Investigation of pre-service mathematics teachers' mathematical connection skills through problem posing activities realized by flipped learning model

Purpose: In this research, we aimed to determine the mathematical connection skills of the pre-service mathematics teachers through the problem posing tasks given to them in the applications using the flipped learning model and to examine their views on the flipped learning model, problem posing skills and connection skills. Method: The research was carried out with the case study pattern, which is one of the qualitative research methods. The study group consisted of a total of 24 teacher candidates, 12 female and 12 male, studying in the third grade of Eskişehir Osmangazi University Elementary Mathematics Teaching in the spring term of the 2020-2021 academic year. In the applications carried out with the flipped learning model, the connection skills of the pre-service teachers were examined through problem posing tasks, and the data collection process took a total of 11 weeks. As data collection tools, before and after the applications, opinion forms and problem posing task forms were used for four categories of connection skills (connecting between concepts, connecting between different representations of the concept, connecting the concept with other disciplines, and connecting the concept with real life). In the opinion forms, the opinions of the pre-service teachers regarding the "flipped learning model, connection skills and problem posing skills" were taken before and after the applications. In the task forms, the pre-service teachers were divided into four categories: (1) connecting between mathematics subjects and concepts, (2) connecting between different representations of the concept, (3) connecting mathematics subjects and concepts with other disciplines, and (4) connecting mathematics subjects and concepts with real life. They were asked to pose problems related to connecting skills. Three different types of problem posing tasks were included in these four task forms: "free, semi-structured and structured". Thematic analysis and content analysis techniques were used in the analysis of the data obtained from the opinion forms and task forms. Findings: When the pre-application views of the pre-service teachers on the connection skill were examined, it was seen that the majority of the pre-service teachers explained the connection skill with its sub-categories. It has been determined that more than half of the candidates do not have knowledge about the flipped learning model and they think that this model has not been used in their lessons before. Considering the views on the sub-categories of the connection skill, it was seen that more than half of the pre-service teachers explained the relationship between concepts by referring to the characteristics of the mathematics course, and they thought that all subjects and concepts in mathematics were related to each other. He explained the relationship between different representations by giving examples of "table" and "algebraic" representations. They gave the examples of "physics" and "chemistry" for the disciplines with the most relations with other disciplines. In the real-life relationship, it was seen that more than half of the pre-service teachers made connections between the "numbers and operations" learning area in mathematics and the "shopping/price/eating" contexts in the real world. Considering the opinions after the applications, it was seen that almost all of the pre-service teachers felt competent to use the connection skill in their own lessons and they wanted to include this skill in their lessons. More than half of the candidates stated that they saw the flipped learning model for the first time in these applications and gained experience about the model. In the research, three types of problems (free, semi-structured and structured) posed by the pre-service teachers for the four sub-categories of connecting skills, according to the characteristics that should be present in an appropriate problem (whether there is a problem situation in the written expression, the characteristics of the problem, the language and expression of the problem conforms to the Turkish grammar rules, the solvability of the problem, the compatibility of the problem with the given association component, the context of the problem) were examined and it was seen that the majority of the candidates posed problems in accordance with the desired characteristics. The following four issues were identified in the problems posed by the candidates in the context of the connection skill: (1) It was observed that they made connections between "different concepts belonging to the same learning area" and "concepts belonging to different learning areas" in the problems they posed for the connection skill between mathematics subjects and concepts. (2) In the problems they posed for the ability to connect between different representations, the most common representation they used was the "verbal" representation, while the "verbal", "algebraic" and "symbolic" representations were used in the solution of the problem. (3) In the problems they posed for the skill of connecting mathematics subjects and concepts with other disciplines, they mostly connected mathematics subjects and concepts with "physics" and "science" disciplines. (4) In the problems they posed for the skill of connecting mathematical subjects and concepts with real life, they mostly connected mathematics subjects and concepts with the daily life contexts of "nature/environment/agriculture/meteorology" and "shopping/price/eating". Conclusion and Suggestions: It was seen that the pre-service teachers did not have knowledge about the flipped learning model before the practices, and after the practices, the majority of the pre-service teachers felt competent about mathematical connection skills, gained experience in problem posing skills, and wanted to use the flipped learning model in their own lessons. This result shows that they have developed a positive perspective towards the flipped learning model. In addition, it was observed that the candidates increased their knowledge and experience about the four sub-categories of connection skills. It was evaluated that the majority of the candidates posed problems in accordance with the criteria. In order to further the research, a longitudinal study can be carried out by examining the connection skills of pre-service teachers from the first year to the fourth year of university. By conducting an experimental study, pre-service teachers can be divided into two groups, and their connection skills can be examined by giving face-to-face training to one group and an online lesson to the other group. Keywords: Mathematics pre-service teachers, Mathematical connection skills, Problem posing skills, Flipped learning model.