Re-conceptualising conceptual understanding in mathematics

In this theoretical paper we explore interrelationships between conceptual and procedural understanding of mathematics in the context of individuals and groups. We question the enterprise of attempting to assess learners’ mathematical understanding by inviting them to perform a (perhaps unfamiliar) procedure or offer an explanation. Would it be appropriate to describe a learner in possession of an algorithm for responding satisfactorily to such prompts as displaying conceptual understanding? We relate the discussion to Searle’s “Chinese Room” thought experiment and draw on Habermas’ Theory of Communicative Action to develop potential implications for addressing the problem of interpreting learners’ mathematical understanding