In this paper, students' cognitive construction of a schema were examined.
The Action-Process-Object-Schema [APOS] theoretical
perspective was used to examine this problem. The question involved
an extensive calculus graphing problem which was worked on by the
students as they explained their anwers to an interviewer. The focus
of this study was to determine the how the students' schema used to solve
this problem had evolved up to that point.
Data consisted of these extensive interviews with students who had
completed at least two semesters of calculus. The complexity
and the non-routine nature of the problem that students attempted to solve
required them to rely on everything they had learned on
graphing functions in calculus. In order to cogently describe
the student responses, we examined two important schema the students were
using. It naturally followed that the interaction of these two schema
was critical. Therefore, there was a two-dimensionality in what we
called their overall "Calculus Graphing Schema".
The calculus graphing problem studied required students to integrate
the properties of the graph of the function with each other, as well as
across contiguous intervals. The triad of schema development - intra, inter,
and trans-aptly describe the data with respect to two dimensions. One dimension
is the "Property Schema" and the other is the "Interval Schema".
Additionally, a number of problems were demonstrated by students consistently
throughout and these problems are discussed in some detail.
This paper is part of a continuing series of research studies of college
students' cognitive development of mathematical concepts by
members of a collaborative group of mathematics education researchers
called the Research in Undergraduate Mathematics education Community [RUMEC].