Students’ understanding of proportional, inverse proportional, and affine functions: two studies on the role of external representations

摘要

We investigated students’ understanding of proportional, inverse proportional, and affine functions and the way this understanding is affected by various external representations. In a first study, we focus on students’ability to model textual descriptions of situations with different kinds of representations of proportional, inverse proportional, and affine functions. Results highlight that students tend to confuse these models and that the representational mode has an impact on this confusion: Correct reasoning about a situation with 1 mathematical model can be facilitated in a particular representation, while the same representation is misleading for situations with another model. In a second study, we investigate students ’ability to link representations of proportional, inverse proportional, and affine functions to other representations of the same functions. Results indicate that students make most errors for decreasing functions. The number and nature of the errors also strongly depend on the kind of representational connection to be made. Both studies provide evidence for the strong impact of representations in students’ thinking about these different types of functions.