Problem Statement: Recent research and evaluation reports show that students are not learning geometry efficiently. One identifier of student understanding related to geometry is teachers' knowledge structures. Understanding what a proof is and writing proofs are essential for success in mathematics. Thus, school mathematics should include proving activities. Proofs are at the heart of mathematics, and proving is complex; teachers should help their students develop these processes in the early grades. The success of this process depends on teachers' views about the essence and forms of proofs. Hence, it is necessary to investigate the classroom teachers' perceptions related to proofs. Purpose of the Study:The purpose of this study is to determine the proof scheme of pre-service teachers when proving a geometry theorem. In this sense, the study is oriented by the research question: which proof schemes do pre-service teachers use when making proofs in geometry?Method: The current case study is a detailed examination of a particular subject. Firstly, an open ended question was asked, and then semistructured interviews were conducted. The three students investigated in this study were selected by considering their Basic Mathematics scores. Two girls having maximum and average scores and a boy having a

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.

It is of critical importance, in particular, for mathematics teachers who will teach future generations to understand and do mathematical proofs. It is important to determine future teachers' beliefs about and difficulties with proofs because their knowledge of this issue affects their teaching. This study aims to determine and compare the proof schemes of prospective mathematics teachers from two state universities, one in Turkey and the other in Spain. The case study was conducted within this study. The participants were 51 prospective teachers at their second year from the department of teaching mathematics education at Huelva University in Spain and 45 prospective teachers from the department of teaching mathematics education at Cumhuriyet University in Turkey. The Proof Test consisted of four questions about proofs for parallelograms. Semi-structured interviews were subsequently conducted to investigate the prospective teachers’ responses in-depth. The findings suggest that prospective teachers from Turkey and Spain indicated affinity in proving. The majority of the prospective mathematics teachers were either unable to complete the proof or completed the proof in an inaccurate way.

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In mathematics the process of visualization requires the process of forming and manipulating images to explore and understand. Indeed, according to mathematicians, it is difficult to understand without visualization. There is consensus that visual proofs are important tools in mathematics education. However, there is no consensus on the effects of the visualization on proof. Therefore, this study aims to reveal the relationship between van Hiele levels of geometric thinking, spatial ability and visual proofs. In this study, relational survey method is used. The study is conducted on 85 pre-service elementary mathematics teachers studying in the Faculty of Education at a public university located in Turkey. The data is analyzed via Spearman correlation. In the study, it was seen that most of the elementary mathematics teachers were at the 3rd level of van Hiele's geometric thinking. Another result is that there is a significant relationship between van Hiele's level of geometric thinking and visual proof skills. However, there is no significant relationship between visual proof skills and spatial ability. The relationship between visual proofs skills and spatial ability can be investigated deeply with qualitative research. Moreover, experimental studies can be done to investigate the effect of training on visual proofs on the level of van Hiele geometric thinking.