Journal Article

  • A simple stay-switch probability game demonstrates the importance of empirically testing our beliefs. Based on intuition, most undergraduate subjects believe that a stay strategy leads to a higher percentage of winning, and most faculty subjects believe that the staying and switching strategies yield equal probabilities of winning. However, a simple in-class experiment proves that switching wins twice as often as staying. Rather than teaching specific probability principles, this demonstration emphasizes reliance on empirically validating our beliefs. A follow-up questionnaire shows that participating in this experiment may increase students' trust in the empirical method.

  • Students in introductory statistics courses often commit similar errors in computation and interpretation. A handout that lists such common errors is described. Students appreciated the handout, thought it reduced errors, and recommended it for future classes.

  • Most statistics teacher strive to make the course material meaningful. This article presents specific examples for elaborating the statistical concepts of variability, null hypothesis testing, and confidence interval with common experience.

  • Although few adults would be able to define probability with any precision, and in fact definitions of probability are a matter for dispute among logicians and mathematicians, most adults are able to behave effectively in probabilistic situations involving quantitative proportions of independent elements. Piaget has studied the behavior of children in a probabilistic situation and from their behavior has concluded that young children (say up to age 7) are unable to utilize a concept of probability. The present study is a demonstration that young children are able to behave in terms of the probability concept under appropriate conditions. It is an experiment in which Piaget's technique for assessing the probability concept in young children is compared with a decision-making technique.

  • 4 experiments administering probability judgment tasks are reported using child Ss ranging in age from preschool to early adolescence. Experiment 1 showed equivalent results with probability and proportionality instructions when judgments were performed between 2 circles with different black and white proportions. Experiment 2 showed that fewer correct probability than proportionality judgments occurred when Ss judged a single circle. It was concluded that the 2-circle task does not require probability concepts, since Ss need not construct probability ratios to succeed. These results confirm those of Piaget and Inhelder. Experiments 3 and 4 modified the 2-circle task to require use of probability concepts and administered a probability task with double arrays of discrete objects. Results were comparable to those found for the single-circle task. Researchers who have claimed that preschool children use probability concepts are criticized since their experimental tasks have been similar to the unmodified 2-circle task of Experiment 1.

  • Previous replication studies have met with discrepant results in their attempts to evaluate Piaget and Inhelder's study of chance and probability concepts in children. Consequently, 56 subjects (aged 5-4 to 17-11) were tested on 2 cognitive tasks taken from the authors' original work. Specific objections to Piaget and Inhelder's experimental and analytic procedures are overcome here by utilizing a scoring procedure which elicits item-type data from relatively standardized interviews. The results of this replication study indicate considerable agreement with Piaget and Inhelder's description of stage-related verbal features while failing to confirm their description of stage-related nonverbal features. Evidence from this study is related to the concept of "stage" in cognitive-developmental theory, and the procedures used here are evaluated vis-`a-vis developmental issues.

  • 2 experiments on the development of the understanding of random phenomena are reported. Of interest was whether children understand the characteristic uncertainty in the physical nature of random phenomena as well as the unpredictability of outcomes. Children were asked, for both a random and a determined phenomenon, whether they knew what its next outcome would be and why. In Experiment 1, 4-, 5-, and 7-year-olds correctly differentiated their responses to the question of outcome predictability; the 2 older groups also mentioned appropriate characteristics of the random mechanism in explaining why they did not know what its outcome would be. Although 3-year-olds did not differentiate the random and determined phenomena, neither did they treat both phenomena as predictable. This latter result is inconsistent with Piaget and Inhelder's characterization of an early stage of development. Experiment 2 was designed to control for the possibility that children in Experiment 1 learned how to respond on the basis of pretest experience with the 2 different phenomena. 5- and 7-year-olds performed at a comparable level to the same-aged children in Experiment 1. Results suggest an earlier understanding of random phenomena than previously has been reported and support results on the literature indicating an early understanding of causality.

  • Functional measurement methodology was used to assess children's attention to the total number of alternative outcomes as well as the number of target outcomes when making probability estimates. In Study 1, first-, third-, and fifth-grade children were given the task of estimating on a simple, continuous but nonnumeric scale the probability of drawing a particular color of jelly bean from a bag containing either 1, 2, or 3 jelly beans of that color, and either 6, 8, or 10 jelly beans total. In Study 2, first- through fifth-grade children were given the task of estimating the likelihood that a bug would fall on a pot containing a flower when presented displays of planters containing either 2, 3, 4, or 5 pots with flowers, and 6, 8, or 10 pots total. In both studies, the children were exposed to each of the combinations of numerator and denominator across 3 replications. The results indicate that all age groups attend to variations in the denominator as well as to variations in the numerator, and, furthermore, that they attend to the interaction between these variables. This finding contrasts sharply with research that requires children to choose which of 2 containers offers the greater chance of yielding a target item in a blind draw. It is suggested that children possess the skill to make accurate probability estimates, but they are unaware that these estimates should always be made and used when comparing the probability of an event across trials. The findings are discussed in relation to the broader issue of the limitations of the choice paradigm as a means of investigating children's thinking.

  • It is not unusual today - even among people who consider probability as a concern of pure mathematics - to start a probability course with an attempt to uncover the experimental roots of the probability concept. In fact, it is not a new feature. The story about tossing a coin with the happy result of a fair distribution of heads and tails in the long run has been the custom for quite a long time. What is new about it, is that the story is dramatized and acted out - I mean, by the author or the textbook. Maybe even the teachers or the students are expected to try out this experiment - following the highly encouraging examples given by the textbook authors. It is a pity that by showing one experiment without asking themselves whether it is typical, textbook authors lead the teachers up the wrong path and help to create wrong attitudes towards probabilistic problems.

  • Not long ago probability was investigated, taught, and published in the way sheep are counted, distances, time and money are accounted for, and physics is pursued, that is tacitly supposing that everybody - investigators, teachers, students, authors and readers - knew what dice, heads and tails, stakes, chances, events, dependency and independency are. There are didactical reasons to believe that at school level probability cannot be taught in any other way, but at any account it may be taken for granted that intuitional knowledge about such tools and concepts is indispensable whenever probability ought to be related to reality.

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