Conference Paper

  • The problem which inspired the present research referred to the relationships between schemata and intuitions. Schemata are defined following the Piagetian line of thought, as programs of processing information and controlling adaptive reactions. Intuitions are defined as self-evident, global, immediate cognitions.<br>Our main hypothesis was that intuitions are generally based on certain structural schemata. In the present research this hypothesis was checked with regard to intuitive solutions of combinatorial problems.

  • Recently there has been a trend towards admitting expert statistical evidence in UK court cases. There have been a number of cases, however, in which outcomes have been distorted by statistical or probabilistic misconceptions and by faulty inference. Typically, lawyers receive no training in these areas apart from their compulsory school mathematical education. In this study, data was taken from five groups of trainee lawyers. This demonstrated that they made errors in assessing likelihoods, irrespective of the level and type of mthematical education that they have received. The typical approaches and content of mathematical education at school or college need to be re-considered. Data from two other groups of subjects (one of statistical educators) with different types of mathematical backgrounds were available for comparison purposes.

  • This paper discusses some characteristic ways of reasoning within the discipline of statistics from the perspective of someone who is both a practising statstician and teaching statistician. It is conjectured that recognition of variation and critically evaluating and distinguishing the types of variation are essential components in the statistical reasoning process. Statistical thinking appears to be the interaction between the real situation and the stiatistical model. The role of variation in staistical thinking and the implications for teaching are also discussed.

  • Concerns about equity in the ways that schools are using the data from the results of their students' state-mandated exams (Confrey &amp; Makar, in press) prompted this mixed-method study, based on the model of Design Research (Cobb et al., 2003). The study was conducted to provide insight into the ways that understanding of the statistical concepts of variation and distribution, developed in the context of learning about equity and assessment, can allow prospective teachers to broaden their understanding of equity and gain experience with conducting an inquiry of an ill-structured problem through the use of data generated by high-stakes tests to investigate equity and fairness in the accountability system. The study took place in an innovative one-semester course for preservice teachers designed to support and develop understanding of equity and fairness in accountability through data-based statistical inquiry (Confrey, Makar, and Kazak, 2004). The prospective teachers' investigations were conducted using Fathom Dynamic Statistics (Finzer, 2001), a learning software built for novice data analysts which emphasizes visualization and building inferential thinking through highlighting relationships between multiple variable displays. Semi-structured investigations during the course led up to a three-week self-designed inquiry project in which the prospective teachers used data to investigate an area of interest to them about equity in accountability, communicating their findings both orally and as a written paper. Results from the study provide insight into prospective teachers' experiences of conducting inquiry of ill-structured problems and their struggle with articulating beliefs of equity. The study also reports how statistical concepts documented in structured settings showed that the subjects developed rich conceptions of variation and distribution, but that the application of these concepts as evidence in their inquiry of an ill-structured problem was more challenging for them. No correlation was found between the level of statistical evidence in the structured and open-ended inquiry settings, however there was a significant correlation between prospective teachers degree of engagement with their topic of inquiry and the depth of statistical evidence they used, particularly for minority students. Implications and suggestions for improving the preparation of teachers in the areas of statistical reasoning, inquiry, equity, and interpreting assessment data are provided.

  • Two studies investigated upper elementary school students' informal understanding of sampling issues in the context of interpreting and evaluating survey results. The specific focus was on the children's evaluation of sampling methods and means of drawing conclusions from multiple surveys. In Study 1, 17 children were individually interviewed to categorize children's conceptions. In Study 2, 110 children completed paper-and-pencil tasks to confirm the response categories identified in Study 1 and to determine the prevalence of the response categories in a larger sample. Children evaluated sampling methods focusing on potential for bias, fairness, practical issues, or results. All children used multiple types of evaluation rationales, and the focus of their evaluations varied somewhat by context and type of sampling method (restricted, self-selected, or random). Children used affective (fairness) rationales more often in school contexts and rationales focused on results more often in out-of-school contexts. Children had more difficulty detecting bias with self-selected sampling methods than with restricted sampling methods because self-selection was initially the most fair (i.e., everyone had a chance to participate). Children preferred stratified random sampling to simple random sampling because they wanted to ensure that all types of individuals were included. When drawing conclusions from multiple surveys, children: (1) considered survey quality; (2) aggregated all surveys regardless of quality; (3) used their own opinions and ignored all survey data; or (4) refused to draw conclusions. Even when children were able to identify potential bias, they often ignored survey quality when drawing conclusions from multiple surveys.

