The development of childrenʹs understanding of intensive quantities

Researchers have been aware for some time of the differences between extensive and intensive quantities but the significance of these differences has not been recognized in mathematics curricula. In England children are provided with many opportunities in their first few years in school to manipulate, measure and reason about extensive quantities but have virtually no opportunity to do the same with intensive quantities. This paper contrasts extensive and intensive quantities and describes the obstacles to primary school childrenʹs understanding of intensive quantities: the need to consider two variables simultaneously and the difficulty of understanding inverse relations between variables. Study 1 shows that children have considerable difficulty in using inverse relations reasoning. Study 2 shows that this form of reasoning is more difficult in the context of intensive than extensive quantities problems. Implications for education are considered and examples of experiences with intensive quantities that could provided in school are presented.