Research

  • 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.

  • The study was conducted to (a) determine the development of children's understanding of seven properties of the arithmetic mean and (b) assess the effects of the material used in the testing (continuous, discontinuous) and the medium of presentation (story, concrete, and numerical). Twenty children were selected at each of the ages 8, 10, 12, and 14 years. Different development courses of the children's reasoning were found on some tasks measuring the properties of the the average. No significant effects were found for the materials used or the medium of presentation. The findings are discussed in terms of their importance for developmental psychology and educational practice.

  • In this study, the schema-theoretic perspective of understanding general discourse was extended to include graph comprehension. Fourth graders (n = 204) and seventh graders (n = 185) were given a prior-knowledge inventory, a graph test, and the SRA Reading and Mathematics Achievement Tests during four testing sessions. The unique predictors of graph comprehension for Grade 4 included reading achievement, mathematics achievement, and prior knowledge of the topic, mathematical content, and form of the graph. The unique predictors for Grade 7 were the same except that prior knowledge of topic and graphical form were not included. The results suggest that children should be involved in graphing activities to build and expand relevant schemata needed for comprehension,

  • This study investigates which formal principles govern subjective probability, and whether the validity of these principles depends on age. Two types of tasks were administered to 144 subjects from 3;8 to 19 years: a gambling task (with objective probabilities) and a sporting task (without objective probabilities). Six formal principles of the mathematical concept of qualitative probability (a non-numerical concept based on ordinal scale properties) were tested. Results indicate that these principles are valid as principles of subjective probability for all age groups. Only the youngest age group (4 years or younger) had a smaller degree of confirmation.

  • Children's understanding of what variables and relations are important in problem structures, and their use of these variables and relations in problem solving, were examined. One hypothesis suggests that knowledge of relevant solution variables is a prerequisite for encoding those variables, which in turn is a prerequisite for learning new strategies that use those variables. An alternative hypothesis holds that knowledge of relevant variables is an outcome, rather than a precursor, of efforts to invent new strategies. In the current studies, children between the ages of 5 and 13 years were given Piaget and Inhelder's (1975, The origin of the idea of chance in children, New York: Norton) two -set alternative choice probability problems. In Experiment 1, problem understanding was assessed by asking children to construct two-set problems that could test whether a learner understood how to solve a model problem type. In Experiment 2, understanding was assessed by asking children to modify model problems to make them harder for a learner to solve. In both experiments, children modified or reproduced only those properties of model problems used either correctly or incorrectly in solving the models. These results partially support both hypotheses, and suggest a mechanism by which problem solving knowledge develops.

  • Research on human judgment demonstrates that people's theories often bias their evaluation of evidence and suggest that people might be more accurate if they were unbiased by prior beliefs. Rather than comparing people's judgments of data when they do or do not have a prior theory, most studies compare people's estimates to conventional statistical standards, even though the status of these measures as normative criteria is controversial. We propose that people's theories may have beneficial consequences not examined in previous research. In two paradigms (the covariation estimation problem and the t-test problem), we compare judgments made by people who have potentially biasing prior information. We vary the quality of the data, presenting subjects with data that are either well-behaved or contaminated with outliers. In both paradigms, people's judgments approximated robust statistical measures rather than the conventional measures typically used as normative criteria. We find the usual biasing effects of prior beliefs but also find an advantage for subjects who have prior theories - even incorrect ones - over subjects who are completely "objective." Potentially biasing beliefs both enhanced people's sensitivity to the bulk of the data and reduced the influence atypical scores had on their estimates. Evidence is provided that this robustness results from the fact that prior theories make judgment problems more meaningful. We discuss the conditions under which prior beliefs are likely to help and hinder human judgment.

  • Several strategies are proposed as bases for judgments of covariation between events. Covariation problems were structured in such a way that patterns of correct and incorrect judgments would index the judgment rule being used by a given subject. In two experiments, 10th-grade or college subjects viewed a set of covariation problems, each of which consisted of a set of observations in which each of two events was defined as present or absent. Subjects were asked to identify the relationship between the events. Subjects' response patterns suggested that the modal strategy was to compare frequency of confirming and disconfirming events in defining the relationship. Response accuracy was influenced by pretraining on the concept of covariation and by response format. Instructions to sort the observations did not influence judgment accuracy.

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