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.
Marjorie Lee Browne
Marjorie Lee Browne was born on September 9, 1914 in Memphis, Tennessee. Her birth mother died shortly after her birth so her father, Lawrence Johnson Lee, and her stepmother, Mary Taylor Lee, raised her (Kenschaft, 3). Lawrence Lee, a railway postal clerk, had a great love for mathematics. He was known around Memphis as a whiz at mental math (Williams, 1). Marjorie has been quoted to say that she knew she loved math even from an early age, mostly because of the enthusiasm her father shared with her (1).
Her stepmother, an elementary school teacher, home schooled Marjorie until eighth grade (1). Marjorie always looked fondly on these lessons, which she commonly referred to as “life lessons” (1). However, when she reached the high school years, her father took it upon himself to make sure she got the best education possible. He enrolled Marjorie in LeMoyne High School, a private school established after the Civil War by the Methodist Church to educate African- Americans (1). Marjorie was a very gifted student and graduated after only three and a half years (Kenschaft, 3). During this time, the depression was hitting America’s economy hard, but Marjorie was able to attend Howard University through a combination of scholarships, job, and loans (Williams, 1). She graduated cum laude in 1935 with a B.S. in mathematics (Kenschaft, 3).
Soon after Marjorie began to think about graduate school. She attended the University of Michigan and received her M.S. in mathematics in 1939 (4). Friends who knew her while she was there say that Marjorie was a completely honest person. As soon as someone showed an interest in math, she stuck with them. Otherwise, she had no time for them at all (4). Upon graduating from the University of Michigan, Marjorie joined the ranks of the very select group of women with master’s degrees in mathematics at this time.
She then joined the faculty at Wiley College in Marshall, Texas and worked toward her doctorate degree during the summers (4). In 1947, she took a leave of absence from the college to become a teaching fellow at the University of Michigan (4). Marjorie wrote her doctorate dissertation on, “On One Parameter Subgroups in Certain Topological and Matrix groups (4). The year prior to her graduation, 1948, she was elected to Sigma Xi and became an institutional nominee for the American Mathematical Society (4). She obtained her Ph.D. from the University of Michigan in 1949, becoming only the second African-American woman to receive a doctorate degree in mathematics (Williams, 1).
Immediately after receiving her Ph.D. she began teaching at North Carolina Central University in Durham, North Carolina (1). She remained at NCCU until her death in 1979 (1). She made many noteworthy contributions during her tenure there. Marjorie was the only faculty member in the mathematics department for the first 25 years of her employment there (Kenschaft, 2). This high achievement allowed her to become the department head beginning in 1951 until 1970 (2).
Marjorie also received numerous awards while teaching at NCCU. She was a Ford Foundation fellow, sponsored by the Fund for the Advancement of Education in the academic year 1952-53 (4). She cared very much for her students’ continuing education. She taught 15 hours a week and was also graduate adviser for 10 Master’s Theses in the Department of Mathematics (2). Marjorie firmly believed that education was a necessity and the key to a better life. She felt her students had to have confidence in themselves and that they needed to believe in themselves in order to be successful (Morrow, 22). Marjorie took pride in being the one to instill this confidence and pride in them.
Marjorie also had a deep interest in continuing education for secondary teachers (Williams, 1). She led NCCU to become the first predominantly Black institute in the United States to be awarded an NSF Institute for secondary teachers of mathematics (Kenschaft, 2). Marjorie also authored four sets of notes explicitly for use in these institutes: “Sets, Logic, and Mathematical Thought” in 1957, “Introduction to Linear Algebra” in 1959, “Elementary Matrix Algebra” in 1969, and “Algebraic Structures” in 1974 (Williams, 1).
