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# Conference Paper

• ### Using the power simulator to help students learn the concept of statistical power

Our purpose was to introduce and explore a dynamic and interactive way of teaching a complex concept in statistics. Our research has used small sample sizes and single class presentations. The findings are far from definitive and conclusive. Future research needs to contrast the use of Power Simulator with instruction where the software is not used. This would be informative for comparison of both classroom presentations and out-of-class assignments. As for now, the results suggest that Power Simulator povides an effective way to introduce students to the complex concept of statistical power analysis. We are encouraged by the results to continue development of the program and find new ways to incorporate the methodology into statistics instruction.

• ### Correlates of students' performance in statistics

This is a descriptive correlational study on students enrolled in a compulsory course in the Teacher Education Program at the Faculty of Educational Studies. Three categories of variables were investigated, namely, the independent variables--learning styles, the variable of prime interest--performance as the dependent variable and several control variables such as gender, age, and program. A moderate positive bivariate relationship existed between learning style and performance or also referred to as cognitive skills of teacher education students.

• ### Students' thinking and writing in the context of probability

This study examined changes in students probabilistic thinking and writing during instruction emphasizing writing to learn experiences. A class of fifth grade students with no previous experiences in writing during mathematics made significant gains in probability reasoning and writing; however the correlation between probabilistic thinking and writing was not significant. Analysis of focus students revealed that their writing changed from narrative summaries to reasoned patterns and generalizations. However, some used invented representations without interpretation and were reluctant to write in mathematical contexts.

• ### Perceptions of unfamiliar random generators--links between research and teaching

This paper reports some findings from two independent studies, one in Northern Ireland and one in South Australia, into young children's understanding of the behaviour of unfamiliar Random Generators (RGs). Our findings indicate that probability understanding is often influenced by the physical properties or appearance of RGs. Sometimes it is their physical arrangement in a container that influences responses. Teaching of the topic depends on teachers being aware of what these misconceptions are before planning for teaching.

• ### Subjective elements in children's comparison of probabilities

In this research work we study the comparison of probabilities by 10-14 year-old pupils. We consider the different levels described in research about these tasks, though we incorporate subjective distractors, which change the predicted difficulty of some items. Analysis of students' arguments serves to determine their strategies, amongst which we identify the "equiprobability bias" and the "outcome approach". Analysis of response patterns by the same pupil serves to show that the coincidence between the difficulty level of probabilistic and proportional tasks is not complete and points to the existence of different types of probabilisitic reasoning for the same proportional reasoning level.

• ### Resampling Methods: Not Just for Statisticians Anymore

Resampling methods in statistics have been around<br>for a long time. Over forty years ago Tukey coined<br>the term jackknife to describe a technique, at-<br>tributed to Quenouille (1949), that could be used to<br>estimate bias and to obtain approximate con dence<br>intervals. About 20 years later, Efron (1979) intro-<br>duced the bootstrap" as a general method for esti-<br>mating the sampling distribution of a statistic based<br>on the observed data. Today the jackknife and the<br>bootstrap, and other resampling methods, are com-<br>mon tools for the professional statistician. In spite of<br>their usefulness, these methods have not gained ac-<br>ceptance in standard statistics courses except at the<br>graduate level. Resampling methods can be made<br>accessible to students at virtually every level. This<br>paper will look at introducing resampling methods<br>into statistics courses for health care professionals.<br>We will present examples of course work that could<br>be included in such courses. These examples will<br>include motivation for resampling methods. Health<br>care data will be used to illustrate the methods. We<br>will discuss software options for those wishing to in-<br>clude resampling methods in statistics courses.

• ### Framework for instruction and assessment on elementary<br>inferential statistics thinking.

The main objective in this paper is to describe a framework to characterize and assess the learning of<br>elementary statistical inference. The key constructs of the framework are: populations and samples and their<br>relationships; inferential process; sample sizes; sampling types and biases.<br>To refine and validate this scheme we have taken data from a sample of 49 secondary students sample<br>using a questionnaire with 12 items in three different contexts: concrete, narrative and numeric. Theoretical<br>analysis on the results obtained in this first research phase has permitted us to establish the key constructs<br>described below and determine levels in them. Moreover this has allowed us to determine the students'<br>conceptions about the inference process and their perceptions about sampling possible biases and their<br>sources.<br>The framework is a theoretical contribution to the knowledge of the inferential statistical thinking domain<br>and for planning teaching in the area.

• ### Statistical thinking: How can we develop it?

Wild &amp; Pfannkuch (1999) stated that statistical thinking comprises four dimensions: an investigative cycle, types of thinking, an interrogative cycle, and dispositions. The four dimensions contain generic and specific statistical thinking habits and are operative within the thinker simultaneously. The five types of thinking that were identified as fundamental elements in statistical thinking were: recognition of the need for data, transnumeration, consideration of statistical thinking models, and integrating the statistical with the contextual. When considering the framework and these types of thinking many questions arise for learning, teaching, and the curriculum such as: How are these types of thinking manifested in beginning students? Are there particular ways of teaching that can elicit such thinking? How does the teacher draw students' attention to notice and to attend to this thinking? How is such a habit of thinking communicated in a curriculum document? The purpose of the framwork was to characterize statistical thinking rather than define students' growth in statistical thinking and was not primarily intended to address teaching.

• ### Students' Conceptual Metaphors Influence Their Statstical Reasoning

Confidence intervals are an attractive means of conveying experiemental results, as they contain a considerable amount of information in a concise format. Two competing metaphors that arise from students' speech and gestures during interviews are examined for their impact on understanind, statistical problem solving, and future learning of mathematics. In the Changing Ring Aroung a Fixed Point metaphor, confidence intervals are moving disks of various diameters covering a fixed but unknown point, like pitching horseshoes of varying widths to capture a fixed stake. Key to this correct conceptual metaphor is that the interval is a property of a sample but not of the population. Here, the diameter of the disk (i.e., the length of the confidence interval) changes from sample to sample, while the location of the stake (i.e., the population paramet, or population mean) is fixed across samples, but generally unknown. In contrast, the Changing Point on a Fixed Disk metaphor conceptualizes confidence intervals as fixed-diameter disks onto which changing points are placed. In this incorrect metaphor, the population parameter can change from sample to sample. The interval is of fixed length and each experiment results in placing a new parameter somwhere onto the fixed-diameter disk. One possible source of this second metaphor is a suspected confusion between acceptance regions in hypothesis testing and confidence intervals, which tend to be taught in close proximity to one another in the statistics textbooks. The Changing Point on a Fixed Disk metaphor will generally support a misinterpretation of the confidence interval that leads to inaccurate problem solving. By better understanding students' mental representations of confidence intervals, and appealing to the metaphors they convey, we can hope to improve both statistics instruction and educational researchers' uses of statistical tests.

• ### Cognitive Task Analysis in Service of Intelligent Tutoring System Design: A Case Study in Statistics

Cognitive task analysis involves identifying the components of a task that are required for adequate performance. It is thus an important step in ITS design because it circumscribes the curriculum to be taught and provides a decomposition of that curriculum into the knowledge and subskills students must learn. This paper describes several different kinds of cognitive task analysis and organizes them according to a taxonomy of theroretical/empirical prescriptive/descriptive approaches. Examples are drawn from the analysis of a particular statistical reasoning task. The discussion centers on how different approaches to task analysis provide different prespectives on the decomposition of a complex skill and compares these approaches to more traditional methods.