TY - JOUR

T1 - Fractions as the coordination of multiplicatively related quantities

T2 - A cross-sectional study of children's thinking

AU - Empson, Susan B.

AU - Junk, Debra

AU - Dominguez, Higinio

AU - Turner, Erin

N1 - Funding Information:
An earlier version of this paper was presented at the 10th International Congress on Mathematical Education in Copenhagen, Denmark. We would like to thank Kevin LoPresto, Luz Maldonado, and Stephanie Nichols for assistance in collecting and coding data for this research, Karen Heinz and Taylor Martin for critical reviews of this work, Jennifer Cook and Chris Bailor for editing, Norma Presmeg and the anonymous reviewers of the manuscript and, last but not at all least, Austin area schools for allowing us to work with their students. This work was supported in part by National Science Foundation grant no. 0138877 to Empson. The views expressed here are the responsibility of the authors and do not necessarily reflect those of the funding agency.

PY - 2006/9

Y1 - 2006/9

N2 - Although equal sharing problems appear to support the development of fractions as multiplicative structures, very little work has examined how children's informal solutions reflect this possibility. The primary goal of this study was to analyze children's coordination of two quantities (number of people sharing and number of things being shared) in their solutions to equal sharing problems and to see to what extent this coordination was multiplicative. A secondary goal was to document children's solutions for equal sharing problems in which the quantities had a common factor (other than 1). Data consisted of problem-solving interviews with students in 1st, 3rd, 4th, and 5th grades (n=112). We found two major categories of strategies: (a) Parts Quantities strategies and (b) Ratio Quantities strategies. Parts quantities involved children's partitions of continuous units expressed in terms of the number of pieces that would be created. Ratio quantities involved children's creation of associated sets of discrete quantities. Within these strategies, children drew upon a range of relationships among fractions, ratio, multiplication, and division to mentally or physically manipulate quantities of sharers and things to produce exhaustive and equal partitions of the items. Additionally, we observed that problems that included number combinations with common factors elicited a wider range of whole-number knowledge and operations in children's strategies and therefore appeared to support richer interconnections than problems with relatively prime or more basic number combinations.

AB - Although equal sharing problems appear to support the development of fractions as multiplicative structures, very little work has examined how children's informal solutions reflect this possibility. The primary goal of this study was to analyze children's coordination of two quantities (number of people sharing and number of things being shared) in their solutions to equal sharing problems and to see to what extent this coordination was multiplicative. A secondary goal was to document children's solutions for equal sharing problems in which the quantities had a common factor (other than 1). Data consisted of problem-solving interviews with students in 1st, 3rd, 4th, and 5th grades (n=112). We found two major categories of strategies: (a) Parts Quantities strategies and (b) Ratio Quantities strategies. Parts quantities involved children's partitions of continuous units expressed in terms of the number of pieces that would be created. Ratio quantities involved children's creation of associated sets of discrete quantities. Within these strategies, children drew upon a range of relationships among fractions, ratio, multiplication, and division to mentally or physically manipulate quantities of sharers and things to produce exhaustive and equal partitions of the items. Additionally, we observed that problems that included number combinations with common factors elicited a wider range of whole-number knowledge and operations in children's strategies and therefore appeared to support richer interconnections than problems with relatively prime or more basic number combinations.

KW - Children's strategies

KW - Distributed thinking

KW - Elementary K-8

KW - Equal sharing

KW - Fractions

KW - Multiplicative structure

KW - Problem solving

KW - Rational number

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U2 - 10.1007/s10649-005-9000-6

DO - 10.1007/s10649-005-9000-6

M3 - Review article

AN - SCOPUS:33748995349

VL - 63

SP - 1

EP - 28

JO - Educational Studies in Mathematics

JF - Educational Studies in Mathematics

SN - 0013-1954

IS - 1

ER -