From verbal descriptions to graphic representations: Stability and change in students' alternative conceptions

The present study investigated students' conceptions and misconceptions relating to the construction of graphs. Participants were 92 eighth-grade students randomly selected from two schools. Students were tested before and after being exposed to formal instruction on graphing. Qualitative analysis of students' responses identified three main kinds of alternative conceptions: (a) constructing an entire graph as one single point; (b) constructing a series of graphs, each representing one factor from the relevant data; and (c) conserving the form of an increasing function under all conditions. In addition, the following kinds of errors were displayed by less than 10% of the subjects: conceiving a generalized, stereotypic idea of a graph, using arrows or stairs to represent the direction of the covariation, and connecting the ticks on the axes by lines or curves. Quantitative analyses of the data indicated that overall students did not enter the learning situation as a tabula rasa. On the pretest, about a quarter of the students constructed correctly graphs representing increasing, constant, curvilinear, and decreasing functions, and many more students represented correctly at least one kind of function. Further analyses showed the stability and change in students' alternative conceptions after students were exposed to formal instruction about graphing. The theoretical and practical implications of the findings are discussed.