The pursuit of economic ambitions sometimes seems to result in school mathematics favouring the performance of skills and procedures rather than nurturing the student’s more intuitive powers of mathematical rationality. This paper argues that this apparent dichotomy conceals a common belief that mathematics can be seen as independent of its past or present applications in everyday life. The paper favours a perspective from where the material points of reference that characterise school mathematics would be seen as supporting a speculative belief in the ideal mathematical entities that make up our mathematical heritage. These privileged entities (e.g. geometric objects, iterations, counts, equations) are seen as deriving from human experience of physical or social worlds and the rationalities that have been created to connect them. As a consequence the paper contributes to recent debates on the politics of mathematics education by arguing that rational mathematical thought necessarily rests on beliefs set within a play of ideological framings. School mathematics then presents not so much a distortion of “genuine” mathematical thought as a particular mode of thinking that enables the inclusion then selection of learners according to arbitrary curriculum or assessment criteria.