Katy Rountree

Olivia Stanley

**Clarence
F. Stephens**

During
a time in history when African Americans were considered inferior to whites,
few men of color were able to achieve educational success. However, Clarence Stephens overcame
many racial barriers to become only the ninth African American to earn a Ph.D.
in mathematics. In addition to
being a superior mathematician, he has also been highly regarded as a pioneer
of African Americans in mathematical education, as his methods of teaching are
recognized as some of the “most profound in producing mathematics majors
(MAD1).”

Clarence
Francis Stephens was born on July 24, 1917 in Gaffney, South Carolina
(SC). He was the fifth of six
children, three girls and three boys.
His parents were Sam and Jeannette Morehead Stephens (MAA). Sadly his mother died when he was two
and his father died when he was eight years old (MAA). All six children went to live with
their grandmother who died two years after Stephens’ father. No relative could take in all six
children, so the had to live with different relatives.

Stephens
lived with his great aunt Sarah in Harrisburg, North Carolina (NC), in a
three-room house. Two of the rooms
were bedrooms, one of which was occupied by the two elementary school
teachers. The other room in the
house was the kitchen, where all homework, dining, and socializing took
place. When Stephens became old enough
to go to high school, he contemplated running away to go to school in the
north, as there were no high schools in Harrisburg. To avoid having her brother run away, Stephens’ oldest
sister Irene arranged for him to go to the Harbison Institute, a boarding
school in Irmo, SC, provided that after she paid the first year’s tuition, he
would pay the subsequent years’ tuition.
Irene also made it possible for Stephens’ brothers to go to Harbison
Institute, with the same condition of tuition payment that Stephens had. A job working on the Harbison farm in
the summer paid for their winter schooling (DAAS). Stephens also worked as a kitchen helper and dusted and
swept the classrooms every day of the school year to supplement his room and
board.

An
indication of Stephens’ intelligence is shown by his placement test into
Harbison. He was thirteen years
old and placed in the eighth grade alongside students who were over twenty
(DAAS). In addition, Stephens’
teacher Dean Robert Boulware would have him present strategies for solving
mathematical problems to the classroom.
Stephens would also tutor his classmates and help them with their math
homework.

Stephens
was very popular in high school.
He played football and was on the varsity baseball team. Stephens was a well-rounded student. During his school career, he was
involved in various extracurricular activities. He held the lead role in his senior play, was elected to
class president his senior year, and was a good debater (DAAS).

Despite
their rough childhood, all six Stephens children graduated college (MAA). Stephens, his brothers, and one sister
attended Johnson C. Smith University (JCSU) in Charlotte, NC. Mathematics was the chosen field of
study for all the boys in the family.
Stephens graduated JCSU in 1938 and then enrolled at the University of
Michigan to begin his graduate studies in mathematics in the fall. Stephens received his M.S. in 1939 and
his Ph.D. in 1943 in mathematics from the University of Michigan (MAA).

While
Stephens was attending school at JCSU, his only plans were to become a high
school math teacher. His decision
to attend graduate school was seemingly impulsive. The Dean of the College of Liberal Studies, T. E. McKinney,
had returned from a visit to the University of Michigan, saw Stephens one day
during his senior year at school, and said to him that the University of
Michigan was where he needed to go.
So Stephens went to the University of Michigan for his master’s degree
(MABPM).

Stephens had not considered getting
his doctorate; he was just going for his master’s. He had considered going to law school. As a lawyer Stephens believed he could
help the poor people he felt were always being exploited. However, he could not become passionate
about law, so he just figured he would teach high school math. Like his decision to obtain his
master’s degree, his decision to further his education with a doctorate degree
was seemingly impulsive. While
talking to one of his professors, George Rainich, one day, Stephens was
encouraged to return to school in the doctoral program. Having a professor assume his return to
further his education opened Stephens’ eyes to his potential. Professor Rainich’s comments were so
inspiring that Stephens set more personal goals that he could achieve in the
mathematical field.

Stephens faced great
difficulty in acquiring his graduate degrees. During the time when Stephens was in school, he had to wait
tables to pay for tuition, as there were no teaching assistant positions for
African Americans (Stephens). He
also worked as a deliveryman for a local drug store that paid six dollars a
week (DAAS). In addition, Stephens
served time in the Navy while working on his degrees.

After completing his tour of duty in
the Navy, Stephens took a position as a professor of mathematics at Prairie
View A. and M. College in Texas in 1946.
He spent little time in this position because he was given the
opportunity to become the department chairman in the mathematics department of
Morgan State University in Maryland.
After fifteen years at Morgan State, 1947-1962, Stephens again took on
the role as a mathematics professor, but this time it was at the State
University of New York (SUNY) at Geneseo.
He stayed at SUNY at Geneseo for seven years before going to the SUNY at
Potsdam (MABPM pp. 63-65).

