Math Stuff related to classes

This page is under construction!

For now, I'm going to work on posting here stuff I've
already written.


* "How to give a 20 minute talk"
See below for my often requested manifesto.

* The Mathematics Bookshelf,
A list of great and cheap math books every undergrad major should get.

*"How to read a math paper"

*"How to get a good letter of reference"

*"How to teach calculus"

How to Give 20 Minute Talk!

Department of Mathematics
University of Pittsburgh

You are welcome to distribute this (not for profit) and use it, cited
appropriately. Please let me know if you have any suggested additions,
modifications or deletions.

1. Introduction.

While we would all like to have a job where we can indulge our
imagination without any noise from the outside world or obligation for our
reverie to have any impact on it, science, engineering and mathematics is
NOT where such jobs exist. Broadly, the job of scientific research can be
summed up as C-cubed= C3:

Creation of new ideas,
Connection of ideas to existing work, and
Communication of your ideas to others.

Communication is a learned skill and is usually done in the format of a 20
minute conference presentation of technical research and a 50 minute
colloquium presentation of technical research plus its larger backdrop. (
Giving a good talk can also be critical to getting a job so your
motivation is not purely altruistic.)

Giving a good talk is a learned skill so the goal is to improve
continually your presentations while (especially at the start of your
career) avoiding making any really bad mistakes. It is most important to
avoid certain mistakes which are guaranteed to make your talk a disaster
(no matter how brilliant the other 19-1/2 minutes were).

The key in this learning process is to think back over all the
talks you have heard: which were good (In particular, pick a topic that
you completely understand and that you think is an exciting and important
topic) , (and why), which were bad or boring (and why). The secrets are
all there. This report gives a few pointers about how to avoid the biggest
mistakes during your learning based upon my own experiences. If there is
someone in your department who gives good talks, ask him or her what their
secrets are (and email their answer to me please).

I will assume that you are giving a technical talk using an
overhead projector. There is a current trend to use power-point. This can
make your talk a lot better but it can also empty your talk of all
scientific content. I suggest you start with the overhead then move to
power-point. Also, if you are using power-point, be sure to bring a copy
on transparencies in case there is a computer glitch. I will also assume
it is a technical talk wherein you really want to communicate a scientific
idea. There are also "Management Style" talks given where the speaker
gives a fast overview of the efforts of a whole research group. You
probably won't be giving one of these at the start of your career!

The first transparency should be prepare with a text editor (such
as MacWrite) and contain



Email Address and Web Page

Graphic or Other Schematic


If the work is joint, put between your email address and the
graphic "joint work with "

Colors are very desirable. If you have one or two keywords in your
title, print them in block letters and fill them in with a red (or other
color marker). Use colors on every page to highlight the most important t
idea of that page.

If you have no idea what picture to include, just get a university
logo and pit it there. It's much better to relate your work to scientific
trends in your field, large scale science, etc. through the picture. For
example, a talk on thermal simulation of a tile could have a photo of the
space shuttle inserted on page 1. The goal of this is to get as many
people as possible interested in your topic immediately. If you have a
very nice color graphic, it's well worth the 99 cents to have Kinko's
color Xerox it onto a transparency. The most common error in talks is
assuming that everyone automatically understands your problem or is
interested in it!

A 20 minute talk means 5 transparencies of scientific interest or
4 plus some figures or schematics which are so clear as to be immediately
understandable. If you need a sentence in your picture's caption then is
counts as a page of text. This scales up with time, e.g., a 50 minute talk
equals 12-13 transparencies. Each transparency must have between 8 and 14 lines on it, absolutely no more than 14 and preferably 8!

For a 20 minute talk, the first (non title) slide should be a
problem statement, as nontechnical as possible and, if space allows, list,
in words, the essential fundamental difficulties in the problem and
questions asked. The second transparency is a survey of the state of
knowledge, list names, year and (nontechnically) what they did. The goal
is not to survey everything but show that your work is the next logical
and important step in the area. For example, the slide might have a list

1937 Hobson and Hobson: Woodchucks discovered.

1937-53 Intensive study of behavior and properties of woodchucks.

