Research

  • This paper discusses the important pedagogical question of by how much experimental probabilities need to deviate from subjective or symmetric probabilities before children consider revising their subjective probabilities. Many children believe that common random generators like coins and dice are subject to mystical or physical powers, or to be inherently opposed to a child's wishes. Even 9% of Year 11 students have been found to believe that a six is the least likely outcome from tossing a die. Providing experiences which encourage children to revise their opinions is difficult. One reason for this difficulty arises from the mathematics of the situation. There are 2500 tosses of a coin necessary to obtain a relative frequency of between 0.48 and 0.52 with 95% confidence. This makes it very difficult to attempt a classroom confirmation of any theory.

  • Two studies were run to determine whether the interpretations of statements or forecasts using vague probability and frequency expression such as likely, improbable, frequently, or rarely, were sensitive to the base rates of the events involved, In the first experiment, professional weather forecasters judged event probabilities in situations drawn from a medical context. In the second experiment, students judged matched forecast scenarios of common semantic content that differed only in prior probability (as determined by an independent group of subjects). Results were (a) the interpretations of forecasts using neutral (e.g., possible) and high probability or frequency terms (e.g. usually) were strong, positive functions of base rate, while the interpretations of forecasts using low terms (e.g. rarely) were much less affected by base rates; (b) in the second experiment interpretations of forecasts appeared to represent some kind of average of the meaning of the expression and the base rate.

  • Can the vague meanings of probability terms such as doubtful, probable, or likely be expressed as membership functions over the [0, 1] probability interval? A function for a given term would assign a membership value of zero to probabilities not at all in the vague concept represented by the term, a membership value of one to probabilities definitely in the concept, and intermediate membership values to probabilities represented by the term to some degree. A modified pair-comparison procedure was used in two experiments to empirically establish and assess membership functions for several probability terms. Subjects performed two tasks in both experiments: They judged (a) to what degree one probability rather than another was better described by a given probability term, and (b) to what degree one term rather than another better described a specified probability. Probabilities were displayed as relative areas on spinners. Task a data were analyzed from the perspective of conjoint-measurement theory, and membership function values were obtained for each term according to various scaling models. The conjoint-measurement axioms were well satisfied and goodness-of-fit measures for the scaling procedures were high. Individual differences were large but stable. Furthermore, the derived membership function values satisfactorily predicted the judgments independently obtained in task b. The results support the claim that the scaled values represented the vague meanings of the terms to the individual subjects in the present experimental context. Methodological implications are discussed, as are substantive issues raised by the data regarding the vague meanings of probability terms.

  • A study to investigate students' facility with proportions has been undertaken by the author and Fay Sharples of the University of Waikato in New Zealand over the period 1989 to 1992. The initial study was done during 1989 and 1990 and 64 students at the University of Waikato and 57 at Brunel University in the UK took part. Some results of this study have been reported elsewhere. We made some changes to our questionnaire after studying the results of the initial study. In the Spring of 1992, 127 students in New Zealand and 29 students in the UK, all of whom were taking statistics as a service course, completed the revised version of the questionnaire. As with the previous study, the results in this follow-up were interesting and not always what we expected. This paper discusses the results on three questions.

  • Perhaps the simplest and the most basic qualitative law of probability is the conjunction rule: The probability of a conjunction, P(A&B), cannot exceed the probabilities of its constituents, P(A) and P(B), because the extension (or the possibility set) of the conjunction is included in the extension of its constituents. Judgments under uncertainty, however, are often mediated by intuitive heuristics that are not bound by the conjunction rule. A conjunction can be more representative than one of its constituents, and instances of a specific category can be easier to imagine or to retrieve than instances of a more inclusive category. The representativeness and availability heuristics therefore can make a conjunction appear more probable than one of its constituents. This phenomena is demonstrated in a variety of contexts including estimation of word frequency, personality judgment, medical prognosis, decision under risk, suspicion of criminal acts, and political forecasting. Systematic violations of the conjunction rule are observed in judgments of lay people and of experts in both between-subjects and within- subjects comparisons. Alternative interpretations of the conjunction fallacy are discussed and attempts to combat it are explored.