Marjorie did a lot to help improve the notoriety of NCCU in both the mathematics and education worlds. For instance, in 1960, she was the principal writer of a proposal for a $60,000 grant from IBM to fund one of the first Electronic Digital Computers to be used for academic computing (Kenschaft, 2). This was a landmark because it was at a predominantly minority university. “Her manifestations of conspicuous attainment and scholarship coupled with her dynamic academic leadership, inspired many high school teachers to receive graduate degrees or advanced training and, thereby, she contributed significantly to the improvement of the quality of Mathematics Education in schools and colleges throughout North Carolina and the South. “ (2). In 1975, Marjorie became the first recipient of the W. W. Rankin Award for Excellence in Mathematics education given by the North Carolina Council of Teachers of Mathematics (3). Her announcement read that she pioneered in the Mathematics Section of the North Carolina Teachers Association helping to pave the way for integrated organizations (3).
Marjorie concentrated her studies on linear and matrix algebra. Her only published work; “A Note on the Classical Groups” explains her ideas on groups and topology more in-depth (Williams, 1). To understand the concepts she used, one must comprehend what a group consist of as well as the ideas presented in linear algebra.
A group is a set G with operations * such that
1. a, b ĪG Ž a*b ĪG “closure”
2. a, b, c ĪG Ž (a*b)*c=a*(b*c) “associative”
3. $ idĪ G such that a*id=id*a for all G “identity”
4. Given aĪG $ bĪG such that ab=ba=id “inverse”.
Browne studied these groups and their properties. In understanding groups, one must also know that it can be divided into subgroups. A subgroup is defined as
“A subgroup H under * of G under * means
1. H Ķ G (subset)
2. H is a group under *, the same operation which makes G a group.
Her concentration was in Linear Algebra which revolves around matrices and
their properties. A matrix is an array such as . This is a matrix of order two: The entries a, b, c, d are the elements of this matrix. They are subject to operations of addition and multiplication, which are generalizations of those on numbers. For two-by-two matrices, the operations are the following:
Listed below are four important properties of matrices:
1. Matrix multiplication is associative. For example, A(BC) = (AB)C where A is an m*n matrix, B is an n*p matrix, and C is a p*q matrix.
2. Matrix multiplication is distributive over addition. The Left Distributive Law is A(B + C) = AB + AC, and the Right Distributive Law is (A + B)C = AC + BC.
3. Scalar multiplication is seen as c(AB) = (cA)B = A(cB).
4. Matrix addition is commutative. We see this in A + B = B + A.
Notice that matrix multiplication is NOT commutative. Therefore, we cannot say AB ¹ BA. An example that shows that matrix multiplication is not commutative is as follows:
Let A = and B = . We see that
AB=. When we commute the problem and say BA,
we get . Therefore, we have shown that AB ¹ BA
and thus proving matrix multiplication is not commutative.
Browne used these definitions to explain classical groups and topological spaces in her article from American Mathematical Monthly, “A Note on the Classical Groups.” This article is the best example of how she tied together groups and matrices and their properties. She broke classical groups into six subtopics: Decompositions as topological spaces; connectivity; deformation retracts; homotopy types; further considerations of SL(n,c), U(n) and SU(n); and further consideration of GL(n,r), O(n), GL(n,r)+, SL(n,r), SO(n).
These six subtopics are all given to be subsets of the group GL(n, C), the set of all nn matrices with complex entries. An example would be GL(n, C) with n=2. This group consists of all 22 matrices with complex entries. It is a group under matrix addition but not matrix multiplication.
The first subset is the group GL(n, R). This consists of all nn matrices that have real entries. This is also a group under matrix addition because 1 through 4 of the definition of a group hold true:
1) To show closure; let A, B GL(2, R). We need to show A+B GL(2, R). Let A= and B= where a, b, …, g, h R. We can show A+B = += by definition of matrix addition. Since
A, BR, then A+BR. Therefore A+BGL(2,R) and has closure.
2) To show associatively; let A, B, C GL(2, R). We need to show (A+B)+C = A+ (B+C). Let A=, B=, and C= where a, b, …, k, l R. We can show (A+B)+C by=
+=. Now we must show A+(B+C).
This can be shown as
. Notice (A+B)+C=A+(B+C) and hence we have shown associatively.
