Understanding rational numbers

Terezinha Nunes

Department of Educational Studies
University of Oxford, Oxford, UK

Research on rational numbers shows that students have to face two distinct types of challenges. First, they are confronted with quantities that involve a relation between two other quantities: therefore, their previous conceptions regarding how transformations change quantities are challenged. For example, if one cake is being shared by three children, and one more child comes, the increase in the number of children results in a decrease in the quantity of cake that each one receives. Addition, previously conceived as increasing quantities, can result in decreasing quantities in the domain of rational number. But it all depends on which quantity is increased: if there is one more cake to be shared, then each child will get more. Finally, because the quantity that each receives is a relation, if there is one more cake and three more children, there is no change in the amount that each child receives. A complex set of understandings, involving reasoning about direct and inverse as well as proportional relations is required when students think about these quantities. The second challenges come from the numbers used to represent these quantities. Natural and rational numbers represent quantities differently. There is a one-to-one correspondence between natural numbers and the quantities they represent. In contrast, because of the relational nature of rational numbers, different numbers may represent the same quantity (1/3; 2/6; 3/9 etc) and the same number might represent two different quantities (1/3 of 12 <> 1/3 of 24). The density of natural and rational numbers also differs: there is only one natural number between 2 and 3 but there are infinite rational numbers. These differences led many to wonder whether it is possible to identify informal knowledge of rational numbers, learned outside school. This presentation will consider the case for informal knowledge of rational numbers, its nature and development.