ORIGINAL ARTICLE Developing conceptions of statistics by designing measures of distribution Richard Lehrer • Min-Joung Kim • Ryan Seth Jones Accepted: 17 June 2011 / Published online: 2 July 2011 Ó FIZ Karlsruhe 2011 Abstract Students often learn procedures for measuring, but rarely do they grapple with the foundational conceptual problem of generating and validating coordination between a measure and the phenomenon being measured. Coordi- nating measures with phenomenon involves developing an appreciation of the objects and relations in each as well as establishing their mutual correspondence. We supported students’ developing conceptions of statistics by position- ing them to design measures of center and of variability for distributions that they had generated through repeated measure of a length. After students invented and explored the viability of their measures individually, they partici- pated in a public (whole-class conversation) forum featuring justification and reflection about the viability of their designed measures. We illustrate how individual invention enticed students to attend to, and to make explicit, characteristics of distribution not initially noticed or known only tacitly. Conceptions of statistics and of relevant characteristics of distribution were further expanded as students justified and argued about the utility and prospective generalization of particular inventions. Teachers supported student learning by highlighting prospective relations between characteristics of measures and charac- teristics of distribution as they emerged during the course of activity in each setting. 1 Introduction Measurement is often conceived as a mundane activity, and in school it typically arrives pre-formed. Students often learn measurement procedures (Lee & Smith, 2011), but rarely do they grapple with the foundational conceptual problem of how to establish coordination between the objects and relations of a particular phenomenon, its structure, and with the corresponding symbolic, and often material, objects and relations of its measure. This ‘‘black box’’ quality of measure is often true of workplaces as well, where the very success of the tools employed tend to obscure the origins of the correspondences established between measure and phenomena (Bakker, Wijers, Joinker & Akkerman, 2011). Yet research in mathematics educa- tion suggests that even young students can be supported to investigate and understand relations between characteris- tics of space, such as length and area, and aspects of their measure, such as unit and scale (e.g., Clements & Bright, 2003). We sought to leverage this emerging tradition of involving students in the conceptual foundations of measurement in the less explored domain of statistical reasoning. Statistics measure characteristics of distribution, but as with other forms of measure, students often treat statistics as matters of procedure (Zawojewski & Shaugh- nessy, 2000). Accordingly, we positioned 10- and 11-year-old students to design measures of characteristics of distributed data, such as their center and variability. Distribution is a key that unlocks much of statistical reasoning (Cobb, 1999). Like other measures, a statistic (a measure of a characteristic of a distribution) represents a commitment to certain aspects of a distribution at the expense of others. For example, the mean statistic partitions the distribution into fair (equal) shares, so that center is measured by the magnitude of a fair share. In The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education or of the U.S. National Science Foundation. R. Lehrer (&) Á M.-J. Kim Á R. S. Jones Vanderbilt University, Nashville, USA e-mail: rich.lehrer@vanderbilt.edu 123 ZDM Mathematics Education (2011) 43:723–736 DOI 10.1007/s11858-011-0347-0