Mathematicians Must Speak:
The DOs and DON'Ts of Giving
Effective Mathematical Presentations
Gerda de Vries
Department of Math & Stat Sciences
University of Alberta

“Mathematical communication, both written and spoken, is the filter through which your mathematical work is viewed.
If you do mathematics purely for your own pleasure, then there is no reason to write about it. If you hope to share the beauty of the mathematics you have done, then it is not sufficient to simple write; you must strive to write well.”
Ashley Reiter, web.mit.edu/jrickert/www/mathadvice.html

Helpful Advice?
Tell them what you are going to say, say it, then tell them what you said.

Let’s hear some of your questions about giving mathematical presentations …

Outline
• Introduction – Focus on essentials
• Main Points – DOs and DON’Ts
• Closing – Concluding thoughts

Why Must Mathematicians Speak?
We must speak for the same reason that we must write:
• To share new research results with the scientific community
• To trigger new discussions, insights, and collaborations
• To let others know what it is that we do, why our work is important or exciting

Some Questions From Me to You:
From the Perspective of the Audience
• How is reading a mathematical text (journal article or book) different from listening to a mathematical seminar presentation?
• What information do you wish to get out of a mathematical text?
• What information do you wish to get out of a seminar presentation?

From the Perspective of the Presenter
• As a presenter, you need to help your audience get the take-home message(s)!
• Focussing on the needs of your audience incidentally directs your energies away from worrying about yourself.

In a Nutshell:
How to Help Your Audience
• Placing your presentation in a scientific/historical context
• Clear statement of the objectives & conclusions
• Signposting to convey presentation structure
• Eliminating unnecessary detail
• Carrying your audience: review definitions and symbology; give illustrative examples of dense statements; talk the audience through graphs, etc.

When Must Mathematicians Speak?
When there is a story to tell …
Stories come in many different forms:
• Literature review on a specific topic
• Announcement of a new result
• Introduction to who you are, what area of research you work on, what methods you use

All Mathematical Stories
• … have a setting
• … have a plot
• … have characters
Setting: scientific and/or historical context
Plot: problem statement and conclusion
Characters: variables, equations, objects such as groups, function spaces, etc.

Questions to Think About
When Putting Your Research
Story Together
• What motivates my research? Why am I doing this? Why should anyone else care?
• What is the significance of my results?
• Who else has done related work? How does my research relate to published works?
How does my work fit in with others?

Outline
• Introduction – Focus on essentials
• Main Points – DOs and DON’Ts
• Closing – Concluding thoughts

Capturing the Audience’s Attention
• You need a “hook” to capture your listener’s attention right from the start.
• Tell them about the important problem you will be discussing.
• Place your presentation in the scientific/historical context that is meaningful for your listeners by providing a concise introduction to the background of your problem.
Scientists Must Speak, page 9.

Clear Statement of the Objectives

Clear Statement of the Conclusions

Signposting: Giving Structure to
Your Presentation
• Signposts are aids that let the listener know where you are in the story and when you’re moving to a new stage in your presentation.
• Signposts can take many different forms: outlines, questions and answers, partial summaries, “where next” type comments, etc.
Image source: www.vanderbilt.edu/HRS/guide/contract/stII2.htm

Where’s the Signpost?

Where’s the Signpost?

Where’s the Signpost?

Simplify Your Slides
Replace
A = randsvd([10,16],1e2) with
A is a 10x16 matrix, with random entries
Excellent examples of slides before and after simplification can be found in Handbook of Writing for the Mathematical
Sciences.

Eliminating Unnecessary Detail
• Resist the temptation to tell the audience everything you know about the subject.
• Don’t bury the key points in detailed descriptions of methods or proofs.
Annoyance: what is the point of this talk?
Bewilderment: what is the point of this talk?
Boredom: what is the point of this talk?
Scientists Must Speak, page 7.

