The Difficulty and Accessibility of Combinatorics Problems: Evidence From Large-Scale Assessments and Student Interviews.

다국어 초록 (Multilingual Abstract)

The objective of this dissertation was to explore the paradoxical nature of Combinatorics as both a difficult and accessible domain in Mathematics, particularly for K-12 students. This paradox in Combinatorics' nature raised questions about how students interact with problems in this domain and the factors influencing their understanding and engagement with mathematics.To investigate these aspects, the study utilized a mixed-methods approach. Quantitative data was derived from the Exame Nacional do Ensino Medio (ENEM), a large-scale nationwide assessment in Brazil. The analysis focused on 28 Combinatorics problems identified across 12 years of the exam, comparing them with non-Combinatorics problems. The study also involved qualitative methods, specifically task-based interviews with Brazilian students, primarily from disadvantaged school backgrounds. These interviews aimed to provide deeper insights into how students approach, understand, and engage with Combinatorics problems.The findings revealed that while the combinatorial domain is notably accessible in the sense that it allows students with varied backgrounds to understand what problems ask, this accessibility does not necessarily translate into students consistently arriving at correct solutions. The study also found that achievement gaps between students of private and public schools in Brazil are smaller in Combinatorics is than in other mathematical domains. Together, these findings point to Combinatorics as a domain that can contribute to issues of equity in mathematics teaching and learning. Furthermore, the research underscored the importance of considering both the 'product' (correct answers) and 'process' (mathematical thinking) aspects in mathematics education, especially in contexts aiming to promote equitable learning opportunities.