Lecturers in mathematics are inclined to accept with resignation that many students will be unable to appreciate any of the finer points associated with convergence of sequences and continuity of functions. They do not, however, question their own method of communication, yet it is that method and its effectiveness which are crucial to the recipients’ mathematical well‐being. This article considers these two areas of conceptual difficulty and concentrates on the problem of the transfer of expertise and understanding. The target group comprises those students for whom the elements of formality are appropriate.