Children's Proportional Judgments: The Importance of "Half"

Abstract

3 experiments are reported, all of which show the crucial importance of the “half” boundary in children's proportional judgments. In the first experiment 4‐, 5‐, 6‐, and 7‐year‐old children had to judge which of 2 boxes of blue and white bricks was represented in a small picture. The proportion of blue to white bricks was different in the 2 boxes, and the question was whether the children could use these proportions to make their choices. The 6‐ and 7‐year‐old children solved the problem much more successfully when the proportions crossed the “half” boundary (e.g., 3/8 blue vs. 5/8 blue). The second experiment showed that discriminations involving half (e.g., 1/2 blue vs. 1/4 blue) are also easier than those that do not cross the “half” boundary for the 6‐ and 7‐year‐olds. The third experiment confirmed the results of the first 2 with pictures of different absolute sizes from each other. We conclude that “half” plays a crucial role in children's early proportional reasoning, and that the “half” boundary is to some extent similar to, though not as powerful as, the category boundaries discovered in speech perception.

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

“…Children's better performance in the continuous than in the discrete conditions is likely related to the use of different strategies in these conditions. Although our results do not provide direct evidence about what kind of strategy children used in the continuous condition, it seems likely that they used a perceptual strategy similar to that used by infants and young children to code the extent of one length or region in relation to another (Duffy et al, 2005;Spinillo & Bryant, 1991). For example, children may have compared target and target areas in each donut (i.e., red versus blue areas [part-part coding] and then chosen the one that had proportionally more red area or have compared the red area in each donut to the total area of each donut [part-whole coding]).…”

“…In stark contrast to these findings, many studies have reported that preschool children can reason about proportions (Acredolo, O'Connor, Bank, & Horobin, 1989;Davies, 1965;Goswami, 1989;Hoemann & Ross, 1971;Huttenlocher, Newcombe & Vasilyeva, 1999;Lovett & Singer, 1991;Mix, Levine & Huttenlocher, 1999;Singer-Freeman & Goswami, 2001; Sophian & Crosby, 1999;Spinillo and Bryant, 1991;Yost, Siegel & McMichael, 1962). Several of these studies have used an information integration approach and found that young children can reason in situations that appear to call for proportional reasoning (Acredolo et al, 1989;Schlottmann, 2001).…”

“…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).…”

“…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).…”

“…Using a different type of proportional matching task, Spinillo and Bryant (1991) presented 4-to 7-years-olds with a picture of a rectangle composed of blue and white regions and asked them to choose which one of two different block models matched the picture. This task required proportional reasoning because the picture and the block models were constructed on different scales.…”