Web Sites for Teaching Undergraduate Mathematics - General

[Encyclopedia]  [Communication Skills]  [History]  [Reasoning and Proof Techniques]  [Mathematics and the Arts] 

Platonic Realms, by Math Academy Online.

This is a collaborative effort to build resources for students, educators, and researchers. They intend to have this evolve by getting contributions from people like us.

Their encyclopedia begins with ``abscissa'' and ends with ``Zermelo Fraenkel set theory''. Each entry has a category (abscissa is CALCULUS, and Zermelo Fraenkel set theory is FOUNDATIONS) and cross reference links (abscissa links to ordered pair, and says ``Compare: ordinate'').

Their Quotes collection is sorted by author, but they will probably add a search by keyword for you to find something relevant to a subject you teach.

They have a nice start on a range of Mini-Texts:

• What is "how many?"

  FOUNDATIONS, NATURAL NUMBERS, INTEGERS, RATIONALS, IRRATIONALS, REALS

• You can't get there from here!

  INTRODUCTION, HISTORY, CANTOR, CARDINALS

• Coping with Math Anxiety

  WHAT IS IT?, SOCIAL & EDUCATIONAL ROOTS, MATH MYTHS, TAKING POSSESSION, STRATEGIES

• Writing for a Math Class

  WHY WRITE?, FOR THE INSTRUCTOR, FOR THE STUDENT

• The Mathematical Art of M.C. Escher

  INTRODUCTION, TESSELATIONS, POLYHEDRA, THE SHAPE OF SPACE, THE LOGIC OF SPACE, SELF-REFERENCE

Their Math Links Library is well organized and artfully presented. Last, their Bookstore offers sales of their ``personal selection,'' which are not necessarily current (their first FEATURED TITLE is Gödel, Escher, Bach: An Eternal Golden Braid, from 1989).

Eric Weisstein's World of Mathematics, by Eric W. Weisstein and Wolfram Research, Inc.

This is an extensive encyclopedia containing thousands of entries. Entries can be reached by search or by browsing an index by subject. Some entries have Java appletts, and most contain a reference for more information.

Elements of Style, by W. Strunk, Jr. at Columbia University.

This is the classic originally written by Strunk and White. It can (and should) be read in an hour, giving great insight to writing.

Grammar and Style Notes, by Jack Lynch at University of Pennsylvania.

This is a comprehensive description of rules, with an index and search engine. It is an excellent resource for students (and everyone), who recognize the value of improving their writing skill.

OneLook Dictionaries, by Study Technologies, Englewood, CO.

This searches more than 250 online dictionaries, allowing wild characters in the word that you enter. The retreivals let you go directly to the entry or to the top of that particular dictionary (or glossary). You can also browse the dictionaries.

Africa and the African Diaspora within the Mathematical Sciences by Scott W. Williams at The State University of New York at Buffalo.

This gives ancient and modern black mathematicians.

African Americans in the Sciences, by Mitchell C. Brown at Louisiana State University.

This tells about black mathematicians (among other scientists) past, present and future.

The Art of Algebra from Al-Khwarizmi to Viète: A Study in the Natural Selection of Ideas, by Karen H. Parshall at University of Virginia.

This is a web version of an article that appeared in History of Science. It's a direct transliteration into HTML, but it is interesting reading.

Biographies of Women Mathematicians, by Larry Riddle at Agnes Scott College.

This illustrates the numerous achievements of women in mathematics.

Earliest Uses of Some of the Words of Mathematics, and Earliest Uses of Various Mathematical Symbols, by Jeff Miller at Gulf High School, Florida.

These companion sites are wonderful for students to satisfy their curiosity. When was "derivative" first used? When was the partial derivative symbol introduced?

Ethnomathematics, by Nancy Casey at University of Idaho.

Ethnomathematics is the study of mathematics that takes into consideration the culture in which mathematics arises. This will give you details. Also see Oceania - Mathematics and the Liberal Arts, by Todd Hammond at Truman State University.

Famous Problems, by Isaac Reed at Swarthmore University (The Math Forum).

This investigates a few famous problems to present a short history of mathematics. For example, the Bridges of Konigsberg presents the beginnings of topology, and the Values of Pi tabulates epochs from the Babylonians through modern computers.

Favorite Math Constants, by Steven Finch at MathSoft.

This has grown into an extensive list, starting with zero and one. The constants are categorized on the home page, but there is also a master "table of constants," and a search utility has recently been added to make access very flexible. Many entries are discussed extensively; for example, Traveling Salesman constants contain several asymptotic bounds. In addition to the html, there are also postscript versions for most entries, which are better quality for printing.

History of Math: Chronology of Mathematicians, from Clark University.

This is a very extensive list of mathematicians, which you can access by era (1700bc - 1940ad). You can use the find button to search for a name, but you can go directly to that site by knowing the syntax of the url:

http://aleph0.clarku.edu/%7Edjoyce/mathhist/name.html

For example, Newton is at http://aleph0.clarku.edu/%7Edjoyce/mathhist/newton.html (There are exceptions; e.g., Emmy Noether's is http://aleph0.clarku.edu/%7Edjoyce/mathhist/e.noether.html)

MacTutor History of Mathematics archive, by John J. O'Connor and Edmund F. Robertson at University of St. Andrews.

This is incredibly complete, giving chronological and topical indexes. Special areas include "Pi through the ages," "Prime numbers," and "Mathematical games and recreations." Of special interest is their "Famous Curves Index," which not only gives students a view with history of dozens, including the cycloid, conics, and serpentine, but also the students can change the parameters (with Java) to see the curve change.

