Differentiate yourself: preparing for Oxbridge Maths Interviews

Written by Aidan; Aidan is a maths teacher at a very academically challenging school in the UK and has prepared many students for Oxbridge interviews.

It’s an old interview horror story: the candidate walks into the room only to discover that the interviewer – a distinguished academic – is sitting reading the day’s newspaper. Its broad pages held aloft in front of the interviewer’s face, the candidate can’t even make eye contact … there’s no acknowledgement; no recognition; no instruction – least of all even a question. The only movement is a faint line of smoke coming off the interviewer’s cigarette, which sits smouldering in an ashtray next to a lighter on a small table alongside the professor’s armchair. So what should the candidate do? What’s the right response? 

Maybe this sort of thing happened in the 1950s – though, even then, I find it hard to believe – but the story is at least a quirky reminder of how terrifying the legendary Oxbridge interview can be. The punchline – if you can call it that – is that the candidate grabs the lighter from the table and sets fire to the newspaper … and is rewarded with a place at the university for their chutzpah. But much like many almost certainly apocryphal stories of bizarre interviews or plucky candidates writing flippant answers to essay questions – it’s not of much use to the student nowadays who wishes to prepare for an admissions interview. 

Fortunately, though Oxbridge interviews are still challenging high-pressure situations, the colleges are highly professional and usually very clear about what they are looking for. And in maths, fundamentally, they are asking two questions:

• • How good is this person already at maths?
  • How much potential do we think they would have to learn more if they came here?

Both of these questions are multidimensional – being ‘good at maths’ means all sorts of things, such as accuracy when solving problems, knowledge and use of terminology, the ability to structure a good written argument, or to articulate ideas clearly. Similarly, potential to learn has many different characteristics, not least the ability to take a hint, or a newly learned idea, and to apply it in an unfamiliar situation.

The interview experience may vary from college to college, but largely Oxbridge maths interviews follow a similar format. Candidates will meet with a couple of academics who will ask some maths questions. There may well be a warm-up: “Tell me a bit about yourself. Why are you interested in studying maths?”, but generally the maths will make up the vast majority of the interview. There may be a test immediately before the interview, in which case the interviewers may start by asking you about the questions you tackled – or the ones that you didn’t – and in most circumstances you can imagine that you’ll be given a fairly gentle question to begin with. Don’t be fooled, though – the interviewers are interested in finding the bounds of your knowledge, and are experts at doing so. They will typically aim to find an area where it is clear that you have reasonable capability and then will typically ask you something that they believe lies just a little bit beyond your current knowledge. For example – let’s imagine they start with a little bit of curve sketching. Some initial questions might be:

• • Sketch

,

showing clearly the coordinates of its stationary point.

• • Sketch

,

indicating the coordinates of a few of the stationary points on your graph.

These should be straightforward questions for an Oxbridge candidate. Then … bang! The interviewer says – “Great, so let’s see if you can use those answers to help you solve the following slightly harder problem”.

• • Explain why the graph

is periodic and find the coordinates of the first minimum point lying to the right of the -axis.

“Slightly harder problem”, you think to yourself. You quickly spot that since the graph is the sum of some functions that you know are periodic then it follows quite quickly that the graph itself should be periodic. And then you start tackling the real problem.

[Perhaps you might want to stop reading for fifteen minutes and see if you can solve this one yourself.]

All in all, this can be quite an unsettling experience, especially if you’re used to being one of the best in your maths class. Probably you usually think you know how to tackle a question; you’re used to questions that increase gradually in difficulty. This is where interview preparation and practice really comes into its own; learning how to deal with problems in this sort of situation comes from being repeatedly exposed to this style of problem – which can be hard to find in a textbook – and from working through these problems conversationally with someone who understands precisely where the limits of your knowledge fall. You’ll know that the preparation is at the right level when you feel like you are having to think hard. The problems might be exhausting but the sense of reward when you complete things will be considerable. 

So what do you do? What is it that all this practice will help you achieve?

The first step is to consider all the possible techniques you might know that are of relevance: perhaps you could use calculus? 

The sensible thing now is to share your ideas aloud with the interviewer. Especially now that an increasing number of interviews happen remotely, it’s very hard for the interviewer to learn anything about your thought processes if you don’t talk them through. So – after a few moments to compose your thoughts – there’s no harm in sharing all the different ideas you’ve got. You bounce through them, trying to reflect on the various things you might do (could it be about using compound angle formulae? Double angle formulae? Would differentiating help – or would the resulting equation be a mess? You said I should use my previous answers … how might they help?). The first function, a quadratic, was easy to minimize by completing the square. And your new function sort of looks vaguely quadratic… it has a squared trigonometric term, after all.

Perhaps if you could somehow complete the square here then you might be able to do something sensible!

“That’s a nice idea”, says the interviewer. “Perhaps you might explore that?”.

And you’re off. You recognise that sort of encouragement, and you’ve dealt with lots of quadratic trig problems before. A quick application of a familiar identity; you complete the square (ln ) for the resulting function and then before too long you have something which clearly has elements very similar to the original warm-up questions:

“So the minimum must be –5”, you confidently exclaim.

There’s a brief pause, and the interviewer asks, “OK – and what would this mean for the value of ?”

You pause, and see your mistake – or at least, your first mistake! But it doesn’t take you long to identify your error and fix it, and before too long you have found the correct solution and explained it to the interviewer.  

So what do we learn about preparing for interviews from this?

• • You need to practice and get used to academic discussion.
    Especially in online interviews, it’s really important that you are used to articulating your ideas, listening carefully to the hints or interventions that others may make, and are ready to review your work in the light of challenges. All of this can be practiced and becomes more comfortable as you work with someone who will challenge you to think hard, and it’s worth starting early. You can certainly start working in this way from the beginning of A-levels/IB. 
  • You need to be ready to tackle challenging problems
    I don’t mean that you need to come up with your own proof of Fermat’s Last Theorem. What you do need is to be ready to take your existing knowledge and to apply it to something a bit more challenging or involved. You need to be ready to take ideas from all the different things you have studied and apply them to one problem: questions will often span many different topic areas.
    Multi-topic questions can be hard to find for practice – Olympiads and maths challenges are certainly a good place to look, but they often focus on slightly niche areas. If you are preparing for interview, you’re probably also going to be looking at tests like the MAT, or STEP, and these can be good starting points. Someone with experience can also steer you towards the right sort of question.
  • You don’t need to try to get ahead on university maths – just be really good at school-level maths.
    They are expecting you to really understand the rules that you use and to be able to apply them thoroughly. Anything on the A-level syllabus is fair game, but you should absolutely make sure that you are really well practiced in key topics such as calculus, trigonometry, graph sketching, combinatorics and proof. 
  • They are interested in potential, not just prior attainment.
    You might not feel like they are trying to teach you in the interview – indeed, they will almost certainly only be asking you questions. But asking questions is a great way to teach and they will be working on the basis that asking you a small question as a stepping stone towards a larger, more difficult, question might enable you to then make a breakthrough elsewhere. So any time they give you a hint – in whatever form – ask yourself how you might use it to unlock something else. And to practice this skill, make sure you are spending time working with people or resources which teach by asking questions, rather than simply by providing solutions. 

The final bit of advice that I would give is that you have to enjoy the journey. What I’ve described here should sound enjoyable for you. Challenging, yes – frustrating, even, at times – but ultimately you should feel that being stretched in this way improves your thinking skills and broadens the range of questions where you can truly find joy in problem solving. If you start to enjoy tackling maths in this way, there’s a great chance that you’ll enjoy – and succeed at – university mathematics.