think aloud method, which lasted for two sessions. It was found that there was no relationship between academic achievement and frequencies of metacognitive behaviour. However, the types of problems could affect these frequencies. Furthermore, there was no pattern in metacognitive behaviour with respect to achievement and type of problem.

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The main purpose of this study is to determine ways of thinking and understanding of eight graders related to generalizing act. To carry out this aim, a DNR based teaching experiment was developed and applied to 9 eight graders. The design of the study consists of three stages; preparation process in which teaching experiment is prepared, teaching process in which teaching experiment is applied, and analysis process in which continuous and retrospective analyses are carried out. Analysing the data, it was found that students' ways of thinking could be determined as relating, searching, and extending. Ways of understanding belonging to generalizing act could be determined as identification, definition, and influence. It was recommended to add two new categories "relating with an authority" and "searching the same piece" to the generalization taxonomy.

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 preservice 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.

The importance of both proof and visualization has been frequently emphasized in mathematics education. Visual proof or nonverbal proofs are defined as diagrams or illustrations that help us to see why a mathematical expression is correct, and how to begin to prove the accuracy of this statement. The aim of this research is to examine nonverbal proof skills of preservice mathematics teachers. The study was carried out with case studies, one of the qualitative research designs. The participants of the study consisted of 53 preservice mathematics teachers at a state university in Central Anatolia, Turkey. The data were collected with a sample of three nonverbal proof samples directed to preservice teachers. The analysis of the data classified the preservice teachers' responses according to their similarities and differences. The findings showed that preservice teachers generally associate images with geometric figures. In addition, it was also seen that those who saw the visual relationship between the given visual and mathematical expression used it to show the expression as correct instead of proofing the visual.

Nonverbal proofs are diagrams or illustrations that will help us see what a mathematical expression means, why it is true, and how it is proved. The aim of this study is to examine the effects of nonverbal proof-based education on preservice mathematics teachers. The study was conducted using case study research methods, one of the qualitative research designs. The participants of the study consisted of 31 preservice mathematics teachers. The data were collected in writing at the beginning and end of the process with questions directed to preservice teachers. These questions in the data collection tool were used to compare the responses of the preservice teachers who had an experience with nonverbal proof in the pre and post assesment and compare the changes. In the analysis of the data, descriptive analysis, which is a qualitative data analysis, was used. Firstly, each preservice teacher's responses in the post assesment are classified according to their similarities and differences and then are categorized. Then, the answers in the pre assesment were examined and the responses were compared whether there was a change or not. The findings of the study showed that the preservice mathematics teachers' experience of nonverbal proof effect on the recognition of the given image and establishing a relationship with different mathematics subjects.

Bu çalışmada matematik eğitiminde ispat ve sözsüz ispat kavramlarının rolü ve önemi üzerine odaklanılmıştır. Sözsüz ispatlar; gerçek ispatlar olarak kabul edilemeyecek fakat özel bir ifadenin niçin doğru olduğunu hatta bir matematiksel ifadenin doğruluğunu ispatlarken nasıl ele alınacağının görülmesine yardımcı olacak diyagram veya resimler olarak tanımlanmaktadır. Sözsüz ispatların gerçek ispatlar olup olmadığına dair tartışmaların olmasıyla beraber sözsüz ispatlara yönelik kullanılan ifadelerde de ortak bir görüş bulunmamaktadır. Sözsüz ispatların gerçek ispatlar olup olmadığına yönelik tartışmalara rağmen sözsüz ispatlar gerek matematik gerekse matematik eğitimi için önemli araçlar olarak görülmektedir. Bu çalışmada sözsüz ispatlarda önemli bir yere sahip olan görselleştirme ve ispatın birbiri ile ilişkileri incelenmiştir. Bu amaç doğrultusunda görselleştirmenin, matematiksel ispatın ve sözsüz ispatın tanımı, matematik eğitimindeki rolü, amacı ve matematik müfredatındaki ispatın rolüne ilişkin açıklama ve tartışmalara yer verilmiştir. Ayrıca çalışmada sözsüz ispat örneklerine ve sözsüz ispatların üstün ve zayıf yönlerine değinilerek sözsüz ispatlar kuramsal olarak ele alınmıştır. Ayrıca sözsüz ispatlarla görselleştirme arasındaki ilişki incelenmiştir.

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