The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities

Abstract

This study examines the development of children's ability to reason about proportions that involve either discrete entities or continuous amounts. Six-, 8-and 10-year olds were presented with a proportional reasoning task in the context of a game involving probability. Although all age groups failed when proportions involved discrete quantities, even the youngest age group showed some success when proportions involved continuous quantities. These findings indicate that quantity type strongly affects children's ability to make judgments of proportion. Children's greater success in judging proportions involving continuous quantities appears to be related to their use of different strategies in the presence of countable versus noncountable entities. In two discrete conditions, children-particularly 8-and 10-year-olds-adopted an erroneous counting strategy, considering the number of target elements but not the relation between target and nontarget elements, either in terms of number or amount. In contrast, in the continuous condition, when it was not possible to count, children may have relied on an early developing ability to code the relative amounts of target and nontarget regions.Studies of children's ability to reason about proportions have yielded a mixed and somewhat confusing pattern of results. Some studies show competence on tasks JOURNAL OF COGNITION AND DEVELOPMENT, 8(2), 237-256

Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

“…Findings suggested that children as young as 8 years old were able to consider both components that constitute a proportion and integrate them in a normative proportional way 2 . These results stand in contrast to previous claims that proportional reasoning does not emerge before the age of 11 years (Moore et al, 1991;Noelting, 1980;Piaget & Inhelder, 1975) and confirm other findings that even younger children are able to reason about proportions (Acredolo et al, 1989;Boyer & Levine, 2012;Boyer et al, 2008;Jeong et al, 2007;Schlottmann, 2001;Singer-Freeman & Goswami, 2001;Sophian, 2000;Spinillo & Bryant, 1991, 1999. In line with previous paradigms showing earlier success in children's proportional reasoning, it is possible that the presentation of continuous proportional quantities and the nature of the response mode (spatial ratings that are more intuitively graspable) led to children's success on our proportional reasoning task.…”

“…That is, children's formal fraction knowledge may have helped them to encode spatial proportions and to reproduce them on the rating scale. Even though this possibility cannot be eliminated by our correlational results, it seems unlikely in light of many studies (Acredolo et al, 1989;Boyer & Levine, 2012;Boyer et al, 2008;Jeong et al, 2007;Schlottmann, 2001;Singer-Freeman & Goswami, 2001;Sophian, 2000;Spinillo & Bryant, 1991, 1999 showing signs of proportional reasoning at an age when understanding of formal fractions is not present (Hecht & Vagi, 2010;Schneider & Siegler, 2010;Stafylidou & Vosniadou, 2004). Nonetheless, future studies using longitudinal designs or training components are needed to pin down the causal direction of the relation we have identified.…”

“…For example, several studies have demonstrated that 5-to 6-year-olds showed successful proportional reasoning when presented with continuous amounts as opposed to discrete amounts (Boyer, Levine, & Huttenlocher, 2008;Jeong, Levine, & Huttenlocher, 2007;Spinillo & Bryant, 1999). Children also showed earlier competence at the age of 3 to 4 years when asked to produce equal proportions, possibly by tapping their ability to reason by analogy (Goswami, 1989;Singer-Freeman & Goswami, 2001).…”

Child proportional scaling: Is 1/3=2/6=3/9=4/12?

“…In fact, research over the past several decades shows that young children, and perhaps even infants, are sensitive to proportional relations (e.g., Denison & Xu, 2010;Jeong, Levine, & Huttenlocher, 2007;McCrink & Wynn, 2007;Sophian, 2000;Xu & Garcia, 2008). However, these studies also show that there are limits to children's early sense of proportion.…”

Children’s multiplicative transformations of discrete and continuous quantities

“…Children's apparent lack of competence could stem not from difficulties in reasoning about discrete quantities per se, but from the tendency to count when a task affords the opportunity (Jeong, Levine, & Huttenlocher, 2007;Boyer, Levine, & Huttenlocher, 2008). …”