Lara Alcock - How to Study for a Mathematics Degree

HOW TO STUDY FOR A MATHEMATICS DEGREE

LARA ALCOCK

Mathematics Education Centre, Loughborough University

Great Clarendon Street, Oxford, OX2 6DP,

United Kingdom

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Lara Alcock 2013

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First Edition published in 2013

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Library of Congress Control Number: 2012940939

ISBN 9780199661329

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E very year, thousands of students go to university to study for single-or joint-honours mathematics degrees. Many of these students are extremely intelligent and hardworking. However, even the best struggle with the demands of making the transition to advanced mathematics. Some struggles are down to adjusting to independent study and to learning from lectures. Others, however, are more fundamental: the mathematics shifts in focus from calculation to proof, and students are thus expected to interact with it in different ways. These changes need not be mysteriousmathematics education research has revealed many insights into the adjustments that are necessarybut they are not obvious and they do need explaining.

This book aims to offer such explanation for a student audience, and it differs from those already aimed at similar audiences. It is not a popular mathematics book; it is less focused on mathematical curiosities or applications, and more focused on how to engage with academic content. It is not a generic study skills guide; it is focused on the challenges of coping with formal, abstract undergraduate mathematics. Most importantly, it is not a textbook. Many transition or bridging or foundations textbooks exist already and, while these do a good job of introducing new mathematical content and providing exercises for the reader, my view is that they still assume too much knowledge regarding the workings and values of abstract mathematics; a student who expects mathematics to come in the form of procedures to copy will not know how to interact with material presented via definitions, theorems and proofs. Indeed, research shows that such a student will likely ignore much of the explanatory text and focus disproportionately on the obviously symbolic parts and the exercises. This book aims to head off such problems by starting where the student is; it acknowledges existing skills, points out common experiences and expectations, and re-orients students so that they know what to look for in texts and lectures on abstract mathematics. It could thus be considered a universal prelude to undergraduate textbooks in general and to standard transition textbooks in particular.

Because this book is aimed at students, it is written in the style of a friendly, readable (though challenging and thought-provoking) self-help book. This means that mathematicians and other mathematics teachers will find the style considerably more narrative and conversational than is usual in mathematics books. In particular, they might find that some technical details that they would emphasize are glossed over when concepts are first introduced. I made a deliberate decision to take this approach, in order to avoid getting bogged down in detail at an early stage and to keep the focus on the large-scale changes that are needed for successful interpretation of undergraduate mathematics. Technical matters such as precise specification of set membership, of function domains, and so on are pointed out in footnotes and/or separated out for detailed discussion in the later chapters of .

To lead students to further consideration of such points, and to avoid replicating material that is laid down well elsewhere, I have included a further reading section at the end of each chapter. These lists of readings aim to be directive rather than exhaustive, and I hope that any student who is interested in mathematics will read widely from such material and thus benefit from the insights offered by a variety of experts.

This book would not have been possible without the investigations reported by the many researchers whose works appear in the references. My sincere thanks also to Keith Mansfield, Clare Charles and Viki Mortimer at Oxford University Press, to the reviewers of the original book proposal, and to the following colleagues, friends and students who were kind enough to give detailed and thoughtful feedback on earlier versions of this work: Nina Attridge, Thomas Bartsch, Gavin Brown, Lucy Cragg, Anthony Croft, Ant Edwards, Rob Howe, Matthew Inglis, Ian Jones, Anthony Kay, Nathalie Matthews, David Sirl, and Jack Tabeart. Thanks in particular to Matthew, who knew that I was intending to write and who gave me related books for my birthday in a successful attempt to get me started.

Finally, this book is dedicated to my teacher George Sutcliff, who allowed me to find out how well I could think.