Mathematics at university is rather different from mathematics at school. There are many different mathematics degree courses available, and there is a great deal of variety within mathematics itself, but university mathematics courses tend to have the following things in common.
1. There is a very strong emphasis on proof, rigour, clarity and accuracy.
2. Questions tend to be much longer and involve multiple steps. Moreover, you won't typically be able to simply look up the method of solution or proof in the lecture notes and change the numbers: you will need to be a confident, independent problem solver.
3. Some topics will significantly extend the results learned at school, some topics will re-present material from school from a rigorous standpoint, and some will be entirely new. Regardless of the topic, new conceptual ideas are encountered frequently in mathematics.
4. Applications of mathematics are complex, requiring many advanced skills of estimation, approximation and the use of techniques from across mathematics. (There are very high levels of mathematics required in subjects other than mathematics too. In particular, physics and engineering courses make extensive use of post-school mathematics.)
For these reasons, there is a very real transition phase between school and university. For the mathematically minded, this is very stimulating and exciting, but it is good to be as prepared as possible.
Fortunately, it is possible to prepare yourself for the new style of thinking that will be required. Below is a list of core problems which will help all students thinking about studying mathematics at university. Similar problems are linked in groups, in order of difficulty. You can find more problems and helpful resources on The NRICH Website and Underground Maths, a fantastic resource for students and teachers.
Preparing for STEP
STEP Mathematics is an additional examination, taken at the end of Y13, which forms part of conditional offers to mathematics applicants by Cambridge and some other universities.
The free online STEP Support Programme, developed at the University of Cambridge, is designed to help you develop your advanced problem-solving skills and prepare for sitting STEP Mathematics examinations.
The STEP Support Programme includes a course of online modules for individual additional study, designed for you to work on from Y12 onwards. Each module consists of a structured mathematical problem-solving assignments, leading to work on STEP questions, with support material and partial solutions to help you assess your progress. You can discuss your work, or ask for additional help and hints, on the accompanying discussion forum, which is mentored by Cambridge staff and students who have taken STEP themselves.
There is very little in the school curriculum which has proof as a focus, whereas proof will take centre stage in all of university mathematics.
These online resources will introduce you to some perhaps familiar (if you study Economics at school) and then unfamiliar concepts in microeconomic
Since university mathematics is founded on proof, developing your ability to construct and analyse logical arguments is critical.