Exploring Student Difficulties in Algebra: Insights from a Four-Item Diagnostic Assessment

Abstract

This study examines undergraduate students’ understanding of four foundational algebraic topics: fundamental algebraic manipulation, systems of linear equations, inequalities, and quadratic functions, using a mid-term diagnostic assessment. Four exam questions, each aligned with a key algebra domain, were analyzed to identify performance patterns and common misconceptions. Error analysis was used to evaluate student responses and interpret the conceptual difficulties underlying incorrect answers. The findings reveal persistent misconceptions consistent with recent literature in algebra education. In fundamental algebra, students frequently miscombine like terms, indicating an incomplete understanding of symbolic structures. For systems of linear equations, many struggled to select and apply elimination or substitution effectively, reflecting difficulties with multi-step reasoning. Inequality errors were dominated by incorrect sign handling when manipulating negative values. In quadratic functions, students demonstrated weak representational fluency, particularly in identifying the vertex, determining intercepts, and sketching parabolas. The study highlights the need for instructional strategies that emphasize conceptual reasoning, multiple representations, and structured error-reflection. Recommendations include integrating formative assessments, strengthening the selection of methods, and enhancing graphical-symbolic connections, particularly for quadratic functions. These findings provide actionable insights for improving algebra proficiency in tertiary mathematics courses.