Making mathematical reasoning visible: a qualitative comparative case study of students’ reasoning, representation, and communication

Abstract

Developing students’ capacity for mathematical reasoning and communication is a central goal of contemporary mathematics education, yet qualitative evidence on how these competencies are externalized during problem solving remains limited. This study investigates how students’ mathematical reasoning, use of representations, and modes of engagement differ in the ways 19 they become visible during problem solving, even when solutions are mathematically correct. Adopting a qualitative comparative case study design, three secondary school students exhibiting contrasting modes of engagement were examined while solving a non-routine algebraic problem designed to elicit representational transitions and justification. Data were drawn from written solution artefacts and follow-up semi-structured, artifact-based interviews. Thematic analysis focused on reasoning documentation, representational coordination, and forms of mathematical communication, emphasizing processes of externalization rather than evaluative correctness. Although all students produced correct solutions, their reasoning became visible in markedly different ways: one case demonstrated systematic written documentation and coordinated use of representations, another relied primarily on verbal explanation with limited written elaboration, and a third engaged in exploratory trial-and-error with minimal explicit justification. Rather than treating individual differences as fixed traits, these cases illuminate distinct pathways through which reasoning is externalized or remains partially concealed. The study contributes an analytic lens for examining the visibility of mathematical reasoning and offers implications for instructional practices that foreground explanation and representation in heterogeneous mathematics classrooms.