This article discusses the piece "Psychological conceptions of randomness".
- Prof Dev
This article discusses the piece "Psychological conceptions of randomness".
The paper analyzes the relationship between the epistemological nature of mathematical knowledge and its socially constituted meaning in classroom interaction.
Much has been written about methods of teaching statistics and about how to assess students' knowledge of statistics, but almost nothing on the extent to which assessment procedures measure whether students understand statistical concepts, or whether they understand what is involved in the application of statistical techniques. There has in fact been very little research on the development of instruments designed specifically to measure statistical understanding. There is, however, work in related areas which has some bearing on the assessment of understanding of statistical concepts. In reviewing this work this paper discusses the extent to which understanding is covered by some classification schemes which have been developed for use in mathematics and looks at ways in which attitude scales investigate understanding. Some alternatives to traditional methods of examining brought about by changes in the method of teaching are also considered.
Previous research indicated that subjects are not very surprised when reading coincidence stories, apparently because they regard the coincidence as one of many events that could have happened. This was true with respect to coincidences written by somebody else. However, there were indications that subjects found their own coincidences more surprising than those of others. The present study examines that egocentric bias and explores it further .
Performance on problems included in the fourth administration of NAEP suggest that roughly half of secondary students believe in the independence of random events. In the study reported here about half of the subjects who appeared to be reasoning normatively on a question concerning the most likely outcome of five flips of a fair coin gave a logically inconsistent answer on a follow-up question about the least likely outcome. In a second study, subjects were interviewed about various aspects of coin flipping. Many gave contradictory answers to closely related questions. We offer two explanations for inconsistent responses: a) switching among incompatible perspectives of uncertainty, including the outcome approach, judgment heuristics, and normative theory, and b) reasoning via basic beliefs about coin flipping. As an example of the latter explanation, people believe both that a coin is unpredictable and also that certain outcomes of coin flipping are more likely that others. Logically, these beliefs are not contradictory; they are, however, incomplete. Thus, contradictory statements appear when these beliefs are applied beyond their appropriate domain.
This article discusses the affect of weighted average.
A series of probabilistic inference problems is presented and base rates will be combined with other information when the two kinds of information are perceived as being equally relevant to the judged case. The base-rate fallacy is then discussed in its relation to the above.
The ability to seek out data, organize it, and interpret it is an empowering skill, and that a person who truly understands data has a source of power to use in influencing the direction of important decisions. The goal of education in statistics and probability should be to impart this sense of power to students, Learning statistics does not mean merely mastering the fomulaic transformations that yield mean, standard deviation, and P value. A true understanding of statistics includes knowing how to use data to discover and evaluate important associations and to communicate these associations to others. It requires learning how to evaluate other people's use of data and to augment or challenge them with additional data. There are NCTM Teaching Standards (NCTM, 1991), which include a new view of pedagogy in mathematics teaching - a focus on understanding the underlying concepts of our number system rather than on memorizing addition and multiplication facts, on facility in spatial visualization rather than on learning formulas for the area of polygons, and on planning and on carrying out data analysis projects rather than on knowing the difference between mean and median. Integrated with these two major changes, researchers and practitioners are looking more to technology to support new approaches to mathematics learning, as "tools for enhancing [mathematical] discourse." (NCTM, 1991, p.52) How does the computer fit into the developing view of statistics education? At first glance, the answer seems obvious: computers free students (and teachers) from the tedious computations that are required to calculate means, standard deviations, confidence intervals, etc. They draw graphs quickly and accurately. They generate multitudes of samples in a single bound. But this list of accomplishments leaves two crucial questions unanswered: 1) Are there other more powerful ways in which computers can facilitate students' learning of statistics? 2) Are there any drawbacks to uses of computers in statistics classes? The remainder of this paper will address both of these questions.
The "Problem of the Three Prisoners," a counterintuitive puzzle in probability, is reanalyzed, following Shimojo and Ichikawa (1989). Several intuitions that are examined represent attempts to find a simple and commonsensical criterion to predict whether and how the probability of the target event will change as a result of obtaining evidence. However, despite the psychological appeal of these attempts, none proves to be valid in general. A necessary and sufficient condition for change in the probability of the target event, following observation of new data, is proposed. That criterion is an extension to any number of alternatives, of the likelihood principle, that holds in the case of only two complementary alternatives. It is based on comparison of the likelihood of the data, given the truth of the target possibility, with the weighted average of the other likelihoods. This criterion is shown to be psychologically sound, and it may be assimilated to the point of becoming a secondary intuition.
For introductory statistics education, several types of software are relevant and in use: custom designed educational programs for a specific educational goal, statistical systems for data analysis (in full professional version, in student version or as a specifically designed tool for students), statistical programming environments, spreadsheets and general purpose programming languages. We can perceive a double dilemma on a practical and on a theoretical level, which is the worse the lower the educational level we have in mind. On the one hand, we have professional statistical systems that are very complex and call for high cognitive entry costs, although they flexibly assist experts. On the other hand, custom designed educational software is of necessity constrained to enable students to concentrate on essential aspects of a learning situation and to make likely certain intended cognitive processes. Nevertheless, as these microworlds, as we will call them here, for short, are often not adaptable to teachers' needs they are often criticized as being too constrained. Their support for flexible data analysis is limited, and to satisfy the variety of demands one would need a collection of them. However, coping with uncoordinated interfaces, notations and ideas in one course would overtax the average teacher and student. This practical dilemma is reflected on a theoretical level. It is not yet clear enough what kind of software is required and helpful for statistics education. We need a critical evaluation and analysis of the design and use of existing educational and professional programs. The identification of key elements of software that are likely to survive the next quantum leap of technological development and that are fundamental for introductory statistics is an important research topic. Results should guide new "home grown" developments of educational programs or, facing the difficulty of such developments, should influence the adaptation and elaboration of existing statistical systems toward systems that are also more adequate for purposes or, facing the difficulty of such developments, should influence the adaptation and elaboration of existing statistical systems toward systems that are also more adequate for purposes of introducing and learning statistics. We will give some ideas and directions that are partly based on results of two projects.