The books on this page are all excellent preparation material for A-level students thinking of studying Mathematics at university. As well as the books below I’d also highly recommend the (free!) notes How do Undergraduates do Mathematics. These are subtitled ‘A guide to studying mathematics at Oxford University’, but are great for mathematics at any university and for a glimpse into it for anyone thinking of applying too.
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Why Study Mathematics? Vicky Neale
Gives a great overview of the reasons why you might want to study maths at university and how it will be different to the maths you’ve done at school, including a brief description of lots of different areas of advanced maths. Written by an Oxford lecturer, but with a broad view of the different maths courses available at other universities, it contains a lot of great advice and things to think about. A fairly quick read and mostly words but also has a few nice mathematical examples for inspiration.
Towards Higher Mathematics – Richard Earl
A true transition textbook that has filled a real gap in the market – written in the language of A-level maths, exploring many key concepts of undergraduate maths. Written by Oxford’s Director of Undergraduate Maths, the book arose out of a course aimed at incoming undergraduates without Further Maths (but very suitable for further mathematicians too!) and includes 1757 exercises to try your hand at too!
How to Think Like a Mathematician – Kevin Houston
Addresses some of the topics that make the transition to undergraduate maths challenging. In particular how to think logically and to write clearly, and how to adapt to the Definition/Theorem/Proof style of mathematics that makes undergraduate study quite different from A-level.
How to Study for a Mathematics Degree / Think About Analysis – Lara Alcock
How to Study for a Mathematics Degree addresses styles of proof, and how to read and write undergraduate mathematics, and about a third of the book is about the often neglected area of study skills and how to get the most out of the teaching experience at university.
Limits, Limits Everywhere – David Applebaum
An excellent, entertaining introduction to the rigorous treatment of sequences and series, and a great warm up to mathematical analysis, one of the hardest parts of undergraduate mathematics to adapt to from A-level.
An introduction to formal logic, using grammatical and linguistic examples as much as mathematical ones. Particularly recommended for students thinking of a course like Maths + Philosophy. Helpful for maths, but also for any other subject that requires a logical analysis of written work (eg law).
Mathematical Beauty: What is Mathematical Beauty and Can Anyone Experience It? Daniel Peracy
Mathematics has turned out to be an incredibly practical and useful subject. This drives a lot of attention and funding towards the subject, but meet any professional mathematician and they will also talk to you about the beauty of mathematical proof, and as mathematics as an art as much as a science. This book explores what it means for mathematics to be beautiful and how to appreciate mathematics as well as much as to apply it.