Evidence shows that appropriate use of technology in education has the potential to increase the effectiveness of, eg, teaching, learning and student support. There is also evidence that technology can introduce new problems and ethical issues, e.g., student privacy. This article maps some limitations of technological approaches that ensure student data privacy in learning analytics from a critical data studies (CDS) perspective. In this conceptual article, we map the claims, grounds and warrants of technological solutions to maintaining student data privacy in learning analytics. Our findings suggest that many technological solutions are based on assumptions, such as that individuals have control over their data ('data as commodity'), which can be exchanged under agreed conditions, or that individuals embrace their personal data privacy as a human right to be respected and protected. Regulating student data privacy in the context of learning analytics through technology mostly depends on institutional data governance, consent, data security and accountability. We consider alternative approaches to viewing (student) data privacy, such as contextual integrity; data privacy as ontological; group privacy; and indigenous understandings of privacy. Such perspectives destabilise many assumptions informing technological solutions, including privacy enhancing technology (PET).