Mathematics FAQ, by Alex Lopez-Ortiz at University of New Brunswick.

This is nearly a book, very structured into subject areas. Fundamental questions include, "What are numbers?" (answered by construction of the number system). Trivia questions include, "What are the names of the numbers (powers of 10) in the U.S. and in Europe?" Besides the usual html with graphics (created by latex2html), this site offers some access (no ps figures) from a text based browser, like Lynx. The entire book is available in dvi and postscript (for downloading or printing).

Women in Mathematics, by Susan Garille at University of Maryland.

This contains some history of women in mathematics, but it is a general resource site. Links to career information and "women's issues" are also included.

Reasoning and Proof Techniques

Axiom of Choice, by Eric Schechter at Vanderbilt University.

This gives a clear introduction and background, with lists of links to other sites, noteworthy books, and other things.

Classic Fallacies from Toronto University.

This has a variety of fallacious proofs, ranging from very simple (1=2 from just elementary algebra) to subtle.

The Glossary of Mathematical Mistakes, by Paul Cox.

This is a list of mathematical mistakes made over and over by advertisers, the media, reporters, politicians, activists, and in general many non-math people. Some of these can be used as examples, such as for Math 2000.

The Most Common Mistakes in Undergraduate Mathematics, by Eric Schechter at Vanderbilt University.

This lives up to its title and would be a good link for any course in which the student can use guidance to avoid common mistakes.

Living Mathematics from University of British Columbia.

This is a collection of Java applets that provides animation and interactive mathematics. One notable entry is Jim Morey's Interactive Proof of Pythagoras' theorem.

Making Geometry Dynamic, by Doris Schattschneider and James King at Swarthmore University.

This is a Preface to the authors' (edited) book, "Geometry Turned On: Dynamic Software in Learning, Teaching, and Research." Here they have a fallacious proof that All triangles are isosceles, which can be instructive for students to see flaws commonly made by misunderstanding logical inference.

Proofs in Mathematics, by Alexander Bogomolny at CTK Software.

This has a collection of proofs annotated to make them more interesting. Some are the standared ones, like Sqrt(2) is irrational, the Infinitude of primes, and the Pigeonhole Principle. He gives 23 proofs of The Pythagorean Theorem, including an Eye Opener Series, which uses Java applets to help understand a proof. Other proofs are more like games, like the Number of vowels in a Lewis Carroll game and the Four Travelers Problem. Besides proofs, the main site uses Java applets and JavaScript for "Interactive Mathematics." There is also a good glossary, worth bookmarking.

Quantitative Reasoning, by Bill Briggs at CU-Denver.

This contains "ideas and information about teaching Quantitative Reasoning courses", notably for Math 2000. Class notes includes 12 modules, exams, and "Just for Fun Problems".

A Survey of Venn Diagrams, by Frank Ruskey at University of Victoria.

This stems from an article in the Electronic Journal of Combinatorics 5(1). After some history and formal definitions (with illustrations for up to 8 sets), attention is given to ``Graphs associated with Venn diagrams,'' starting with the planar dual.

Techniques of Proof, by John Lindsay Orr at University of Nebraska--Lincoln.

This is part of the author's Analysis WebNotes. He gives help for showing two sets are equal, a set is closed, and a sequence converges. He also illustrates how to use convergence of sequences to prove other things.

The Technique of Proof by Induction, by David Sumner at University of South Carolina.

This gives an introduction to induction with a variety of examples.

Understanding Mathematics, by Peter Alfeld at University of Utah.

This has a useful introduction, especially for students who have math anxiety (or phobia). In his first demonstration of "An Example of Logical Construction," he illustrates how to build on simpler mathematical concepts.

Writing Proofs, by Leslie Lamport at Digital Equipment Corporation.

This has two insightful papers: How to Write a Proof and How to Write a Long Formula. The first bears directly on an element of mathematical reasoning, "hierarchical structuring." The second is more about how to write clearly to make a long expression clearer, but it also bears on how to prove a theorem by suggesting a way of thinking about logical expressions.

Mathematics and the Arts

Escher Gallery.

You will see thumbnails. Click on one to see an enlargement.

Mathematical Connections, an electronic journal from Augusta State University.

This contains short articles (more like extended abstracts), such as

Numerical Music in Shakespeare's Sonnets (Vol. I, No. 1)
Mathematical Duality in Literature (Vol. II, No. 2)

Mathematics and Music, by Dave Rusin at Northern Illinois University.

This has some information that relate the two, such as "Why are there 12 tones in an octave?"

Using Chaos to Create Choreographic Variations, by Liz Bradley at University of Colorado (Boulder).

This does for dance what others have done for music. The idea is to demonstrate how variations of a theme can be produced in way that resembles what an artist might do, but with computer algorithms that are based on the dynamics of chaotic systems. This contains postscript papers that describe the technical elements, and (more interestingly for the students) you can view prepared animations with QuickTime movies, each accompanied by an explanation. (There is a link to create new animations and to generate new variations, but these are restricted to those working with Professor Bradley.)

Strange Music Archives, by Yo Kubota.

This plays a "Mandelbrot suite" immediately, and you can listen to other fractal music. His own articles are in Japenese, but he has links to explain fractal music and provide software you can download for offline use. The key concept is self similarity. This is fundamental in fractals and chaos, and it is what we like in music. There are many fractal music sites that have appeared in just the past year. I list this one partly because it is from Japan; that gives students a sense of global community.

Back to Web4Teaching home page.