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.