Defragmentation of construction holes in students with high mathematical ability: Addressing “skipping steps” errors using scaffolding
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- Defragmentation, Construction holes , High mathematical ability , Scaffolding
Abstract
High-ability students generally excel in mathematics but frequently make "skipping steps" errors during complex problem-solving. These errors occur due to cognitive construction holes, where specific schemas are unconsciously bypassed, leading to incomplete final solutions. This study aims to identify and defragment these construction holes in high-mathematical-ability students using Level-3 scaffolding to restore complete cognitive processing. A qualitative multiple-case study was conducted involving three high-ability eighth-grade students from three different middle schools. Data were comprehensively collected through written tests, semi-structured interviews, and observations. The analysis utilized cognitive mapping based on Polya’s problem-solving stages to pinpoint specific cognitive gaps. Each student exhibited a "Skipping Steps" construction hole, omitting one or two essential schemas in the final stages. The application of Level-3 scaffolding successfully guided students to recover these missing schemas. It enabled them to self-detect errors, reconstruct their mathematical reasoning, and achieve accurate solutions while fully maintaining their procedural independence. Targeted conceptual scaffolding effectively remediates light construction holes in high-ability learners, ensuring problem-solving completeness without compromising their learning autonomy.
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