Literature Index

In statistics, and in everyday life as well, the arithmetic mean is a frequently used average. The present study reports data from interviews in which students attempted to solve problems involving the appropriate weighting and combining of means into an overall mean. While mathematically unsophisticated college students can easily compute the mean of a group of numbers, our results indicate that a surprisingly large proportion of them do not understand the concept of the weighted mean. When asked to calculate the overall mean, most subjects answered with the simple, or unweighed, mean of the two means given in the problem, even though these two means were from different-sized groups of scores. For many subjects, computing the simple mean was not merely the easiest or most obvious way to initially attack the problem; it was the only method they had available. Most did not seem to consider why the simple mean might or might not be the correct response, nor did they have any feeling for what their results represented. For many students, dealing with the mean is a computational rather than a conceptual act. Knowledge of the mean seems to begin and end with an impoverished computational formula. The pedagogical message is clear: Learning a computational formula is a poor substitute for gaining an understanding of the basic underlying concept.

This paper examines how preservice teachers and middle school students reason about distributions as they consider graphs of two data sets having identical means but different spreads. Results show that while both subject groups reasoned about the task using the aspects of average and variation, relatively more preservice teachers than middle school students combined both aspects to constitute an emerging form of distributional reasoning in their responses. Moreover, these emergent distributional reasoners were more likely to see the data sets as fundamentally different despite the identical means used in the task.

This study was undertaken with the goal of inferring an informal approach to probability that would explain, among other things, why subject responses to problems involving uncertainty deviate from those prescribed by formal theories. On the basis of an initial set of interviews such as an informal approach was hypothesized and described as outcome-oriented. In a second set of interviews, the outcome approach was used to successfully predict the performance of subjects on a different set of problems. In this chapter I will elaborate on the importance of understanding that subjects' performance in situations involving uncertainty is based on a theoretical framework that is different in important respects from any formal theory of probability. Additionally, I will argue that the outcome approach is reasonable given the nature of the decisions people face in a natural environment. To this end, I will review research which suggests some reasons why causal as opposed to statistical explanations of events are salient and functionally adaptive.

There are two parts to this literature review. The first part includes bibliography directly focusing on variation,: meaning of variation, role of variation in statsitcal reasoing, researchon conceptions of variation, as well as literature discussing the neglect of variation. The second part lists references belonging to four bodies of literature which, although not having the study of intuitions abaout variation as their main object of study, do offer rich insights into people's thinking about variation: literarure on sampling and centers, on intutioons about the stochastic, on the role of technology, and on the effect of the formalist mathematics tradition on statistics education.