  • College business statistics students (N=84) participated in a spreadsheet CBT tutorial. Noncontent related students-instructor interaction (present/absent) was studied in two learning settings: individual study and paired study. Student-instructor interaction led to higher achievement in the individual setting. However, when subjects studied in pairs, instructor interaction was less influential. The lowest scoring subjects were those who studied individually and who received no instructor interactions. The results demonstrate that what the instructor does, and how the students are arranged can affect achievement in CBT.

  • This study attempts to identify the relevant mental model for hypothesis testing. Analysis of textbooks provided the identification of the declarative and procedural knowledge that constitute the relevant mental models in hypothesis testing. A cognitive task analysis of intermediates' and experts' mental models was conducted in order to identify the relevant mental models for teaching novices. Of interest were the steps taken to arrive at the solution and the representations that were used in the problem solving process. Results indicate that experts and intermediates differ in their conceptual understanding. In addition, diagrammatic problem representation was useful in for all particularly for the intermediates. On this basis, the intermediate models were deemed relevant for instructing novices. Two instructional strategies were investigated: presentation sequence (concepts and procedures taught separately or together) and presentation mode (diagrammatic vs. descriptive). Based on their findings, the researchers conclude that meaningful learning occurs when conceptual instruction is provided prior to the procedures, that is, when they are taught separately rather than concurrently, and when a diagrammatic strategy was utilized rather than a descriptive method. This facilitates development of representational ability for understanding hypothesis testing. In short, using separate and diagrammatic representation strategies are effective for teaching novices in the area of hypothesis testing. The researchers conclude that by developing relevant mental models through this type of instruction, the learner's knowledge can be more accessible (awareness of knowledge), functional (predict or explain), and improvable.

  • Ten subjects were asked to think aloud while solving two statistical problems. Ten subjects were instructed after each substep of his/her problem solving, to check in various ways the solution of the previous substep. The subjects detected 25 out of a total of 56 errors when they solved the problems. About half of the detected errors were computational errors. Nine errors were eliminated in response to the checking instructions. The think aloud data indicated that subjects' most common way of detecting their own errors was by noting that computations resulted in extreme values. Subjects also detected errors by (a) "spontaneous discovery"; (b) discontent with other aspects of a solution than the numerical value of the answer; (c) repeating a solution. The last mentioned type of error detection only occurred when subjects responded to the checking instructions. Finally it was found that subjects had a strong tendency to respond to the checking instructions either in a routinized or in a non-elaborated way. It was discussed how the formulation of checking instructions can be improved in order to avoid this effect.

  • In statistical problems, the differential effects on training performance, transfer performance, and cognitive load were studied for 3 computer-based training strategies. The conventional, worked, and completion conditions emphasized, respectively, the solving of conventional problems, the study of worked-out problems, and the completion of partly worked-out problems. The relation between practice-problem type and transfer was expected to be mediated by cognitive load. It was hypothesized that practice with conventional problems would require more time and more effort during training and result in lower and more effort-demanding transfer performance than practice with worked-out or partly worked-out problems. With the exception of time and effort during training, the results supported the hypotheses. The completion strategy and, in particular, the worked strategy proved to be superior to the conventional strategy for attaining transfer.

  • Previous studies of correlational reasoning have focused on the interpretation of 2 x 2 tables. The research in this article examined age trends in responses to problems involving more than 2 continuous variables. Instruments were developed and administered to Ss from Grade 4 through postgraduate (n = 20 in each grade) to produce multidimensional profiles of student growth. Experiment 1 found that correlational reasoning skills increased with age. Experiments 2 and 3 found that students performance could be improved through instruction. Evidence of convergent and discriminant validity of the instruments was obtained. Although there were similarities between results obtained with 2 x 2 data problems and results on continuous data problems, the evidence in support of a single correlational schema underlying both was ambiguous. There was no transfer, and correlations between the 2 types of performance were weak.

Pages

register