Integrating digital technologies to benefit teaching and learning has long been driving higher education. The uptake of technology has been supported by teacher training focused on developing teachers' capabilities to design for learning. However, in this paper, we raise the point of moving towards teaching-sensitive technology as a clear alternative to current strategies focusing on teachers' mental processes. To develop this point, the paper offers a qualitative study that explores teaching to identify critical features of technology supporting teachers' work. Analysing teaching from a hermeneutic perspective, we arrive at six fundamental dynamics within which teachers operate. Based on the factors identified, we present three principles to guide future design of technologies for teaching and two approaches to designing technology sensitive to teachers' values.