Research

  • "Variation is the reason why people have had to develop sophisticated statistical methods to filter out any messages in data from the surrounding noise" (Wild &amp; Pfannkuch, 1999, p. 236). Both variation, as a concept, and reasoning, as a process, are central to the study of statistics and as such warrant attention from both researchers and educators. This discussion of some recent research attempts to highlight the importance of reasoning about variation. Evolving models of cognitive development in statistical reasoning have been discussed earlier in this book (Chapter 5). The focus in this chapter is on some specific aspects of reasoning about variation.<br>After discussing the nature of variation and its role in the study of statistics, we will introduce some relevant aspects of statistics education. The purpose of the chapter is twofold: first, a review of recent literature concerned, directly or indirectly, with variation; and second, the details of one recent study that investigates reasoning about variation in a sampling situation for students aged 9 to 18. In conclusion, implications from this research for both curriculum development and teaching practice are outlined.

  • Covariation concerns association of variables; that is, correspondence of variation. Reasoning about covariation commonly involves translation processes among raw numerical data, graphical representations, and verbal statements about statistical covariation and causal association. Three skills of reasoning about covariation are investigated: (a) speculative data generation, demonstrated by drawing a graph to represent a verbal statement of covariation, (b) verbal graph interpretation, demonstrated by describing a scatterplot in a verbal statement and by judging a given statement, and (c) numerical graph interpretation, demonstrated by reading a value and interpolating a value. Survey responses from 167 students in grades 3, 5, 7, and 9 are described in four levels of reasoning about covariation. Discussion includes implications for teaching to assist development of reasoning about covariation (a) to consider not just the correspondence of values for a single bivariate data point but the variation of points as a global trend, (b) to consider not just a single variable but the correspondence of two variables, and (c) to balance prior beliefs with data-based observations.

  • In this chapter we present results from research on students' reasoning about the normal distribution in a university-level introductory course. One hundred and seventeen students took part in a teaching experiment based on the use of computers for nine hours, as part of a 90-hour course. The teaching experiment took place during six class sessions. Three sessions were carried out in a traditional classroom, and in another three sessions students worked on the computer using activities involving the analysis of real data. At the end of the course students were asked to solve three open-ended tasks that involved the use of computers. Semiotic analysis of the students' written protocols as well as interviews with a small number of students were used to classify different aspects of correct and incorrect reasoning about the normal distribution used by students when solving the tasks. Examples of students' reasoning in the different categories are presented.

  • Although reasoning about samples and sampling is fundamental to the legitimate practice of statistics, it often receives little attention in the school curriculum. This may be related to the lack of numerical calculations-predominant in the mathematics curriculum-and the descriptive nature of the material associated with the topic. This chapter will extend previous research on students' reasoning about samples by considering longitudinal interviews with 38 students 3 or 4 years after they first discussed their understanding of what a sample was, how samples should be collected, and the representing power of a sample based on its size. Of the six categories of response observed at the time of the initial interviews, all were confirmed after 3 or 4 years, and one additional preliminary level was observed.

  • This chapter presents a series of research studies focused on the difficulties students experience when learning about sampling distributions. In particular, the chapter traces the seven-year history of an ongoing collaborative research project investigating the impact of students' interaction with computer software tools to improve their reasoning about sampling distributions. For this classroom-based research project, three researchers from two American universities collaborated to develop software, learning activities, and assessment tools to be used in introductory college-level statistics courses. The studies were conducted in five stages, and utilized quantitative assessment data as well as videotaped clinical interviews. As the studies progressed, the research team developed a more complete understanding of the complexities involved in building a deep understanding of sampling distributions, and formulated models to explain the development of students' reasoning.

