These are actual student papers that were not designed to be web pages. They may contain historical, grammatical, mathematical, or formatting errors. These papers were graded using the criterion mentioned in the paper directions, and the writing checklist. The test review sheets and the WebCT tests are good indicators of the mathematics that was discussed in class during and/or after each presentation.
Olivia Stanley


 


Larien Chilton

Olivia Stanley

 

Dudley Weldon Woodard

 

 

     “The widespread American belief that blacks and females cannot learn mathematics as easily as white males is a self-perpetuating myth (Journal, p.270).”  This myth was certainly evident in the early part of the twentieth century when racial and gender inequities were widespread across the United States.  However, if one looked at the life and achievements of Dudley Weldon Woodard you would probably conclude that there is nothing to this myth.  Woodard was a brilliant individual that overcame the racial barriers of his time, and became only the second African American to receive a Ph.D. in mathematics.  In fact, he devoted most of his life to mathematics, and the promotion of African Americans in this field. 

Dudley Weldon Woodard was born October 3, 1881 in Galveston, Texas.  Not much is known about Woodard’s life except he was an African American who was growing up in a time in which racial inequities and segregation were in full force in the United States, especially in the south. However, Woodard remembered not feeling disadvantaged as he was growing up, as his father had a good job with the United States Postal Service (Journal, p.173).

 He attended Wilberforce College in Ohio and received a bachelor’s degree in 1903.  Woodard then moved to the University of Chicago where he received his second BS and an MS degree in 1906 and 1907 respectively (mad).  Then in 1928 Woodard received his Ph.D. in mathematics from the University of Pennsylvania.

Not much is mentioned about Woodard’s family life.  We do know however that he had a wife, who was the dean at Minor’s Teacher’s College in Washington, DC. Together Woodard and his wife had a son who, like his father, became a member of the mathematics department at Howard University (Journal, p.173). 

Woodard was a very dedicated and diligent worker.  After receiving his master’s degree and before entering graduate school, he taught at the collegiate level for two decades (upenn).  During this time he worked for seven years at the Tuskegee Institute, six years as a member of the faculty at Wilberforce College, and was the Dean of the College of Arts and Sciences at Howard University (upenn). 

Upon receiving his Ph.D. in 1928 from the University of Pennsylvania, he returned to Howard University and worked on establishing a graduate program in mathematics from 1929-1947 (mad).  He also started a mathematics library and sponsored many professorships and scholarly seminars while at Howard (upenn).

Although we could find nothing specific in our research of Woodard regarding why he chose mathematics as a career path, we can make an inference as to the reason.  As previously stated, Woodard did not recall feeling disadvantaged growing up (Journal, p. 173).  Using this and the fact that his father had a good job during this time period, we inferred that Woodard’s father probably supported his son’s desire to further his education.  This support from his family could have encouraged Woodard to become a mathematician. 

We also concluded that although Woodard probably had his family’s support, he might not have had a lot of support from society.  We arrived at this conclusion by examining the time period, which was full of racial tension, in which Woodard received his education and started his work as a mathematician.  This time period did not typically support African Americans and their desire to further their education.

 We did, however, find statements that other mathematicians had made about Woodard.  He seemed to be a highly regarded individual.  Deane Montgomery, former president of the American Mathematical Society and the International Mathematical Union described Woodard as, “an extremely nice man, well balanced personally (Journal, p.173).”  He was also regarded by Leo Zippin, who was an internationally known specialist in Woodard’s field as, “one of the noblest men I’ve ever known (Journal, p.173).” 

Although we do not know if Woodard worked with other mathematicians to write papers, we do speculate that he did collaborate with other mathematicians.  We arrived at this conclusion given that Woodard was a professor for many years, as well as the Dean of the School of Liberal Arts.  He basically had to collaborate with his colleagues in order to be an effective teacher and leader. 

Woodard was an independent researcher, who published several mathematical papers.  When he was working on his master’s degree at the University of Chicago, Woodard’s master’s thesis was published.  It was entitled Loci Connected with the Problem of Two Bodies (mad).  After receiving his Ph.D., he wrote and published a research paper entitled The characterization of the closed N-cell.  This paper was described as, “Being too far ahead of its time to receive the recognition it deserved (Journal, p.174).”  Woodard also published On two dimensional analysis situs with special reference to the Jordan Curve Theorem.  This was “the first research paper published in an accredited mathematics journal by an African American (mad).” 

Based on the fact that Woodard worked at the college level for many years, we can infer that he probably had many students throughout his career.  However he had at least one master’s student, William Waldron Schieffelin Claytor.  Claytor was “the most promising student in the inaugural year of Professor Dudley Weldon Woodard’s new graduate mathematics program at Howard University (upenn).”  Claytor later went on to provide a significant advance in a branch of point-set topology (upenn).

Dudley Woodard was once described by his colleague as, “severe and commanding in coffee color (Journal, p.173).”  He broke racial barriers to become only the second African American to receive his Ph.D. in mathematics.  He was also very “aware of his own dignity (Journal, p.173).”  He used the phrase “Black is Beautiful” in the 1930’s when this type of expression was considered wrong by most whites (Journal, p.173).  He often ignored the color signs and visited any men’s restroom that he wanted.  He also ate at many “nice” restaurants and the theatre in New York (Journal, p.173). 

Probably the best description of Woodard’s pride comes from a story in which he has to defend his house against an angry mob of white people.  This occurred when Woodard and his wife bought a house in a formerly all-white neighborhood near his place of employment, Howard University.  The cause of this mob was not mentioned, but we do know that Woodard was prepared and willing to kill in order to protect his family and home (Journal, p.174).