  • Graph comprehension is considered a basic skill in the curriculum, and essential fo rstatistical literacy in an information society. How do students interpret a graph in an authentic context? Are misleading features apparent? Responses to questions about a graph-based advertisement suggest that students commonly did not appreicate scaling difficulties, relate a grph as relecant int he context of a standard interpretation task , or apply numeracy skills for calculations based on data in graphical represtations.

  • Variation is a key concept in the study of statistics and the understanding of variation is a crucial aspect of most statistically related tasks. To express this understanding students need to be able to describe variation. Students aged 13 to 17 engaged in an inference task set in a real world context that necessitated the description of both rainfall and temperature data. This research qualitatively analysed the student responses with respect to the descriptions of variation that were incorporated. A Data Description hierarchy, previously developed for describing variation in a sampling task, was found to be appropriate to code the better responses and was extended to accommodate a range of less statistically sophisticated responses identified. The SOLO Taxonomy was used as a framework for the hierarchy. Two cycles of U-M-R levels, one for more qualitatively descriptions and the other for more quantitative descriptions, were identified in the responses. Task and implementation issues that may have affected the descriptions, as well as implications for research, teaching and assessment, are outlined.

  • Although research has been done on students' conceptions of centers (averages, means, medians), there has not been a corresponding line of research into students' conceptions of variability or spread. In this paper we describe several exploratory studies designed to investigate school students' conceptions of variability. A sampling task that was a variation of an item on the 1996 National Assessment of Educational Progress (NAEP) was given to 324 students in Grades 4 - 6, 9, and 12 from Oregon, Tasmania, and New South Wales. Three different versions of the task were presented in a Before, and in an After setting. The Before and After students did the task both before and after carrying out a simulation of the task. Responses to the sampling task were categorized according to their centers (Low, Five, High) and spreads (Narrow, Reasonable, Wide). Results show a steady growth across grades on the center criteria but no clear corresponding improvement on the spread criteria. There was considerable improvement on the task among the students who repeated it after the simulation. Students' growth on the center criteria may be due to the emphasis that instruction places on centers in school mathematics. Similarly, the lack of clear growth on spreads and variability, and the inability of many students to integrate the two concepts (centers and spreads) on this task, may be due to instructional neglect of variability concepts.

  • Reflecting on a body of research work can sometimes lead to the recognition of areas of opportunity for research that have gone largely unnoticed. In this paper we consider three such opportunities in the area of research on the teaching and learnig of probability and statistics: i) Following up on students' initial thinking to watch for future transitions; ii) Investigating students' thinking on variability; and iii) Posing research questions that begin with what students can do rather than pointing out what they cannot do. Situations from research tasks, past and future, are used as starting points for the discussion.

  • This paper discusses the premise that young children do not perceive accurate relationships between the behavior of different, but related random generators. Data for this preliminary study has been collected from suburban primary school students aged from 7 - 12 years, who were questioned about their perceptions of the behavior of dice, coins, raffle tickets and a range of different and unusual random generators in identical situations. The findings indicate that children predict different result depending for example, on whether tickets rather than dice are used in a game. Their predictions appears to be based on the observation of the physical differences between dice and raffle tickets. Owing to the size of the preliminary study no tests of significance have been carried out and results are given in simple percentages.

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