When time is an implicit variable: An investigation of students’ ways of understanding graphing tasks

ABSTRACT

In this study, we investigate students’ ways of understanding graphing tasks involving quantitative relationships in which time functions as an implicit variable. Through task-based interviews of students ages 14–16 in a summer mathematics program, we observe a variety of ways of understanding, including thematic or visual association, pointwise thinking, and reasoning parametrically about changes in the two variables to be graphed. We argue that, rather than comprising a hierarchy, these ways of understanding complement one another in helping students discover an invariant relationship between two dynamically varying quantities, and develop a graph of the relationship that captures this invariance. From these ways of understanding, we conjecture several mathematical meanings for graphing that may account for students’ behavior when graphing quantitative relationships.

Notes

¹ When classifying graphs, we consider a student’s response to be the graph of a function if one variable can be viewed as a function of the other variable, whether that variable is on the vertical or horizontal axis.

² We distinguish here between direction-of-change thinking as a mathematical meaning for contextual graphing and gross coordination of values in Thompson and Carlson (2017) framework for covariational reasoning. The latter refers to the mental action of forming an image of two quantities’ values varying together and need not correspond to a graphing action. While gross coordination of values may support direction-of-change thinking about a graph, an individual may engage in direction-of-change thinking by envisioning two quantities varying together or by reasoning intra-parametrically about two quantities that each vary with time or some other parameter quantity.