### Focus of a paper or presentation

It’s difficult for even the best communicator to write or present well without a clear focus. For example, a presentation titled “Hadamard Matrices” that presents everything the student knows about Hadamard Matrices will be difficult to structure well, and the audience is likely to wonder why details are being presented. A more specific focus like “A Method for Constructing Quasi-Cyclic Generalized Hadamard Matrices” is likely to be easier to structure.

If a paper draft seems choppy or poorly structured it’s often helpful for the student to outline what he or she has written and then analyze how each section fits within the focus of the paper. Restructuring or refocusing the paper will often help the student to improve the exposition.

A handout about how to choose a focus for a term paper is here.

### Structuring a paper into sections

In a longer paper, sections typically include “Introduction”, “Preliminaries”, “Main Theorem”, etc. Outlining the paper before writing may be helpful to some students, but others may prefer to write a draft first, outline what they’ve written, and then restructure as needed.

If the paper is particularly long some sections may include subsections, but sometimes students rely too much on subsections to communicate the structure of the paper. If short subsections make the paper feel choppy, you might suggest that the student instead use guiding text to communicate the section’s structure.

Guidance about how to handle different sections of the paper is offered in

- Chapter 3 of Stephen Maurer’s
*Undergraduate Guide to Writing Mathematics*(includes sample introductions and sample papers). - The MAA’s student resource A Guide to Writing an Abstract
- Haynes Miller’s Notes on Writing Mathematics
- Paul Seidel’s presentation slides Writing a Paper, which include samples of good and better writing for an introduction, a background section, and a main theorem (topic: knots).
- Steve Kleiman’s How to Write a Math Phase Two Paper (See Section 2)

Annotated introductions help students see how to structure an introduction:

- An annotated journal article, “Maximum Overhang” by Paterson et al,
*American Mathematical Monthly*116, December 2009. - Annotated student paper, from the
*Undergraduate Journal of Mathematics*: “Finding Generators for Invariant Rings” by Qingchun Ren (annotated)

### Section and proof structure

We often provide structure by using “Theorem,” “Proof,” and “Example” environments. Students benefit from discussion of the following:

- What is a “Theorem” vs. “Proposition” vs. “Lemma” vs. “Fact”, etc.?
- A brief explanation with examples is given in this lesson plan about proof structure from M.I.T.’s communication-intensive offering of
*Real Analysis*. - Examples illustrating the differences between these statements are given in the annotated journal article “Maximum Overhang” by Paterson et al. See in particular page 772 (page 10 of the pdf).
- Here is an example of a long proof that is improved by pulling out a lemma.

- A brief explanation with examples is given in this lesson plan about proof structure from M.I.T.’s communication-intensive offering of
- How do you choose good examples and counterexamples? What is the value of both examples and counterexamples?
- A description of common elements of mathematics prose is given in the article “Varieties of Mathematical Prose” by Bagchi & Wells,
*PRIMUS*vol. 8, 1998, pp. 116-136. The article describes various kinds of statements, various kinds of examples, various roles for exposition, etc.

### Communicating the structure of a presentation

The structure of a long presentation can be communicated by giving an outline after the presentation’s introduction and then revisiting the outline at each transition within the talk. It’s helpful to present the outline after rather than before the talk’s introduction, so the audience understands the topic and relevance of each section when the outline is presented. Revisiting the outline at transitions enables the audience can keep track of where they are within the structure.

In long presentations, transitions provide an opportunity to bring back on board those who have become lost, and transitions also carry the risk of losing people who miss the transition, great care should be taken with transitions: e.g., gain the audience’s attention to ensure that noone misses the transition, recap, ask for questions, pause, indicate the purpose of the next part of the talk.

The structure of the talk can also be communicated by choosing titles for the boards or slides to make clear the purpose of the details on each board or slide. Examples of board/slide titles include “Theorem:…” “Proof Sketch” “Why do we care?” “Example” “Why is *k* prime?” “Connection to Linear Algebra” etc. If using boards, leaving the boards in reading order (top down and left to right) enables the audience to see the structure and find information if they become confused.

More specific guidance for presentations is available on the page of presentation resources.