  Although Woodard wrote many mathematical papers, the one concept that we are going to focus on is the Jordan curve theorem. This was the basis of one of his three papers, On two dimensional analysis situs with special reference to the Jordan-curve theorem.  The idea behind the Jordan curve theorem is that every simple closed curve divides the plane up so that there is one inside component and one outside component.

In Axiom 8, R.L. Moore assumes that “every simple closed curve is the boundary of a region, that is, that every simple closed curve defines a bounded connected set of connected exterior having further properties implied by certain other axioms of the three systems (Woodard1, p.121).” Woodard’s chief purpose for his investigation of this theorem is to “replace Moore’s Axiom 8 by another axiom of such nature that no property of the simple closed curve is assumed (Woodard1, p.121).”  

Woodard also omitted some of Moore’s axioms that were not necessary to prove the others, and changed the wording in some of the ones that remained to make them clearer.  After making these changes Woodard wrote, “It is of interest to note that under these circumstances, the proof of the Jordan curve-theorem is based upon a set of axioms which contain no assumption as to the character of the boundaries of regions (Woodard1, p.122).”  Basically, Woodard’s work on the Jordan curve theorem was geared toward eliminating all assumptions that Moore had made about the boundaries of these closed regions.

An explanation of this theorem must begin with the definition of a simple closed curve.  A curve that is simple means that the line that defines its boundaries does not cross at any point along the curve.  In order for a curve to be defined as closed, it must begin and end at the same point.  So, the curve defined in the Jordan curve theorem begins and ends at the same point and does not cross itself at any point along the curve. 

Some examples of why it is hard to prove the Jordan curve-theorem are ideas related to the Koch’s Snowflake Curve.  Here is a picture that we will refer to in our discussion (scidiv).


As you can see, the snowflake curve is drawn by starting with an equilateral triangle.  On the middle third of each of the three sides, construct an equilateral triangle with sides that are 1/3 the length of the previous sides.  Erase the base of each of the three new triangles. In the picture above, the dotted lines represent where the bases of the triangle that have been erased. Continue this process on each side of every new triangle.  Notice that continuing this process infinitely many times forms the snowflake.

 As you can see the snowflake formed has infinite perimeter and finite area.  The infinite perimeter comes from adding infinitely many triangles to the outside edge of the curve.  The aspect of finite area is a bit more interesting.  If a circle is drawn around the original figure, the area of the figure remains inside the circle no matter how large the perimeter gets (rice).  These are two interesting aspects of the Koch Snowflake.

Another interesting fact is that this image has no derivative at any given point.  Most importantly, this snowflake is a simple closed curve because it does not cross itself, and you can pick a beginning and ending point that are the same point. 

     Embedded in the definition of a curve for the Jordan curve-theorem, is the criterion that you have a finite length curve.  If we do the snowflake iteration 1000 times and compare it to the result of doing the iteration 1001 times, one would observe that it’s really hard to tell when you are inside or outside of the curve.  In fact, some points switch. 

     Based on this snowflake curve as our example, we can explain why the proof of the Jordan curve theorem is nontrivial.  As the number of sides of the curve approaches infinity, it becomes more and more difficult to determine if any given point near the edge of the curve is located on the inside of the curve or on the outside of the curve. 

     Through Woodard’s attempts to solidify Miller’s assumptions about the Jordan curve theorem, he was able to develop a concrete proof of the theorem.  He was also one of the first to call this concept by the name Jordan curve-theorem.  Although the Jordan curve theorem seems obvious, Woodard showed us that the proof was nontrivial.  After all, his proof for this theorem was twenty-four pages long!  This proof turned out to involve complex analysis and a lot of paper, work, and time! 

     Dudley Weldon Woodard was a man of intelligence and dignity.  Not only did he break many racial barriers in his lifetime, but he also made history by becoming only the second African American to receive a Ph.D. in mathematics.  Woodard devoted his entire life to education and the advancement of students in mathematics.  His work on the Jordan curve theorem was evidence of his intelligence, determination, and dedication to mathematics.   He showed that something that seems so obvious, like the Jordan curve theorem, might actually have a nontrivial proof that deals with complex mathematics.  He also showed us that with a little drive and will power, anything is possible.

 

References

http://math.rice.edu/~lanius/frac/koch3.html  (Denoted rice in paper) Useful in describing the finite area of our Koch Snowflake.

 

http://www.math.buffalo.edu/mad/PEEPS/Woodard_dudleyw.html  (mad)  Useful for Woodard’s educational background, and for listing publications of Woodard.  

 

http://www.math.upenn.edu/History/bh/text99.html (upenn) Useful in tracking Woodard’s work.

 

http://www.scidiv.bcc.ctc.edu/Math/snowflake.html (scidiv)  Helped illustrate and understand the snowflake curve.

 

Journal of Black Studies.Vol.18 No.2,December 1987,170-190.

(Journal)  Useful for colleague’s comments, familial information, and multicultural aspects of Woodard’s life.

 

Priestly, H.A., Introduction To Complex Analysis, Revised Edition, 28-31.  Priestly  Not very useful in our research.

 

Woodard, D.W., On two dimensional analysis situs with special reference to the Jordan curve-theorem, Fundamenta Mathematicae 13 (1929), 121-145.  (Woodard1)  VERY useful for trying to interpret Woodard’s work on the Jordan curve-theorem (i.e. why he was working on it and what he was trying to do with it).

 

Woodard D.W., The characterization of the closed N-cell, Trans. Amer. Math.Soc.42 (1937), no.3, 396-414. (Woodard2)

Not useful in our research.

 

Special thanks to Dr. Rhoads and Dr. Sarah for helping us understand the mathematics a little better!

 

Web address for List of Published papers from MathSciNet from lab:

http://www.ams.org/msnmain?fn=200&SELECTED=View+All+Items+by+Selected+Author&author=651986%7CWoodard%2C+D.+W.%7C20