You wouldn't actually get the student to memorise a canned response, that would be silly. But you have to assume that the candidates they are competing against are as academically capable as they are (i.e top UMS marks also).
Applying to Oxbridge? You have to stand out, simple as that.
Do you have potential? There are certain key things that are common in maths students that have great potential. And these are what Cambridge and Oxford they will be looking for. And these are easy to spot.
How would someone who is extremely interested in maths behave? Chances are he/she would have an insatiable appetite for anything maths, and would be reading around the subject and learning more than what is taught, spending majority of your spare time thinking about and doing maths.
What interesting maths can you talk about?
Do you understand the Monty Hall problem and the maths for it?
If they ask "Why Maths?" or "Why do you want to study this?" etc Do you talk about how your amazed at all the myriad of ways to prove
[br]ζ(2)=∑n=1∞n21=6π2[br], which span every area of maths? How you can barely understand one of the proofs you've read, but are desperate to understand every one of them all??
Can you talk about how you would one day want to contribute to the human understanding of maths?
Do you have an overwhelming desire to learn maths?
Can you talk for hours about maths, how interesting it is, what it all means and it's importance, well beyond the typical student?
Can you convey that you are self-driven? Low maintenance? Put in a lot of effort? Do you use perfect mathematical notation? Do you do LOTS of maths questions?
And bear in mind, all the top independent schools have Oxbridge preparation programs for their students, and I think it's safe to say they know what they are doing.