One of the major challenges in (research on) elementary mathematics education is to develop in children a disposition towards adaptive rather than merely routine use of strategies when doing various kinds of arithmetic tasks. Such a disposition towards the development and efficient, flexible and insightful use of a variety of strategies has become a major goal of elementary school curricula all over the world. But there are several major issues related to the learning and teaching of such a disposition, such as: What do we exactly mean by efficient, flexible and insightful use of an arithmetic strategy? When to teach for adaptive expertise? How to teach for it? Is it a valuable and feasible goal for all children, including the weaker ones? In this talk these issues will be addressed, making exemplarily use of a line of studies in which I am involved, together with a number of colleagues and collaborators from the KU Leuven, wherein we are investigating the theme of routine vs. adaptive expertise in the domain of multi-digit subtraction.