The potential of a ‘data-driven life’, together with the realization of Weiser’s vision of ‘the disappearing computer’ have been embraced by many. However, the increasing invisibility, virtuality and complexity of data systems also come with a variety of concerns, such as issues of sensemaking, ownership, representation and control. Although data and technology is all around us, its virtual and invisible nature, thereby its lack of material and tangible forms has implications on the way data systems are (mis) used, understood, experienced and perceived. This paper presents craft-based approaches for physical sense making –widely ranging from physical artefacts to show and hide from monitoring, to crafting data physicalizations for critical thinking, communication and creativity. In doing so, this paper discusses how crafting physical forms can be used as a way to grasp and understand ‘invisible’ data systems.