Summary: Mathematics in schools exists substantially as pedagogical material crafted for supposed modes of apprehension. But of course such apprehension depends on how we understand mathematical objects and how we understand human subjects. The apprehension of mathematical objects is examined through sessions with student teachers researching their own spatial awareness from a pedagogical point of view. The paper is guided by the recent work of Alain Badiou whose philosophical model develops a Lacanian conception of human subjectivity and defines a new conception of objectivity. In this model the conception of subjectivity comprises a refusal to allow humans to settle on certain self-images that have fuelled psychology and set the ways in which humans are seen as apprehending the mathematically defined world. The assertion of an object, meanwhile, is associated with finding a place for it in a given supposed world, and it may reconfigure that world. The composite model understands learning as shared participation in renewal where there is a mutual dependency between the growth of human subjects and of mathematical objects. Renewal is referenced to a diversity of ever shifting discursive parameters that enable learning through negotiating the spaces within which we operate and the objects those spaces allow. Learning to teach then comprises developing sensitivity towards the discursive spaces that allow others to build objects. The paper provides examples from teacher education activities centred in addressing these concerns.