The best books for studying undergraduate mathematics

Lara Alcock Author Of How to Study as a Mathematics Major
By Lara Alcock

The Books I Picked & Why

How to Read and Do Proofs

By Daniel Solow

Book cover of How to Read and Do Proofs

Why this book?

This book provides a systematic account of how to understand and structure mathematical proofs. Its approach is almost entirely syntactic, which is the opposite of how I naturally think – I tend to generate arguments based on examples, diagrams, and conceptual understanding. But that difference, for me, is precisely what makes this book so valuable. Solow gives a no-nonsense, practical, almost algorithmic approach to interpreting logical language and to tackling the associated reasoning. His book thereby provides the best answer I know of to the “How do I start?” problem so often encountered when students begin constructing proofs.  

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How to Think Like a Mathematician

By Kevin Houston

Book cover of How to Think Like a Mathematician

Why this book?

Many undergraduate mathematics books – even those aimed at new students – are dense, technical, and difficult to read at any sort of speed. This is a natural feature of books in a deductive science, but it can be very discouraging, even for dedicated students. Houston’s book covers many ideas useful at the transition to proof-based mathematics, and he has worked extensively and attentively with students at that stage. Consequently, his book maintains high mathematical integrity and has lots of useful exercises while also being an unusually friendly read.

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Mathematical Writing

By Franco Vivaldi

Book cover of Mathematical Writing

Why this book?

Mathematics requires accurate calculation, and students sometimes think that getting the right answer is enough. But mathematics is also about valid logical arguments, and the demand for clear communication increases through an undergraduate degree. Students, therefore, need to learn to write professionally, with attention to general issues like good grammar, and mathematics-specific issues like accuracy in notation, precision in logical language, and structure in extended arguments. Vivaldi’s book has a great many examples and exercises, and students could benefit from studying it systematically or from dipping into it occasionally and reflecting on small ways to improve.

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Understanding How We Learn: A Visual Guide

By Yana Weinstein, Megan Sumeracki, Oliver Caviglioli

Book cover of Understanding How We Learn: A Visual Guide

Why this book?

Research in cognitive psychology has revealed a lot about human learning and how to make it more effective. Most mathematics students – and indeed their professors – know very little about this research or how to apply it. Weinstein and Sumeracki’s book explains how psychologists generate evidence on learning, gives a basic account of human cognitive processing, explains some strategies for effective learning, and gives tips for applying them. It is not about mathematics and it certainly will not make advanced mathematics simple, but I think that we would all have an easier time if we were more aware of some common misunderstandings about learning and effective ways to improve it.  

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Getting Things Done: The Art of Stress-Free Productivity

By David Allen

Book cover of Getting Things Done: The Art of Stress-Free Productivity

Why this book?

Succeeding in a mathematics degree requires not only intelligence but also organization. Many students are not great at organization because they have comparatively little experience in taking responsibility for time management and because they are, after all, just people. This sometimes causes them a lot of stress. I think that the stress is largely avoidable, and Allen agrees: one of his main points is that stress comes from the nagging sense of important things not being done, so that it is useful to have both a grip on what is important and realistic plans for when important things will be done. His book was recommended to me by a much-respected mentor, and I now operate more-or-less exactly the system it recommends.

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