The Venn diagram is suggested as a graphical solution to conjunction fallacies and a modification of it is suggested to more fully communicate set relations.
The Venn diagram is suggested as a graphical solution to conjunction fallacies and a modification of it is suggested to more fully communicate set relations.
The least squares method of fitting a line is not one that naturally occurs to students. We present three tasks to understand student views on how lines may be fit.
This article addresses an important problem of graphing quantitative data: should one include zero on the scale showing magnitude? Based on a real time series example, the problem is discussed and some recommendations are proposed
Bayes's theorem - a difficult concept for many students - can be introduced through simulated data, expected frequencies, and probabilities.
Although much attention has been paid to issues around student<br>assessment, for most introductory statistics courses few<br>changes have taken place in the ways students are assessed. The<br>assessment literature describes three foundational elements -<br>cognition, observation, and interpretation - that comprise an<br>"assessment triangle" underlying all assessments. However,<br>most instructors focus primarily on the second component:<br>tasks that are used to produce grades. This article focuses on<br>three sections written by leading statistics educators who describe<br>some innovative and even provocative approaches to rethinking<br>student assessment in statistics classes.
Based on a synthesis of the literature and observations of young children over two years, a framework for assessing probabilistic thinking was formulated, refined and validated. The major constructs incorporated in this framework were sample space, probability of an event, probability comparisons, and conditional probability. For each of these constructs, four levels of thinking, which reflected a continuum from subjective to numerical reasoning, were established. At each level, and across all four constructs, learning descriptors were developed and used to generate probability tasks. The framework was validated through data obtained from eight grade three children who served as case studies. The thinking of these children was assessed at three points over a school year and analyzed using the problem tasks in interview settings. The results suggest that although the framework produced a coherent picture of children's thinking in probability, there was 'static' in the system which generated inconsistencies within levels of thinking. These inconsistencies were more pronounced following instruction. The levels of thinking in the framework appear to be in agreement with levels of cognitive functioning postulated by Neo-Piagetian theorists and provide a theoretical foundation for designers of curriculum and assessment programs in elementary school probability. Further studies are needed to investigate whether the framework is appropriate for children from other cultural and linguistic backgrounds.
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.
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.
As in other areas of the school curriculum, the teaching, learning and assessment of higher order thinking in statistics has become an issue for educators following the appearance of recent curriculum documents in many countries. These documents have included probability and statistics across all years of schooling and have stressed the importance of higher order thinking across all areas of the mathematics curriculum. This paper reports on a pilot project which applied the theoretical framework for cognitive development devised by Biggs and Collis to a higher order task in data handling in order to provide a model of student levels of response. The model will assist teachers, curriculum planners and other researchers interested in increasing levels of performance on more complex tasks. An interview protocol based on a set of 16 data cards was developed, trialed with Grade 6 and 9 students, and adapted for group work with two classes of Grade 6 students. The levels and types of cognitive functioning associated with the outcomes achieved by students completing the task in the two contexts will be discussed, as will the implications for classroom teaching and for further research.