A link between students' discernment of variation in unidirectional change and their use of quantitative variational reasoning

In this design experiment study, we investigated the problem: What conditions might foster students' discernment of a critical aspect-variation in unidirectional change (e.g., discerning a "decreasing" increase)? At a public middle school in a large US city, we led a sequence of three days of whole class lessons, followed by task-based, clinical interviews with 14 seventh grade students (similar to 13 years old). Students interacted with researcher-developed dynamic computer environments, which linked filling polygon animations with dynamic graphs. We report results to demonstrate a range of students' work along with detailed analyses of student interview episodes. We found connections between students' predictions of graph type (linear, "mostly" linear, nonlinear), students' conceptions of attributes represented by a graph, and students' discernment of variation in unidirectional change. Students who discerned variation in unidirectional change also engaged in quantitative variational reasoning. To foster students' discernment of variation in unidirectional change, task designers should provide opportunities for students to conceive of attributes as capable of varying and possible to measure.