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.
- Prof Dev