Students’ problem-solving patterns in arithmetic series tasks viewed through reflective thinking

Abstract

Abstract Reflective thinking plays an important role in mathematical problem-solving because it enables students to monitor their reasoning, adjust strategies, and integrate prior knowledge when confronting non-routine tasks. This study aimed to identify the problem-solving patterns students employed while engaging in reflective thinking to solve non-routine arithmetic series problems. Using a descriptive qualitative approach, research was conducted with high school students. Data was collected through non-routine mathematics problems tests on arithmetic series. The students solved the problems while thinking aloud and recorded them using audiovisual methods. Subsequently, the constant comparison method analyzed observations from answer sheets, observation sheets, and recordings. The findings revealed three distinct problem-solving patterns: Employ Variable, Visual Representation, and Direct, which reflect how students engaged reflective thinking when solving non-routine arithmetic series problems. These patterns differ in how students manage confusion, represent information, and select problem-solving strategies. Students in the Employ Variable and Visual Representation patterns activated all reflective thinking components, whereas those in the Direct pattern engaged only partially, particularly lacking in articulating the question, restructuring information, revisiting the problem when confused, and integrating conceptual understanding. Overall, this study contributes to a deeper understanding of how reflective thinking manifests through different problem-solving patterns and offers insights into the complexity of reflective engagement in mathematical problem-solving.