Literature Index

Most statistics educators would agree that statistical inference is both the central objective of statistical reasoning and one of the most difficult ideas for students to understand. In traditional approaches, statistical inference is introduced as a quantitative problem, usually of figuring out the probability of obtaining an observed result on the assumption that the null hypothesis is true. In this article, we lay out an alternative approach towards teaching statistical inference that we are calling "informal inference." We begin by describing informal inference and then illustrate ways we have been trying to develop the component ideas of informal inference in a recent data analysis seminar with teachers; our particular emphasis in this article is on the ways in which teachers used TinkerPlots, a statistical visualization tool. After describing teachers' approaches to an inferential task, we offer some preliminary hypotheses about the conceptual issues that arose for them.

Concerns about the importance of variation in statistics education and a<br>lack of research in this topic led to a preliminary study which explored<br>pre-service teachers' ideas in this area. The teachers completed a written<br>questionnaire about variation in sampling and distribution contexts.<br>Responses were categorised in relation to a framework that identified levels<br>of statistical thinking. The results suggest that while many of the students<br>appeared to acknowledge variation, they were not able to provide adequate<br>explanations. Although the pre-service teachers have had more real-life experiences<br>involving statistics and have been involved in the study of statistical<br>concepts at secondary school level, they still demonstrated the same misconceptions<br>as those of younger students reported in research literature. While<br>more students showed competence on the sampling question, they were less<br>competent on the distribution task. This could be due to task format or<br>contextual issues. The paper concludes by suggesting some implications for<br>further research and teaching.

This study investigated introductory statistics students' conceptual understanding of the standard deviation. A computer environment was designed to promote students' ability to coordinate characteristics of variation of values about the mean with the size of the standard deviation as a measure of that variation. Twelve students participated in an interview divided into two primary phases, an exploration phase where students rearranged histogram bars to produce the largest and smallest standard deviation, and a testing phase where students compared the sizes of the standard deviation of two distributions. Analysis of data revealed conceptions and strategies that students used to construct their arrangements and make comparisons. In general, students moved from simple, one-dimensional understandings of the standard deviation that did not consider<br>variation about the mean to more mean-centered conceptualizations that coordinated the effects of frequency (density) and deviation from the mean. Discussions of the results and implications for instruction and further research are presented.

The Birthday Problem is "How many people must be in a room before the probability that some share a birthday (ignoring the year and ignoring leap days) becomes at least 50%?" Multiple approaches to the problem are explored and compared, addressing probability concepts, problem solving, modelling assumptions, approximations (supported by Taylor series), recursion, (Excel) spreadsheets, simulation, and student preconceptions. The traditional product representation that yields the exact answer is not only tedious with a regular calculator, but did not provide insight on why the answer (23) is so much smaller than most students' predictions (typically, half of 365). A more intuitive (but slightly inexact) approach synthesized by the author focuses on the total number of "opportunities" for matched birthdays (e.g., the new "opportunities" for a match added by the kth person who enters are those that the kth person has with each of the k-1 people already there). The author followed the model of Shaughnessy (1977) in having students give predictions in advance of the exploration and these written data (as well as interview data) collected from students indicated representative multiplier or representative quotient effects, consistent with the literature on misconceptions and heuristics. Data collected from students after the traditional and "opportunities" explorations indicate that a majority of students preferred the opportunities approach, favoring the large gain in intuition over the slight loss in precision.