Preface: Our Pedagogical Philosophy

or, How to Teach This Course

Click on the Chapter or Subchapter you wish to read. 
    Preface for Teachers 
  1. Introduction to Orientability:  A Fable 
  2. The Math of Non-Orientable Surfaces 
    1. Surface and Manifold
    2. Non-Orientable Surface 
    3. Orientable Surfaces:  Sphere, Torus
    4. Möbius Band
    5. Klein Bottle 
    6. Real Projective Plane 
    7. And Beyond: 3-Manifolds 
    8. What Would it Be Like to Live on a...?
    9. Homework Exercises about Math
  3. The History and Philosophy of Non-Orientability 
    1. The Original Topological Tyrant
    2. Klein Bottles and Kant
    3. Homework Exercises about History and Philosophy
  4. Literature 
    1. "The No-Sided Professor"
    2. "A Subway Named Moebius"
    3. Extra Short Stories
    4. The Bald Soprano
    5. The Gift
    6. Homework Exercises about Literature
  5. Music
    1. Bach and Schoenberg
    2. The Moebius Strip Tease
    3. If You're Musically Inclined...
    4. Homework Exercises about Music
  6. Other Topics 
    1. Knit Hats and Scarves
    2. Fun Toys on the Internet
    3. Non-Orientable Housing
    4. The Marvelous Moebius Molecule
    5. Moebius Mistakes
    6. Non-Orientable Surfaces in Art
    7. Homework Exercises on These Topics
We are all undergraduate college students at Yale University. While only one of us is a "math person," we all remember our middle and high school math courses well enough to recall the good and bad parts of them, and of secondary education in general. None of us have had formal pedagogical training, but we are confident that our experience as students qualifies us to offer some advice. We offer the following guidelines and suggestions not to tell you as the teacher exactly how to teach or exactly how not to teach--in fact, you'll note that in one of the suggestions below--but to offer you hints of how we think our curriculum would be best used. 

"Math that Makes you go Wow" is just that. We hope that by using a multi-disciplinary approach to something as abstract--and yet very "cool"--as non-orientable surfaces, students who are otherwise uninterested in more traditional algebra will be enthused by math. Few things are worse, either for the teacher or the student, for a child to sit through a boring math class utterly uninterested in understanding. Too many math classes teach the "how" and not the "why." For those not necessarily mathematically inclined, there is nothing fascinating about memorizing the cosine of 30°; we challenge even the least mathematically-inclined student not to find beauty in a Klein bottle or be fascinated by the idea of living on a Moebius band. A student who sees math all around--in music, in literature, in art--will use her interest in those subjects as a door to the world of mathematics. 

Part of this philosophy of allowing a student's other interests to guide him into math governs our homework problems. They are intentionally open-ended. We encourage you to have your students choose the problem or problems that interest them. A student will learn more investigating a problem that interests her than one that was foisted upon her by a teacher. Equally, not all students are as oriented toward math as are others. Again, some homework problems are designed for the "techies" in your class, others for the "fuzzies." We have also included external links to other web pages for those particularly interested in a specific subject to explore. Students who intend to spend a short time on their math homework may very well find themselves surfing the web looking at Moebius bands, Klein bottles, and projective planes. They may actually learn something while playing on their computer! 

Unlike some textbooks, we have not created a teaching guide to tell you which problems to assign when. We trust you as teacher to know your own students and to know the material well enough to know which problems are appropriate for which students. Most of our homework exercises have no right or wrong answers. They can be graded on the ingenuity of the response or the creativity shown. This may be harder to grade (or, for the student, to complete) than a typical math worksheet, but we are convinced that that the time will pay off in increased enthusiasm for nontraditional math topics. 

This course is designed to be included as a short segment in a late middle school (e.g. 8th grade) or early high school (e.g. freshman or sophomore level) math course. We assume that class periods are about three-quarters of an hour and that homework assignments should take students as long to complete as the class for which they are assigned. Therefore, we encourage to you assign reading one night and responses or exercises as a separate night's assignment. This approach may be particularly useful in the literature section. 

Topology is not often (if ever) included in secondary school curricula. It may however serve as a useful introduction to "higher" math, especially for those who wouldn't otherwise try math beyond the level of trigonometry or calculus. We encourage you to try. 

Every math student's favorite question seems to be "When are we ever gonna have to know this?"  If you can't convince your students that non-orientable surfaces are worth knowing because of the history, literature, and music surrounding them, try a more tangible approach.  According to Moebius and His Band, the Moebius band was the basis of patents for: 
  • an endless sound record, filed in 1920 by Lee de Forest 
  • an abrasive belt, in 1949
  • a conveyor of hot material, in 1952. 

A note on spelling: The preferred way to spell the type of band is the foundation of all non-orientable surfaces is Möbius. Since in German the "ö" is equivalent to "oe", when it was difficult or impossible type a "ö", we have replaced it with "oe". Some sites to which we have external links may spell it Mobius; we encourage you and your students to use either an "e" or an umlaut. If nothing else, Möbius is far more fun to say that merely Mobius. 

And a note on browsers: "Math that Makes You Go Wow" uses tables, Java, Javascript, and animated .gifs; not all the pictures have ALT tags. Therefore, students and classes should used relatively advanced browsers while viewing the pages.  For the music section, computers should be able to play MIDI sound files.  Some of the pages are very graphics-intensive.  For best results, the pages should be cached on a local computer before students try to read them to avoid delays.

This section written by JACR. 

For information on sources and other ideas for further reading, see the bibliography.

 Created 981125 by jacr. Updated 981203 by jacr. URL is ./preface.htm