Making connections: Compartmentalization in pre -calculus students' understanding of functions
Students develop knowledge constructs that they build into concepts through their experiences. Students demonstrate compartmentalization when they understand a construct or concept within one representation, but not another, or when they do not connect mathematically related ideas. For instance, a student may understand f(x) to mean plug x into the function within a symbolic representation, but the same student may understand f(x) to mean f times x within a tabular representation. A student with these understandings has a compartmentalized understanding of function notation. A two-month study was conducted with a class of pre-calculus students enrolled in a parochial high school. The class was observed and a subset of students (n = 7) were given a series of tasks in an interview setting in order to determine their understanding of functions and in particular periodicity within the three representations: equations, graphs, and tables. The researcher studied compartmentalization in the students' understanding. Three of the seven students showed compartmentalization. All three had a compartmentalized understanding of function notation within the tabular representation. In addition, two had compartmentalization within representations in their understanding of periodicity. Students with compartmentalization in their understanding, had the greatest difficulty in solving the interview tasks. Furthermore, those students who could not translate between representations had an automatic compartmentalization in their understanding and lacked flexibility in problem-solving. All seven of the students preferred the symbolic representation. The students used this representation overwhelmingly in their classwork and homework. Six of the seven students attempted to find equations for the functions in the interview tasks before trying any other solution strategy. However, only one student was able to solve the interview tasks in this representation. Some interesting conceptions of periodicity emerged in the students' understandings. The students used symmetry, familiarity, and continuity to determine whether a function was periodic. The students did not work from a conventional definition of period. Instead, they constructed their own definition of periodicity by generalizing sinusoids and other familiar functions. The generalizations that the students made were often inconsistent with the conventional definitions. These unconventional understandings imply that they need experiences with more than just sinusoids.
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