The best books for studying undergraduate mathematics

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

Who am I?

I am a Reader in the Mathematics Education Centre at Loughborough University in the UK. I have always loved mathematics and, when I became a PhD student and started teaching, I realized that how people think about mathematics is fascinating too. I am particularly interested in demystifying the transition to proof-based undergraduate mathematics. I believe that much of effective learning is not about inherent genius but about understanding how theoretical mathematics works and what research tells us about good study strategies. That is what these books, collectively, are about.


I wrote...

How to Study as a Mathematics Major

By Lara Alcock,

Book cover of How to Study as a Mathematics Major

What is my book about?

My book is about the knowledge and skills needed to succeed in undergraduate mathematics. I wrote it because research in mathematics education has revealed a lot about what students struggle with when mathematics shifts in focus from calculations to definitions, theorems and proofs. But the information was all in articles for other researchers – no-one had produced a version suitable for students. I thought that someone should, because mathematics students are intelligent and perfectly capable of understanding the relevant ideas. My book has two parts: the first is about how mathematics changes at the transition to proof, and the second is about independent study skills for mathematics. How to Study as a Mathematics Major is the North American version – the equivalent for UK-like systems is How to Study for a Mathematics Degree.

The books I picked & why

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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.  

How to Read and Do Proofs

By Daniel Solow,

Why should I read it?

1 author picked How to Read and Do Proofs as one of their favorite books, and they share why you should read it.

What is this book about?

This text makes a great supplement and provides a systematic approach for teaching undergraduate and graduate students how to read, understand, think about, and do proofs. The approach is to categorize, identify, and explain (at the student's level) the various techniques that are used repeatedly in all proofs, regardless of the subject in which the proofs arise. How to Read and Do Proofs also explains when each technique is likely to be used, based on certain key words that appear in the problem under consideration. Doing so enables students to choose a technique consciously, based on the form of the…


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.

How to Think Like a Mathematician

By Kevin Houston,

Why should I read it?

1 author picked How to Think Like a Mathematician as one of their favorite books, and they share why you should read it.

What is this book about?

Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many…


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.

Mathematical Writing

By Franco Vivaldi,

Why should I read it?

1 author picked Mathematical Writing as one of their favorite books, and they share why you should read it.

What is this book about?

This book teaches the art of writing mathematics, an essential -and difficult- skill for any mathematics student.

The book begins with an informal introduction on basic writing principles and a review of the essential dictionary for mathematics. Writing techniques are developed gradually, from the small to the large: words, phrases, sentences, paragraphs, to end with short compositions. These may represent the introduction of a concept, the abstract of a presentation or the proof of a theorem. Along the way the student will learn how to establish a coherent notation, mix words and symbols effectively, write neat formulae, and structure a…


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.  

Understanding How We Learn: A Visual Guide

By Yana Weinstein, Megan Sumeracki, Oliver Caviglioli

Why should I read it?

1 author picked Understanding How We Learn as one of their favorite books, and they share why you should read it.

What is this book about?

Educational practice does not, for the most part, rely on research findings. Instead, there's a preference for relying on our intuitions about what's best for learning. But relying on intuition may be a bad idea for teachers and learners alike.

This accessible guide helps teachers to integrate effective, research-backed strategies for learning into their classroom practice. The book explores exactly what constitutes good evidence for effective learning and teaching strategies, how to make evidence-based judgments instead of relying on intuition, and how to apply findings from cognitive psychology directly to the classroom.

Including real-life examples and case studies, FAQs, and…


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.

Getting Things Done: The Art of Stress-Free Productivity

By David Allen,

Why should I read it?

5 authors picked Getting Things Done as one of their favorite books, and they share why you should read it.

What is this book about?

The book Lifehack calls "The Bible of business and personal productivity."

"A completely revised and updated edition of the blockbuster bestseller from 'the personal productivity guru'"-Fast Company

Since it was first published almost fifteen years ago, David Allen's Getting Things Done has become one of the most influential business books of its era, and the ultimate book on personal organization. "GTD" is now shorthand for an entire way of approaching professional and personal tasks, and has spawned an entire culture of websites, organizational tools, seminars, and offshoots.

Allen has rewritten the book from start to finish, tweaking his classic text…


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