3) To show identity; we need to produce an IGL(2, R) ' " A GL(2, R) and I+A=A. Define I = . Let AGL(2, R). We will show that I+A = A. Let A= where a,…, dR. Thus, I+A = += ==A. Therefore A=A as desired and we have shown identity.
4) To show inverse; let A GL(2, R). We need to produce a B GL(2, R) ' B+A=I=. We know that A= where a,…, dR. We can define B=. Notice B+A =+= ==I. Therefore GL(2, R) has an inverse.
We have shown 1-4 of the definition of a group hold true for GL(2, R) for matrix addition.
The second subset SL(n,C) consists of all nn matrices with determinant equal to one. This subset is not a group under matrix addition since the identity matrix, , added to itself does not have a det = 1. However, it is a group under matrix multiplication since det = 1 matrices have inverses. To show this, we can take two det = 1 matrices and show det (AB) = det (A) det (B). Let A= and B= . Notice det (AB) = det = det = (ae+bg)(cf+dh)-(af+bh)(ce+dg)= acef+adeh+bcfg+bdgh- (acef+adfg+bceh+bdgh)= acef+adeh+bcfg+bdgh-acef-adfg-bceh-bdgh. Through reducing by cancellation, we obtain the det (AB) = adeh+bcfg-adfg-bceh. Now we must show “det A det B”. Notice “det A det B” = det det = (ad-bc)(eh-fg) = adeh-adfg-bceh+bcfg. Hence, det (AB) =det A det B.
The third subset U(n) is the set of all matrices of size n where AŽ =. This means that the inverse of the matrix A is equal to the transpose of its conjugate. The fourth subset, SU(n), is the subset of U(n) with determinant 1 and satisfies =. The fifth subset, O(n), is the set of all matrices of size n where A*=I. For example, let O(2) with A= and = . To show A*=I, notice * = . The sixth subset, SO(n), is the subset of O(n) with determinant 1. The seventh and last subset, , is a group of nn matrices whose determinants are positive. We know is the topological product of O(n) * triangular square matrix with positive entries on the diagonals.
There are many real life applications for matrices. Some such examples are computer graphics, geometry, biology, graph theory, forest management, genetics, fractals, etc. Matrices are very interesting mathematically.
Marjorie was motivated by her love of math, her desire to instill in her students the same passion that she had for math, and her desire to see her students achieve not only in the math classroom, but also in life (Morrow, 21). She had a genuine love not only for math but for her students as well. Her philosophy was that “You appreciate those things in life that you earn. People can not take those things away from you.” (24). Shortly before her death she was quoted as saying, “If I had my life to live again, I wouldn’t do anything else. I love mathematics.” (Kenschaft, 2).
Marjorie Lee Browne died of a heart attack on October 19, 1979 in Durham, NC (2). She was memorialized by her former students and friends with a tribute that states, “She was a teacher, scholar, author, leader in her profession, humanitarian, builder of character in young men and women, and a friend who first showed us that math could be a delightful, creative pursuit. Her forbearance and encouragement made life challenging and brighter when we were beginning students.” (Morrow, 25).
Even after her death Marjorie is still giving to those who wish to further their educations in the mathematics field. The Marjorie Lee Browne Scholarship was set up in designed to give full scholarships to students wishing to major in mathematics at North Carolina Central University (Kenschaft, 4).
Browne, Marjorie. “A Note on the Classical Groups.” North Carolina College at Durham.
Kenschaft, Pat. “Marjorie Lee Browne: In Memoriam.” American Women in Mathematics Newsletter. 1980 Sept/Oct Issue. 8 pages.
Morrow, Charlene and Teri Perl, ed. Notable Women in Mathematics: A Biographical Dictionary. Westport, CT: Greenwood Press, 1998. pp. 21-25.
Williams, Scott. “Marjorie Lee Browne.” Black Women in Mathematics. 2 pp. Online Internet. Available
Marjorie Lee Browne
Anna Wright and Melissa Shoaf
Dr. Sarah Greenwald
March 24, 2001
 All math computations and explanations were taken from Marjorie Browne’s article “A Note on the Classical Groups”