# Tackling Oxford Mathematics Interviews (Undergraduate)

Oxford Mathematics Interviews may feel daunting. Beyond the basic format of the interviews, here are some of my advice on tackling them.

## Goals

There are a few things you are aiming for through the interview.

### Show that you can be taught

Show that you can improve from your tutor’s teaching and that it’s satisfying to teach you.

This means that communication with the tutors is crucial. You should place great care on how you communicate with your tutor—How you explain yourself and how you respond to your tutors.

This is often overlooked. Think like a tutor interviewing applicants. Imagine admitting a student who aces all the interview questions but communicates poorly. What if they start struggling in year 1 or 2? How can they explain what they’re struggling with? How can you understand their struggles? How can you help them? As such, good communication is far more important than momentum solving the questions.

### Show your mathematical ability

You need to show that you have mathematical intuition and you have the skills to put intuition into words through rigorous explanation.

Mathematical intuition and rigour are both learnable skills. Intuition comes from practice and experience. When someone sees a problem and is able to tackle it quickly, most often it’s because they have seen a variation of that problem somewhere and can alter the techniques slightly and apply it to this new problem rather than innate talent. If you like maths and have gone through some MATs, be confident in your intuition! You should have seen enough problems and have done enough mathematics to be able to have a lot of ideas on a wide range of mathematical problems.

Rigour comes from experience doing mathematics rigorously. This may be a little bit harder to obtain through secondary school or Olympiad mathematics alone. I would recommend reading some maths books (e.g. “Book of Proof”) that places emphasis on proofs just to get a taste of how mathematicians generally construct mathematical arguments logically.

### Show enthusiasm, persistence, flexibility and confidence

Tutors will like to see if you have the enthusiasm needed to get through 4 years of constant mathematics. It’s much more satisfying to teach someone who’s enthusiastic anyways.

Show that you’re persistent and flexible. Show that you’re persistent in advancing your mathematical understanding but flexible enough to consider any idea or method in pursuing that goal.

All in all, think like a tutor interviewing yourself. What qualities would you be looking out for if you were a tutor?

## Before the interview

### Talk about maths with people

It could be you helping a fellow classmate understand a problem, or you asking a fellow classmate for help. Take great care ensuring that they understand your argument. The priority should be on being as clear as possible, not on showing that you got the answer right. Think about what others find confusing in your mathematical arguments. Think about what you often find confusing in other people’s arguments as well.

During interviews, even if you got the right answer, the tutors may ask further questions to make sure your logic holds water. Sometimes you may also be confused by what the tutors say and have to ask questions. The more you talk about maths, the better you get at this.

### Know the common identities

Try to remember the common identities (calculus, trigonometry …) before the interview. It would be unfortunate if you had to waste valuable time deducing them on the spot. Make sure you understand them thoroughly so you’re not just mechanically applying them.

## During the interview

### Be enthusiastic, persistent, flexible and confident

Be enthusiastic about the interview question! If the question seems easy, think about the mathematical truths they are hinting at, think about how you can explain it to a 5 year old. What’s the essence of the question?

Be persistent and flexible! Be persistent in seemingly complicated equations and calculations and not be afraid of them. If your methods aren’t working, be flexible. Don’t be afraid of starting over and attacking the problem with a different angle. If the tutors raise doubt about your methods, it’s best to pause and re-examine your strategy. But keep talking about what you’re thinking!

Persistence and flexibility can be in contradiction and uncertainty is unavoidable. At the end of the day, you need to be confident in yourself regardless of your choice. Give every idea your best shot before considering others.

### Generalise your thinking

The first question they ask you might be very easy. While you are answering it, think about the complications they could throw at you. If they were asking about the $n = 1$ case, what about $n = 2, 3 …$? Could you figure out a general solution?

A corollary of the above is that you should not rely on brute force for simple questions. You should not use excessively time-intensive strategies as time is limited. Brute-force also will not work if they complicate the question slightly later on.

### Connect the subparts

Previous subparts should aid you in solving the full question, just like the MAT. Whenever you’re stuck or confused, try and think about how the previous parts of the question connect with each other.

## Explaining yourself

If you think of the interview as a process of communicating with your tutors (instead of just solving a mathematical problem), you’re generally either explaining what’s going in your mind or responding to what your tutor has said.

As such, the first step to effective communication is to explain yourself.

### Relentlessly communicate

Use all the communication methods at your disposal all the time. Constantly talk about what you are thinking and doing (except when your tutors speak). Write down anything that is important / prone to mistakes, but not everything so you’re efficient with your time.