1962 Smith, Reilley, Subramanian and Lee: Transportation of
wood elements by woodchucks first proposed.

1994 (Your Initials): First quantitative studies of wood transport
abilities of woodchucks: "How many chucks could a woodchuck chuck?"


Do not use "et al." in listing names. Et or al might be in the
audience. They might have been the person that really did the work. They
might also be refereeing your next paper or grant proposal!

The third slide should be a statement of your principal result,
the one to which you have been building up! Be careful how you do this
though. You can only really state 2 of the 37 underlying assumptions of
your theorem or experimental result. Pick the two key ones and summarize
the others with statements like:

"Under a smoothness condition"



Be prepared, after your talk, to give a precise form of your
result. Prepare an extra slide with all the conditions and only use it if
someone asks about the other 35 assumptions.

If there is space remaining (if not this goes onto the last
transparency), give a summary of your methodology, techniques and tools
used. For example, your next slide might resemble:

Theorem. (Under a regularity condition on the wood elements) The woodchuck chucking ability of a "generic" woodchuck is proportional to latitude found.

Ingredients: (or other such word)

" Extensive field tests.

" Modeling and Simulation upon CMS Supercomputer.

" Duality argument in L.C.T.V.S.


If you're working in applied areas you need to delineate which
parts are empirical and which can be rigorously justified.

A schematic or interesting example can follow on the last
transparency together with a summary of your contributions (what did you
do after all!).

2. The Rules for Avoiding Disaster.

If you violate any of these rules you will end your academic career.


A1:WRITE BIG!!! 8-14 lines per transparency. Use colors whenever possible
to draw peoples' eyes to the one key idea on that page.

which don't require a lot of explanation. This scales linearly so 50

A3: Focus on the key ideas, big pictures, etc, not eh details.

A4: NEVER Xerox a normal typed page onto a transparency.

A5: STRESS the geometry, use schematics.

A6:FIRST PAGE: Title, name, plus picture connecting your work to "big

SECOND PAGE: Problem statement, why is it difficult and important?

THIRD PAGE: Previous work leading up to your great step forward.

FOURTH PAGE: Your main result plus methodology.

FIFTH PAGE: Interesting example or application or explanatory schematic.

How to Avoid Disasters: Seven Deadly Errors.

DE1: ADHERE strictly to A1-A8.

DE2: LOOK at the audience, speak slowly and don't mumble. Focus upon
friendly or sympathetic (looking) people. Don't make eye contact with
people who appear hostile - this invites an interruption or hostile

DE3: DON'T read from a script of a talk. Use a written version as a
prompter not as a script.

DE4: SUPRESS all nervous habits. A common example if you talk with your
hands in your pockets, remove all change and keys before the talk.

DE5: DON'T go too fast. Your talk with transparencies should be paced as
if you were writing them on the blackboard (practice your talk once as a
"chalk talk").

DE6: REMEMBER that four issues must be addressed in your talk:

(a) Importance: Why should anyone care about your problem?
(b) Novelty: What new grounds are you breaking?
(c) Nontriviality: What are the difficulties of this problem and
how were they overcome?
(d) Validations: How do you know your results are correct?

DE7: If you plan to use technology in your talk, go to the room beforehand
and try the machines.

3. How to Handle Questions.

Q1: All questions:

(1.1) Look at the person while he or she asks it.
(1.2) Don't interrupt, let them state it themselves. Repeat it to
the audience, perhaps with minor restatements, explanations and
background material.
(1.3) Then answer it graciously!

Remember this might require an explanation (short) of the question
you are answering to the whole audience not just the questioner.

Q2: Hostile Questions:

Repeat 1.1, 1.2, and 1.3 above then don't look at that person
again! Don't get involved in a back and forth quibbling contest,
offer to discuss it late. As you keep your calm, the questioner
will look worse and worse and you will come off better and better.

Q3: If you don't understand the question, say so and ask to discuss it

Q4: Questions which are really long rambling statements:

Just let them talk themselves out, and then give a succinct
Question: "But in 1906, I studied woodchuckery using
(blah-blah-blah) and we found (blah-blah-blah). Why didn't you do
it my way?"
Answer: "I considered that method. It is certainly valid but I
took another approach." Say this with a smile, immediately call on
someone else.

