The development of proportional reasoning and the ratio concept Part I ? Differentiation of stages

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

“…Similar to Piaget and Inhelder's marble studies, children did not succeed until they were 12 years of age, typically responding on the basis of the number of cups filled with orange concentrate without considering the number of cups filled with water. Consistent with these findings, many other researchers have provided evidence that proportional reasoning develops quite late (Chapman, 1975;Hoemann & Ross, 1971;Karplus, Pulos, & Stage, 1983;Noelting, 1980;Siegler & Vago, 1978). Children's difficulty in all of these studies was related to focusing on only one aspect of the quantities (e.g., number of red marbles), rather than coding the relation between quantities (e.g., the number of red marbles in relation to the number of white marbles [part-part reasoning] or the number of red marbles in relation to the total number of marbles [part-whole reasoning]).…”

“…The demands of our ordinal judgment task may have led the 6-year-olds to try to use their early developing per-ceptual strategy in lieu of counting, although they were unable to apply this strategy effectively in the discrete conditions. It is also possible that the 6-year-olds found it difficult to count the discrete elements because they were arranged in a circle with no separation except a thin black line rather than in a more typical aligned configuration with blank space between each element (e.g., Acredolo et al, 1989;Noelting, 1980). Fuzzy trace theory (Brainerd & Reyna, 1990;Reyna & Brainerd, 1993) provides another possible explanatory framework for children's poor performance on the proportional reasoning problems involving discrete quantities.…”

“…Until age 11, children tend to select the set with the greater absolute number of red (target) marbles rather than the set with the higher proportion of red marbles. A subsequent study by Noelting (1980) also showed that young children have difficulty relating two relative quantities. Noelting presented 6-to 16-year-old children with two sets of cups.…”

“…We expected that the 6-year-olds might be able to perform at above chance levels in the continuous condition, given prior evidence of early sensitivity to proportional relations on tasks involving continuous quantities (e.g., Duffy et al, 2005;Huttenlocher et al, 1999;Spinillo & Bryant, 1991). In contrast, we expected that even 10-year-olds might struggle in the discrete conditions, given prior findings that school-age children have difficulty on proportional reasoning tasks involving discrete quantities (e.g., Noelting, 1980;Piaget & Inhelder, 1975). Participants were middle-class children in Seoul, Korea.…”

“…Mix, Huttenlocher, and Levine (2002) pointed out that a salient difference between many studies reporting early versus later success on proportional reasoning tasks is the kind of quantity involved. In particular, tasks reporting early success involved relations between continuous quantities (e.g., Goswami, 1989;Huttenlocher et al, 1999;Lovett & Singer, 1991;Sophian, 2000;Spinillo & Bryant, 1991), whereas tasks reporting later success involved relations between discrete quantities (e.g., Noelting, 1980;Piaget & Inhelder, 1975). Mix et al suggested that continuous quantities may facilitate proportional reasoning because such quantities are often seen in containers that can serve as a perceptual referent (e.g., the glass is half full).…”