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.

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The use of short assessment tasks can provide valuable information about undergraduates' knowledge and understanding. However, it is known that there are gender-related di erences in performance on certain types of objective tests, both among school pupils and university undergraduates. This article focuses on undergraduate learning, using a mixture of tasks presented in either an open or closed form. Although the success rate in performing the given tasks seems to depend on the degree of openendedness inherent, more unexpected is the consistent di erence in achievement between men and women students.

For many sixth-formers and undergraduates, the method of proof-by-induction seems to resemble nothing so much as a confidence trick. They appear to have little or no appreciation of the logical foundation of the method; frequently, they have been given a mysterious recipe which might just as easily be “eye of newt and toe of frog”. They slavishly follow the rules given and so it is little wonder that they are confused when, having been cajoled incessantly not to beg the question, they are now seemingly allowed to assume the answer. However, it is not my intention in this article to present yet another justification for the principle of mathematical induction, important as that might be; in any case, perfectly good articles on the subject have already appeared in the Gazette [1,2].

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