This article examines 9-11 year olds' attempts to beat the computer at a game of strategy and asks how far and under what circumstances they developed a strategic approach to the game. Extracts from videotape transcripts are analysed to reveal the components of children's particular approaches, evident even amongst children who did not develop an overall strategy for beating the computer. We analyse children's mathematical reasoning in depth and identify circumstances conducive to valid reasoning. It is argued that use of appropriate discourse conventions is a significant factor in developing valid approaches, helping children to develop appropriate linguistic tools of their own, provided that teachers use the conventions flexibly. (Contains 6 figures.)