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. Coordinating 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 positioning 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 participated 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 characteristics of distribution as they emerged during the course of activity in each setting.

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ABSTRACT:The effect of Na-alginate content on the gas permeation properties of water-swollen membranes prepared by varying Na-alginate and poly(vinyl alcohol) (PVA) content in membranes was investigated. The influences of water content and crystallinity of the membranes on the gas permeation performance of the water-swollen membranes were studied. The gas permeation rate and selectivity of Na-alginate/PVA water-swollen membranes were compared with those of the dry membranes. The permeation rates of nitrogen and carbon dioxide through water-swollen membranes were in the range of 0.4 -7.6 ϫ 10 Ϫ7 to 3.7-8.5 ϫ 10 Ϫ6 cm 3 (STP)/cm 2 s Ϫ1 cmHg Ϫ1 , which were 10,000 times higher than those of dry-state membranes. The permeation rates of mixture gases through water-swollen Naalginate/PVA membranes were found to increase exponentially with the increase of Na-alginate content, whereas carbon dioxide concentration in permeates was decreased linearly. It was found that the gas permeance of the waterswollen membranes increased with increasing the Na-alginate content in the membrane. Gas permeation rates of the water-swollen Na-alginate/PVA membranes increased with increasing the water content in the membrane and decreasing the crystallinity of the membrane.

Although variability and structure are often considered as antonyms in many everyday settings, a mathematically disciplined view contradicts fllis opposition. To initiate fifth-(10 years old) and sixfll-grade (11 years old) students in fllis disciplinary view, we engaged students in practices of modeling data. These practices included inventing and revising data displays, inventing and revising measures of centre and variability, and inventing and revising models of chance to account for variability. Here we focus on prospective correspondences between students' invented measures (statistics) of variability and those favoured by the discipline. We suggest that inventing measures positions students to transform their vision of variability from mere difference to more structured forms, some of which coordinate centre and spread. By tracing interactions among an inventor, her classmates, and the teacher, we trace how structuring variability and constituting its measure cooriginated during the course of negotiations about the meaning of the measure. Consideration of the coherency, transparency and generalisability of a statistic, all of which are valued by the discipline of statistics, emerged during the course of invention.

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