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## Mathematics of Symmetry

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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**Rigid Motions of the Plane**Translation Rotation**Rigid Motions of the Plane**Translation Rotation (translate and reflect across line of 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**M.C. Escher**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**Mathematics is mostly facts and procedures that have to be**memorized.**In mathematics you can be creative and discover things by**yourself.**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.