It has been common in recent years, if not entirely obligatory, for your President to address you on this occasion about the difficulties and opportunities he has found in the practice of teaching, and this year will be no exception. But following the excellent example of all past Presidents, as far back as I have investigated, I intend to confine my address to that part of the activity about which I am practically conversant. So while I hope that the general ideas and conclusions of my talk will be enjoyable and possibly useful to everyone here, I shall deal in detail only with the narrow problem of first-year undergraduates. I say simply ‘problem’ because I believe that the various different questions (the ‘interface’, the content of first-year pure/applied courses, how should the sixth form teacher best prepare the future university entrant,…) are all aspects of a single problem. For the sake of precision I shall deal with only one aspect of this problem, as it arises in analysis. Knowing me, you might have expected rather that I would have discussed the difficulties of teaching mechanics, or of persuading young people of 18 plus (although many of them have been brought up since five years of age on a practical approach to mathematics!) that there is any possibility at all of making an application of mathematics in the external world. My depression about that aspect is too strong, and my involvement in it is probably too deep, for me to see it clearly. But my experiences supervising small groups of students who are making their first genuine acquaintance with the real number field suggest to me the possibility of improving matters.