Two case studies were carried out to explore students’ understanding of concepts in calculus. The first study explored engineering students’ understanding of limit and integral. The second study involved students within a mathematics programme, working on a challenging task including the concepts function and derivative, requiring proof by induction. Drawing on a theory of contextualisation data were analysed within a constructivist research framework following the principles of intentional analysis.  The results reveal that the students in the mathematics programme expressed their understanding in a formal context where also intuitive ideas played an important role. The engineering students expressed their understanding in an algorithmic context, in which procedural knowledge was predominant. The studies display how students may have a potential for developing a formal understanding of a mathematical concept previously viewed within an algorithmic context.