Refining Progressions and Tasks: A Case Study of Design Cycles to Support Covariational Reasoning

Design-based research is a common tool mathematics educators use to study student learning and to generate high-level learning progressions and sequences of concrete mathematical tasks through iterative research cycles. There is a need for more transparent accounts of how researchers make decisions during the generation of such progressions and tasks. We address this need by describing the results of a case study of our design decisions, leveraging the constructs of local instruction theories and hypothetical learning trajectories to frame our decisions to promote students’ quantitative and covariational reasoning. We describe four considerations that influenced our re-design both of mathematical tasks and of learning progressions to support students’ covariational reasoning across seven teaching cycles with middle school students in the U.S. (ages 12-14). The four considerations that repeatedly influenced our task design decisions between cycles are: 1) supporting student thinking towards our goals, 2) eliciting student thinking, 3) keeping instruction efficient, and 4) exploring new possibilities. We discuss the importance of these considerations in our own work. We also highlight ways this work informs research on students’ quantitative and covariational reasoning and provide implications for task design. Through this report, we intend to provide an account that examines the task design process with reflexivity in a way that is useful for other researchers.