Mathematicians Must Speak:
The DOs and DON'Ts of Giving
Effective Mathematical Presentations
Gerda de Vries and Thomas Hillen
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
• 1. Introduction – Focus on essentials
• 2. Main Points – DOs and DON’Ts
• 3. Closing – Concluding thoughts
1.1 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 topic of your talk and
the needs of the 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.
1.2 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
• 1. Introduction – Focus on essentials
• 2. Main Points – DOs and DON’Ts
• 3. Closing – Concluding thoughts
2.1 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.
2.2 Clear Statement of the
Objectives
2.3 Clear Statement of the Conclusions
2.4 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?
2.5 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.
2.6 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.
2.7 Dense Statements:
Losing your Audience
Theorem
k is an even perfect number if and only if it
has the form 2n-1(2n-1) and 2n-1 is prime.
Proof
Suppose first that p = 2k-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
2n-1(2n-1), where 2n-1 is prime.
We can check that the statement of the theorem
holds for the first two perfect numbers:
6 = 2(3) = 21(22-1) = 2n-1(2n-1) for n=2
28 = 4(7) = 22(23-1) = 2n-1(2n-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?
2.8 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.
2.9 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!
2.10 Abbreviations?
• Avoid abbreviations as much as possible.
• If you must use abbreviations in your slides,
at least use the full name in your speech.
2.11 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
• 1. Introduction – Focus on essentials
• 2. Main Points – DOs and DON’Ts
• 3. 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?
• What are meaningful questions I could ask?
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.
The Smile
• Conclusions should answer the objectives
from the beginning.
• Audience should feel good, they should
have the impression that they learned
something interesting.
• Audience should burst with questions to
ask.
• You should feel good too.
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