A Transition to Advanced Mathematics: A Survey Course

William Johnston and Alex McAllister
OXFORD UNIVERSITY PRESS USA 2009, 768 PAGES
PRICE (HARDBACK) £50 ISBN 978-0-195-31076-4

This text is intended to aid students in the move from a calculational, or operational, approach to mathematics to a more conceptual, proof based approach, and also to provide a broad overview of undergraduate mathematics. Chapters cover, in turn, the basics of mathematical logic, an introduction to abstract algebra via group theory, number theory, real analysis, probability and statistics, graph theory and complex analysis. Each of these topics is introduced clearly, with a wealth of examples and exercises, ranging from straightforward computations through conceptual questions forcing the reader to show an understanding of the definitions and statements, to problems requiring the construction of rigorous proofs. The exposition also includes some history of mathematics, placing it into a useful context. Each chapter concludes with a large final collection of exercises and notes providing additional contextual material and pointers to further reading on the topics.

I found the selection of material interesting and stimulating, in both the selection of general topics and the particular items chosen within each topic. It would provide useful background or supplementary reading for a student progressing from A-level to undergraduate study, where the new emphasis on proof and understanding as opposed to computation can prove something of an obstacle to many. Although survey courses of this nature are not common in the British undergraduate system, the book nevertheless contains material that would be useful as background, introductory, or supplementary reading for many courses that a mathematics student would undertake.

The book certainly has a place in any undergraduate mathematics teaching collection; and although most undergraduates will probably prefer to target their spending on texts which closely match a particular course, it would also be a highly suitable gift for a student who is just about to start, or has just started studying a mathematics degree.