Examining middle school students' processes of genaralizing and justifying linear figural patterns

This study aims to examine students' processes of generalization and justification related to linear patterns. A case study model, one of the qualitative research designs, was adopted in the study. The participants consisted of 10 students from 5th, 6th, 7th, and 8th grade levels. For this purpose, a worksheet comprising 10 open-ended questions involving linear figure patterns was developed. During the implementation phase, data were collected through documents, worksheets, and individual interviews. The data obtained were analyzed using descriptive analysis through the MAXQDA software and presented in a visualized report format.The findings revealed that students reached generalizations using strategies such as modeling, counting, extending to the whole, contextual reasoning, linear reasoning, prediction-control, and iterative approaches. To justify their generalizations, they employed numerical, explanatory, figural, and algebraic reasoning strategies. The data indicate that over time, students' generalization and justification strategies became more diverse, complex, and abstract.Furthermore, it was observed that students who adopted linear generalization strategies supported their reasoning with numerical checks, explanations, figural representations, and algebraic arguments, while other strategies were associated with more limited forms of justification. In terms of representational levels, although students were able to reach algebraic representations, their rates of justification remained low.Based on these findings, the study highlights the multilayered nature of early algebraic thinking, making it more visible through aspects such as transitions between strategies, matches between strategies and justifications, and the fluidity between different forms of representation.