What I learnt about ‘Writing Maths’ at University (Undergrad)

Travelling from Bangalore, India to Coventry, UK in September 2019, I had the basics of my high school maths at the tip of my fingers as I charged ahead to begin my 3 year undergraduate course of Maths at the University of Warwick.

The transition from the type of math problems I solved in the AISSCE (All India Senior School Certificate Examination) to my first assignment submission at university (which I remember was in Analysis I), was a huge, shocking step. The difficulty, the way to write proofs and solutions, the expectations, the marking scheme, the style of teaching and of course…the sudden change in accent and culture.

Honestly, it wasn’t very easy. Of course, it took time to understand and get used to the system there. And COVID was definitely a cherry on top. My first year exams got cancelled, second year exams was open book and online (hence, had a completely different question paper pattern compared to previous years and increased difficulty of questions) and my third and final year was the first time I wrote regular British exams.

So, in this article, I will focus on how to write maths well at University, that is, in the coursework, assignment submissions and exams. (Disclaimer: This is based on my experience at Warwick University). This is what I learnt through my 3 year experience. This may help future undergrad math students gain some useful tips or advice and be ready for a great university experience.

Clarity

Let’s begin with ‘clarity’. During my first year, 5 marks out of the total was always based on ‘clarity’ which means ‘clarity of expression’. This was given to encourage us to develop good writing habits. But, from second year onwards, there weren’t any official marks for clarity, but, every assignment handed in and exam written was evaluated based on it — if a marker cannot understand our solution or proof, we can’t expect any marks even if the idea in our head was good.

Presentation

Clarity also means presenting answers clearly. I had a habit of scribbling randomly everywhere on the paper while solving questions. As mathematicians, this is inevitable as we (at least me) do this while focusing on following a flow of thought and exploring various ideas to solve the questions. This makes the page look messy. So does scratching out a lot of things on the page if we think something written is unsatisfactory. Then, it’s hard for a marker to see the solutions within the mess.

So, for assignments it’s helpful to create multiple drafts before writing a neat final draft for submission. For exams, this obviously isn’t possible. So, make sure to spread out the writing. For each question, use a lot of area and lines in the booklet. If you are stuck and want to return to a question later, leave a lot of space, maybe even a few pages, and then start the next question as jumbling the order of questions sometimes may confuse a marker. You can always ask for extra booklets. There is nothing called ‘shortage of paper’ during an exam. For rough work or as I mentioned, ‘exploring ideas’, draw a separate section in the page for working out the solution or do it in pencil so that it can be erased. Although, the difficulty of questions or the need to explore ideas in an exam is not as much as assignments where we have a week or two for solving questions and submitting them.

Writing complete sentences

Math is filled with alphabets, symbols and a never ending amount of equations, maps, vectors and matrices. So, it is of course easier to quickly write solutions using just the mathematical symbols, which may be perfectly understandable to you. But, chances are, it won’t be same for a marker. One of the many reasons is not everyone agree on the same symbols for various concepts in math.

For example, the notation of Dihedral Groups is different in Geometry and Abstract Algebra. In Geometry, it is denoted as D_n where ‘n’ is the total number of sides of the polygon. But, in abstract algebra, the same is denoted as D_2n, where 2n is the order of the group. So, if you write D_8 for an octagon, but, the marker takes it as a quadrilateral polygon with order 8, the confusion can cause you to lose marks.

Thus, when writing down a logical step in a solution, it is good to explain in words what was done in that step.

Not Writing Essays

This may seem like it’s contradicting the previous part. Explaining steps in a few words is good. But, writing answers in the form of large paragraphs or page long essays, is not. Answers with no words are hard to understand, but, answers with nothing but words is imprecise. It helped me to translate the complicated statements of theorems, definitions or proofs into simpler english to understand them better. But while writing them on paper, it is better to write the proper statements instead of the intuitive translations in our heads.

Writing as neatly as possible

This is about how the answers are arranged on the page and handwriting. The handwriting should, of course, be as good as you can possibly make it. But, if you’re like me and have naturally poor handwriting, one thing that may help is to write in short capitals. Do whatever you can to make your writing as readable as possible. The main thing about the arrangement of on the page is that it should flow in a logical order. If you’re making additions above or below the main line of an argument, then it’s not your final draft. Your work should read from left to right and from top to bottom, just as non-mathematical writing does.

Concise answers

Only write down what’s necessary to answer the question. When I was uncertain about what is required in an answer, I used to think, ‘let me just write down everything I know in order to not lose marks for missing something in the answer’. I know there are many who may think the same. But, it isn’t helpful to make the marker go through a load of irrelevant details in the solutions. It’s good to know in advance what points you want to write down (hence the multiple drafts).

If you are using a result, state it.

This is important. If you use some result from the course, such as a particular theorem, lemma or proposition, but don’t state it when you use it, it looks like you’ve written a logical step with no justification. Therefore, when you use a result from the notes or any book, and you know you’re allowed to use it without proof (this usually mentioned at the start of a question), always state the result, preferably by name, if it has one. If you can’t remember the name of the result, the best option is to make it clear that you’re using a known result, and write the statement of the result in full, along with where you’ve read it. (For example, ‘As mentioned in the lecture notes provided by the professor,…’). This is crucial because often in marking schemes many marks is given just for stating results before the clarity of the solution is even considered.

Carefully use logical notations

Quantifiers are the symbols,

which mean ‘for all’ and ‘there exists’, respectively. You’ll notice these used a lot in the coursework. It is perfectly acceptable to use them in solutions. But, if utilised, be sure to understand them well, because there are some subtle points about their proper use.

To give one example, the order matters. The statement

is not logically equivalent to the statement

The first statement says that for every epsilon > 0, we can associate an N such that it fulfils the condition given in the rest of the statement. The second statement says that there’s a single, unique N that fulfils the condition for every epsilon > 0. That is, the quantifier that comes first takes precedence.

You should also be very aware of the difference between

which mean respectively ‘implies’ and ‘therefore’. Many tend to use this more or less interchangeably, writing them down whenever taking a logical step.

Actually, their meanings are distinct. Say we have two statements and we denote them by P and Q. Then,

means ‘P is true, so Q is true.’ On the other hand,

means ‘if P is true, then Q is true (key word: if). ‘Implies’ has a hypothetical sense, but ‘therefore’ does not. When we write ‘therefore Q’, it means P is definitely true, therefore Q is true. When we write implies Q, P may not always be true. But if it is, then Q is true.

Going from high school maths to university maths, I realised how much more important ‘getting to the solution’ is than the solution itself. The techniques and results used to solve a question or maybe even using a new method devised on your own, is what actually matters. How we have proved something by smartly choosing the accurate results from the notes or recommended books rather using a longer, more time consuming method. University maths suddenly broadened my perspective of the subject so wide, and deepened the already known school math concepts with so much more depth and detail, it felt like I barely learnt anything in school. Just the tiny tip of an enormous ice berg. And yet, there is so much more to read and explore.