Reasoning about Covariation

Abstract

OVERVIEW Covariation concerns association of variables; that is, correspondence of variation. Reasoning about covariation commonly involves translation processes among raw numerical data, graphical representations, and verbal statements about statistical covariation and causal association. Three skills of reasoning about covariation are investigated: (a) speculative data generation, demonstrated by drawing a graph to represent a verbal statement of covariation, (b) verbal graph interpretation, demonstrated by describing a scatterplot in a verbal statement and by judging a given statement, and (c) numerical graph interpretation, demonstrated by reading a value and interpolating a value. Survey responses from 167 students in grades 3, 5, 7, and 9 are described in four levels of reasoning about covariation. Discussion includes implications for teaching to assist development of reasoning about covariation (a) to consider not just the correspondence of values for a single bivariate data point but the variation of points as a global trend, (b) to consider not just a single variable but the correspondence of two variables, and (c) to balance prior beliefs with data-based observations. THE PROBLEM Covariation, in broad terms, concerns correspondence of variation. The nature of the covariation may be categorized according to the variation possible in the measure of each variable involved. For logical variables, which can be either True or False, the logical statement A = NOT(B) expresses logical covariation between A and B, since varying the value of A from True to False entails a corresponding variation in the value of B from False to True to maintain the equation as true. The equation y = 2x expresses numerical covariation between real-number variables x and y, since a variation in the value of either x or y entails a corresponding variation in the value of the other variable. Other polynomial and piecewise functions also express numerical covariation. In all of these cases, the values of the variables may