Concept definition or concept i̇mage?: Preferences of teachers in mathematical problem solving process

The aim of this study is to reveal whether middle school mathematics, high school mathematics and other subjects' teachers use concept images or concept definitions in solving mathematics questions. The study is a qualitative research and holistic multiple case design was used as a model. For the purpose of the study, an open-ended data collection tool consisting of 13 questions related to the concepts of square-trapezoid, ratio and function was developed and applied to 5 middle school mathematics teachers, 5 other subject teachers and 5 high school mathematics teachers in 9 different schools. The data were analysed using Tall and Vinner's (1981) concept image - concept definition theoretical framework. As a result of the analyses, it was found that all of the teachers had difficulties in explaining the relationship between trapezoid and square, all of the high school mathematics teachers used the concept definition for the concept of square, while 60% of the middle school mathematics teachers used the concept definition. As in the case of the relationship between square and trapezoid, the definition cells of the other subjects' teachers were almost empty for each concept. Especially not knowing the definition of the trapezoid concept plays a decisive role in not determining the square-trapezoid relationship. In the questions related to the ratio, it was observed that all participant groups mainly utilised concept images in their solutions. Especially the use of concept definition cells by other subjects' teachers is completely empty. In function questions, high school mathematics teachers made use of concept definition at a rate of 90%, while this rate was 60% for middle school mathematics teachers. In other subjects' teachers, 5%, i.e. only 1 out of 20 answers, used concept definition. This situation reveals that mathematics teachers give more weight to the use of the formal definition of concepts, while other subjects' teachers give more weight to the concept image. The nature of the concepts directly affects the form and number of responses given by teachers in mathematical problem solving processes. For example, the fact that the concept of function has a formal and precise definition causes teachers to provide more definitional and systematic approaches towards this concept, while the fact that the ratio has a more flexible definition and can be associated with daily life causes teachers to resort more to concept images. This shows that the nature of concepts plays a critical role in determining teachers' definition and image preferences.