What 'Picture in Mind' do Secondary Students have about Defining, Proving, and Modelling?

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.