I was asked recently by a first-year maths undergrad student if I could recommend any books on problem-solving, as they were hoping to develop their problem-solving skills. Asking around some maths communication colleagues has resulted in an impressive list of recommendations for books for maths undergraduate students, which I’m sharing here.

All of these should be available to buy from bookshops and are mostly in paperback – but many will also be available in university libraries, and it’s worth a check before shelling out for expensive copies!

Problem Solving

The classic text on mathematical thinking is George Pólya’s How to Solve It. Originally written in 1945, the book has been reprinted dozens of times, and includes some great advice on how to break down problems into steps or restate them more simply, make a plan for solving, and assess your work afterwards. Paperback £10.44

Fields Medalist Terence Tao’s Solving Mathematical Problems has lots of practical strategies for problem solving, accompanied by examples. Paperback £29.49

People also suggested Peter Eccles’ An introduction to mathematical reasoning : numbers, sets, and functions (paperback £36), and Techniques of problem solving by Steven G. Krantz (available to borrow from the Internet Archive).

Another classic of the genre is Keith Devlin’s Introduction to Mathematical Thinking – written to accompany a ‘transition course’ from school to university, it aims not to give a crash course in mathematical topics, but instead to get people thinking in more mathematical ways.

And on that subject, how could we forget friend of the site Kevin Houston’s How to think like a mathematician: a companion to undergraduate mathematics. Kevin’s website also has more information and solutions. Paperback £31

For some practice at problem-solving, Stephen Siklos’ Advanced Problems in Mathematics is published using OpenBook, and is available as in PDF or HTML format – containing many great examples of maths problems, along with comments and solutions. It’s particularly good for STEP Practice, but also for general problem-solving.

Proofs

More on the side of proofs, we had a recommendation for How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, by Daniel Solow, which gives an outline of proof methods and exercises to work through. Paperback £75.95

Other recommendations going more into proof included Lakatos’ Proofs and Refutations – published in four parts in The British Journal for the Philosophy of Science (requires institutional login). It’s written as a dialogue between teacher and students, and focusses on a single problem for the entire book, conjecturing, refuting with counterexamples, then studying the counterexample to systematically improve the next conjecture. It was actually part of his PhD thesis!

More General Advice

If you want to help someone prepare for studying maths at uni – or advise someone who’s thinking about doing that – Vicky Neale wrote an excellent book called Why Study Mathematics?, which gives an idea of what to expect from maths at uni, and some wonderful advice. It’s available in paperback, and Vicky’s website has some links to excerpts, and her interview on the Numberphile podcast which covers the same content (transcript). Paperback £12.99

Another recommendation in this area is Lara Alcock’s How to study for a mathematics degree (one of her many excellent books). Paperback £22.99, and available in all good university libraries.