The present study tries to identify the ‘picture in mind’ that students’ have about defining, proving and modelling. We access to it through external representations of their mental representations; in particular, from the way in which these representations are expressed in written form (sentences and situations). In the study we have identified different ways of considering defining, proving and modelling: superficial, utilitarian and intrinsic, which were maintained by some students for the three elements in the different written representations. This coherence can be perceived as a characteristic of students’ mathematical understanding. Our results provide an analytical scheme whose domain goes beyond the way of students’ approaching to the mathematical elements considered in the study.