# Tips for preparing for an Oxbridge undergraduate maths interview

*First published September 12, 2024*

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```education

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```admissions

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From my experience as an interviewer, some advice for students preparing for an undergraduate admissions interview in maths at Oxford or Cambridge.

Over time, I intend to occasionally update this post with new resources and suggestions.

### What’s the structure of the interview?

Most interviews are about 30 minutes in length. The contents of the interview will be almost entirely mathematical, with maybe some softer questions at the start to serve as a warm-up.

From the point of view of the college, the interview is a chance for them to see first-hand how you think when faced with a challenging and unseen problem. For the most part, the interviewers will sit back and observe your working, but will occasionally step in and provide suggestions if they feel it could be helpful to you.

Since the pandemic, almost all interviews at Oxford are currently held online. At Cambridge, some colleges are once again holding some interviews in-person. See here for details.

### Where to find interview practice questions?

Finding practice questions is probably one of the trickiest parts of preparing for an interview. There are many websites out there which present huge lists of ‘practice questions’. Some of these can be helpful, though my opinion is that many of these practice questions are, quite frankly, a bit rubbish. I suggest also looking at the following resources.

#### 1. MAT papers

Questions from the Maths Admissions Test (MAT) are a good source of practice. If there are any past papers questions which you have not seen before, then take the time to imagine they’re interview questions and answer them accordingly.

MAT past papers are available here.

#### 2. TMUA papers

Similar to the MAT, the TMUA can also serve as a good source of practice.

Note that there are two TMUA papers. I specifically recommend **Paper 1**, as they can be very similar to MAT questions. Paper 2 covers more specific knowledge related to logic and proofs which is less relevant to the interview (though still interesting).

TMUA past papers can be found here.

#### 3. STEP (1) past papers

It is also worth looking at some STEP papers for practice problems, though some care is advised here. Remember, STEP stands for ‘Sixth Term Examination Paper’, and is an exam intended to be sat alongside your final A-level exams. Therefore, the prerequisite knowledge for some STEP questions is much higher than a standard interview question.

There are three different STEP exams, STEP 1, 2, and 3. Note that STEP 1 is now defunct and not used. However, I think that the questions in STEP 1 are pitched at a very good level for interview practice. Other STEP exams can also sometimes serve as helpful practice, but do be mindful of the issue with prerequisite knowledge.

STEP past papers can be found here.

#### 4. Siklos’ *Advanced Problems in Mathematics*

This book was written with the goal of helping students who are intending to apply to a mathematical subject at university. Particularly those who need to take STEP. It is a compendium of 75 challenging problems, complete with solutions and discussions.

For the most part, these problems are pitched at a suitable level for interview questions. However, since this book was written to prepare students for STEP the reader should also take care when using this book as some questions might involve prerequisite beyond what is expected at interview.

#### 5. Maths competitions

Finally, I would recommend looking towards maths competitions, particularly those run by the UKMT which are aimed at those in Sixth Form, such as the BMO or the SMC. There are countless past papers available for these on the UKMT website.

Though not a perfect match to what you might be asked about in an interview, I do think that the problem solving experience working through these provides can serve as very good practice.

UKMT BMO past papers can be found here.

### Practice tips

Actually preparing for the interview can sometimes feel a bit daunting, and it can be hard to know where to start.

#### 1. Talk through any ideas out loud.

I cannot stress enough the importance of this. During a real interview, it is paramount that the interviewer can see and understand your thoughts and ideas. You should practice this skill from the very beginning, as it doesn’t always come naturally.

If you say an idea and realise it can’t be right, then that’s absolutely fine. It is worth then vocalising *why* your idea doesn’t work, perhaps disproving your own suggestion. Remember, nobody will mark you down for having a bad idea, but being able to critique and verbally refute any bad ideas can leave a good impression.

#### 2. When you solve a problem, practice explaining your solution back.

Presenting mathematics is a very different skill to doing mathematics. Some of the great mathematicians of our age are terrible speakers, and conversely there are many ‘average’ mathematicians who give the most incredible and engaging talks.

After arriving at a solution in some practice problem, practice explaining your solution to some listener (real or imagined). You might also find that explaining your working outloud can lead to further insight into the problem.

#### 3. Reread your personal statement.

Not always, but some interviews may begin with some questions about your personal statement. This can serve as a bit of an icebreaker, and sometimes a chance for you to elaborate on something interesting you have written about.

It is a good idea to re-familiarise yourself with the contents of your personal statement before an interview. A question like this is a good opportunity for you to talk confidently about something you really do understand before the actual interview begin, which might help provide some confidence for the real questions.

#### 4. If you get stuck, don’t immediately reach for the solutions.

I think this one is really important.

While practicing, you will almost certainly come face to face with some problems which feel intractable. However, before you go to reach for the solution I ask that you consider the following quote, commonly attributed to Paul Halmos.

'’The only way to learn mathematics is to do mathematics.’’

In mathematics, the real learning happens when you wrestle with difficult question and understand things by yourself. By reading the solution to a problem you are denying yourself the opportunity to internalise the concepts present. As far as ‘practice’ goes, this problem has now basically been wasted.

Good practice problems are a finite resource. It can be quite difficult to find problems which are pitched at the right level for interview practice, and so they should seldom be wasted like this.

Put simply, if you take the chance to really work through a problem yourself then next time you come across a similar problem, you will have ideas and intuition for the problem. However, if you just read a solution then another similar problem might look completely alien to you, because you haven’t given your brain a chance to internalise anything.

Instead of reading the solutions when you are stuck, I suggest putting the tricky problem aside, and treating it as a ‘project’. You’ll be surprised how often a bit of sleep and time away from a problem can lead you to new ideas and perspectives.

#### 5. Stuck? Specialise.

When a mathematician is stuck with a problem and she isn’t sure how to proceed, then the first technique she might try is *specialisation*.

Specialisation refers to the opposite of *generalisation*, and involves considering a simpler, more specific version of the problem. Try adding extra assumptions or hypotheses to your problem and seeing if you can make any progress, or perhaps consider some specific examples. More often than not, the intuition garnered from specialising can be taken and adapted to the more general case.

### Some final advice

I will close with a few suggestions to prospective candidates.

#### 1. Keep talking

It’s completely normal (and expected) to feel a bit stuck during the interview. If this happens then it’s a good idea to keep verbalising your thoughts and talking through your ideas.

When you’re stuck, the interviewer is likely to give you a hint. If you’ve verbalised your thoughts well then the interviewer should understand exactly *why* you’re stuck and be able to give you a very effective hint which will (hopefully) enable you to proceed. Conversely, if you remain silent then this hint may not help you at all.

#### 3. Don’t give up

If you really are stumped, then one of the worst things you can do is simply give up without presenting any ideas. Try first considering a few simple examples or see if you can break the problem down into easier steps.

Many an initially strong interview has been let down by a student’s attitude to a harder problem. Remember, one of the key purposes of the interview is to see how you respond to new mathematics and new challenges. If you simply answer what you do know and shy away from anything new to you then this can leave a bad impression.

#### 3. Try to have fun!

The best candidates are normally those that really enjoy the interview and have a lot of fun. Don’t beat yourself up for getting stuck, the hard truth is that maths is sometimes just very hard.

In some sense, the interviews are designed to mimic the tutorial environment. If you really hate the interview experience then you may find tutorial-based teaching similarly challenging.