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# Conceptual development of prospective elementary teachers the case of division of fractions /

Abstract (Summary)
iii The purpose of this study is to investigate the understandings and the conceptual development of prospective elementary teachers in the area of division of fractions. The participants were two prospective elementary teachers enrolled in a teacher certification program at a northeastern U.S. university. The current study is based on teaching experiment as a vehicle to investigate the conceptual development of the participants. The investigation conducted on the two prospective elementary teachers revealed that they had a compartmentalized understanding of division of fractions. The results of the study suggest that the participants viewed division of fractions as a sequence of arithmetic relationships and they did not have an abstraction of quotitive situations. This study also shows that the participants made an abstraction of quotitive situations by reflecting on their mental activities in a diagrammatic world. In addition, this study investigates the mathematical structure of division of fractions as seen through these two participantsâ€™ work. As a result, division of fractions concept is based on two main operations: partitioning and quantification. In this regard, the overall goal for the division of fractions is to determine number of divisor groups through partitioning a given dividend quantity. Therefore, one needs to understand that in a given division of fractions problem, a given quantity needs to be partitioned based on a second quantity and then this partitioning needs to be quantified. This study also shows that the participants did not have an abstraction of divisor as an intensive quantity that connects the two extensive quantities, the dividend and the quotient. Lacking such an abstraction caused the participants not be able to coordinate the quantitative relationship between the divisor, remainder and fractional part of the quotient under the guidance of the overall goal they set for the given division of fractions iv problems. What also arises from this study is a developmental trajectory for creating an algorithm that is based on the participantsâ€™ spontaneous activity. This algorithm is common denominator algorithm as opposed to the commonly known invert-and-multiply algorithm. Using diagrams for thinking about the common denominator algorithm helped the participants make conceptual advances in their thinking.
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