Do third‐year mathematics undergraduates know what they are supposed to know?

The paper describes a test, covering 155 third‐year undergraduates in 15 different institutions, to examine the extent to which certain core first‐year material is retained and understood. The topics considered explore ideas such as the definition of differentiability, the principle of mathematical induction, understanding of basis and dimension, and perceptions about the real numbers. Also included are a question in mechanics and one on elementary aspects of group theory. As well as recording the students’ results (according to the criteria set out below), the paper includes an; lysis of the answers given by the students to each question and also some of their comments, as recorded on their scripts. Their misconceptions indicate that the foundations laid in the first year, on which their subsequent knowledge is built, is often very flimsy. Data collected include information on the final degree award of most of the test group. It is found that even among those students who subsequently achieved good degrees, the retention of first‐year material is demonstrably weak and suggests some cause for concern among those who teach them.