Leveraging Contextual Worksheet to Reveal Students' Proofing in Counting Rules

Abstract

This study aims to explore how contextualized combinatorics tasks set within a cinema-themed worksheet support undergraduate students’ transition from concrete enumeration strategies to formal combinatorial reasoning through a qualitative case study design. Participants engaged with four scaffolded tasks that first had them list seating arrangements for three friends and subsequently required them to explain the principles governing ordered versus unordered selections, while their written responses and verbal protocols were analyzed using thematic coding of representational approaches and justification quality. Findings reveal that embedding tasks in a familiar narrative and providing multimodal supports such as tables and color-coded diagrams helped students explicitly identify variables, assumptions, and logical steps, enabling them to construct factorial-based arguments without reliance on rote computation. Although all participants ultimately derived correct combinatorial solutions, the depth and clarity of their proofs varied in relation to individual verbal reasoning skills, and common misconceptions were effectively addressed when scenarios contrasted ordered and unordered cases. This single-paragraph abstract highlights that progressively reducing concrete scaffolds within real-world contexts enhances both engagement and conceptual understanding in discrete mathematics, suggesting that culturally relevant narratives and adaptive supports merit further investigation in larger-scale, mixed-methods studies.