The SUNY at Potsdam was
predominantly a training school for secondary mathematics education majors when
Stephens arrived. In addition, in
1969, there was no faculty in the math department who had received their Ph. D.
from SUNY at Potsdam. Furthermore,
no student who had graduated with a B. S. in mathematics from Potsdam had gone
on to earn a Ph. D. in mathematics.
Stephens believed this should be changed. Thus, he devoted much time and effort into developing
methods and strategies of teaching mathematics to encourage students to pursue
advanced degrees in mathematics.
His methods were very successful in that at least eleven of his students
went on to earn a Ph. D. in mathematics (MABPM).

Because of his drive and ability to
learn math and his desire to educate others in math, Stephens has received many
honors and awards. He received the
Julius Rosenwald Fellowship in 1942.
In addition he received a Ford Fellowship and had the honor of being a
member of the Institute for Advanced Study in Princeton, New Jersey from
1953-1954, working with such well-known people as Dr. Albert Einstein. He received an honorary Doctorate of
Science from JCSU in 1954. In
1962, Stephens was honored for his many distinguished contributions to
mathematical education by Governor J. Millard Tawes of Maryland, and again in
1987 by Governor Mario Cuomo of New York (NY) (MAA). He received the SUNY Chancellor’s Award for Excellence in
Teaching in 1976-1977 while he was working in Potsdam, NY. He was inducted and
permanently placed in the National Museum of American History, Smithsonian
Institute, as a part of the “A Living History Project on Black Americans In the
Sciences” (MAD). In the past
eleven years he has received three more honorary doctorates: from the
University of Chicago in 1990, from SUNY in 1996, and from Lincoln University
in 2000 (MAA and MAD).

Today
Stephens lives with his wife Harriette on their farm in Conesus, NY. His mathematical influenced is evident
in that his daughter, H. Jeanette Stephens, and his son, Clarence F. Stephens
Jr. both have advanced degrees in mathematics. Even in his retirement Stephens is often asked to speak at
colleges and universities in the United States and Canada about mathematics
education (MAA).

Dr.
Stephens chose the topic of difference equations for his doctoral project. His advisor thought this topic was too
difficult for him, but the fact of the matter was, once Stephens got the
problem into his head, it took only two weeks to complete his work (DAAS,
p.299). Perhaps Stephens’ reason
for choosing this topic was discussed in his paper, “Nonlinear Difference
Equations Analytic in a Parameter.”
In this paper, Stephens quotes, “Up to the present time but little
progress has been made in the development of a systematic theory of nonlinear
difference equations from the point of view of general function theory
(Stephens2, p. 268).” The main
purpose of this particular paper is to investigate the “solutions of nonlinear
difference equations analytic in a parameter (Stephens2, p.268).”

A
student of mathematics could better understand the nature of Stephens’ work
with a deeper knowledge of nonlinear difference equations. These particular equations have been
described as, “very complicated equations which, however, have many practical
uses (DAAS, p. 299).” Difference
equations can be used to describe things such as changes of the texture of oil
under the differing conditions of heat and pressure in an engine (DAAS, p.
299). They also can be used to
model certain interactions of individuals such as the spread of rumors and sicknesses,
and natural growth such as the growth colonies in species populations
(Bauldry). Difference equations
are used to model discrete data such as traffic patterns and temperature
measurements.

In
order to understand a model of a difference equation, one must first understand
the definition. A difference
equation is an equation written in terms of changes of a particular function.
These functions, which take the form F(n, a_{n+2}, a_{n+1},…,a_{n}),
define a rule to transform whole numbers into elements of a sequence.

The
rule F, gives a linear equation if the terms are multiplied only by a constant.
For example, the function N_{t+1} – N_{t} = F – G, is a
difference equation representing the traffic patterns on and off a particular
highway. In the general form the traffic equation looks like F (t, N_{t+1},
N_{t}), where the rule F gives the form N_{t+1} – N_{t}
= F – G combined with one initial condition, makes the generated sequence
unique. This equation is a linear equation because the terms, N_{t} and
N_{t+1}, are multiplied only by constants. If the terms are multiplied
by one another the equation is nonlinear. If F (n, a_{n+1}, a_{n})
takes the form of a_{n}^{2} + a_{n+1} = a_{n,}
then it is nonlinear because a_{n} is a term and is multiplied by
itself. The factors that make a difference equation nonlinear are powers,
products, roots, or functions. Here is a difference equation that is nonlinear
by all these factors: F (n, a_{n+3}, a_{n+2,} a_{n+1},
a_{n}) with the rule, Ö(a_{n+3)} * a_{n+2}! + tan (a_{n+1}) = a_{n}^{5
}(Moorefield).

The graph of a linear difference
equation is not necessarily a line. For example, some graphs have been known to
be parabolas. Stephens worked on
nonlinear difference equations for his doctoral dissertation. One can tell that these equations are
nonlinear because their forms contain aspects of nonlinear equations mentioned
above.

An initial model of a nonlinear
difference equation can be formed by using the potential spread of a rumor
throughout a specific population.
To do this, one would need algebraic representations of time (n which
represents one more than the actual number of time intervals passed), the
number of people who know the rumor after n-1 time intervals (a_{n}),
and the number of people who do not know the rumor. One will obtain this representation by taking the total
population (P) and subtracting the number of people who do not know the rumor
(a_{n}). The result is P –
a_{n} equaling the number of people who do not know the rumor.

