Investigation of middle school students' and teachers' use of representation in mathematical problem solvi̇ng

The purpose of this research is to examine the ways in which students and teachers use representation in solving mathematical problems. For this purpose, the types of representations used in problem-solving and the purposes of using representations have been revealed. In addition, the representations used by teachers and students in the problem-solving process and the usage purposes of representation were compared. The research was carried out with 4 secondary school teachers working in various public schools in Denizli in the 2021-2022 academic year and 16 students of these teachers (4 students of each teacher) attending the 7th grade. In the research, qualitative research method was used to understand the situations and events to be studied from the perspectives of the participants. As a research design, a case study, which allows to obtain in-depth information about events and situations with data collection tools (observation, interview, document, visuals, etc.) containing multiple sources, was used. Research data was collected from documents obtained from the solution of 14 open-ended mathematical problems and from semi-structured interviews. In the interviews, 14 open-ended mathematical problems involving 7th grade topics and requiring the use of representation were used. Data was analyzed using content analysis. In the data analysis process, it was tried to determine whether the teachers and students used similar representations, at which stages of the questions or in which types of questions the representations differed from each other, and what kind of close similarities there were between the representations used by the students. According to the findings of the study, it was observed that teachers and students used numerical representation more than other mathematical representations. Considering the usage purposes of representation, it is noted that representations were mostly used for understanding the problem and then for recording purposes. It has been determined that the teachers used the representations for exploratory purposes at least, whereas the students are used for evaluation purposes. It has been found out that, the minimum usage of the representations for the teachers is for exploratory purpose, whereas it is for evaluation purpose for the students. Considering the most used mathematical representations according to their purpose of use, it was observed that teachers preferred symbolic representation and students preferred visual representation to understand the problem, but both teachers and students preferred numerical representation for exploration and recording, and algebraic. It has been observerd that the use of similar representations by teachers and students according to their purpose of use varies according to the expectation of the teacher, previous experiences, whether they have encountered a similar problem before or not. In addition, it has been observed that teachers and students cannot always reach the right solution to the problems. Some of the reasons for this are misuse of representations, not being able to make the appropriate transition between representations, not being able to associate the problem situation with the representation used, being insistent on using some representations. It is thought that the results of the research will contribute to the development of teacher training programs and will be a roadmap for the use of representation in the teaching process.