In this article, which is in criticism of a previous paper by Eric Blaire in this journal, I suggest that one's perspective of mathematics teaching is a logical consequence of one's epistemological commitment in relation to mathematical knowledge, and not merely one of expediency in response to societal pressures, or of pedagogical convenience. I suggest that from a consideration of the schools of thought in philosophy of mathematics, two distinct movements emerge. These are the Euclidean programme, an attempt to base mathematics on firm foundations, and the quasi‐empirical programme, the recognition that mathematics progresses heuristically, and through the re‐transmission of falsity. As a consequence, two teaching perspectives can be identified, knowledge‐centred and problem‐solving respectively. In describing these styles, I criticize the former and propose that the latter can initiate substantial changes and advances for school mathematics programmes.