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Research

• How does Group Composition and Gender Influence the Learning of Statistics?

The purpose of this study was to examine how group composition and gender influences the learning of statistics for eighth grade mathematics students. Students participated in an extended grup project where they designed a research question, collected, analyzed, and interpreted data using the computer. Same-gender mixed ability groups were randomly assigned to two exemplar conditions (text vs. video and text). Assessment criteria were described to students prior to their project development through exemplar conditions, video and text was thought to be more explict. An Analysis of Variance of Condition (text or text and video) by Gender (male or female) by Test (pre, post) revealed no condition effect, but females performed higher than males on a post-test of statistical knowledge. Gender differences were also noted in journal keeping. Implications of these results are discussed.

• Using cooperative learning to teach statistics

Mosteller noted the paucity of studies on "how to improve collegiate or university teaching in a behavioral way," and offered examples of techniques that he had found effective in the classroom. Stimulated by Mosteller's suggestions and by research results on "cooperative learning", I adopted new procedures for teaching an introductory undergraduate course in psychological statistics and compared results with those from the course that I had taught by more conventional methods in prior years. A Letter to the Editor in The American Statistician provides a summary of findings. That letter is reproduced (with the permission of the publisher) on the facing page. The present report provides a more complete description of the methods employed, with the intent that it may facilitate their use by others.

• Cognitive Models and Problem Spaces in "Purely Random" Situations

As part of a study on the natural interpretations of probability, experiments about elementary "purely random" situations (with dice of poker chips) were carried out using students of various backgrounds in the theory of probability. A prior study on cognitive models which analyzed the individual data of more than 600 subjects had shown that the most frequent model used is based on the following incorrect argument: the results to compare are equiprobable because it's a matter of chance; thus, random events are thought to be equiprobable "by nature". In the present paper, the following two hypotheses are tested: 1) Despite their incorrect model, subjects are able to find the correct response. 2) They are more likely to do so when the "chance" aspect of the situation has been masked. An experiment testing 87 students showed, as expected, that there is a way to induce the utilization of an appropriate cognitive model. However, the transfer of this model to a classical random situation is not as frequent as one might expect.

• A potential for learning probability in young children

Sixty-one children, from 4 to 11 years old, were presented with two sets, each containing blue and yellow elements. Each time, one colour was pointed out as the payoff colour (POC). The child had to choose the set from which he or she would draw at random a POC element in order to be rewarded. The sets were of varying sizes with different proportions of the two colours. The problem was to select the higher of the two probabilities. Three kinds of materials were used; Pairs of urns with blue and yellow beads, pairs of roulettes divided into blue and yellow sectors, and pairs of spinning tops, likewise divided into two colours. Roughly around the age of six, children started to select the greater of the two probabilities systematically. The dominant error was selecting the set with the greater number of POC elements. Verbal concepts of probability and chance were explored and some 'egocentric' thought processes were described. The study indicates that probability concepts could be introduced into school teaching even in the first grades. The deterministic orientation in the instruction for young ages should be attenuated, permitting concepts of uncertainty right from the beginning.

• The conjunction fallacy: A task specific phenomenon?

The present investigation adopted a debiasing approach to the judgmental error known as the conjunction fallacy. Such an approach was used to determine the extent to which the conjunction fallacy reflects task specific misunderstanding of particular judgment problems. The results suggest that (a) subjects' misunderstanding of conjunction problems is indeed somewhat task specific, and (b) a debiasing approach can effectively lower but not eliminate the conjunctive error rate for problems that do not strongly implicate representativeness thinking. Educational strategies based on statistical and probabilistic knowledge are discussed as an approach to debiasing inferential errors like the conjunction fallacy.