  • This study offers a descriptive qualitative analysis of one third-grade teacher's statistical reasoning about data and distribution in the applied context of classroom-based statistical investigation. During this study, the teacher used the process of statistical investigation as a means for teaching about topics across the elementary curriculum, including dinosaurs, animal habitats, and an author study. In this context, the teacher's statistical reasoning plays a central role in the planning and orchestration of the class investigation. The potential for surprise questions, unanticipated responses, and unintended outcomes is high, requiring the teacher to "think on her feet" statistically and react immediately to accomplish content objectives as well as to convey correct statistical principles and reasoning. This study explores the complexity of teaching and learning statistics, and offers insight into the role and interplay of statistical knowledge and context.

  • The importance of distributions in understanding statistics has been well articulated in this book by other researchers (for example, Bakker &amp; Gravemeijer, Chapter 7; Ben-Zvi, Chapter 6). The task of comparing two distributions provides further insight into this area of research, in particular that of variation, as well as to motivate other aspects of statistical reasoning. The research study described here was conducted at the end of a 6-month professional development sequence designed to assist secondary teachers in making sense of their students' results on a state-mandated academic test. In the United States, schools are currently under tremendous pressure to increase student test scores on state-developed academic tests.<br>This chapter focuses on the statistical reasoning of four secondary teachers during interviews conducted at the end of the professional development sequence. The teachers conducted investigations using the software Fathom&trade; in addressing the research question: "How do you decide whether two groups are different?" Qualitative analysis examines the responses during these interviews, in which the teachers were asked to describe the relative performance of two groups of students in a school on their statewide mathematics test. Pre- and posttest quantitative analysis of statistical content knowledge provides triangulation (Stake, 1994), giving further insight into the teachers' understanding.

  • Recent research has been aimed at finding out how precollege students think about variation, but very little research has been done with the prospective teachers of those students. Absent from the literature is an examination of the conceptions of variation held by elementary preservice teachers (EPSTs). This study addresses how EPSTs think about variation in the three contexts of sampling, data and graphs, and probability situations.<br><br>A qualitative study was undertaken with thirty students in an elementary teachers' mathematics course. The course included three classroom interventions comprised of activities promoting an exploration of variation in each of the three contexts. Written surveys were completed by all students both before and after the class interventions, and six students participated in pre and post interviews.<br><br>Collective results from the survey data, interview data, and class observations were used to describe components of an evolving framework useful for examining EPSTs' conceptions of variation. The three main aspects of the framework address how EPSTs reason in expecting, displaying, and interpreting variation. Each of the three aspects is further defined by different dimensions, which in turn have their own constituent themes. The depth in describing the evolving framework is a main contribution of this research.<br><br>Particular tasks created or modified for this research proved useful in examining EPSTs' conceptions of variation. One kind of task asked students to evaluate supposed results of experiments and decide if the results were genuine or not. Another kind of task provided specific arguments to which subjects could react. A third kind of task involved a comparison of data sets that were displayed using different types of graphs.<br><br>The framework was used to compare the thinking of the six interviewees from before to after the class interventions. Changes included richer conceptions of expectations of variation, more versatile understanding about displays of variation, and better interpretations of variation. The most notable changes were the overall depth in maturity of responses and an increased sophistication in communication during the post interview. Evidence suggests that the class interventions, and the survey and interview tasks, stimulated changes in the way students thought about variation.

  • The Sampling Distributions program and ancillary instructional materials were<br>developed to guide student exploration and discovery. The program provides graphical,<br>visual feedback which allows students to construct their own understanding of sampling<br>distribution behavior. Diagnostic, graphics-based test items were developed to capture<br>students' conceptual understanding before and after use of the program. An activity<br>which asked students to test their predictions and confront their misconceptions was<br>found to be more effective than one based on guided discovery. Our findings demonstrate that while software can provide the means for a rich classroom experience, computer simulations alone do not guarantee conceptual change.

  • Presents an experimental study on students' strategies and association judgments when faced with comparison of a numerical variable in two different samples. Classifies the strategies from a mathematical standpoint to identify theorems in action and two types of misconceptions about association.

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