In an accompanying study, Alibali et al. [Alibali, M. W., Ockuly, K. M., Fischer, A. D. (2009) "Learning new problem-solving strategies leads to changes in problem representation." "Cognitive Development, 24," 89-101.] present an important experimental result: introducing a new strategy can affect conceptual aspects of children's problem representation. In this commentary, we briefly explore implications of this finding for theories of the development of mathematical cognition and of the development of cognitive structures, more generally. We argue that Alibali et al.'s results, taken in the context of previous findings, imply that children's actions during mathematical problem solving affect their subsequent understanding of the problem. We further suggest that results such as these may be best accommodated by reconceptualizing how cognitive structures achieve both stability and flexibility.