Journal Article

  • We distinguish two conceptions of sample and sampling that emerged in the context of a teaching experiment conducted in a high school statistics class. In one conception 'sample as a quasi-proportional, small-scale version of the population' is the encompassing image. This conception entails images of repeating the sampling process and an image of variability among its outcomes that supports reasoning about distributions. In contrast, a sample may be viewed simply as 'a subset of a population' - an encompassing image devoid of repeated sampling, and of ideas of variability that extend to distribution. We argue that the former conception is a powerful one to target for instruction.

  • In this article data from Old Faithful geysers are introduced to highlight the important role that variation should play in our teaching of statistics. Our past teaching may have overemphasized the role of centers to the neglect of issues of spread and variability.

  • Are you increasing your emphasis on probability and statistics with students? Are more of your students studying statistics or probability during secondary school? Are your students improving in their performance in data and chance? If you answered yes to any of these three questions, you are not alone, according to the most recent National Assessment of Educational Progress (NAEP). The NAEP is administered approximately every four years to students attending a representative sample of schools across the United States.

  • This paper examines the role of variation in statistics education and describes a chance and data unit with a focus on variation that has been conducted with three high school mathematics classes.

  • An appreciation of variation is central to statistical thinking. In this study, four students from each of grades 4,6,8 and 10 were interviewed individually on aspects of variation present in three settings. The first setting was an isolated random sampling situation, whereas the other two settings were real worlds sampling situations. Four levels of responding were identified and described in relation to developing concepts of variation. Implications for teaching and future research on variation are considered.

  • This study follows two earlier studies of school students' abilities to draw inferences when comparing two data sets presented in graphical form (Watson and Moritz, 1999; Watson, 2001). Using the same interview protocol with a new sample of 60 students, 20 from each of grades 3, 6 and 9, cognitive conflict was introduced in the form of video clips of reasoning expressed by students in the earlier studies. This methodology was intended to mimic the type of argumentation that might take place in the classroom but in a controlled setting where identical arguments could be presented to different students. Interviews were videotaped and analysed in a similar fashion to the earlier studies in order to document change associated with the presentation of cognitive conflict. Change was documented with respect to the levels of observed response for two parts of the protocol and for the use of displayed variation in the graphs. Implications of the methodology for future research and teaching are discussed.

  • This study follows an earlier study of school students' abilities to draw inferences when comparing two data sets presented in graphical form (Watson and Moritz, 1999). Forty-two students who were originally interviewed in grades 3 to 9, were subsequently interviewed either three or four years later. The results for individual student development add to the credibility of the cross-age observations, as well as support the hierarchical framework suggested by the original study. Changes in levels of performance and strategies for drawing conclusions are documented. A further step from the original study is the consideration of how students used the variation displayed in the graphical presentation of the data sets as a basis for decision-making. Implications for teaching and for further research are discussed.

  • This study explores school students' understanding of variation within a probabilistic setting involving spinners. Sixty-six students in Grades 3 to 9 answered survey questions involving a single 50/50 spinner and then were interviewed using a protocol involving compound events with different types of spinners (50/50 and 25/75). Of interest in interviews were students' initial responses and changes in response and reasoning that occurred after experimentation with the spinners. Because there was the possibility of variation occurring in the experiments that could be considered contrary to expectation, responses following experimentation were analysed based on their appropriateness in terms of the observed experimental outcome. Data were used to determine a developmental progression among the students in the study and data from survey responses were considered in relation to the interview data. Differences were found between primary and secondary students. The outcomes of the study lead to suggestions for the classroom and for future research in relation to expectation and variation in probabilistic settings.

  • Deals with a study which explored the ways in which school students evaluate numerically the chance expressed in various common language phrases as they appear in newspaper headlines in Tasmania. Exploration on how students interpret verbally a percent expression of chance in the context of risk of a side-effect from a medicine; Significance of appreciating chance language; Implications for classroom teaching.

  • A developmental model involving four response levels is proposed concerning how students arrange pictures to represent data in a pictograph, how they interpret these pictographs, and how they make predictions based on these pictographs. The model is exemplified by responses from three related interview-based studies. In Study 1, examples of each response level are provided from 48 preparatory- to tenth-grade students. Students from higher grades were more likely to respond at higher levels. In Study 2, 22 students were interviewed longitudinally after a three-year interval; many improved in response level over time, although a few responded at lower levels. In Study 3, 20 third-grade students were interviewed and then prompted with conflicting responses of other students on video; many improved their initial responses to higher levels after exposure to the conflicting prompts. Associations among levels of representing, interpreting, and predicting were explored. Educational implications are discussed concerning reasonable expectations of students and suggestions to develop these skills in students at different grades.

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