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Terezinha Nunes: Understanding rational numbers

Although there are different subconstructs or meanings for rational numbers (see, for example, Behr, Harel, Post, & Lesh, 1992; Kieren, 1988), it seems reasonable to seek the origin of children’s understanding of rational numbers in their understanding of division. Our hypothesis is that in division situations children can develop some insight into the equivalence and order of quantities that would normally be represented by fractions, even in the absence of knowledge of representations for fractions, either in written or oral form. In our analysis we reviewed the empirical literature of this field. The mathematics education literature traditionally considers two types of division problems: partitive and quotative division. This classification distinguishes two ways in which children use the same scheme of action, which will be referred to here as partitioning. Although partitioning is the scheme that is most often used to introduce children to fractions, it is not the only scheme of action relevant to division. Children also use correspondences in division situations when the dividend is represented by one measure and the divisor is represented by another measure. Empirical results show that both partitioning and correspondences could help children understand something about the equivalence between quantities. However, the reasoning required to achieve this understanding differs across the two schemes of action. Children can use the scheme of correspondences to establish equivalences between sets that have the same ratio to a reference set (Piaget, 1952); re-distribute things after having carried out one distribution (Davis and colleagues); to reason about equivalences resulting from division both when the dividend is larger or smaller than the divisor (Bryant and colleagues; Empson, 1999; Nunes and colleagues); and to order fractional quantities (Kieren, 1993; Kornilaki & Nunes, 2005; Mamede, 2008). In using partitioning, students can develop insight into the inverse relation between the number of parts and the size of the parts through the partitioning scheme but there is no evidence that they realize that if you cut a whole in twice as many parts each one will be half in size. Finally, improper fractions seem to cause uneasiness to students who have developed their conception of fractions in the context of partitioning; it is important to be aware of this uneasiness if this is the scheme chosen in order to teach fractions.

MAGYAR PEDAGÓGIA 108. Number 1. 5-27. (2008)

Levelezési cím / Address for correspondence: Terezinha Nunes, Department of Education, University of Oxford, 15 Norham Gardens, Oxford, OX2 6PY


Magyar Tudományos Akadémia