Math 247: Elementary Real Analysis,
Fall Semester 1997
- Course Description
- A careful and rigorous examination of the elementary theory of calculus
through the study of properties of the real line.
- Calculus III and an introductory course in linear algebra. The purpose
of the linear algebra course is to assure some experience with mathematical
- Text: Foundations of Analysis by
D. F. Belding and Kevin J. Mitchell, Prentice Hall, 1991.
- Unfortunately, this book is out of print. The local bookstore was able
to find an adequate number of copies for my class, but I cannot rely on
this for future classes. The text is not ideal for my purposes, but it
is the best that I have found.
- My aims for the course this semester.
- My goals are to do a careful and rigorous but simple "stripped
down" introduction to the real analysis of the line with an emphasis
on the writing of proofs. Whenever possible I avoid the more abstract
concepts such as compactness and connectedness . This occasionally requires
me to prepare handouts with alternative proofs. For example, the text incorporates
the modern notion of compactness (although the authors avoid the actual
term) to prove that a continuous function defined on a closed interval
is uniformly continuous. I will devise a proof using the Bolzano-Weierstrass
- My method of teaching this course.
- I use the "lecture" method. I think it is important for students
at this stage to see proofs as well as explanations of how one can discover
and write proofs in analysis. I use a "Brechtian" approach. For
example, in proving properties of limits I talk about the problem of finding
a proof, show how it is possible to "think backwards" in order
to discover a rigorous proof, and then give the rigorous proof based on
my explorations. I emphasize the importance of distinguishing between the
preliminary thinking in order to discover a proof, and the actual proof.
I assign a lot of homework in which students are required to give rigorous
proofs. One of the linguistic problems that students in analysis must face
is the appropriate use of quantifiers. This adds a dimension to the difficulties
of writing. To a great extent I think of myself as a writing teacher as
well as a mathematics teacher. In fact, I believe, based on many conversations
with students over many years, that students who do well in English composition
classes are more likely to succeed here.
There is some movement in the department to replace this course, which
is required for math majors, with a computer based modeling course. I don't
think this is a good idea because I think the students do need more experience
with rigorous mathematics with some depth. They also need the writing experiences
that normally accompany this course.