Teachers of mathematics face a particular tension, which the authors call the planning paradox. If teachers plan from objectives, the tasks they set are likely to be unrewarding for the pupils and mathematically impoverished. Planning from tasks may increase pupils' engagement but their activity is likely to be unfocused and learning difficult to assess. By seeking inspiration from research in the areas of curriculum design, the nature of authenticity in the classroom and the use of tools, and by looking retrospectively at the design of computer-based tasks that have underpinned their research for many years, the authors recognise a theme of purposeful activity, leading to a planned appreciation of utilities for certain mathematical concepts. The authors propose utility as a third dimension of understanding, which can be linked to purpose in the effective design of tasks. The article concludes with a set of heuristics to guide such planning.

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