Project 2: What is a Mathematician?
(Presentation and Handout)
In this segment we will examine the way that mathematicians do research
and the kind of problems that they work on.
While we will mainly focus on the mathematics,
you should try and identify with the mathematicians and their
struggles and relate this to the way that you
do mathematics, and you should also continue to think about what
mathematics is, and the useful problem solving techniques that arise from its
study.
We will highlight the validity of diverse styles and diverse mathematical
strengths and weaknesses.
We will see that there are lots of different ways that people are
successful in mathematics.
We will also examine the changing roles of women
and minority mathematicians over time.
I have worked hard to accumulate
good references for you.
I will give you both web and paper references.
Step 2: 7 - 10 minute
PowerPoint Presentation and Classroom Handout
Prepare a PowerPoint presentation and classroom handout by using the
related checklists.
The presentation computer file (whatever.ppt)
must be sent to Dr. Sarah as an attachment on WebCT
before
3pm for a Monday presentation and
11:30am for a Tues/Thur presentation.
It is your responsibility
to make sure that this is received by Dr. Sarah
and runs correctly. Do not expect to load
your presentation from a disk - the computer does not have a disk drive -
but do also have the presentation backed up on disk so that you could
send it from school if necessary.
Be sure to follow the
presentation checklist and to prepare great presentations!
Oral presentations my be summed up as follows:
"Tell them what you're going to tell them. Tell them. Then tell them what
you told them".
Don't be scared of this repetition. Sometimes repetition is the only way to clarify misconceptions. Naturally, this means that you should repeat things in different ways, and not quote yourself verbatim.
Bring copies of a related handout.
These are DUE at the beginning of class on the
day of your presentation, but I am happy to make the photocopies for you
if the completed handout is given to me
a day in ADVANCE during office hours.
Otherwise you must make the copies yourself. Follow the
handout checklist.
Group Work
Group work on major
assignments will be
self-evaluated and these evaluations will be
taken into account in the determination of the final
grade. So, your job is to make sure that you do your
part to make sure you are working in a
group effectively.
Inequalities in group work WILL
be addressed.
Step 3: Dr. Sarah Will Answer the Following Questions on the
Mathematics that is Related to your Mathematician
Thomas Fuller (1710-1790)
Speed of Mental Calculations, Calculator and Computer Time
How would we do Fuller's calculation of the number of seconds
a man who is seventy years, seventeen days and twelve hours old has lived
by hand?
What affects calculator and computer calculation time?
Compare Fuller's times to various calculator and computer times
(the first calculator, the first computer, eniac, and modern calculators
and computers).
Is there a limit to how fast a computer can calculate?
Maria Agnesi (1718-1799) Witch of Agnesi and Calculus
What is the witch of Agnesi? How did it receive that
(derogatory) name?
How do you construct it geometrically?
If the generating point
is dragged far enough to the right,
why won't the point generated on the curve be located at y=0?
How does it relate to Agnesi's mathematics?
How is it used
in real life?
What are some of the applications of calculus to real-life?
Sophie Germain (1776-1831)
Sophie Germain Primes and RSA Coding
What is a Sophie Germain prime?
Why did she come up with Sophie Germain primes?
What is the largest Sophie Germain prime that has been found?
How many digits does this have?
What is the definition of a mod b?
How do modular arithmetic and Sophie Germain primes relate to
RSA coding?
Carl Friedrich Gauss (1777-1855) Non-Euclidean Geometry
What is Euclid's 5th postulate?
What is the form of the
5th postulate that we learn in
high school? What is the negation?
What two geometry possibilities does this negation give rise to?
How does this relate to Gauss?
What are some other areas Gauss worked on?
Georg Cantor (1845-1918) The Size of Infinity
What are the natural numbers?
What are the real numbers?
Using an argument that is similar to Dodgeball,
show there more real numbers than natural numbers.
Srinivasa Ramanujan
(1887-1920)
Chebyshev's Theorem and
Estimating the Number of Primes Less than a Given Number.
What is a prime number? Why are they of interest?
What is the statement of Chebyshev's Theorem? How does this
relate to Ramanujan?
What were Gauss' estimates of the number of primes less than or
equal to a given number? How does this relate to Ramanujan?
Why are people still interested in Ramanujan's notebooks?
Paul Erdos
(1913-1996) The Party Problem
What is the statement of the party problem? How does this relate
to Erdos?
Why does a 6 sided polygon colored red with a 6 sided embedded
star with edges colored blue prove that 5 people at a dinner party is not
enough to ensure that there are at least 3 people who are either complete
strangers or acquaintances?
To show that if there are 6 people at a
dinner party then there are at least 3 people who are either
complete strangers or acquaintances,
once we reduce to just looking
at 4 of those people (3 who all either know or don't know the first person),
how do we complete the argument from here?
David Blackwell (1919-) What is Game Theory?
What is the prisoner's dilemma? How does this relate to what
David Blackwell worked on?
Give an array of payoffs filled with numbers that
represents the prisoner's dilemma, and explain how to read the array.
If a person is deciding what to do, why does it make sense
(when looking at the possible cases) for him
to confess?
What are some applications of game theory to real life?
Mary Ellen Rudin (1924-) What is Topology?
What is topology? How is it different from geometry and projective
geometry?
Why are a basketball and a football the same in topology?
Why are an iron and a mug with one handle the same in topology?
Why is a vest and a mug with two handles the same in topology?
Why are objects with different number of holes
different from each other in topology?
Frank Morgan (195?-) The Double Bubble Problem
Why is the sphere the least area way to enclose a given volume for
a package?
If a sphere and a box have the same surface area, then which
will have the largest volume?
What is the Double Bubble Problem?
What did Frank Morgan prove about it?
Did he have to check every possible double bubble?
Ingrid Daubechies (1954-) Wavelets
How can images be stored on a computer?
What is image compression? Why do we need it?
How are wavelets related to image compression?
Why are wavelets better then JPEG?
What are other applications of wavelets to real life?
What do wavelets have to do with Ingrid Daubechies?