Maths at university
School mathematics is mostly about how to solve particular problems. For example, examinations usually consist of questions which are solved by selecting a method from a list of methods you have been taught.
This approach changes when you study mathematics at university: when you go into the world with a degree which says you are a mathematician, you may be asked questions with no obvious method of solution. For example, here at Sheffield there are mathematicians developing ways of answering the following questions:
- How can you tell whether a new treatment for a disease offers value for money when the very ideas of value and cost are uncertain?
- How can you investigate the internal structure of the sun when you see only its surface - and from a very long way off?
- How can you relate particle trajectories in space-time to the various ways you can knot pieces of string?
To work on such problems you need to know how mathematicians have tried to answer similar questions in the past - sometimes successfully, sometimes not!
You need to know methods which might give an approximate answer good enough for practical purposes because many real-life problems cannot be given definitive answers! And perhaps, above all, you need to be able to give a logical argument which guarantees that your conclusions follow from the information you began with. These conclusions may only be approximate, so your argument should tell you how good your approximation might be.
The main point here is that mathematics not only produces results, but produces results which can be relied upon!