Frameworks for mathematical learning have been updated to make sense of digital tools, leading to the notion of mathematical digital competencies. In particular and in parallel, the computational thinking (CT) construct has gained relevance, and the mathematics curriculum takes part in this wave of interest. We offer a theoretical discussion on the meanings and implications of enabling technomathematical discourse at the intersection of CT and mathematical digital competencies. We do so by networking theories anchored in episodes from an implemented geometry task. A computermathematical fluency is not a particular case of techno-mathematical discourse, and it must entail combining and disambiguating computer and mathematical knowledge. It relates to the artefactinstrument duality to communicate through and with the computer environment through instrumented actions to construct computer models in mathematical situations.