ABSTRACT. The present study examined the reasoning strategies and arguments given by pre-service school teachers as they solved two problems regarding fractions in different symbolic representations. In the first problem, the pre-service school teachers were asked to compare between two different fractions having the same numerical representation, and in the second problem, they were asked to compare between different notational representations of the same fraction. Numeration systems in bases other than ten were used to generate various representations of fractions. All students were asked to provide justifications to their responses. Strategies and arguments relative to pre-service teachers' concepts of fractions and place value were identified and analyzed based on results of 38 individual clinical interviews, and written responses of 124 students. It was found that the majority of students believe that fractions change their numerical value under different symbolic representations..
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] subconstruct) and 3/4 of the size of each bundle (stretcher/shrinker [SS] subconstruct). 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 over a uniting operation. With these SS strategies, rational number is conceptualized as a rate. However, the SS strategies were used less often than the DPR strategies. Detailed cognitive models of these strategies in terms of the underlying conceptual units, their structures, and their modifications, were produced, and a “mathematics of quantity” notational system was used as an analytical tool to describe and model the embedded abstractions.
Third and fifth graders were given a sequence of number pairs and asked ro discover the function rule relating them, test the hypothesis, and generalize it to other instances. The fifth graders were better than the third graders on all performance criteria. Task difficulty depended on the funcuon. Some functions were easier when presented in a graphic mode, others in a symbolic mode. Whether inferences made use of abscissa-ordinate pairs, in contrast to a sequence of ordinate values, appeared to depend on both function and mode of presentation. Unlike adults, the children seemed ready to relinquish a hypothesis in the face of disconfirming evidence.
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