The worst is to sit in silence and not write anything. Your tutors aren’t mind readers and they would not understand what you’re going through. Remember that your ultimate goal is to communicate and not only to complete the mathematical problem presented to you.

Make sure you keep talking. In reality, talking is a brilliant way of mathematical communication. While we are used to writing as the main way of mathematical communication through exams and homework, writing everything out often takes up too much time and it’s hard to convey intuition through mathematical writing alone. Long sentences are better spoken than written.

You could talk about what you’re doing right now. What is the immediate goal of what you’re doing right now? What results do you expect to show up? Why is it relevant to the problem at hand? When you introduce a variable, what is it supposed to represent? Why is it relevant?

You could also talk about your strategy for tackling the problem. Why do you think this strategy would work? Is it because it was used in the previous problem / some other problems you have seen before?

Explaining your overarching strategy is especially important if you are unsure whether you have the right strategy. They would usually say something to indicate whether you’re on the right track. If they affirm your plan, then great! You can focus on the calculations / nuts and bolts of the argument. If they dispute it, then at least you didn’t waste time exploring a dead end.

The tutors may also be more lenient on your careless mistakes if they know that you have the right ideas in mind. If you didn’t talk about your plans, they may think that you had the entirely wrong idea in the first place.

### What if you’re stuck?

You may immediately get stuck when you first hear the problem. It’s important to give yourself some time to process what’s been asked. You should almost always write down important parts of the question carefully and slowly to ensure you’re answering the right question / not have to remember the question in your head. Very often after long calculations, people are so confused about what the question was about they didn’t realise they just answered it!

After spending a bit of time thinking (3 seconds), you should try and say some of your initial ideas on the problem, or what methods you would like to use to tackle the problem, even if they are not polished. You could speak in broad terms / talk about strategies, but you should avoid saying something concrete if it’s not thought through. Muttering concrete statements without much justification may reflect that you’re blindly guessing which does not reflect well.

For example, if it’s a graph sketching question, you can say that you would try and find the intercepts by writing down the relevant equations, but don’t immediately guess that the intercepts are positive or negative. If you find those equations too tricky to solve explicitly, you could mention some observations, or simply say this seems difficult and move on to other aspects that you can consider like derivatives or asymptotic behaviour.

You could also consider simpler problems which often reveal insight into the problem you’re dealing with. For example, if you’re asked to sketch $\sin(1/x)$, try plotting $\sin(x)$ and $1/x$ carefully and see if it helps. You could also segment the problem, such as sketching the graph at particular intervals first. You could also find approximation / bounds to the graph you’re asked to sketch. For example, $-1 \leq \sin(1/x) \leq 1$ simply from the properties of the sine function.

However, you should not dwell on low-hanging fruits for too long. For example, you shouldn’t find the values of the graph at $x=1, 2, 3, \dots$ and brute force your way. If you recognize that there’s a complicated but important idea that you have to try and understand, try to explain it and give it your best go! It shows that you have an awareness of what’s important and that you have a strategy in mind. The tutor would see you struggle but at least you would be on the right track and they may offer some help.

You should also constantly explain why you think you’re stuck. Why are your methods not working? What results are you expecting? Why do you think they are not showing up? What’s the main difficulty of the problem?

Since this is something you would constantly face in lectures / problem sheets / tutorials, tutors are eager to see if you’re able to tackle difficulties. Remember that getting stuck is as an opportunity to display your problem-solving and communication skills.

### Explain in detail but do not overdo it

When you’re explaining your workings, take care in explaining both your intuition and the technical know-how. Remember to do both and not only one of them.

For example, if you are asked to find the maximum of a function over some interval, you could first briefly that the maximum is located at a point that has a tangent line with slope 0, otherwise you could “move” slightly to the left or right to get a higher value. Alternatively, the maximum could be at the endpoints of the interval (That’s the intuition). Then you could explain that we could calculate the derivative, which is the slope of the tangent line, and briefly explain the chain / product rules if they are used (That’s the rigour).

You should also use technical terminology if possible, such as “tending to”, “asymptotes”, “periodicity”. You should give both an intuitive explanation of why they’re relevant and a somewhat rigorous definition.

However, don’t explain things for the sake of explaining it / the pretense of rigour. Your time in the interview is limited and you should spend it wisely. For example, you don’t need to explain or write out all the steps taken in basic algebraic manipulation. Again, long sentences are better spoken than written.

