Areas of Interest:

Functional Analysis

Non-linear functional analysis; fixed point theory; differential equations; dynamical systems

Role of Technology in Mathematics

Mathematica, Maple, etc.; use of computer algebra systems in the undergraduate classroom

Curriculum Reform

Calculus Reform; Calculus and Mathematica; NCTM Standards; Rocky Mountain Teacher Education Collaborative (RMTEC)

Education:

Ph. D.

University of Kansas, 1975

Dissertation:

Fixed Points of Differentiable Maps in Ordered Locally Convex Spaces

M.A.

University of Kansas, 1973

B.A.

College of Wooster, 1966

Curriculum Vitae

Current Events:

1. My take on the Ward Churchill controversy.

Selected Papers & Presentations:

1. "Simpson's Rule is Exact For Quintics," American Mathematical Monthly, 113(2006), 144–155. Abstract: In this article, we use tools accessible to freshman calculus students to develop exact—though usually uncomputable—expressions for the error that results in replacing a definite integral with its midpoint rule, trapezoidal rule, or Simpson's rule approximation. Among the tools we use is an extended version of the first mean value theorem for integrals. We obtain not only the classical estimates that appear in calculus books, but estimates for functions less smooth than the classical results require. We show, in particular, how to compute the exact error for a Simpson's rule approximation to an integral of a quintic polynomial.
2. "Extending Some Elementary Numerical Quadratures," a 20-minute talk delivered to the Rocky Mountain Section of the Mathematical Association of America on April 9, 2011, in Boulder CO.
3. "Surfing the Web for Mathematics," a short talk delivered on April 16, 2010, at the annual meeting of the Rocky Mountain Section of the Mathematical Association of America in Ft. Collins, CO.
4. "Knowing Your Asymptote From A Hole in the Graph," a colloquium talk delivered to the Faculty of the Department of Mathematical and Computer Sciences of Metropolitan State University of Denver on November 14, 2008. I gave a shorter version of this talk at the Annual Meeting of the Rocky Mountain Section of the Mathematical Association of America in Golden, CO, on April 17, 2009.
5. "The Mother of All Calculus Quizzes," a colloquium talk delivered to the Faculty of the Department of Mathematical and Computer Sciences of Metropolitan State University of Denver on February 22, 2008.
6. "Questions My Calculus Teacher Couldn't Answer," a short talk delivered on January 6, 2008, at the Joint Mathematics Meetings in San Diego, CA.
7. "Simpson's Rule is Exact for Quintics," a preliminary report delivered to the Rocky Mountain Section of the Mathematical Association of America on April 16, 2004. A short version of the Monthly article in an item above—of which it was the predecessor.
8. "Some (Almost) Rational Thoughts," a preliminary report delivered to the Rocky Mountain Section of the Mathematical Association of America on April 20, 2001.
9. A Remarkable Concurrence; a short note in Portable Document Format (PDF). In May of 1999, Steve Sigur posted a note to the College Geometry Forum in which he mentioned a conjecture of Adam Bliss, who was then one of his students. A few days later, I put this proof up on this site (LaTeX and PDF being a much easier way to communicate the symbols than an ASCII post to a list-serve). In it I establish the main part of Bliss' conjecture. I'd actually removed the link from this site until someone sent me a note (on March 5, 2001) saying that Floor van Lamoen had cited the note in his paper "Morley Related Triangles on the Nine-Point Circle" [Amer. Math. Monthly, Vol. 107, 2000, pp 941–945], and asking how to get access. So I thought I probably ought to put it back. Here it is as of March 6, 2001. Addenda: (1) I learned on July 21, 2001, that Bruce Shawyer had cited my proof in his note "Some Remarkable Concurrences" [ Forum Geometricorum, Vol. 1, 2001, pp 69–74]. (2) More recently, Charles Thas cited this note in his paper "Projective Generalizations of Two Points of Concurrence on the Nine-Point Circle" [Amer. Math. Monthly, Vol. 110, 2003, pp 624–627].
10. "An Unfortunate Metaphor," Notices of the American Mathematical Society, 40(1993)
11. "Weekly Journal Entries—An Effective Tool for Teaching Mathematics," Using Writing To Teach Mathematics, MAA Notes, No. 16, Mathematical Association of America, 1990

Fun Stuff

1. Mathematics Animated is a collection of QuickTime animations illustrating mathematical topics from the undergraduate curriculum. I originally intended them for use by instructors in classrooms (or in private) who could give full explanations of what's going on. But they turned out to be of much more interest to general audiences than I anticipated, and I have tried to provide some basic explanations along with the moving pictures.
2. The Teacher's Guide to Calculus is a work in progress; designed to answer your questions about the theoretic underpinnings of elementary calculus. This is v0.3, posted on February 2, 2003. Chapter 7 is now substantially complete—though there are still some things that I may decide to put in it. I have had a report of files that won't open properly; I don't know whether it was a problem with my files or a problem with somebody else's system. I would be grateful if you would report such problems to me; otherwise I have no way of knowing.
3. My AP Calculus Resource page (under construction).
4. Dream-Catcher Mandalas: Some interesting art produced by applying mathematical transformations to the Dream-Catcher—a figure from Native American art.
5. For what real x does the "continued exponential" x^(x^(x^(x^...))) converge? Available in PDF format (464K), in TeX DVI format (19K), and in PostScript format (79K).

Miscellanea

1. A paper entitled "Four Years of California Mathematics Progress", by Wayne Bishop. When he said he was having trouble finding a place to publish it, I offered to put it here. Disclaimer: I offer here no judgment on the merits of this paper.

Current Schedule