As someone going through this process at the moment, I find it unlikely that a candidate who could naturally cope with the Oxbridge courses would struggle with the further maths content in isolation. For all its flaws, the course is largely a disjointed collection of fundamental problem solving methods and concepts which are reintroduced at University level anyway.

The exams themselves however are an entirely different matter, and I know of several capable students who have had their university prospects diminished by poor performance in the exam room - whether due to anxiety, poor 'exam technique' (less relevant for maths of course), or other factors.

Notably, Cambridge interviews a high proportion of candidates who apply for maths, which presumably enables them to take some of the emphasis away from the exam results on paper, and give much more consideration towards the logical thinking and problem solving skills useful for the Tripos.

It's important to note the distinction between breezing through course material and breezing through an exam.

I have taught plenty of students who possessed a deep understanding of mathematics but could not perform within the confines of a structured exam. They may be berated for performance or even lack of preparation, but given that problem-solving is the main criteria for admissions this surely reflects the limitation of the measuring tool.

Similarly, many students who breeze through A Level exams may have thought they'd mastered topics like calculus, but it turns out the exams only called on their computational fluency. The moment they are pushed conceptually and asked to formulate their thinking with the rigour that undergraduate mathematics, they come unstuck.

My Cambridge interview tested for this explicitly. They gave you 45 minutes to prepare answers to 8 'further maths' type questions, then you had to solve some creative problems at the end. These creative problems required knowledge of further maths concepts but were unusual so you needed time to think about them and develop a mathematical framework for the solution, this therefore tested that you could do the further maths questions easily and quickly and that you could solve new problems.

The 'creative' problem I had was this: A mouse walks along an elastic band at 2 meters per minute. The elastic band stretches by 1 kilometer every hour. How far along the band does the mouse get? If you got that right, then they gave you a stretch question where the band doubles in length every hour.

Am I wrong to think that the stretching of the band does not have anything to do with "how far along the band" the mouse will travel? Maybe it's my English as a second language, but my intuition is that the mouse still travels along the band as it expands, and you would probably need a point of reference, e.g. How far along the gap that the elastic band bridges to make the question more challenging.

The mouse is walking along the band, while the end of the band moves away, however the mouse is also being carried along by the expansion of the band. I think the best approach is to think in terms of the fractional coordinate of the mouse along the band, and work out its effective velocity in this coordinate.

The elastic band could expand in both directions or either. If I assume the band expands so fast that I remain at the same point with regards to a point of reference, have I not still moved by 2m along the band after a minute? I guess it's more about the semantics of how moving along the band applies.

Also how the expansion carries the mouse has to do with its movement. E.g. if it's galloping it would not carry it.

Correct, if you structure the problem as the proportion completed at time t relative to the proportion at time t-1, then you can sum in the limit as t goes to infinity to show surprisingly that he gets to the end under both scenarios.

I think you meant to say "if you got that right, then they gave you a stretch stretch question" :-)

My experience is that there are three levels of maths ability among people who breezed A-level, or at least passed it well enough to do university maths:

- fundamentally does not grok University-level maths at all;

- gets "ordinary" university maths (ie redbrick and "below"), but would not be able to cope with Oxbridge level;

- Oxbridge level

There are stark step-jumps between these three, and exam results are not a good indicator of which one an individual actually falls into. In particular, I believe you can be a good jobbing middle-level mathematician by learning the relevant stuff, providing you have the basic aptitude for it, but you need a real gift to be able to cope with the top level, and that's something that an exam format cannot elicit.

Speaking personally, I did an interview at Cambridge, which served the excellent function - for both parties - of determining that I was not in the top group.

Liked the tone of your post. Did not detect any bitterness. Does your mathematical background serve a use in your current career, which I assume is as a dev of some sort since you are on Hacker News?

Thanks. Dev: yes; maths useful: not so much, except I guess in the broader "mental training" sense.

What does "redbrick" mean in this context?

Mea culpa, UK cultural assumption: https://en.wikipedia.org/wiki/Red_brick_university

Totally a possibility. I know lots of people who are skilled problem solvers, but sometimes do poorly on exams because they simply aren't motivated enough. That was definitely the case for me entering college. I had work experience that proved I could do complicated computer science work, but I didn't have great exam grades, so I had a hard time getting into the department.

That's not to say I couldn't have done well on the tests, I just didn't care enough to study hard for them because I had more interesting things taking up my time.

Now, that's not to say there aren't problems with that philosophy too - but it's certainly one reason.