As another example, consider proofs by induction. It’s unnecessary to write out the formalities required in an exam setting. You could write “Base case:” and “$n \geq 1:$”. You should however explain why you’re using induction in the first place (You want to show something is true for all positive integers, and that there’s a connection between the $n$th and the $n+1$th case?).

### Write neatly

Writing neatly is worth the time investment. It makes your content more readable to you and the tutor. You will actually trust what you’ve written down instead of constantly wanting to re-do it all again.

Write from left to right, from top to bottom. Don’t make insertions all over the place for the sake of convenience. If you need to cross something out, you might as well write it all over again for the sake of clarity.

Keep your font size consistent. In Miro you can check the scale of your board (bottom left corner) and ensure it’s always around 100%. Try writing more on lined paper to become better at this.

If they ask you to draw a graph, take some time drawing your cartesian plane! The axes should be straight lines with appropriate ticks and arrows.

Writing out basic arithmetic (addition / multiplication / division of two-digit numbers) quickly instead of doing it in your head is generally a good idea as well. It ensures you don’t make mistakes while not taking up much time.

If possible, use tables. They are often the clearest way to display information Here are some examples.

No. of sides | Angle sum |
---|---|

$3$ | $180\times(3-1) = 360$ |

$4$ | $180\times(4-1) = 540$ |

$n$ | $f(n) = n(n+1)/2$ | $f(n+1) - f(n)$ |
---|---|---|

$1$ | $1 \times 2 / 2 = 1$ | $2-1 = 1$ |

$2$ | $2 \times 3 / 2 = 3$ | $3-2 = 2$ |

In conclusion, time is limited in an interview. Even if you know the full solution to a problem, you may only have a few minutes to explain it or else there might not be time for other more advanced questions. Find a balance between accuracy, ease of understanding and time taken. Find what works best for you!

## Listening and Responding

The second step to effective communication is to listen and respond well to the tutors.

### Stop and Listen

Whenever the tutor says something, stop and listen! Being ignored is extremely frustrating. Try to understand what they’re saying and try as hard as possible to come up with an appropriate response.

You should also avoid interjecting. It’s best to let the tutor finish their sentences (which are usually short anyway) so that you have more information to work with.

There are generally a few scenarios that play out when the tutor says something to you which I will describe below.

### You don’t understand what they said

If it’s due to some specific terms that they used, don’t be afraid to ask for definitions! It’s perfectly acceptable to forget / not know some mathematical terminology, especially if you’re an international student. You can also make an educated guess. Regardless of if you’re right or not it shows that you have thought about it and have a rough idea of the mathematical concepts at play. The tutor could give you a better answer if they know what you already know.

If the idea they’re suggesting is too complicated, try to paraphrase and internalise what they have said first. Reproduce the idea they have given you in the simplest way. It could be writing out an equation, setting up some variables or drawing a graph. Do the most that you can. It shows what you currently understand (you could be closer than you think!).

### They asked why you are doing this or that

Be calm and confident! Don’t immediately think that you’re wrong. It may be due to a lack of explanation and the tutors want to know that you really know what you’re doing. Slow down and explain more about your goals / strategy.

### They are not convinced by your explanation or suggested that you’re on the wrong track

It may feel very distressing to be wrong. As such the most important thing is to keep your cool, don’t get immediately defensive about your methods. Take at least a few seconds to organise your thoughts.

Then, look out for any careless mistakes/oversights. If you spot some, take your time to explain and correct them. It’s important to take your time so that the tutors can see that you truly understand what you got wrong. It reflects poorly on you if you hastily make some corrections only to be wrong again. To explain yourself effectively you must be confident in yourself.

Also, consider whether there was any miscommunication. Sometimes the tutors might have lost your train of thought. If you are certain that you were right, explain more about the exact point of confusion your tutors had. You could start over your argument carefully in hopes of finding any overlooked mistakes.

If they still insist that you are wrong or should try something else, it’s very important to be flexible. Don’t be fixated on the method that you’ve been using. Don’t be afraid to start over completely and attack the problem from a different angle.

## Conclusion

The interview is very much a communication process. Just as you communicate with your tutors through written work, it’s important to excel at communicating verbally in tutorials as well. Think about how you communicate effectively with others and apply those lessons in the interview!

While a lot of the qualities I’ve mentioned may seem like an act, those qualities are genuinely needed to succeed at mathematics. Try to internalise those qualities so that the interview process not only helps you get into Oxbridge but also helps you become a better mathematician.