In this article, we seek to understand how university students learn to use programming as an instrument for 'authentic' mathematical investigations. We use the instrumental approach as a framework, focusing on how the transformation of the programming language into an instrument requires that the user develops or mobilizes multiple schemes at different stages of the task development. Moreover, we propose to adapt Assude's instrumental integration model for teachers and shift its focus to students to describe four instrumental stages of student development (i.e. instrumental genesis) of their schemes. These four stages are illustrated by examining two undergraduates' engagement at different times during a first-year, programming-based, mathematics course. The proposed approach takes into account not only individual scheme development, but also the development of a complex web of schemes. It incorporates the concept of schemes in the different instrumental stages for students and also unfolds Brennan and Resnick's three-dimensional computational thinking development framework (for mathematics) as students appropriate programming as an instrument.