Ty W. Boyer

Development of Proportional Reasoning: Where Young Children Go Wrong

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 4th-graders in Study 1, and to 1st- and 3rd-graders in Study 2. Both studies involved 4 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; 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.

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

The current experiments examined the role of scale factor in childrens proportional reasoning. Experiment 1 used a choice task and Experiment 2 used a production task to examine the abilities of kindergartners through fourth-graders to match equivalent, visually depicted proportional relations. The findings of both experiments show that accuracy decreased as the scaling magnitude between the equivalent proportions increased. In addition, childrens errors showed that the cost of scaling proportional relations is symmetrical for problems that involve scaling up and scaling down. These findings indicate that scaling has a cognitive cost that results in decreasing performance with increasing scaling magnitude. These scale factor effects are consistent with childrens use of intuitive strategies to solve proportional reasoning problems that may be important in scaffolding more formal mathematical understanding of proportional relations.

True or False: Do 5-Year-Olds Understand Belief?

In 3 studies (N = 188) we tested the hypothesis that children use a perceptual access approach to reason about mental states before they understand beliefs. The perceptual access hypothesis predicts a U-shaped developmental pattern of performance in true belief tasks, in which 3-year-olds who reason about reality should succeed, 4- to 5-year-olds who use perceptual access reasoning should fail, and older children who use belief reasoning should succeed. The results of Study 1 revealed the predicted pattern in 2 different true belief tasks. The results of Study 2 disconfirmed several alternate explanations based on possible pragmatic and inhibitory demands of the true belief tasks. In Study 3, we compared 2 methods of classifying individuals according to which 1 of the 3 reasoning strategies (reality reasoning, perceptual access reasoning, belief reasoning) they used. The 2 methods gave converging results. Both methods indicated that the majority of children used the same approach across tasks and that it was not until after 6 years of age that most children reasoned about beliefs. We conclude that because most prior studies have failed to detect young childrens use of perceptual access reasoning, they have overestimated their understanding of false beliefs. We outline several theoretical implications that follow from the perceptual access hypothesis.