Preservice middle school mathematics teachers' proportional reasoning knowledge resource using skills in measuring the areas of rectangles

This study aims to explain how preservice middle school mathematics teachers understand the area of rectangles and how they use their proportional reasoning skills in problems that require calculating the area of rectangles. Considering the content and purpose of the research, the holistic multiple case design, one of the qualitative research designs, was used. The sample of the study consists of eight pre-service middle school mathematics teachers selected by purposive sampling method, enrolled in a teacher education program at a university in the Central Anatolia region in the fall semester of 2021-2022. In the research, semi-structured interviews were conducted with preservice middle school mathematics teachers. A worksheet consisting of six open-ended questions about the area of rectangles was prepared by the researcher and used in the data collection process. The data collected from the pre-service teachers' worksheets and the observer's notes of the researcher were analyzed using content analysis and document analysis of the worksheets. The transcripts obtained as a result of the interviews and the notes of the researcher were analyzed using content analysis, and the preservice teachers' answers to the questions in worksheets were analyzed using document analysis. Weiland et al. (2020) presented a theoretical framework consisting of 10 categories and 19 operationalized knowledge resources pointing to the sophistication of teachers' proportional reasoning. The data collected from preservice teachers were analyzed in the light of these knowledge resources. As a result of the research, we found that the preservice teachers were formula-oriented while calculating the area of the rectangle, had difficulties in finding the area of the irregular shape and did not pay attention to the units of length and area. However, it was determined that they had difficulties in proportional reasoning questions, they reasoned additively in multiplicative comparison questions, and they could not distinguish between direct and inverse proportions. When the information sources used by the pre-service teachers are examined, it is seen that 8 pre-service teachers used 5 types of information sources as "proportional situation", "partitioning and tiling", "covariance", "productive comparison" and "ratio between opposite sides", while "enlargement/reduction", "fixed ratio", "unit ratio", "equivalence", "collective thinking", "ratio within itself" and "horizontal content knowledge" made a difference. In the light of the findings, after the basic concepts are formed in the teaching of the topic, it would be more beneficial to gain relevant formulas based on conceptual knowledge in terms of meaningful learning, and for the development of proportional reasoning skills, the use of daily life problems that the student can reason, reasoning and verifying are the suggestions of the research.