This article is concerned with cognitive aspects of students' struggles in situations in which familiar concepts are reconsidered in a new mathematical domain. Examples of such cross-curricular concepts are divisibility in the domain of integers and in the domain of polynomials, multiplication in the domain of numbers and in the domain of vectors, and roots in the domain of reals and in the domain of complex numbers. The article introduces a polysemous approach for structuring students' concept images in these situations. Post-exchanges from an online forum and excerpts from an interview were analyzed for illustrating the potential of the approach for indicating possible sources of students' misconceptions and meta-ways of thinking that might make students aware of their mistakes.