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### Mathematics of Symmetry

An Experimental Core Mathematics Course

Blake Mellor

Department of Mathematics

October 26, 2006

Outline of the Talk

- Goals of the course
- Content of the course
- Data collected
- Survey questions
- Exams
- Projects
- Changes for next time
- Conclusions: Symmetry and the Core

What is Mathematics?

- "Mathematics is the gate and key to the sciences." -- Roger Bacon
- "Mathematics, in the common lay view, is a static discipline based on formulas...But outside the public view, mathematics continues to grow at a rapid rate...the guide to this growth is not calculation and formulas, but an open ended search for pattern.” -- Lynn A. Steen
- “Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.” -- John William Navin Sullivan

What were the goals of the new course?

- I hoped to convince students that mathematics was not just about calculation, formulas and procedures, but also about identifying and generalizing patterns, critical thinking, classification and proof.
- I also wanted to convince them that mathematics had connections to the arts and humanities, as well as the sciences.

Current Math Core

Math 102 - Quantitative Methods for the Modern World

- Percentages
- Large Numbers
- Probability
- Statistics
- Financial Mathematics
- Voting

Emphasis is on calculations and formulas.

Important, but gives a narrow view of mathematics

Math 198Mathematics of Symmetry

Text: Groups and Symmetry: A Guide to Discovering Mathematics (Farmer)

Topics:

- Definition of symmetry
- Classification of rigid motions in the plane: reflections, rotations, translations, glide reflections
- Classification of patterns in the plane: finite figures, strip patterns, wallpaper patterns
- Connections to other mathematics

Assignments:

- Discovery based
- Final project on symmetry

What is symmetry?

Merriam-Webster Dictionary:

- Beauty of form arising from balanced proportions.
- The property of remaining invariant under certain changes.
- A rigid motion of a geometric figure that determines a one-to-one mapping onto itself.

For this class, a symmetry of a figure or pattern is a rigid motion which leaves it looking the same before and after.

Rigid Motions of the Plane

Translation

Rotation

Translation

Rotation

Rotation

Rotation

Rotation

Translation

Translation

Finite Figures

Starfish

Finite Figures

Rotation Symmetry (1/5 turn)

Finite Figures

Rotation Symmetry (1/5 turn)

Finite Figures

Reflection Symmetry

Finite Figures

Hubcap

Finite Figures

Rotation Symmetry (1/4 turn)

Finite Figures

Rotation Symmetry (1/4 turn)

Finite Figures

No Reflection Symmetry

Wallpaper Patterns

Alhambra

William Morris

Day to Day

- Text organized as a series of Tasks
- Students worked on Tasks in groups; some were turned in as homework (~50%)
- Lecture and discussion (~50%)

Data Collected

- Online Survey (anonymous) at beginning and end of class
- Asked students to define mathematics at beginning and at end of class.
- Copies of all Homeworks, Exams, Projects. (For future analysis and comparison.)

Online Survey

- Student Assessment of Learning Gains (http://www.wcer.wisc.edu/salgains/fac/default.asp)
- Anonymous - students assigned random number, so can match pre- and post-survey responses.
- 36 Likert-scale questions (1 = strongly disagree, 5 = strongly agree)
- 4 or 5 short response questions
- 31 responses for Pre-Survey
- 21 responses for Post-Survey
- 17 matched responses

Categories of Questions

- Beliefs about mathematics
- Interest in mathematics
- Usefulness of mathematics
- Confidence in doing mathematics

I have a lot of self-confidence when it comes to mathematics.

Summary of Quantitative Results

- Most questions did not show a statistically significant change.
- Appears to be some success in changing view of what mathematics is, and even (to a lesser extent) the attitude towards mathematics.
- Little or no change in questions about the usefulness of mathematics, or confidence in doing mathematics.

What is mathematics?

- Short-response question on survey.
- Classified responses into 7 categories (a response can appear in multiple categories):
- (N)umbers and equations
- (L)isting of topics
- (A)pplications
- (G)eometry, shapes and patterns
- (P)roblem-solving, logic, critical thinking
- (S)tructures, abstraction, generalization
- (O)ther

Example

“I believe that mathematics is the study of numbers, shapes, formulas and patterns and how they relate to the world around us. I think the main goal of math should be to recognize its presence in our world.”

Classified as N, L, A, G

Final Exam

- In-class and take-home
- Take-home: Classify colorings of a square grid, up to rigid motions.

Final Projects

- Group or Individual
- Research or Creative Project
- Proposal (with references)
- Rough draft
- Final draft
- Class presentation

Schindler House

- Islamic Textiles
- African Kente Cloth
- William Morris
- M.C. Escher
- Symmetry in 2001: A Space Odyssey
- Optical Art
- Symmetry and Facial Beauty
- Celtic Knots
- Symmetry in Bodybuilding
- Symmetry in Animation
- Elementary School Lesson Plans

Student Comments - The Good…

- It was the most interesting math course I have ever taken, it has definitely changed my view of math.
- I experienced a entirely new side of math that I did not even know existed and that was very interesting to me. It was very difficult, but when I figured out the problem it felt that much better.
- Good art major choice for core requirement.
- I look at math in a more practical, problem solving way than just trying to do the proper steps to solve a problem
- This course has changed my view of math, because it was taught from a new dimension; I never knew that symmetry was so complex.
- It taught me that math is not simply numbers and equations but also shapes, patterns and using logic and creativity to solve problems.

…the Bad…

- This course has given me a more discouraging view of mathematics. I came into this class with high hopes and am very disappointed in the lack of things I have learned.
- I still find mathematics to be the same dull and boring class with numbers; the only thing is it also has dull and boring pictures as well.
- Worst class I ever took.

…and Others.

- I didn’t realize that geometry could be visibly applied, however I do not consider it mathematics based on the definition I gave. The class was more of a creative, classification, visual type class, rather then a mathematical numbers class.

Did the course achieve the goals?

- Students overall did broaden their view of mathematics to include visual patterns.
- Some (though not as many) seem to see math as less about formulas and more about critical thinking, and saw that math could be creative.
- A few saw math as connected to their studies in the arts and humanities.
- The idea of proof in mathematics was not well-understood.
- Ability to do critical thinking in mathematics was mixed.

The next time…

- I would definitely like to teach it again, but with substantial revisions.
- Given the success of the projects, integrate more (smaller) papers into the class, and (if possible) field trips.
- Possibly team-teach with someone in art?
- New textbook and/or organization of material
- Discuss patterns earlier
- Intersperse pure math with “applications”
- More discussion of the idea of proof and justification in mathematics.

Symmetry and the Core

- Does a course like this belong in the core curriculum?
- It does not have the same goals as Math 102
- Quantitative Literacy
- Is this entirely the purview of Mathematics?
- But it has other advantages
- Interdisciplinary
- More abstract mathematical thinking
- Involves writing
- Provides a new perspective on a field
- I believe a course like this should be in the Core, but perhaps with a prerequisite to ensure students have basic quantitative skills.

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