We ground Cultural-Historical Activity Theory (CHAT) in studies of workplace practices from a mathematical point of view. We draw on multiple case study visits by college students and teacher-researchers to workplaces. By asking questions that "open boxes", we "outsiders and boundary-crossers" sought to expose contradictions between College and work, induce breakdowns and identify salient mathematics. Typically, we find that mathematical processes have been historically crystallised in "black boxes" shaped by workplace cultures: its instruments, rules and divisions of labour tending to disguise or hide mathematics. These black boxes are of two kinds, signalling two key processes by which mathematics is put to work. The first involves automation, when the work of mathematics is crystallised in instruments, tools and routines: this process tends to distribute and hide mathematical work, but also evolves a distinct workplace "genre" of mathematical practice. The second process involves sub-units of the community being protected from mathematics by a division of labour supported by communal rules, norms and expectations. These are often regulated by boundary objects that are the object of activity on one side of the boundary but serve as instruments of activity on the other side. We explain contradictions between workplace and College practices in analyses of the contrasting functions of the activity systems that structure them and that consequently provide for different genres and distributions of mathematics, and finally draw inferences for better alignment of College programmes with the needs of students.