Young children's understanding of multiplicative concepts : a psychological approach
This thesis investigates the origins of children's understanding of multiplication and division and how they progressively become coordinated in young children. The hypothesis of the study is that the origins of these operations are in children's schemas of action. This leads to the prediction that children can understand about multiplicative relations before they can solve computational multiplicative problems. This hypothesis is contrasted with Fischbein's et al. (1985) hypothesis that multiplication originates from repeated addition and quotitive division from repeated subtraction, which leads to the prediction that children must be able to quantify in order to understand multiplicative relations. To test these predictions a series of studies was carried out analysing children's performance in relational, non computational problems and in computational problems, involving discontinuous and continuous quantities which could not be quantified. The first study explored children's use of one-to-many correspondence reasoning to solve a variety of problems. Children aged 4 to 7 were asked: a) to order different sets on the basis of correspondence relations; and b) to indicate the size of the corresponding sets. Even the 5 year olds were able to order the size of the sets when they could use correspondence relations. The same children had great difficulty in indicating the size of the sets. This demonstrated that the ability to quantify is not a prerequisite for understanding multiplicative relations. The second and third studies explored children's understanding of the inverse relation between the number of the quotas and their size in sharing problems. The children (aged 4 to 7) were tested in partition and quotition problems. In partitive problems more than half of the 6 year olds and the majority of the 7 year olds showed an understanding of the inverse divisor-quotient relation. In the quotitive problems, although there was a drop in the level of performance, it was not found to be significant across the same age groups. Children's's ability to reflect on the relations involved in a sharing situation before being able to quantify the problems challenges the hypothesis that quantification is the origin for understanding division. The fourth study explored whether multiplication and division develop as independent or coordinated operations. Children were asked to quantify a set of multiplication and division problems in which it was possible to model the problem directly using their schemas of action and a second set where this direct modelling was not possible because a crucial piece of information was missing. It was expected that children who have coordinated their multiplicative schemas of action would be able to deploy an action usually associated with division to solve a multiplication problem and vice versa. More than half of the children were able to quantify the problems that matched their actions directly, but their performance decreased in the missing value multiplication problems that were solved by an action related to division. The discrepancy in the children's performance suggests that the two operations have different roots and initially develop independently of each other. The findings support the hypothesis that the understanding of multiplication and division is constructed from children's schemas of action. The two operations have distinct roots and develop independently before they become coordinated at a later stage.