Journal Article

  • The purpose was to investigate the development of probabilistic thinking as outlined by Piaget and Inhelder, to clarify interrelations of different probabilistic tasks, and to examine the effects of IQ and gender on probabilistic thinking from 5 to 12 years. There were 240 subjects, boys and girls, medium in socio-economic status (SES) and IQ, of two cultural backgrounds, and of three age groups: 5 to 6 years, 8 to 9 years, and 11 to 12 years. Each subject was administered individually the Stanford- Binet and four Piagetian tasks assessing randomization, distribution of multiple discrete elements, odds estimation and permutations. The responses were evaluated globally and in terms of specific measures. The results showed an increase in probabilistic performance that was mostly larger between the first two groups than the last two, positive interrelations among the tasks, partly different factorial structures in each age group with increasing factorial differentiation, more interrelations of IQ with global than specific measures and more in older than younger children, and more differences in favor of boys in older than younger groups. Discussion focuses on the degree of support for Piaget's claims and implication concerning the structure and development of probabilistic cognition.

  • Artificial data sets are often used to demonstrate statistical methods in applied statistics courses and textbooks. We believe that this practice removes much of the intrinsic interest in learning to do good data analysis and contributes to the myth that statistics is dry and dull, In this article, we argue that artificial data sets should be eliminated from the curriculum and that they should be replaced with real data sets. Real data supplemented by suitable background material enable students to acquire analytic skills in an authentic research context and enable instructors to demonstrate how statistical analysis is used to model real data into applied statistics curricula, we identify seven characteristics that make data sets particularly good for instructional use and present an annotated bibliography of more than 100 primary and secondary data sources.

  • In the last few years a new consensus on the nature of learning has begun to emerge, stimulated by research in the field that has come to be known as cognitive science. The emerging conception of learning has a direct bearing on how science and mathematics can be taught most effectively. I will sketch here a few examples of recent findings in cognitive science, many of which support the the intuition of our most thoughtful teachers.

  • 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 study focused on the relations between performance on a three-choice probability-learning task and conceptions of probability as outlined by Piaget concerning mixture, normal distribution, random selection, odds estimation, and permutations. The probability-learning task and four Piagetian tasks were administered randomly to 100 male and 100 female, middle SES, average IQ children in three age groups (5 to 6, 8 to 9, and 11 to 12 years old) from different schools. Half the children were from Middle Eastern backgrounds, and half were from European or American backgrounds. As predicted, developmental level of probability thinking was related to performance on the probability-learning task. The more advanced the child's probability thinking, the higher his or her level of maximization and hypothesis formulation and testing and the lower his or her level of systematically patterned responses. The results suggest that the probability-learning and Piagetian tasks assess similar cognitive skills and that performance on the probability-learning task reflects a variety of probability concepts.

  • This paper describes a situation where systematic use is being made of data collected by students as part of a class project and advocates the wider use of such projects. The immediate learning benefits to the students involved in carrying out projects have been widely canvassed recently, and this paper reports some experiences with a particular type of project. Advantage is also taken of these projects as a source of material for problem-based learning in applied statistics at all levels, and some specific reasons for the potential importance of such material are advanced.

  • The p-value can be introduced with a coin flipping exercise. The instructor flips a coin ten times and has a student call each flip. The students record their thoughts after each flip. The instructor reports that the caller calls every flip correctly. In this exercise students intuitively reject a null hypothesis because the p-value is too small. Students are reassured to learn from this concrete example that they intuitively followed the logic of statistical inference before they studied statistics.

  • This article explores the use of multimedia in an introductory business statistics course through a new computer vehicle called Teacher 2000. Traditional educational processes are reviewed and reinterpreted in light of technological advances in computing, video, and software. These advances provide new opportunities to educators. To highlight the potential of a multimedia approach in statistics, an example is developed that explains how professors and students might interact and use this new technology. Software developed by one of the authors is used to showcase multimedia potential.

  • The likelihood of a statement is often derived by generating an explanation for it and evaluating the plausibility of the explanation. The explanation discounting principle states that people tend to focus on a single explanation; alternative explanations compete with the effect of reducing one another's credibility. Two experiments tested the hypothesis that this principle applies to inductive inferences concerning the properties of everyday categories. In both experiments, subjects estimated the probability of a series of statements (conclusions) and the conditional probabilities of those conclusions given other related facts. For example, given that most lawyers make good sales people, what is the probability that most psychologists make good sales people? The result showed that when the fact and the conclusion had the same explanation the fact increased people's willingness to believe the conclusion, but when they had different explanations the fact decreased the conclusion's credibility. This decrease is attributed to explanation discounting; the explanation for the fact had the effect of reducing the plausibility of the explanation for the conclusion.

  • The attempt to structure a curriculum which balances professional demands with intellectual aspirations induces academic quarrels that ballots do not assuage. In what follows I will address these issues.