STUDENTS' UNDERSTANDING AND REASONING ABOUT SAMPLE SIZE AND THE LAW OF LARGE NUMBERS AFTER A COMPUTER-INTENSIVE INTRODUCTORY COURSE ON STOCHASTICS

A post-course study of students who participated in an introductory course on stochastic (proba- bility and statistics) for future mathematics teachers is the focus of this paper. In this course the software FATHOM was used. Students learn to use FATHOM as a (cognitive and culturally mediated) tool for exploratory data analysis, for simulation and for inferential statistics as well as a tool for experimenting with statistical methods. We will focus on the analysis of three interview-tasks and report on the students' intuitions on the role of sample size in two tasks that were analogous to the maternity ward problem and about their understanding of the role of n (sample size) and N (num- ber of simulations) in sampling distributions in a coin-tossing context. Although the students had encountered similar tasks before in the course, our analysis will show the fragility and partly contradictory nature of their knowledge. As a background we refer briefly to the contents of the "probability part" of the course. Contents of the course were: sample space, three approaches of probability: classical approach (Laplace), statistical approach (relative frequency of events), and (computer-) simulation, random variables and their probability distributions and expected value, multistage random experiments, independence of random experiments, sampling distributions, hypergeometric and binomial distri- bution, standard deviation, variance, 1/� n-law, exactness of simulation results, and confidence intervals. In all themes, computer simulation was combined with theoretical approaches. During the course students had to work themselves with FATHOM on 15 homework assignments. In the course a final examination as well as pre- and post-tests with open response format items for assessing students' knowledge were used. In addition we used post-course structured interviews with 13 voluntary students. These post-course structured interviews contained 10 tasks concerning methods on determining probabilities, independence, precision and uncertainty of simu- lation results, intuitions about sample size, and about sampling distributions. The students in the interviews had different levels of competence: the test score of the final examination of the course varied from 47% to 92% in this group. The interviews were designed for about 60 minutes and lasted 25-70 minutes, depending on how much students were talking and how confident they were in their statements. The interviews were videotaped and transcribed. In difference to other remarkable studies on understanding sampling distributions (e.g., Sedlmeier, 1999; Saldanha & Thompson, 2007) students worked on simulations of random ex- periments and gained experience with sampling distributions in our course over half a year. Thus the setting of the study was not a training program or a specially conducted design experiment, which was only devoted to sampling distributions. In addition, these interviews took not place di- rectly after the course, but a few weeks after the course ended. Therefore the interviews rather show a long-term understanding. For this paper we focus only on two aspects: the understanding of the empirical law of large numbers and of sampling distributions and discuss only related tasks of our interviews.