From Rule to Reason: Grade 8 Students’ Understanding of the Distributive Property under the Field Axioms of Real Numbers

Abstract

This qualitative phenomenological study explored how Grade 8 students in Nepal conceptualize and apply the field axioms of real numbers, with particular attention to the distributive law. Data were collected through classroom observation, task-based interviews, and students’ written work from fifteen instructional sessions. Thematic analysis revealed five major patterns of understanding. Most students demonstrated procedural fluency without conceptual depth, relying on memorized steps such as “multiply both” rather than logical reasoning. Many confused distributions with expansion, viewing factorization as a division process instead of a reversible property of equivalence. The connection between the axiomatic structure of real numbers and their operations was poor, as students performed operations correctly but could not relate them to closure, commutativity, or identity. The teacher-centered classroom culture reinforced procedural learning and limited opportunities for reflective dialogue. Nevertheless, a few students showed emerging conceptual shifts when supported by visual and reflective activities, linking distributivity to area models and recognizing its reversibility. The study concludes that explicit engagement with axiomatic reasoning and dialogic instruction can help students move from instrumental to relational understanding. It recommends integrating visual representations, reasoning prompts, and reflective discussion to strengthen structural comprehension of the field axioms in lower-secondary mathematics.