# Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions

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## Abstract

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.

## Keywords

Prospective elementary teachers Partitive division Division with fractions## Notes

### Acknowledgments

We would like to thank the mathematics education community at the University of Delaware for feedback at our Friday seminar presentation about this work during its early stages. The first author also appreciates the support of her women’s writing retreat group (AnnaMarie Conner, Amy Ellis, Beth Herbel-Eisenmann, Heather Johnson, and Eva Thanheiser) who talked with her about ideas for the initial draft of this paper.

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