Analysis of Students’ Difficulties on Understanding Integral Concepts

Abstract

Integral concepts are among the most challenging topics in mathematics learning because they require students to understand symbolic forms, graphical representations, and the relationship between integrals and derivatives. Many students still experience persistent difficulties due to weak conceptual foundations and limited representational skills. This study aims to analyze the types and causes of students’ difficulties in understanding integral concepts in high school mathematics. A qualitative descriptive method was used, supported by diagnostic tests, semi-structured interviews, and classroom observations. The diagnostic test consisted of three questions designed to measure students’ conceptual, procedural, representational, technical, and affective difficulties. Interviews were conducted with five students selected based on varying levels of difficulty to explore their thought processes, while observations were carried out to validate the behavioral patterns that emerged during the test and interview sessions. Data were analyzed using the Miles and Huberman model, consisting of data reduction, data display, and conclusion drawing. The results show that conceptual difficulties are the most dominant, particularly in understanding integrals as areas, accumulation processes, and their relationship to derivatives. These difficulties contribute to procedural and technical errors, including incorrect steps, algebraic mistakes, and inaccurate limit substitutions. Representational and affective difficulties, such as anxiety and low confidence, also negatively impact students’ performance. The study concludes that students’ integral difficulties are multidimensional and interconnected, suggesting the need for instructional approaches that emphasize conceptual reinforcement, multiple representations, and supportive learning environments.