Middle school students' strategies and cognitive comprehension types in the process of generalizing figural patterns

The generalization of figural patterns plays a significant role in the development of algebraic thinking. Due to the geometric nature of figural patterns, the engagement of geometric reasoning processes is inevitable. In this context, the aim of this study is to explore the strategies and cognitive apprehension processes employed by seventh-grade middle school students during the generalization of figural pattern tasks. The study adopts a case study design, one of the qualitative research methods. The participants consist of eight seventh-grade students. Data were collected through task-based interviews using the Figural Patterns Form, which includes two linear figural pattern tasks. The data were analyzed through deductive thematic analysis. For analyzing students' cognitive comprehension processes in pattern tasks, the apprehension processes outlined in Duval's cognitive model were utilized. The findings reveal that students employed various strategies in the generalization of figural pattern tasks. Notably, strategies such as contextual reasoning, counting parts, modeling, and vertical-horizontal grouping were frequently observed. It was identified that students generally made generalizations by converting figural representations into numerical expressions, although some students struggled with generalization. In the process of generalizing figural pattern tasks, students exhibited various types of cognitive apprehension. When students' cognitive comprehension processes were examined within the framework of Duval's cognitive model, it was found that discursive and operative apprehension processes were particularly prominent. Based on the findings, recommendations are provided for the development of instructional materials and the enhancement of algebraic thinking in the context of mathematics education.