Look Forward to University

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.

Problem	Why do this?	Style
Picture this	This problem will introduce you to an area of university-level mathematics. It's also a great way of working on problem solving and proof. Challenge yourself to find interesting questions to explore using the interactivity -- what conjectures can you make and prove/disprove?	IR, PS, NC
IFFY logic Direct logic Proof Sorter Mind your Ps and Qs	You will find that proof is absolutely central to much of university-level mathematics. These related problems will get you thinking hard about proof and clarity of thought in mathematics	IR
The clue is in the question	First-year mathematics undergraduates meet the idea of studying a topic axiomatically: we agree on a set of rules (`axioms'), and then see what we can prove about objects that satisfy those rules. This problem gives a great taste of this.	IR, PS, NC
What is a group?	Group theory is one of the key topics introduced in university mathematics -- it plays a vital role in pure mathematics, but also in areas such as theoretical physics. This problem will introduce you to the idea of a group.	IR, PS, NC
Take three from five Few and far between	Simply because it's a great problem! You'll need to experiment, to conjecture, and to prove. When you can solve the problem as stated, can you generalise?	PS
Taking trigonometry series-ly	This problem introduces you to different ways of representing functions that are common at university. They also provide the opportunity to familiarise yourself with graphics and calculation packages.	NC, ESC
A long time at the till	This problem requires you to study two lengthy solutions and decide which approach would be more useful when solving similar problems. Working through this problem (which makes use of elementary topics in number theory) will help to develop the stamina required to make sense of lecture notes and lengthy problems	IR, ESC
The not-so-simple pendulum 1 The not-so-simple pendulum 2	This pair of problems show how simple ideas from calculus and mathematical modelling encountered at school can be extended in various clever ways to solve more difficult problems.	NC, ESC, MA
Dodgy proofs	This is a set of incorrect proofs of seemingly absurd statements, such as 1=0. Trying to pin down the precise points where the proofs break down will give many insights into proof and will help you to develop your mathematical reasoning.	IR
Trig reps	This problem will show that there are different ways of representing the same mathematical concepts. You will realise that some representations are easier to use than others in different scenarios. Moreover, particular representations often given simple ways to generalise the underlying concepts.	ESC
Calculus Countdown Operating machines	This pair of problems will allow you to see concepts in calculus in a different light, using the language of 'differential operators'.	NC, ESC
Phase space Predator-prey systems	These problems will allow you to see differential equations in a different light and will help you to develop the skills needed to relate a differential equation to a physical scenario.	NC, ESC, MA