CONCEPTUAL ISSUES IN UNDERSTANDING SAMPLING DISTRIBUTIONS

Patrick W. Thompson

Vanderbilt University

Pat.thompson@vanderbilt.edu

Luis A. Saldanha

Vanderbilt University

Luis.a.saldanha@vanderbilt.edu

This study investigated students’ abilities to conceive the ideas of sampling distribution and margin of error. Twenty-seven 11^th- and 12^th-grade students participated in a teaching experiment addressing ideas of sample, sampling distributions, and margins of error. Our aim was to produce epistemological analyses of these ideas (Glasersfeld, 1995; Steffe, 1996; Thompson, in press) – ways of thinking about them that are schematic, imagistic, and dynamic – and hypotheses about their development in relation to students’ classroom engagement.

Better performing students and students exhibiting coherent discourse during class had developed a multi-tiered scheme of conceptual operations centered around the image of repeatedly sampling from a population, recording a statistic, and tracking the accumulation of statistics as they distribute themselves along a range of possibilities. These operations seemed to be grounded in an image of samples as quasi-proportional, mini-versions of the sampled population. Poorer-performing students (1) tended to view samples simply as some of the population, (2) did not extend their sense of variability to ideas of distribution. Instead, variability meant only that if we were to draw more samples and compute statistics from them, those statistics would differ from the ones of previously drawn samples, and (3) had difficulty coordinating the various levels of activity (drawing one sample and calculating a statistic, repeating this process many times, analyzing outcomes from the second-level process, etc.).

Note

Research reported in this paper was supported by the National Science Foundation Grant No. REC-9811879. Any conclusions or recommendations stated here are those of the authors and do not necessarily reflect official positions of NSF.

References

Glasersfeld, E. v. (1995). Radical constructivism: A way of knowing and learning. London: Falmer Press.

Steffe, L. P. (1996). Radical constructivism: A way of knowing and learning [Review of the same title, by Ernst von Glasersfeld]. Zentralblatt fur Didaktik der Mathematik [International Reviews on Mathematical Education], 96(6), 202-204.

Thompson, P. W. (In press). Radical constructivism: Reflections and directions. In L. P. Steffe & P. W. Thompson (Eds.), Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld (pp. 412-448). London: Falmer Press.