Proofs?
• The beautiful details of a clever proof may detract from the main story line of your presentation.
• Remember, beauty is in the eye of the beholder.
• The appropriate place for the exposition of a a proof generally is a journal article.

Did I Say You Cannot Include Proofs?
No! If you need to talk about a proof of a theorem in the course of your presentation, you have a number of options:
• Outlining the steps of a proof, skimming over common steps, and highlighting those steps that required clever insight on your part.
• Verbally sketching the main ideas of the proof.

Dense Statements:
Losing your Audience
Theorem k is an even perfect number if and only if it has the form 2 n -1 (2 n 1) and 2 n 1 is prime.
Proof
Suppose first that p = 2 k -1 is a prime number, and …

Carrying Your Audience:
Review Essential Definitions
Recall:
A positive integer n is called a perfect number if it is equal to the sum of all its positive divisors, excluding n itself.
The first two perfect numbers are 6 and 28:
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14

Carrying Your Audience:
Include Illustrative Examples of Dense Statements
The statement of the theorem pertains to the factoring of even perfect numbers into the form
2 n-1 (2 n -1) , where 2 n -1 is prime.
We can check that the statement of the theorem holds for the first two perfect numbers:
6 = 2(3) = 2 1 (2 2 -1) = 2 n -1 (2 n 1) for n =2
28 = 4(7) = 2 2 (2 3 -1) = 2 n -1 (2 n 1) for n =3

Carrying Your Audience:
Describe Graphs, Charts, and Tables
• Label all your axes.
• Describe the axes verbally as well.
• Discuss the interesting features of the graph: what can we conclude from the graph?

Funky Colour Schemes?
• People have no trouble reading dark text on a light background.
• Some people have trouble reading light text on a dark background.
• Also be consistent in your background – changing the background picture from slide to slide can be distracting.

Use Color Effectively

Do I Need to Use Fancy Visual Aids?
No!
Use whatever medium you are comfortable with – blackboard, overheads, powerpoint, slitex, beamer, etc., or a combination thereof.
As long as your take-home message comes through loud and clear, the medium you use is irrelevant.

Can or Should I Read My Slides?
• Preferably not.
• If you really must, then at least have the text in point form, and give the audience time to absorb the information in each point.

This is Not a Good Idea!

Abbreviations?
• Avoid abbreviations as much as possible.
• If you must use abbreviations in your slides, at least use the full name in your speech.

Summarize Your Presentation
End your presentation strongly:
• State a concise, positively worded, solid conclusion that summarizes and reinforces your take-home message.
Scientists Must Speak, page 14.

Outline
• Introduction – Focus on essentials
• Main Points – DOs and DON’Ts
• Closing – Concluding thoughts

Advice 1: Learn From Others
After you’ve attended a presentation, ask yourself the following questions:
• What was the speaker’s main result/point?
How did the speaker support this main result/point?
• Did I get lost? When, and why?
• In the presentation that I just heard, what worked for me and what did not?

Advice 2: Developing the Story of Your Research
• Talk to your peers, your supervisor, other professors about the context of your research.
• Create one-page summaries of papers you read, abstracting the essence of the paper, and writing a few lines about how the research relates to your own research.

Advice 3: Start Early, and Practice, Practice, Practice!
• Storyboard your presentation.
• Practice the first draft of your presentation in front of other members of your group.
• Ask for feedback, and be open to feedback.
• Revise your presentation.
• Practice your talk in privacy, with a timer.

Talks Should Be Stimulating
• Pay attention to the setting: what is the scientific or historical context of your topic?
• Pay attention to the plot: every presentation must have an objective, or take-home message.
• Pay attention to the characters: carry your audience by properly introducing listeners to variables, definitions, etc.

References
• Scientists must speak: bringing presentations to life, D.E. Walters and G.C.
Walters, Taylor and Francis, 2002
• Handbook of Writing for the Mathematical
Sciences, N.J. Higham, SIAM Press, 1998