In this paper we show the results of a study in relation to the mathematical work and practice of two novice teachers, when the classroom topic involved the differing views related to synthetic and analytic Euclidean geometry. We present the ideal mathematical working space (MWS) for each teacher, specifically, the movements between the components of the epistemological and cognitive planes, and the privileged geometric paradigm. The results provide empirical information regarding the teaching practice when using different geometric methods and on the possibility of complementing them, and furthermore the topic is presented as a new way of studying through the MWS.