Research

  • A long-standing belief is that statistical rules helpful in solving practical problems do not transfer beyond the subject matter domain in which they were learned. Recent research by G. T. Fong, D. H. Krantz, and R. E. Nisbett (1986) challenges this belief. Fong et al. showed that instructing learners about abstract rules, such as the law of large numbers, improved reasoning about ill-defined problems and transferred to solving everyday statistical problems that involved probabilistic relations. Fong et al.'s research is extended in 3 experiments with 276 university, secondary, and middle school students. The law-of-large-numbers heuristic was taught in regular classroom settings and students' abilities to solve ill-structured, everyday problems were tested. Students learned a good deal about how to reason statistically, and these gains generalized over different structures of problems and topics. The results support a revival of formalist views of transfer, that teaching formal rules about inference making can improve reasoning and support transfer.

  • This study examined changes in students probabilistic thinking and writing during instruction emphasizing writing to learn experiences. A class of fifth grade students with no previous experiences in writing during mathematics made significant gains in probability reasoning and writing; however the correlation between probabilistic thinking and writing was not significant. Analysis of focus students revealed that their writing changed from narrative summaries to reasoned patterns and generalizations. However, some used invented representations without interpretation and were reluctant to write in mathematical contexts.

  • Based on a synthesis of research and observations of middle school students, a framework for assessing students thinking on two constructs--conditional probability and independence--was formulated, refined and validated. For both constructs, four levels of thinking which reflected a continuum from subjective to numerical reasoning were established. The framework was validated from interview data with 15 students from Grades 4-8 who served as case studies. Student profiles revealed that levels of probabilistic thinking were stable across the two constructs and were consistent with levels of cognitive functioning postulated by some neo-Piagetians. The framework provides valuable benchmarks for instruction and assessment.

  • People possess an abstract inferential rule system that is an intuitive version of the law of large numbers. Because the rule system is not tied to any particular content domain, it is possible to improve it by formal teaching techniques. We present four experiments that support this view. In Experiments 1 and 2, we taught subjects about the formal properties of the law of large numbers in brief training sessions in the laboratory and found that this increased both the frequency and the quality of statistical reasoning for a wide variety of problems of an everyday nature. In addition, we taught subjects about the rule by a "guided induction" technique, showing them how to use the rule to solve problems in particular domains. Learning from the examples was abstracted to such an extent that subjects showed just as much improvement on domains where the rule was not taught as on domains where it was. In Experiment 3, the ability to analyze an everyday problem with reference to the law of large numbers was shown to be much greater for those who had several years of training in statistics than for those who had less. Experiment 4 demonstrated that the beneficial effects of formal training in statistics may hold even when subjects are tested completely outside of the context of training. In general, these four experiments support a rather "formalist" theory of reasoning: People reason using very abstract rules, and their reasoning about a wide variety of content domains can be affects by direct manipulation of these abstract rules.

  • The purpose of this research was to investigate the evolution, with age, of probabilistic, intuitively based misconceptions. We hypothesized, on the basis of previous research with infinity concepts, that these misconceptions would stabilize during the emergence of the formal operation period. The responses to probability problems of students in Grades 5, 7, 9, and 11 and of prospective teachers indicated, contrary to our hypothesis, that some misconceptions grew stronger with age, wheras others grew weaker. Only one misconception investigaged was stable across ages. An attempt was made to find a theoretical explanation for this rather strange and complex situation.

  • People attempting to generate random number sequences usually produce more alternations than expected by chance. They also judge overalternating sequences as maximally random. In this article, the authors review findings, implications, and explanatory mechanisms concerning subjective randomness. The authors next present the general approach of the mathematical theory of complexity, which identifies the length of the shortest program for reproducing a sequence with its degree of randomness. They describe three experiments, based on mean group responses, indicating that the perceived randomness of a sequence is better predicted by various measures of its encoding difficulty than by its objective randomness. These results seem to imply that in accordance with the complexity view, judging the extent of a sequence's randomness is based on an attempt to mentally encode it. The experience of randomness may result when this attempt fails.

  • The effects of a visible author (one who reveals aspects of him- or herself) on women's experience reading statistical texts were examined among 47 female college students who read texts that differed in the extent to which the author revealed attitudes and personality. Data included "think-and-feel aloud" protocols, measures of concentration, mood, level of perceived challenge, and readers' images of the author. Women reading the visible author text interacted with the author while reading: this relationship appeared to influence the relations among comprehension, motivation, and affective response. For these women, author image and initial self-efficacy for statistics were related to cognitive engagement, feelings of accomplishment, and intrinsic motivation. Implications for text construction and methodology in research on the interaction of cognition and affect during learning tasks are discussed.

  • The purpose of this study was to explore the basis of test anxiety expressed when taking a statistics course using a structural modeling approach. The study involved 219 university students. The data indicated that statistical test anxiety was different from general test anxiety. The females expressed more general and statistical test anxiety than males, and students who had taken more prior math courses had higher math self-concept scores. Math self-concept and achievement in statistics were negatively related to statistical test anxiety, and the students who reported high levels of general anxiety also reported high levels of statistical anxiety. The structural model revealed variables not studied previously to be important in understanding statistical test anxiety.

  • This study examined the calibration techniques from the adult judgment and decision making literature for the purposes of assessing the adequacy of children's subjective probability judgments. Two hundred eighty-eight children from an inner-city school participated in the study. In accordance with the adult decision making literature, the children were consistently overconfident in their subjective probability judgments Gender and culture were each found to have a significant effect on the degree of overconfidence.

  • This paper reports some findings from two independent studies, one in Northern Ireland and one in South Australia, into young children's understanding of the behaviour of unfamiliar Random Generators (RGs). Our findings indicate that probability understanding is often influenced by the physical properties or appearance of RGs. Sometimes it is their physical arrangement in a container that influences responses. Teaching of the topic depends on teachers being aware of what these misconceptions are before planning for teaching.

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