Abstracts - Earli

KN4 201 September 2007 12:30 - 13:30Room: HarmóniaInvited PaperUnderstanding rational numbersTerezinha Nunes, University of Oxford, United KingdomChair:Erik De Corte, University of Leuven, BelgiumResearch on rational numbers shows that students face two distinct types of challenges. First, theyare confronted with quantities that involve a relation between two other quantities: therefore, theirprevious conceptions regarding how transformations change quantities are challenged. Forexample, if one cake is being shared by three children, and one more child comes, the increase inthe 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 thedomain of rational number. But it all depends on which quantity is increased: if there is one morecake to be shared, then each child will get more. Finally, because the quantity that each receives isa relation, if there is one more cake and three more children, there is no change in the amount thateach child receives. A complex set of understandings, involving reasoning about direct and inverseas well as proportional relations is required when students think about these quantities. The secondchallenges come from the numbers used to represent these quantities. Natural and rational numbersrepresent quantities differently. There is a one-to-one correspondence between natural numbersand 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 mightrepresent two different quantities (1/3 of 12 1/3 of 24). The density of natural and rational numbersalso differs: there is only one natural number between 2 and 3 but there are infinite rationalnumbers. These differences led many to wonder whether it is possible to identify informalknowledge of rational numbers, learned outside school. This presentation will consider the case forinformal knowledge of rational numbers, its nature and development.– 866 –