• Usefulness of a balance model in understanding the mean

The study tested whether improving students' knowledge of balance rules through experience with a balance beam promotes understanding of the mean. The study consisted of three sessions. In the pretest session, subjects' levels of balance knowledge and abilities to calculate the solutions to a variety of problems dealing with the mean were assessed. Subjects were classified as nonbalancers if they performed at a a level below Siegler's Rule IV and as noncalculators if they were unable to calculate solutions to weighted mean problems. In the second session, half the nonbalancers were given balance training and the other half were asked to solve unrelated control problems. In the transfer session, subjects were given a set of five problems to assess their understanding of the mean. Significant transfer was found: Subjects classified as nonbalancers on the pretest performed significantly better on the transfer problems if they had been given balance training rather than assigned to the control condition.

• A comparative study of a computer programming and a textbook approach in teaching the concept of variable

The purpose of this study was to investigate whether there were significant differences in understanding the concept of variable and in attitudes toward mathematics among sixth-grade students (n=89) who use a Logo graphics approach, students who used a textbook-based approach, and students who received no instruction on the concept of variable. The Test of Logical Thinking (TOLT), Comprehensive Test of Basic Skills (CTBS), and Robustness Semantic Differential (RSD) were administered as pretreatment measures. The Understanding of the Concept of Variable Instrument (UCVI) was administered immediately and three weeks after the experiment ceased. Although the results indicated there was no significant difference between computer and textbook-based groups with respect to understanding the concept of variable immediately after treatment, there was a significant difference (p &lt; .01) between the two groups with respect to long-term retention (three weeks after treatment ceased). There were significant positive correlations between CTBS and TOLT scores and UCVI scores.

• Evaluation of three simulation exercises in an introductory statistics course

This study evaluates the effect of participation in three computer simulation exercises on performance of students enrolled in an introductory statistics class. One-half of the students were required to participate in three computer exercises: means, normal curve, and correlation coefficient estimation. At a later date all students were given a paper-and-pencil test of their ability to quickly estimate statistics. Results demonstrated that the students who participated in the exercises attempted significantly more exercises with greater success than those who did not. Questionnaire results indicated that the students felt that the exercises were useful.

• Rote versus conceptual emphases in teaching elementary probability.

"Forty-eight sutdents with no previous exposure to probability or statistics read one of three texts that varied in the emphasis placed on rote and conceptual learning of basic concepts of elementary probability. A qualitative analysis of errors on a postinstruction test employed a model of problem solving whose stages were categorization of the problem, retrieval of the appropriate formula, and translation of values from the problem into the formula. Variations in performance on problems requiring the same formula for solution were largely attributable to surface features that affected the ease of categorization and translation. Students who read a text that emphasized conceptual learning tended to be less sensitive to surface features, which parallels results regarding novices and experts in various content domains."

• The role of an evaluation exercise in the resolution of misconceptions of probability

The differential effect of two activity-based instructional treatments on subjects' concepts of probability was investigated. The concepts of interest were a classical/frequentist interpretation of probability and three misconceptions cited in the literature: law of averages, law of small numbers and availability. All subjects completed a workbook which presented a long-run frequency interpretation of probability. After completion of the workbook, subjects participated in a probability-matching activity where the task was to predict correctly the outcomes for 100 coin tosses of a fair coin. Half the subjects ( the No-Evaluation group) recorded only the outcome of each toss but not their guess. After the 100 tosses, No-Evaluation subjects were presented statements which pointed out congruities between the observed outcomes and the theory presented in the workbook. Evaluation subjects recorded both their guess and the outcome of each toss. In addition to the same statements presented to the No-Evaluation subjects, Evaluation subjects were asked questions which pointed out incongruitities between their recorded data and the three misconceptions. Evaluation subjects showed an increase in classical/frequentist responses from pretest to posttest. In contrast, No-Evaluation subjects showed an increase in law of averages responses with a consequent decrease in classical/ frequentist responses. These findings support the idea that misconceptions formed before or during instruction can be reinforced by experience with stochastic phenomena since subjects may be biased to attend information which confirms the misconceptions.