Development of proportional reasoning: Where young children go wrong.

Abstract

Previous studies have found that children have difficulty solving proportional reasoning problems involving discrete units until 10-to 12-years of age, but can solve parallel problems involving continuous quantities by 6-years of age. The present studies examine where children go wrong in processing proportions that involve discrete quantities. A computerized proportional equivalence choice task was administered to kindergartners through fourth-graders in Study 1, and to first-and third-graders in Study 2. Both studies involved four between-subjects conditions that were formed by pairing continuous and discrete target proportions with continuous and discrete choice alternatives. In Study 1, target and choice alternatives were presented simultaneously and in Study 2 target and choice alternatives were presented sequentially. In both studies, children performed significantly worse when both the target and choice alternatives were represented with discrete quantities than when either or both of the proportions involved continuous quantities. Taken together, these findings indicate that children go astray on proportional reasoning problems involving discrete units only when a numerical match is possible, suggesting that their difficulty is due to an overextension of numerical equivalence concepts to proportional equivalence problems. KeywordsProportional Reasoning; Continuous and Discrete Quantities; Mathematical Development; Intuitive and Explicit Processing Proportional reasoning involves understanding the multiplicative relationships between rational quantities (a/b = c/d), and is a form of reasoning that characterizes important structural relationships in mathematics and science, as well as in every day life (Cramer & Post, 1993;Lesh, Post, & Behr, 1988). As Ahl, Moore, and Dixon (1992) emphasized, "Proportional reasoning is a pervasive activity that transcends topical barriers in adult life." Proportional information is crucial in dealing with such diverse topics as economic values, relational spatial contrasts, temperatures, densities, concentrations, velocities, chemical compositions, demographic information, and recipe formulation (Karplus, Pulos, & Stage, 1983;Moore, Dixon, & Haines, 1991;Siegler & Vago, 1978;Sophian & Wood, 1997;Spinillo & Bryant, 1999). For example, when baking, one needs to think proportionally about the relative measures of each ingredient (e.g., 2-and-½ cups flour, 1/3 cup sugar, and ¼ cup butter), and must maintain these proportions whenever deviating from the recipe (e.g., whenever doubling or halving the intended amount). In chemistry, proportionality is central to balancing chemical equations. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author Manuscript geographical locations based on the proportion of the population represented by specific demographic target groups. Understanding of proportionality is also central to mathematics; it is the basis of rational number operations, unit partitioning, and basic algebra and geometry problem solving (Empson, 1999;Fuso...

Reasoning strategies with rational numbers revealed by eye tracking

“…This finding is particularly striking because adults should easily be able to ignore the segments and view the images as two continuous subsections, given that the similar colored elements are grouped together. The fact that discrete images favored counting not only for fractions but also for decimals is consistent with evidence (e.g., Boyer et al, 2008) that older children and adults are strongly predisposed to count whenever possible, whether or not counting is the optimal strategy. Counting does not align naturally with decimals, and estimation is less accurate than counting; hence, neither strategy is particularly well-suited for evaluating relations when decimals are paired with discrete images.…”

“…Children must first learn the number words and map them onto counting procedures (Gelman & Gallistel, 1978;Rips, Bloomfield, & Asmuth, 2008). Counting provides more precision, and once children have mastered it, it seems to replace estimation as the dominant strategy (Boyer, Levine, & Huttenlocher, 2008;Mix, Levine, & Huttenlocher, 1999;Wynn, 1997).…”

From rational numbers to algebra: Separable contributions of decimal magnitude and relational understanding of fractions

“…Gray and Tall (1994) found that students who become proficient in arithmetic at an earlier age show greater ability to move back and forth flexibly between an arithmetic expression and the result of that expression. The difficulty in understanding a fraction as a relational expression may explain why young children appear to understand quantitative relations such as part-whole or proportions with visual displays (Goswami, 1989;Mix, Levine & Huttenlocher, 1999;Boyer, Levine, & Huttenlocher, 2008;Sophian, 2000), but not when they have to answer comparable questions with fraction notation (Ball & Wilson, 1996;Mack, 1995). For example, Ni and Zhou (2005) found that most children could answer the question, "How much is one third plus one third?"…”

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). …”