THE UNIT CONSTRUCT IN RATIONAL NUMBER REASONING
Ong, W.M. and Reeve, R.A., University of Melbourne, Australia
The research explored Vergnaud's (1994) and Steffe's (1992) hypothesis that common mathematical structures guide children's thinking across different rational number domains (fractions, ratios, proportions etc).Steffe argues that children's knowledge of units is central to their understanding of rational number.Steffe's claim is that the flexible formation and decomposition of unit quantities is central to an understanding of rational number and quantity.However, little research has explored the relevance of children's unit understanding for their rational number problem solving. Sixty-six Grades 5 and 6 children (Mean age: 11 years 3 months) individually solved 36 pictorially presented "sharing" problems in which the number/quantity relations between and within measures was systematically varied. Each problem represented two 'whole' quantities partitioned into equal shares in which the nature of the quantities (e.g., discrete versus continuous quantity) were systematically varied to assess conceptual reasoning about units.Children were asked how many shares from one whole equaled a specified number of shares from the other whole.Children's reasoning was recorded for later analysis. Of interest was the way in which children referred to units in their problem solving.Solution strategies were classified and analysed using cluster analytic techniques.Analyses revealed that (1) the concept of a unit is central to children's reasoning about rational number, and (2) there is a developmental progression in unit understanding, independent of age, that affects all aspects of problem-solving (namely a progression from one-to- one to one-to-many, to many-to-many unit relations).