Term paper, outline

In this assignment you will write an outline for your term paper, a first version of a writing plan.

As a reminder, from the syllabus, this paper is:

a term paper (6–15 pages) about a topic of the student’s choice within real analysis.

What is an outline

1. An outline is an organized list of sections and topics that you are planning for your term paper.
2. Usually it’s presented as a nested list,
   1. starting with the main sections of your paper;
   2. within them, lists of
      • subsections
      • and topics;
   3. possibly, sub-topics within those.
3. It can be very helpful to write some short information (1-2 sentences) for each topic: a capsule summary, description, or other information such as relevant sources.

For more tips and examples, see for example

• https://wikihow.com/Outline-a-Term-Paper

• Ten Simple Rules for Math Writing

  (At this stage, the “small rules” are not important and the “broad rules” are only semi-important. The “composition rules” are the most important!)

What you are being asked to do

• Prepare an outline for your term paper

• You can use LaTeX, Microsoft Word, Google Doc, or any other system (but your outline has to be typed, not handwritten).

• Upload your PDF to Gradescope.

If you use LaTeX, then you can use things like:

\begin{enumerate}
    \item Introduction
    \item Background
    \item First Main Theorem
    \item ...
\end{enumerate}

and they can be nested. For example:

\begin{enumerate}
    \item Introduction
        \begin{enumerate}
            \item A surprising example (this is the hook to get reader into the topic)
            \item It raises a question, more discussion
            \item Describe the main theorem and summarize some of the ideas
        \end{enumerate}
    \item Background
        \begin{enumerate}
            \item Review of some background knowledge
        \end{enumerate}
    \item First Main Theorem
    \item ...
\end{enumerate}

For something a little fancier you can look at https://tex.stackexchange.com/questions/12279/outline-of-style-i-a-i-a-1.

Specific issues for mathematics papers

• Mathematics papers virtually never have a “Methods” section.

  (Most science papers have a “Methods” section. But in mathematics the method is almost always proofs, so you don’t need a “Methods” section.)

• Mathematics papers very rarely have a “Conclusion” section.

  Instead, the conclusion is the “highlights” or main theorems/results/examples. We will talk later about how to emphasize them, and how to show the reader which theorems/whatever are the main ones.

  (Basically, this can be laid out in the Introduction, and again later on when the theorems come up. In both those places the paper can talk about why those items are important.)

• There can be a section at the end to talk about “further questions”, directions for future research, etc. However for your term paper this is optional.

• You should probably start with an “Introduction” section where you lay out the subject of your paper and set up the rest of it.

• Have a section for each topic (each highlight, or main theorem).

  One section can split into two if there’s a lot to say; or two sections can merge together, if highlights are closely related. But usually one main theorem or topic per section is a good rule.

• Between the introduction and the main sections, there can be 1-2 sections of setup like reviewing background information; preliminary work before the main work; history; etc.

• These are all suggestions. Please don’t ignore them without a good reason, but if you have a good reason to change the order or the setup, then it is okay!

What I’ll be looking for

Not all of the following elements need to be present at this stage, but try to have as many as possible of the following ideas:

Outline

Outline is presented as an organized, clear list of topics and sub-topics with some appropriate descriptions

Highlight

What will be the highlight of the paper—a theorem, an application, etc. Where in the paper will it be

Reaching the highlight

An organized, clear plan to get to the highlight; necessary background and buildup; not too much extra/unnecessary/digression along the way

Introduction

Plan for how to introduce the topic and how to begin the paper

First part of body

Clear, understandable; setting foundation for later/advanced part of the body

Later part of body

Building on introductory material in a connected way. Moving beyond course content in some way.

Purpose

• The purpose is for you to plan your term paper, and to think through how you will arrange your topics.

• Especially, you should select one or two “highlight” ideas such as main theorems, examples, algorithms, etc., and start planning how you will present them. That includes planning how you will build up to them, what is the needed background, etc.

• You will be allowed to change your outline when you write your drafts and final version! So this is just a tentative outline.