Variation in a Chance Sampling Setting: The Lollies Task

The motivation for the sequence of studies of which this one is a part arose from a survey item used in the National Assessment of Educational Progress (NAEP) in the United States, known as the "Gumball Task". Although it was set in the context of drawing a number of gumballs in a chance setting, the wording of the question encouraged an exact answer (the expected value), instead of a likely range (Zawojewski & Shaughnessy, 2000). The task was rewritten in three alternative forms by Shaughnessy, Watson, Moritz, and Reading (1999) in an attempt to explore more deeply students' understanding of variation and to move away from the strong emphasis on centres. In association with trials of the rewritten tasks in surveys with 324 students in the United States and Australia, Shaughnessy et aI. developed a two-dimensional coding system based on a "centring scale" and a "range scale". The "centring scale" for a set of six predictions, for example, classified responses using the mean to determine whether the response was "low", "five" (mean), or "high". The "range (r) scale", informed by a simulation of 1000 trials, resulted in a classification scheme of "narrow" (r':::; 1), "reasonable" (2 < r < 7), or "wide" (r ~ 7). Each response was then classified based on centre and spread, with the optimum classification being a "five-reasonable" prediction. Two small studies followed the surveys, where students were interviewed to gain more appreciation of their understanding of the "Lollies Task", as it was renamed for Australia. Torok and Watson (2000) used a similar protocol to that used here as well as several others involving variation and conducted 16 interviews with students in grades 4, 6, 8, and 10. The responses were categorised and clustered into similar groups that formed a four-tiered hierarchy demonstrating an increasing sophistication in the understanding of the proportional ideas and the variation involved in the tasks. The categories they found were those displaying (i) "weak appreciation of variation", (ii) "isolated appreciation of aspects of variation and clustering", (iii) "inconsistent appreciation of variation and clustering", and (iv) "good consistent appreciation of variation and clustering" (p. 155). Reading and Shaughnessy (2000) also interviewed 12 students using a similar protocol to this study and reported on four case studies from one student in each of grades 4, 6, 9, and 12. These students reflected many of the characteristics of the four levels observed by Torok and Watson, with the grade 12 student expressing conflict in choosing between multiple choice responses representing strict probability and sampling variation. Following these four studies, this report focuses on two research questions. First, following the survey work of Shaughnessy et aI. (1999) what are the distributions of students' responses given in an interview setting with respect to centres, spreads, repeated values in predictions, and change in predictions after experimentation? Second, following