Differentiation and integration played large roles within classic developmental theories but have been relegated to obscurity within contemporary theories. However, they may have a useful role to play in modern theories as well, if conceptualized as guiding principles for analyzing change rather than as real-time mechanisms. In the present study, we used this perspective to examine which rules children use, the order in which the rules emerge, and the effectiveness of instruction on water displacement problems. We found that children used systematic rules to solve such problems, and that the rules progress from undifferentiated to differentiated forms and toward increasingly accurate integration of the differentiated variables. Asking children to explain both why correct answers were correct and why incorrect answers were incorrect proved more effective than only requesting explanations of correct answers, which was more effective than just receiving feedback on the correctness of answers. Requests for explanations appeared to operate through helping children notice potential explanatory variables, formulate more advanced rules, and generalize the rules to novel problems.