As Director of Studies at Newnham College, and through teaching many first year mathematicians, I experience quite a few students making the transition from school maths to university maths.
Mathematics at university is not the same as mathematics at school. Most people who have not experienced maths at university think of maths as calculation, as working things out following certain rules and recipes. There is the expectation that there is always a “right answer”, a very specific end result of a calculation. However, maths at university is not (just) like that. Yes, there are still calculations to be done, but one of the main aspects of mathematics is a logical way of thinking and proving general results. Mathematics students learn to work and think on a more abstract level. They learn to make rigorous arguments leading from some very specific assumptions to a general result. They learn to question these assumptions and to find connections between different mathematical situations. This often has nothing to do with calculation at all.
Meeting such a different flavour of mathematics for the first time is hard. Of course some students will have met glimpses of this way of working before, either through some first proof methods in school or in exploring mathematics for themselves through books, public lectures or competitions. But still many people meet this very rigorous way of proving results for the first time in their mathematics degree.
The question is: did they know what they were signing up for when they chose to study mathematics? We hope of course that most people do. Many students really enjoy this aspect of mathematics and relish the opportunities to develop their logical thinking in such ways. But it can happen that a student who is very good at calculations and the kind of things they have met in school mathematics ends up not liking this business of proving general results. Fortunately, mathematics is a very broad subject. In applied mathematics, students do still need to calculate. There is also some amount of theory behind which ensures that certain calculations make sense and lead to the desired results. In the first few years of university, maths students will usually meet so-called methods courses where they learn to apply certain calculational techniques, and (at least in Cambridge) there are also physics courses such as Quantum Mechanics or Electromagnetism, where mathematics is used in the context of a physical theory. In such courses, the emphasis is usually not so much on proof, and a good proportion of students prefer this side of mathematics.
Mathematics also crops up in many other subject areas. Physics is not really possible without the use of quite a lot of different mathematics, and physicists need to develop mathematical skills as well as physical intuition and understanding. In the first few years of a physics degree, mathematics is a tool which helps to solve physical problems; physics students have to be able to apply calculational techniques and understand what physical processes are being described by the mathematics, but they are usually expected to trust that the techniques they are taught do indeed work, without needing to investigate the mathematical theory behind them for themselves. Engineering crucially relies on mathematical calculations as well; we all need engineers to be able enough mathematicians to work out for example how much weight a bridge can take before it collapses, and so to build bridges that won’t collapse under a reasonable weight of vehicles and people that are using them. Economics is another area where a lot of mathematics is needed, and many economics students find this the hardest part of their course.
If you are good at maths at school, does that necessarily mean you want to study maths at university? If you enjoy the way of thinking that I described above, proving general results and finding out connections between different mathematical situations, then yes, you will probably like a mathematics course. But perhaps your aptitude in mathematics is more related to solving physical problems, in which case you might really enjoy a physics or engineering course at university.
At Newnham College, we are going to run a residential where we give around 30 girls from maintained-sector schools and colleges who are good at mathematics the opportunity to explore these different possibilities, so they can make a more informed choice about what they might want to study at university. Newnham is an all-female College, and apart from learning about what might await you in a mathematics or physics or engineering course, the participants will also get to know the College and what it has to offer. We will also include admissions information for applications to Cambridge, and some fun activities to get to know Cambridge. For more information on this Joan Clarke Maths Residential at Newnham, please email firstname.lastname@example.org, with Joan Clarke Maths Residential as the title of your email. Details of how to apply will then be sent by the end of May.