Spring 2008 office hours & class information: Topology , Calculus 1
I am a math professor at American University. Our department has a lot to offer students in mathematics and statistics, at
both the undergraduate and graduate levels. Prospective students who wish
to find out more should visit the
department webpage.
My main interests include developing new courses or ways to teach courses, developing web-based instructional computer activities, and of course, studying and exploring mathematics. Before joining the faculty at American, I worked for nearly a decade as an applied mathematician and software engineer in the aerospace industry in Southern California. Prior to that I taught mathematics in a variety of places, including the University of Wisconsin in Green Bay, Lawrence University in Appleton, Wisconsin, and Augustana College in Sioux Falls, South Dakota. As suggested by my return from industry, the academic setting holds my greatest professional interest and attraction. However, my experiences in industry were broadening and valuable, and influenced both my mathematical interests and my approach to teaching.
More information about my interests and some of the things I am working on can be found below.
Classroom Resources
including Mathwright Books
and Mathwright Webpages
My Book: Elementary Math
Models (Also visit the website
for teachers)
MATHEMATICS
I have pretty wide interests in mathematics, and most of my published work involves topics related to the undergraduate curriculum. A couple of examples are described below. The actual papers are available for download in PDF format. This can be read or printed using Adobe's Acrobat Reader software, which is available for free. Download Adobe Acrobat Reader Software. For a complete list of papers available for download click here. For a shorter list of selected reprints of published papers click here.
SATELLITE COMMUNICATIONS, GPS
When I worked in the aerospace industry,
I got involved in many interesting problems. One example concerns the design
of a satellite communications system, and an unexpected use for an existence
result from graph theory. Usually, existence results are mainly of theoretical
interest, because they indicate whether a solution to a problem exists, but
do not tell how to actually find a solution. In this case, there is a practical
application that depends on knowing about the existence of a solution, even
though the solution is never found. View the
article. A more recent paper
on a related topic concerns the Global Positioning System (GPS). This paper
shows how space/time triangulation calculations can be analyzed using methods
of linear algebra. View the article.
MATRIX METHODS
I really enjoy studying properties of matrices, and using matrix theory in connection with other areas of mathematics. In one recent example matrix methods give a novel way of thinking about irrational numbers. View the article.
In another example matrix methods lead
to general solutions for cubic and quartic equations. These solutions are
similar to the well known quadratic formula, and have been known since the
16th century. But the matrix approach offers a new (and I think simpler)
way to think about the solutions. View the article.
AUTOMATIC DIFFERENTIATION
This is an area that concerns computer algorithms for automatically computing derivatives of functions as a by-product of the computation of values of the functions. As a simple example, in a computer language like FORTRAN, C, or BASIC, lines like these
X = 2
Y = X*SIN(X)
assign X a value of 2 and then compute the value of Y as 2*sin(2). Think of this as computing f(2) for the function f(x) = x sin(x). In an automatic differentiation system, the very same lines would not only compute f(2), but would also compute f'(2). This makes derivative information available to the programmer as a service of the programming language, thereby saving the programmer the labor of defining and coding the computations needed to find derivatives.
While working in the Aerospace industry,
a co-worker and I discovered an amazingly simple way to make this process
work for partial derivatives. This discovery was the subject of a paper in
Mathematics Magazine and an invited address to the national winter meeting
of the American Mathematical Society and the Mathematical Association of
America, January 1997, in San Diego. View the
article.
Classroom Resources
In this section are links to some materials that I have developed for my classes. The most complete collection of materials goes with the Elementary Models course I developed. This collection is available at a website for teachers.
The Elementary Models materials includes computer activities developed using software called Mathwright, as described below. I have used mathwright to develop activities for most of my classes. Click here for a list of these activities and instructions for downloading and using them.
The newest Mathwright development is a webpage compatible version called Mathwright Web. I am starting to use this to develop activities on webpages. Click here for links to these webpages.
I also sometimes write supplementary
materials for my students that might interest other teachers. One example
is a brief overview of determinants defined in terms of measure and orientation.
This is available as a pdf file.
MATHEMATICS EDUCATION
There are two main areas that I am currently interested in. The first has to do with the college curriculum that precedes calculus courses. As part of this concern, I developed a new kind of course for entry level college students. It involves a novel arrangement of topics and takes an innovative approach to the material that should be covered at this level. The material is presented in a book that I have written, and which is described in greater detail below. I am currently trying to make other teachers aware of the objectives and philosophy of this course, so that others who might have an interest can try teaching similar courses.
My second area of interest is a software
package called MATHWRIGHT. This is a tool that teachers can use to create
highly structured computer activities for students. I use MATHWRIGHT for
activities in the classes I teach, and also contribute to an internet
library of MATHWRIGHT activities. You can find out more about the educational philosophy behind
these activities in a recent paper: Virtual Empirical Investigation (web version).
Mathematical Poetry
From time to time, I have been inspired
to write poetry about mathematics. Some of it is supposed to be humorous.
Here are a few samples:
Nomenclature
A subject with a lot of plusses, has
to do with abacusses:
Calculating e and p upon the beads
of abaci.
The naming game is lots of fun, especially
when there's more than one
The choices may be stated thus: to abaci
or abacus.
Math Anxiety Treatment
Proof and theorem: don't fear 'em.
A Lambda Slaughter (A Lambda's Laughter)
Mary had a little lamb-da, curled and
curved for show.
And every where that lambda went, the math
came out just so.
It followed her to calculus with multiplier
rules,
which show the way to optimize as in Lagrange's
school.
In matrix class it proved itself to be
a trusty pal, whose
assistance could be counted on for writing
eigenvalues.
So keep an eye on Mary's friend - its
uses transcend measure.
Beyond a doubt her lambda is a character
to treasure.
