Literature Index

In the research reported in this paper, we address two major sets of questions about children's understanding of average. 1) When they are working with data sets, how do children construct and interpret indicators of center? It's important to examine how children learn to describe data sets in a meaningful, useful, and flexible manner. In particular, we are concerned with the development and use of the idea of "representativeness" in the context of real data sets. The second major question we are addressing deals with the use of the mean in a precise mathematical sense: 2) How do children develop their thinking about the mean as a mathematical relationship? This question moves into the important more general question of how children develop mathematical abstractions, and how they map (or fail to map) these abstractions onto their informal understanding of a concept.

Over the years I've strongly supported the federal government's role in gathering and circulating educational data because we can't make intelligent decisions without knowing how public education is doing. But getting adequate funding for data collection and research hasn't been easy. The Congress has been suspicious that this Administration, in particular, was more interested in cooking the facts to fit its ideology than in doing objective research, especially on the subject of education spending. This article shows proof that such suspicions are justified.

This article contains two cautionary tales based on my experience working with students, adults, and teachers on research and professional development projects involving data and chance. The first arose from observing two grade 9 boys who ignored instructions and then tried to supplements samples of size two with other samples of size two in order to make samples of size four. The second was related to several observations of students and adults expressing beliefs about dice tossing that were contrary to my expectations: either expecting peaks in distributions that should be uniform or expecting uniformity in distributions that should be peaked. The solution to the dilemmas presented in these two tales would appear to be the creation of cognitive conflict to illustrate forcefully the importance of sample size and the difference between equally likely and non-equally likely outcomes. To handle the situations however, teachers need to be aware that these beliefs may be abroad, to experience the activities that can lead to conflict resolutions, and then plan their own strategies.

Five studies examined Kahneman and Miller's (1986) hypothesis that events become more "normal" and generate weaker reactions the more strongly they evoke representations of similar events. In each study, Ss were presented with 1 of 2 versions of a scenario that described the occurrence of an improbable event. The scenarios equated the a priori probability of the target event, but manipulated the ease of mentally simulating the event by varying the absolute number of similar events in the population. Depending on the study, Ss were asked to indicate whether they thought the event was due to chance as opposed to (a) an illegitimate action on the part of the benefited protagonist, or (b) the intentional or unintentional misrepresenation of the probability of the event. As predicted, the fewer ways the events could have occurred by chance, the less inclined Ss were to assume that the low-probability event occurred by chance. The implications of these findings for impression-management dynamics and stereotype revision are discussed.