Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Focusing and structuring longer communication

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

Annotated introductions help students see how to structure an introduction:

Section and proof structure

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

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.

License: CC BY-NC-SA Page content licensed by MAA MathDL Mathematical Communication under the license:
CC BY-NC-SA (Attribution-NonCommercial-ShareAlike)