CONCEPTUAL UNITS ANALYSIS OF PRESERVICE ELEMENTARY-SCHOOL TEACHERS STRATEGIES ON A RATIONAL-NUMBER-AS-OPERATOR TASK

This study explores preservice teachers' understanding of the operator construct of rational number. Three related problems, given in 1-on-1 clinical interviews, consisted of finding 3/4 of a pile of 8 bundles of 4 counting sticks. Problem conditions were suggestive of showing 3/ 4 of the number of bundles (duplicator/partition-reducer [DPR] subcons truct) and 3/4 of the size of each bundle (stretcher/shrinker [SS] sub construct). This study provides confirming instances that students use these 2 rational number operator subconstructs. The SS strategies are identified when the rational number, as an operator, is distributed o ver a uniting operation. With these SS strategies, rational number is conceptualized as a rate. However, the SS strategies were used less of ten than the DPR strategies. Detailed cognitive models of these strate gies in terms of the underlying conceptual units, their structures, an d their modifications, were produced, and a ''mathematics of quantity' ' notational system was used as an analytical tool to describe and mod el the embedded abstractions.