preparing for university maths
The links below are designed primariy for those who are planning to study maths, computer science or physics at university. There's a mixture of items - some are designed to help with the transition to the higher expectations of rigour and formality that are required at this level; others aim to introduce you to some of the content you will meet on university maths and physics courses. I would recommend sticking to the order below; jumping straight in to the later courses will be daunting! Don't worry if you find some of the material hard - it is sophisticated, but seeing some of the ideas for a first time will pay dividends when you're on a university course.
There's also a reading list afterwards - do have a look at some of the books.
Nrich has a couple of packages of problems and notes for preparing for university - these are both useful starting points:
This Introduction to University Maths from Oxford University aims to establish some of the basic language and notation of university mathematics, and formalise the idea of mathematical proof.
On similar lines, this OU short course looks at rigorously exploring the number system - the backbone of formal work on calculus, known as Analysis. For many the first university analysis course is a real shock to the system; see also some of the reading below.
If you want to hone your higher-level problem solving skills, try some STEP questions.
The courses below look at some formal linear algebra, a key component of first year maths programmes and building on the work you've already done on matrices and vectors:
University of Texas at Austin's Linear Algebra - Foundations to Frontiers course.
MIT's linear algebra course
This course from MIT is designed for computer scientists, but contains a lot of useful material for all maths and physics students, and covers a lot of analysis and discrete mathematics.
Stewart, I. and Tall, D. The Foundations of Mathematics. Oxford, second edition 2015. An excellent introduction to the way maths is taught at university. Read this and be ready for your first analysis course.
Stewart, I. Concepts of Modern Mathematics. Pelican, 1975; Dover reprint, 2012. A great overview (easy-going but rigorous) of some of the 'modern' ideas in mathematics, from groups to analysis to topology to computing.
Alcock, Lara, How to Study for a Mathematics Degree. Oxford, 2012.
Courant, R. and Robbins, H., What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford, 1996. A brilliant book, containing an awful lot of material on higher level mathematics. Great stuff - I wish that I had discovered it before or during my own maths degree.
Alcock, Lara, How to Think About Analysis. Oxford, 2014. Does what it says on the tin: an introduction to formal calculus.
Smith, G. C. Introductory Mathematics: Algebra and Analysis. Springer-Verlag, 1998. Covering the key concepts from first year Pure Maths courses.
Burkill, J. C. A First Course in Mathematical Analysis. Cambridge, 1978. The course text for Analysis I when I was a student. Concise and thorough.
Körner, T. W. The Pleasures of Counting. Cambridge, 1996. Excellent. Lots of examples of mathematics applied to real-life situations. Gives a good insight into what mathematics at a higher level is like, without being too technical.
Polya, George, How to Solve it. Penguin, 1990. A classic text on the art of problem solving.
The books above are all useful for 'making the transition' to university mathematics. Below is a more eclectic selection, but all worth dipping in to.
Davis, Donald, The Nature and Power of Mathematics. Dover, 2004. A superb book, covering interesting ideas from the world of mathematics. Not much except basic algebra is required, but topics are treated in a rigorous fashion. Highly recommended.
Wells, David, You are a Mathematician. Penguin, 1995. Lots of examples and problems to solve.
Acheson, David, 1089 and All That - A Journey into Mathematics. Oxford, 2002. A brief introduction to some advanced topics and what the study of mathematics is all about.
Gowers, Timothy, Mathematics: A Very Short Introduction. Oxford, 2002. An overview of mathematics at higher level. Not too technical.
Dunham, William, Journey Through Genius: The Great Theorems of Mathematics. Wiley, 1990. An excellent book. Takes major theorems from the history of mathematics and gives details of their proofs.
Ball, Keith, Strange Curves, Counting Rabbits and other Mathematical Explorations. Princeton, 2003. Investigations of various mathematical topics beyond the A level syllabus, each with explanations, exercises and suggestions for further reading.
Nelsen, Roger, Proofs without Words: Exercises in Visual Thinking, 3 volumes. MAA, 1997, 2000 and 2016. Exactly what the title suggests: often beautiful visual proofs of theorems from various mathematical disciplines.
Dunham, William, Euler: The Master of Us All. MAA, 1999. Chapters on various aspects of Euler's work. Each topic is explored in detail, with plenty of mathematics to get one's teeth into.