We conduct a critical review to explore how research on mathematics classroom assessment has positioned students (127 studies, 2015-2020). Our analysis shows how research has positioned students as passive recipients of assessment by portraying assessment through discourses of measurement and cognition. Conversely, students are positioned as active agents in their own learning through discourses of empowerment and monitoring. Finally, a discourse of performativity portrays classroom assessment as a way to promote results in large-scale assessments. These five discourses summarize how research produces knowledge about mathematics assessment and, in doing so, positions students as social agents with certain roles and responsibilities. Our review challenges assessment research communities to rethink how students are positioned in mathematics assessment.