Investigation in terms of cognitive of service and pre-service mathematics teacher' proving process

Proving is a mathematical skill that requires complex mental processes. However, the biological structure of the human mind is handle in such a way that it can understand and develop these complex processes. The understanding of these complex processes is make possible by the manageable of multi-cognitive structures in the human mind. Two of these cognitive structures are cognitive and metacognitive skills. In this context, it is important to examine the cognitive structures in the proving process in the dimensions of cognitive and metacognitive skills in order to understand the proving process. The aim of this study is to examine the cognitive structures of mathematics teachers and pre-service mathematics teachers in the proving process of geometry and algebra in the context of cognitive and metacognitive skills. In this study conducted using the explanatory sequential design from mixed method research, "Proof Making Ability Diagnostic Test" was applied to 25 math teachers and 48 pre-service math teachers at different levels. 6 math teachers and 12 pre-service math teachers selected from these participants at different levels carried out a think aloud protocol. Participants were asked to perform four different proving tasks on the activity cards in the think aloud protocol. During this process, we was observe participants have exhibited skills through the observation form. Descriptive and inferential statistics were used in the analysis of quantitative data collected in the study; descriptive analysis and content analysis were used in the analysis of qualitative data. Result of the analysis of the quantitative data show that the ability of proving the mathematics teachers and the last grade pre-service math teachers were significantly higher than the grade year pre-service math teachers. The results of qualitative data analysis showed that there are seven dimensions -reading the proof, drawing the figure, underlining the suggestion, evaluating the correctness, determining the strategy, operation process and heuristic shortcut thinking strategy- in cognitive skills and eight dimensions – facilitation the proof, questioning, awareness, planning, strategy determination, analogical reasoning, checking, relationship- in metacognitive skills. In addition, it was determined that the number of cognitive and metacognitive skills displayed in the proving process is in parallel with the "Proof Making Ability Diagnostic Test" scores. Key Words: Mathematical Proof, Proving, Algebraic Proof, Geometric Proof,Cognitive Skill, Metacognitive Skills, Service and Pre-service Math Teacher