The instrumental approach is a useful theoretical lens for understanding students' learning processes with a main focus on the transformation of an artefact (a human design for a goal-directed activity) into a meaningful instrument (i.e. an artefact and schemes developed by the student). In this paper, we articulate the instrumental approach for computer programming technology use in the context of pure or applied mathematical investigation. We illustrate it by examining the engagement (scheme development) of an undergraduate student engaged in ‘authentic’ programming-based mathematical inquiry work, i.e. complete inquiry work as mathematicians would do. Two schemes are elaborated, namely the scheme of validating the programmed mathematics and the scheme of articulating in the programming language a mathematical process. Our discussion evidences some specific features of the programming language as artefact associated with a general aim of mathematical inquiry, in particular in terms of articulation between mathematics and programming.