Developing Purposeful Mathematical Thinking: a Curious Tale of Apple Trees

In this paper I explore aspects of the ways in which school mathematics relates to the "real" world, and argue that this relationship is an uneasy one. Through exploring the causes of this unease, I aim to expose some problems in the ways in which context is used within mathematics education, and argue that the use of context does not ensure that the purposes of mathematics are made transparent. I present and discuss a framework for task design that adopts a different perspective on mathematical understanding, and on purposeful mathematical thinking.