English The concept of a mathematical function is applied widely in science to describe phenomena in which time, frequency, distance, temperature, and other continuous variables depend on one another. Two tasks were designed to test students' conceptualization of such relations. Each task involved one independent and one dependent variable in a real-world context (bacterial growth, spacecraft design). Information about the function was provided in the form of a table of paired values that exhibited a clear non-linearity. The almost 400 subjects, ranging in age from 11 to 18 years, were required to make interpolations between the given values and to explain their procedure. A brief demonstration of graphical curvilinear interpolation was given between the presentation of the two tasks. Student responses were classified into four categories according to the method of interpolation: curvilinear, combined curvi- and rectilinear, rectilinear, and intuitive (estimates, guesses, unsystematic or erroneous calculations). Most of the youngest subjects used the intuitive approach, while most of the older subjects used the rectilinear approach (either in the form of arithmetic averaging or straight lines on a graph). Only a small percentage of the subjects used curvilinear interpolation, considered to be the most appropriate procedure. The numbers of students using a systematic interpolation procedure was increased modestly by the demonstration. Interviews of some students revealed that many were imitating procedures they had seen in their classes, but they did not understand the reasons behind these procedures.
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