Fractions as the Coordination of Multiplicatively Related Quantities: A Cross-Sectional Study of Children's Thinking

Abstract

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 strategies involved children's partitions of continuous units expressed in terms of the number of pieces that would be created. Ratio strategies 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.

Variations of Reasoning in Equal Sharing of Children Who Experience Low Achievement in Mathematics: Competence in Context

“…Specifically, children's activity within equal sharing situations-equally sharing some number of the same-sized objects among some number of people, where the result is a fractional quantity-has provided observable evidence of children's fractional reasoning in previous research [12][13][14][15][16]. Additionally, the partitioning activity that children use in these situations is well established as the root of children's knowledge of fractions [15].…”

“…Children's activity within equal sharing situations first appears as representations of acts of partitioning [12][13][14][15]18]. Literature suggests that in early experiences with fractions children may see the problem as unsolvable, possibly because they do not yet see wholes as divisible [14].…”

“…For example, when sharing five sticks of licorice equally among four people, some children may begin by adding more whole objects to obtain a whole number result (i.e., add three more sticks such that each person receives one whole stick of licorice) or they may create unequal shares (i.e., give one person two sticks of licorice, [13]). Empson and her colleagues [13] refer to this way of reasoning as "no coordination" because the children are attending to either the necessity to make shares equal or the need to share everything (unequally), but not both at the same time.…”

“…In fact, for these children, the parts may not be differentiated from the whole. Empson et al [13] calls this way of reasoning "non-anticipatory" because, while the student is now attending to both the need to exhaust all items to be shared and make the shares equal, the partitioning and subsequent naming of the fractional quantity produced is not associated with a relation, or coordination, between the number of sharers and the amount being shared.…”

“…The child begins to use the number of sharers as an a priori plan to create a predetermined number of parts to exhaust the shared items [13]. Unitizing one whole, the child may eventually plan to cut each stick into four parts [14].…”