Research

  • Statistics is the science of modeling the world through theory-driven interpretation of data. Models of chance and uncertainty provide powerful cognitive tools that can help in understanding certain phenomena. In this chapter, we consider the development of children's models of chance and uncertainty by considering their performance along five distinct, albeit related, components of a classical model of statistics: a) the distinction between certainty and uncertainty, b) the nature of experimental trials, c) the relationship between individual outcomes (events) and patterns of outcomes (distributions), d) the structure of events (e.g., how the sample space relates to outcomes) and e) the treatment of residuals (i.e., deviations between predictions and results). After discussing these five dimensions, we summarize and interpret the model-based performance of three groups as they solved problems involving classical randomization devices such as spinners and dice. The three groups included second-graders (age 7-8), fourth/fifth-graders (age 9-11), and adults. We compare groups by considering their interpretations of each of these five components of a classical model of chance. We conclude by discussing some of the benefits of adopting a modeling stance for integrating the teaching and learning of statistics.

  • This chapter focuses on our work in middle schools. We describe the implementation and evaluation of a three-week instructional unit that was adapted to three different subject-matter contexts (science, social studies, and mathematics). In all contexts, the purpose of the unit was to improve students' abilities to think and reason statistically about real-world issues.

  • Five studies examined Kahneman and Miller's (1986) hypothesis that events become more "normal" and generate weaker reactions the more strongly they evoke representations of similar events. In each study, Ss were presented with 1 of 2 versions of a scenario that described the occurrence of an improbable event. The scenarios equated the a priori probability of the target event, but manipulated the ease of mentally simulating the event by varying the absolute number of similar events in the population. Depending on the study, Ss were asked to indicate whether they thought the event was due to chance as opposed to (a) an illegitimate action on the part of the benefited protagonist, or (b) the intentional or unintentional misrepresenation of the probability of the event. As predicted, the fewer ways the events could have occurred by chance, the less inclined Ss were to assume that the low-probability event occurred by chance. The implications of these findings for impression-management dynamics and stereotype revision are discussed.

  • A well-substantiated, surprising finding is that people judge the occurence of an event of low probability as less likely when its probability is represented by a ratio of small numbers (e.g., 1 in 20) than of larger (e.g., 10 in 200) numbers. The results of three experiments demonstrated that this phenomenon is broadly general and occurs as readily in pre- as in postoutcome judgments. These results support an interpretation in terms of subjective probability, as suggested by the principles of cognitive-experiential self theory, but not as an interpretation in terms of imagining couter-factual alternatives, as proposed by norm theory.

  • Twelve students answered questions involving the distribution of sample means both before and after an instructional intervention. Correct performance improved on these problems but dropped on problems having to do with the distribution of samples.

  • A detailed, multisite evaluation of instructional software design to help students conceptualize introductory probability and statistics yielded patterns of error on several assessment items. Whereas two of the patterns appeared to be consistent with misconceptions associated with deterministic reasoning, other patterns indicated that prior knowledge may cause students to misinterpret certain concepts and displays. Misconceptions included interpreting the y-axis on a histogram as if it were a y-axis on a scatter plot and confusing the values a variable might take on by misinterpreting plots of normal probability distributions. These kinds of misconceptions are especially important to consider in light of the increased emphasis on computing and displays in statistics education.

  • This paper reports on technological aspects of an ongoing international study in which secondary students engage in authentic data inquiries involving posing, sharing, and critiquing of statistical word problems. This is part of a larger study, which aims to (1) investigate developments in students' statistical understanding and reasoning processes as they engage in authentic data investigations involving data modeling, statistical problem posing, and problem critiquing, (2) foster students' awareness and appreciation of the influence of cultural factors in the statistical understandings of their international peers, (3) investigate the use of Web-based Intranets to enable schools connected to the Internet to conduct collaborative statistical investigations with students from other countries, and (4) use the findings of the study to develop a conceptual model of students growth of statistical understanding. In this paper, we focus on aim #3 and consider our developing experience in using semiprivate sites on the World Wide Web to facilitate activities in which students both publish mathematics problems that they have created and provide structured comments on problems posed by their local or international peers. To date, we have conducted a series of exploratory case studies in classrooms in England, Australia and Canada in which students posed and shared problems involving measures of central tendency (mean, median and mode). The problems were based on the results of an authentic, international dataset which they helped to create. The purpose of this paper is to discuss issues that have emerged in our present application of computer-mediated communication for fostering mathematical problem posing and critiquing. More specifically, we consider the following issues: the emergence of the Intranet design over the initial phase of the research project, the impact of the Intranet design on the effectiveness of computer-mediated communication in the key stages of the project, and the implications for subsequent Intranet designs for networked collaborative problem-posing activities.

  • This paper will describe the initial phases of a three-year project whose purpose is to devise assessment instruments and follow the implementation of the Chance and Data content of A National Statement on Mathematics for Australian Schools. A discussion of the rationale used in devising items will be followed by descriptions of the four types of instruments developed. Finally, the results of pilot trials carried out in recent months will be presented.

  • 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.

  • In this paper the responses of 247 secondary students to 8 test items used in classical studies of probabilistic reasoning (representativeness, equiprobability bias and outcome approach) are analyzed. The study was designed to assess the quality of probabilistic reasoning of two levels of secondary students (14 and 18 year-old students). These groups are compared revealing few differences in their responses.

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