Supporting a robust understanding of square root for middle school students

There is a need for middle school students to develop an understanding of square root as a functional relationship. Examining two continuously changing quantities with the square root relationship can promote students to engage in both covariational reasoning and correspondence reasoning. In this study, we conducted a small-scale, exploratory teaching experiment with one middle school student, Ivan. Ivan engaged in a task sequence examining the relation between two continuously changing distances, which had the potential to support the development of smooth covariational reasoning. We found that Ivan was able to examine the continuously changing quantities to reason covariationally, a meaningful accomplishment for a middle school student, and then leveraged his reasoning to develop a correspondence relation. We identify shifts in Ivan's covariational reasoning as he constructed an understanding of the square root relationship, and discuss how covariational and correspondence reasoning can mutually support one another to engender functional thinking.