Q5: Incredibly stupid questions:

These are much more common than hostile ones. Don't laugh or
ridicule them. Try to explain why it doesn't apply but then relate it to
a good question, answer that and thank the questioner for bringing
up the topic.
Dealing with these questions is harder than with hostile

Q6: If a question makes you nervous:

Anytime you feel nervous, take a deep breath, slow down, let out
half the air, bear down and begin talking. This will lower your
voice, slow it down and no one will even suspect you are nervous.


How to Make It a "Good" Talk.

GT1: Your talk should be personal (show how you view the area). Be
positive and look at the big picture. Don't assume though that anyone
understands or cares about your problem. You must first relate it to
something in which they are interested.

GT2: First do the universal aspects of the problem, then slightly more
detailed aspects, etc. Don't overestimate your audience. Try to begin with
a phrasing of it that is so general that many people have seen special
cases of it in their own work.

GT3: Stress the geometry; try to illustrate your key conditions with

GT4: It's good (though difficult) to involve the audience in a positive
way at the start of the talk. A humorous antidote from the history of the
area, leading in to your work, is often a good way to begin.

GT5: (P.T. Barnum's maxim) "Tell them what you are going to do, do it,
then tell them what you did." (I don't really know what this means but it
is repeated so often that it must be important.)

GT6: Charisma:

The more abstract and technical the subject the more important
it is to use concrete terms and action words. For example,
compare the following presentation of Monte Carlo
"The main program calls a random number generator with a
specified distribution. The first or second index is incremented by one unit depending upon the subinterval to which the random number which is generated belongs"

"We roll a dice (gesture). Depending upon the number that comes
up, we jump one step North, South, East or West on the grid".

Charismatic people tend to use sensory words a lot (touch, feel,
taste, smell, hear and see). For example:

"We have certain doubts about the techniques"

"It just doesn't smell right because"

Using a concrete, sensory description of an idea is a good way
to introduce preface or motivate a technical discussion.

GT7: Pacing: practice your talk so that it's paced correctly. You
shouldn't have to jam everything into the last 5 minutes. Try to have
something in the middle of your talk as a change of pave (to jar the
sleepers from lethargy). Above all, don't drone on in a monotone, be a
little unpredictable.

5. If You Are Nervous.

Remember that it's normal to be excited and a bit nervous
before a talk. Be nervous, but avoid such things as taking fast in a high
squeaky voice, shaking all over and distracting nervous gestures.
Practicing your talk a few (e.g. 2) times is a very good way to
overcome the worst of the effects of nervousness.

N1: Don't write on your slides or point to things on the transparency if
you are nervous. If your hands are a little shaky, the projector will
magnify their shakes onto the screen. Write on the nearby blackboard and
point to your screen instead!

N2: Don't go back and recycle through your slides after each question.

N3: Focus only on friends (or friendly face) in the audience. Smile; enjoy
your subject and the opportunity to talk about it.

N4: If you start to speak quickly or in a squeaky voice, take a deep
breath, let out about half, bear down and begin speaking. This will slow
and lower your voice.

N5: Repeat (N4) before answering every question. Begin your answer by
explaining to the others in the audience why the question is interesting.

N6: Anticipate your nervous habits before your talk and do something to
make them impossible. For example, if you are nervous and like to talk
with your hands in your pockets, remove all your coins and keys
beforehand. I recall attending one very good talk that was ruined because
the speaker was jingling his keys in his pockets. Everyone sitting around
me was discussing how much change he had ("What do you think-about a buck
forty?") rather than the mathematics of the presentation.

N7: Preparing your slides. If you are giving an important talk and you are
nervous about it, it is very easy to put too much information on each
transparency. Here's one way to avoid that and yet have all you want to
say at hand. First write your talk on paper (just like each sheet is a
transparency). There will be way too much detail. Then practice your talk.
As you do, underline on each page the things you stress. When you prepare
your final transparencies, only put the underlined parts on them. Keep the
original "script" next to the overhead projector to use as a prompter. Do
not read off it though!