Based
on this information, one can build the difference equation a_{n} = a_{n-1}
+ c * a_{n-1 }* (P - a_{n-1}), where a_{1} = b. Our values that remain constant are c,
which is the constant of proportionality that governs the rate at which the
rumor actually spreads, P, the population size, and b, which is the number of
people who start the rumor. This
leaves a_{n }and n to be the variables. Because the a_{n-1 }terms are multiplied together,
the resulting degree of this difference equation will be two, thus making it
nonlinear.

Since
the difference equation that is formed has three parameters, c, P, and b, it is
necessary to estimate the values of these constants. The values chosen in the book by Bauldry on page 27 imply
that the population size should be 1.00 because a large population size is
unlikely to affect the way a rumor spreads. The initial value of b is taken to be seven percent of the
population, and c is taken to be 0.05, which assumes the number of people who
learn the rumor equals approximately five percent of the interaction between
people who know and people who are unaware of the rumor (Bauldry, p. 27).

Substituting
the indicated values for the appropriate parameters forms the new difference
equation which is of the form a_{n }= a_{n-1 }+ 0.05 * a_{n-1
}(1 - a_{n-1}). The
graph of this difference equation looks like the following:

(Bauldry, p. 28).
This graph can be obtained by using a graphing calculator and plugging
the equation into the “y =” menu.
In order to do this, one must substitute the variable “x” in for a_{n-1}. Thus, the equation entered into the
graphing calculator will be of the form y = x + 0.05 * x (1 – x). The graph of this difference equation
is nonlinear because the equation itself is nonlinear. It is sometimes hard to tell the
difference between a linear and a nonlinear difference equation just by
examining their graphs. In fact,
it is necessary to look at the equations because parts of the graphs could look
similar.

These nonlinear difference equations
are similar to the equations Stephens worked on, only these are much
simpler. The previous example
provides just an introduction and overview of nonlinear difference
equations. Stephens wrote papers
on “Difference equations having no linear terms in dependent variables and
parameter (Stephens2, p.268).” An
example of a system of difference equations that he worked on in his
dissertation is the system: y_{i
}(x + 1) = f_{i }(y_{1} (x), . . . , y_{n }(x); P
(x), x), f_{i }(0, . . ., 0; 0; x) = 0 and (i = 1, . . . , n), where
P(x) represents a parameter that is independent of y_{i}(x), and (i =
1, . . . , n) is periodic of
period one with respect to x (Stephens2, p. 268). He formulated properties of these equations and proved their
convergence. Stephens displayed an
in-depth knowledge of such equations as he published two papers based on
nonlinear difference equations.

Clarence Francis Stephens is a very
intelligent and dedicated man. He
had to surpass many racial obstacles and unfortunate circumstances to become
the ninth African American to receive a Ph.D. in mathematics. Stephens’ love for mathematics
developed at a very young age and he has devoted most of his life to the
field. He was once quoted as
saying, “More than fifty years ago I came to the conclusion that every college
student who desired to learn mathematics could do so. I spent my entire professional life believing this was the
case (MAD1).” His many awards and
honors in mathematics further prove that Stephens is truly a great African
American pioneer in the field of mathematics.

Bauldry,
Ellis, Fiedler, Giordano, Judson, Lodi, Vitray, West. 1997. __Calculus: __

** Mathematics
& Modeling**. Addison Wesley
Longman, Inc., Reading

Massachusetts.
(Denoted *Bauldry* in paper) Used
in math part of paper to

obtain
rumor example of nonlinear difference equation.

Datta,
Dilip. ** Math Education at its Best: The Potsdam Model.** Rhode Island

University
Press, Kingston, RI, pp 61-65. 1993. *(MABPM)* Good source for

his
teaching methods.

__ __

**Discussion
with Dorothy Moorefield.** (*Moorefield*) Very helpful with the math.

__ __

** Distinguished
African American Scientists of the 20^{th} Century.** “Clarence

Stephens.” P.296-301. (*DAAS) *The best source for biographical,

educational,
and career highlights.

http://www.maa,org/summa/archive/Stephn_c.htm
(*MAA)* Good overview of life

and
work.

http://www.math.buffalo.edu/mad/morgan-potsdam_model.html
(*MAD2)* Included

repeated
information.* *

http://www.math.buffalo.edu/mad/PEEPS/stephens_clarencef.html
(*MAD1)*

Excellent
biographical overview of Stephens’ life and work.

http://www.potsdam.edu/MATH/quotes.html
(*POTSDAM)* Contained

quotes
from Stephens.

**Interview
with Clarence Stephens **on February
21, 2001. (*Stephens) *Provided

an
all around good basis of information.

* *

*Nonlinear
difference equations containing a parameter**, *Proc.
Amer. Math.

Soc.1
(1950), 276-281. (*Stephens3*)* *Used to examine Stephens’ work on

difference
equations.

** Nonlinear
difference equations analytic in a parameter, **Trans. Amer. Math.

Soc.64**
**(1948), 268-282.** **(*Stephens2) *Also used to
examine Stephens’

work
on difference equations.