Writing an essay on a mathematical topic is likely to be different from anything you have done before. However you have written lots of mathematics during your education and have probably had to produce an extended piece of writing at some point in the past, albeit probably on a non-mathematical topic. Writing a good essay requires you to combine the two different sets of skills needed in writing short pieces of mathematics and more extended but not necessarily mathematical pieces of writing.

Your first task should be to think about the global structure of your essay. Decide what topics you want to include, make sure that you understand them thoroughly and think about how they should be ordered so that your essay is a coherent piece of work, ideally moving neatly from one topic to the next in a logical way. Once you have done that you can start to think about the content of each section. What are the main results that you want to discuss? What are their consequences? What results do they depend on? What is the history of the problem? What is the motivation for studying the problem? What notation and concepts do you need to define? When you can answer these questions, you should have a clear picture of how your essay will be structured and you can begin writing. It is likely that the structure will need some tweaks as you start writing, but good planning will keep these to a minimum.

Here are some tips on writing a good essay once you have finished the planning stage and are ready to start writing:

- Give your essay a structure of sections and possibly subsections, but don’t go too far with dividing things into subsubsections. It’s often tempting to provide lots of titles, headings and subheadings to indicate the structure of an essay, but you should observe that professionally written mathematics doesn’t do this. Instead you should try to make the text flow by indicating to the reader where you are going next. It’s much more pleasant to read what will come next and see where the essay is going than to be jolted by a sudden change of direction signalled by a heading.
- You should imagine that the reader of your essay is a fellow student who understood the key material in the relevant lecture, but might need reminding of it. So you don’t need to describe standard undergraduate material.
- Any notation not commonly used in undergraduate mathematics should be carefully defined and used consistently. Different sources may use different terminology and notation, so if you quote results from more than one source, you might need to adapt the notation so that it is consistent with the rest of your essay. You should include definitions of notation or a concept whenever its meaning is not universal. For example, you should state whether your graphs are allowed to have parallel edges or loops, and whether or not 0 is a natural number.
- You should include plenty of examples, wherever possible examples that you have invented yourself. It helps the reader to confirm that they have understood what you are saying. It also means that if there is a slip somewhere in a definition, such as a missing minus sign, its correct use in an example will show the examiners that the slip was just a slip, and not an error of understanding on your part.
- Use simple, clear language, forming proper sentences grouped carefully into paragraphs. Each sentence should generally express one idea. You may need to think very carefully about the best way to explain a complex idea: it is much better to explain it precisely, briefly and accurately just once, than to describe it in multiple different but imprecise ways.
- Abbreviations such as i.e., etc., wlog, iff and the implies sign should be avoided. Although they are often used by lecturers when writing on the board, they should not appear in a formal essay.
- Treat mathematical symbols as part of the text, and do not overuse them. It is much easier to read “Let
*x*and*y*be strictly positive real numbers” than to read the same thing written with lots of notation. All of the mathematics, whether displayed or written inline, should be parts of sentences and properly punctuated. Try reading back sentences including mathematics with words replacing the symbols and ensure that they make sense as English. - Include a bibliography, listing all the texts that you have cited. Label and describe the items clearly and consistently, by putting them in alphabetical order by the first author.
- When using a reference, always cite it properly, and include guidance as to where in the reference to look. If you state a theorem and then say that the proof is in [3], when [3] is a 600 page book, it’s not very helpful! It’s much better to say that the theorem is proved as Proposition 4.6 of [3].

The school has three learning development tutors who can provide support. Their role is not to provide higher level mathematical support, but they can help with skills such as essay writing. See http://www.bbk.ac.uk/business/current-students/learning-co-ordinators for information about them and who can help with particular ski

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