Math 120 Writing in the Major Paper

Clear writing is essential to mathematical communication, as you probably realize from reading better and worse mathematical texts. Good exposition is an acquired and important skill. Throughout this class, you'll receive feedback on solutions to your problem sets, and you should use this to refine your ability to communicate your ideas clearly and effectively. This writing project will give you an opportunity to focus on your exposition, as opposed to absorbing new mathematical content.

Francois Greer (the WIM grader) will be available for help with the writing project. His office is 381-A, on the first floor of the math building, and his office hours for WIM are simultaneous with his regular office hours for 120.

The project. Your goal is to present a readable and complete discussion of the proof of the impossibility of the doubling of the cube and/or the trisection of a general angle with straightedge and compass. This was discussed in class on Tuesday, October 14. It is also discussed in the book in section 13.3, except at a crucial stage of the argument, Dummit and Foote use degrees of field extensions (which we have not yet collectively seen), and instead you should use the argument using cubics. (You can discuss this argument with Francois in more detail.) Write your own discussion of this topic. In particular, be careful to avoid copying the book or any other source. Be sure to give motivation, as it can make ideas much clearer!

The mathematical content of this assignment is not intended to be the primary challenge. The point of this project is to concentrate on the exposition of the topic. Your paper should be 4-7 pages long; quality, not length, is what matters. Past papers in this class have gone on to win university writing awards, and I hope to nominate at least one paper this year too.

What you should do. Your target audience is a typical Math 120 colleague who has not yet read this section. Your target audience is not me or Francois. If you have been frustrated by reading mathematical writing in the past (which you undoubtedly have), this is your chance to show how it should be done!

In the introduction, you should describe the problem, and why people cared. Put it in some larger context. Give enlightening examples. By the end of the introduction (but not at the start!) there should be a precise statement of what you will prove. Prepare the reader for what follows by succinctly describing the main ideas and techniques you will use. Then give a detailed dicussion. Be clear what results you are quoting, and try to use as little as possible from earlier in the text. The less self-contained the paper is, the less useful it is to the reader. Do not just say something like "by Theorem 4.2 of the book" --- state any invoked theorem precisely or else give it a descriptive name (such as "the First Isomorphism Theorem"). Your paper should be readable by someone who is familiar with the material we have discussed, but who learned it from a different source.

You may want a brief conclusion, in which you highlight the key points of the exposition, so your reader can remember them. This is an opportunity to make sure your reader has a big picture in mind. Ask yourself: what should the reader remember after reading this paper?

Define your notation ("let G be a group..."). You don't need to define "group", "abelian group", etc.; your target audience is familiar with these notions, and can be assumed to have read everything up until where we are in the course.

Use complete sentences. Use paragraphs to organize your ideas into logical chunks. Do not use shorthand symbols and words when possible ("iff", right arrows, three dots for therefore, etc.) --- these shorthand symbols are useful for the author, and sometimes necessary during a lecture when time is in short supply, but they needlessly slow down the reader. But definitely use "usual" mathematical notation (of the sort used in the text).

Run your draft by someone else (ideally in the class).

Suggestions (from Daniel Murphy). Although there is a degree of subjectivity in what constitutes good writing, the following suggestions are common practice, and while they do not necesarily need to be followed, they are provided in case some resonate with your idea of style. It would be unfortunate if points of good style were not followed only for lack of consideration. A helpful resource with several suggestions is the article with writing guidelines for MIT math majors written by Professor Kleiman, which is available here. (At the end there are several other references to guides on writing.) Here are some helful suggestions:

  • Often one can avoid "I" or "we" and write more concisely by using imperative form. For example (from Halmos in Mathematical Writing available as notes from, instead of "All we need to do to get the answer is to replace x by 7 throughout." you could write "Replace x by 7 throughout."
  • Separate symbols with words which can help convey meaning (such as "S_n for n=1,2") instead of punctuation (such as "S_n, n=1,2") because the latter could be ambiguous.
  • Adding to the above comment under "What you should do": Words are often easier to read than symbols. "The set of real numbers with absolute value less than 1.'' is preferred to "The set of real numbers with absolute value <1.'' Avoid combining phrases with symbols as in the second sentence above.
  • Don't use commas for appositives with symbols. For example, "The subgroup N is normal." is preferred to "The subgroup, N, is normal."
  • Don't start sentences with symbols or variables.
  • Use complete sentences. (This was mentioned above under "What you should do", but is important enough to get a second mention.)


  • Friday, October 31, noon: drafts handed in to Francois Greer's mailbox by anyone who wants feedback first.
  • Friday, November 7, noon: drafts handed in to Francois Greer's mailbox. Drafts may not be handwritten.
  • Tuesday, November 11: drafts returned with comments (you can pick them up in class, after the midterm).
  • Friday, November 21, noon: final papers handed in to Francois Greer's mailbox.
  • Monday, December 1: final papers available for pick-up outside Francois Greer's office.


    You can use whatever word processing or typesetting program you wish. The standard one used in mathematics, statistics, and other parts of science and engineering is called LaTeX, a version of Donald Knuth's famous TeX typesetting program. Implementations of LaTeX are available for free on all operating systems. A not-so-short introduction to LaTeX is available here.

    Here is a link to an article about writing mathematics well. Click on "In His Own Words".

    You might want to be aware of the Hume Writing Center, which offers its services for any stage of the writing process. You can use their online schedule to make appointments of 30 or 60 minutes with one of their tutors in the main location of the Center (open M-Th 10-6 and F 10-4). You can also use their drop-in schedule to see tutors without an appointment either in the Center or at one of many remote locations, most of which operate in the evening and on the weekend. They have a good reputation, and if you use them, I'd be interested in hearing about your experience.

    Acknowledgments. Thanks to Joan Licata, Kamil Szczegot, Tracy Nance, and (very much) Daniel Murphy (who provided many of the ideas on this page). (Subliminal message: give appropriate credit when writing!)

    Back to the course home page.