Students' Conceptual Metaphors Influence Their Statistical Reasoning About Confidence Intervals
WCER Working Paper No. 2008-5
Timothy Grant and Mitchell J. Nathan
August 2008, 22 pp.
ABSTRACT: Confidence intervals are an attractive means of conveying experimental results, as they contain a considerable amount of information in a concise format. This paper examines two competing metaphors for confidence intervals—as revealed in students’ speech and gestures during interviews—for their impact on understanding, statistical problem solving, and future learning of mathematics. In the first metaphor—Confidence Intervals Are Changing Rings Around a Fixed Point (Changing Ring metaphor)—confidence intervals are moving disks of various diameters covering a fixed but unknown point, like horseshoes of varying widths pitched at 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, whereas the location of the stake (i.e., the population parameter or population mean) is fixed across samples but generally unknown. In contrast, the second metaphor—Confidence Intervals Are Changing Points on a Fixed Disk (Fixed Disk metaphor)—conceptualizes confidence intervals as fixed-diameter disks onto which successive points are placed. In this incorrect metaphor, thepopulation parameter can change from sample to sample. The interval is of a fixed length, and each experiment results in placing a new parameter somewhere 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, concepts that tend to be taught in close proximity to one another in statistics textbooks. TheFixed Disk metaphor will generally lead to a misinterpretation of the confidence interval that results in 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 education researchers’ uses of statistical tests.
keywords: Conceptual Metaphors; Statistics; Confidence Intervals; Gesture
