Generating different lesson designs and analyzing their effects: The impact of representations when discerning aspects of the derivative

This paper reports results that concern the way in which the design of instruction may influence students' opportunities to discern the relationship between a graph and its derivative graph. Two studies were conducted that together included 144 Swedish upper-secondary students who were enrolled in an introductory calculus course. In both studies, all students participated in a 120-min intervention. The first study used a qualitative approach and aimed at generating, and analyzing the outcomes of, three different lesson designs. The second study used a quantitative approach and aimed at testing the validity of the results from the first study. The results of the studies are compatible and suggest that teaching should initially focus exclusively on graphs and also include a variety of graphs. In contrast, using graphs in conjunction with formulas and/or using only graphs of polynomial functions decreased students' opportunities to discern graphical aspects of the derivative.