This article contributes to current debates on progressive, knowledge-based approaches to the curriculum by addressing the question of what it is that students are entitled to learn in school mathematics. From the outset it recognizes progressive arguments that teaching should be reconnected with the emancipatory ambitions of education. In doing so, it takes the notion of "powerful knowledge" as a starting point, based on what knowledge school students have the right to have access to. In turn, it considers this as a question of "epistemic" quality. This is elaborated as a concept by drawing on outcomes from a recent study arising from the Developing Mathematical Thinking in the Primary Classroom (DMTPC) project. This concept is founded on the analysis of a distinction between "mathematical fallibilism", based on a heuristic view of mathematics as a human activity, and "mathematical fundamentalism", which reflects an authoritarian view of the subject as being infallible, absolutist and irrefutable. The relation between powerful knowledge and epistemic quality is considered further by framing it within a sociological theory of knowledge. This helps to highlight a further distinction between "knowing that and knowing how", which is used to illustrate examples of high and low epistemic quality in school mathematics. The first example of "high epistemic quality" is drawn from the DMTPC project. The second example is of "low epistemic quality" and comes from the highly promoted Core Knowledge Foundation that has recently been imported into English schools from the USA. Finally, the article considers the role of teachers as curriculum makers at the classroom level where curriculum and pedagogy effectively merge. In conclusion, the implications for both policy and practice are considered, in particular proposals are made in relation to the role and place of subject didactics in teaching and teacher education.