A dynamic approach to functions and their graphs: a study of students' discourse from a commognitive perspective

The purpose of this research is to study how a set of mathematical activities designed within a Dynamic Interactive Environment (DIE) influence a class of high school students’ learning of functions. The study builds on findings of previous research on students’ difficulties in working with real functions; these include difficulties in managing covariation of the two variables involved, that is the relationship between variations of the independent variable and variations of the dependent variable, as well as difficulties in dealing with the graphs of functions. The study involved designing a sequence of activities in a DIE that introduce functions through a dynamic approach, and it takes a discursive approach in analyzing students’ interactions in this setting over a nine-week period. Data collected included: audio and video recordings of what happened in the classroom, video recordings of what happened on the students’ computer screens, and the students' work on paper that was produced during the lessons and during a set of subsequent interviews. The analyses of these data lead to a model describing how dragging mediates students’ discourse on functions; moreover, they shed light onto important features characterizing such discourse on functions and their graphs. The study has implications for the design of activities based on the use of DIEs for introducing real functions, highlighting the covariational aspect behind the functional dependency between the two variables.