Counterexample Search in Diagram-Based Geometric Reasoning

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object-wise, could underlie diagram-based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions.     

Children's reasoning about responses to peer aggression: Victim's and witness's expected and prescribed behaviors

This research examined children's reasoning about expected (i.e., what a peer would do) and prescribed (i.e., what a peer should do) responses to unprovoked, intentional aggressive actions in two contexts: as a victim of such a transgression and as a witness to the incident. Physical harm and property damage items were used in a structured interview format. There were 90 subjects drawn from three elementary school grades (2nd, 4th, and 6th). Children differentiated between the expected and prescribed responses of peers and significant developmental differences in children's evaluations were found. Although the majority of the subjects in all grades denounced retaliation on the basis of concerns about others' welfare, older children stated that peers were likely to retaliate against the perpetrator nonetheless. Across different contexts, older children's responses appeared to reveal a greater independence from authority in negotiating peer interactions. In evaluating the witness's responses to aggressive acts, younger children's expected and prescribed responses were less disparate than that of the older children. The utility of including different vantage points of the child in examining children's social reasoning about aggression and the application of the present findings to social information-processing models are discussed. © 1996 Wiley-Liss, Inc.     

The interaction approach to early intervention

A brief summary of research, supportive of the idea that quality of parent-infant interaction has a powerful effect on infant development, introduces an early intervention model based on that proposition. Also, central to the interaction model is the knowledge that (1) parent and infant both contribute to what happens between them and that (2) a multitude of factors, environmental and internal to parent and infant, affect the parent, the infant, and their interaction. Ten sets of principles and strategies that convey the essence of the interaction model constitute the core of this article. The principles and strategies address the importance of identifying sources of support and stress in the family, of developing relationships with both parents, and of supporting positive interactions between parents and the infant's siblings. The limitations of this model are also addressed, especially in cases where the parent is the primary risk factor. The infant's exposure to nurturing adults in quality day care programs and a more clinical approach with the parent and infant, such as Selma Fraiberg's model, may be more effective with such families.