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Engaging Undergraduates in Geometry Courses


Session #1 - Friday, August 3, 2012, 1 PM - 3:15 PM

Session #2 - Saturday, August 4, 2012, 1 PM - 2:55 PM




There are a variety of geometry courses: some take an intuitive, coordinate, vector, and/or synthetic approach; others focus on Euclidean geometry and include metric and synthetic approaches as axiomatic systems; and still others include topics in Euclidean and non-Euclidean geometries and provide opportunities for comparisons and contrasts between the two.


In this session, we invite presentations that address the following questions:

  • What approaches and pedagogical tools are best?
  • What are particularly good topics with which to begin geometry courses?
  • What are some of the most enjoyable proofs to share with students?
  • What are the best ways in which to explore polyhedra, tessellations, symmetry groups and coordinate geometry?
  • How can we help students to develop the visualization skills for two and three dimensions as well as to help them to develop the mathematical reasoning skills that are important for studying/exploring/applying geometry at any level?
  • What are the best ways in which to compare and contrast Euclidean and non-Euclidean geometry?
  • How can we best convey the beauty of geometry to students?


Presenters are welcome to share interesting applications, favorite proofs, activities, demonstrations, projects, and ways in which to guide students to explore and to learn geometry. Presentations providing resources and suggestions for those teaching geometry courses for the first time or for those wishing to improve/redesign their geometry courses are encouraged.



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Session #1 -  Friday, August 3, 2012, 1 PM - 3:15 PM, Hall of Ideas F


1 PM -  1:15 PM

The Pizza Theorem and the Joy of Discovery

Michael Nathanson, Saint Mary's College of California


The best mathematics course I took as an undergraduate was Tom Banchoff’s student-driven class in geometry. This course began with a list of ten challenging questions and evolved organically based on student efforts at solution. This experience was my first opportunity to explore and research mathematics and had a profound impact on me both as a student and as a teacher. It also introduced me to one of my favorite geometry problems, the Pizza Theorem, which was recently written up in Mathematics Magazine. I will demonstrate this theorem and its generalizations; and discuss how I have used problems like this to recreate Professor Banchoff’s active, exciting classroom culture which I enjoyed as an undergraduate.


1:20 PM -  1:35 PM

Two Geometry Problems

Aaron Hill, University of North Texas


A geometry teacher might ask: “How can I help my students to develop the visualization skills (or the reasoning skills) that are important for studying/exploring/applying geometry?”  We’ll discuss two important aspects of an answer to the above question: Rich mathematical problems and substantive student engagement.  Then we’ll discuss two geometry problems that are simple to state and naturally interesting (increasing the likelihood that students would be substantively engaged) and that require important visualization and reasoning skills (so in some sense they are mathematically rich).


1:40 PM -  1:55 PM

Elementary and Advanced Coordinate Geometry Exercises on a Single Triangle, with Euclidean Connections

J Bradford Burkman, Louisiana School


In my teaching, I make extensive use of triangles in the plane, and connect the techniques back to Euclidean geometry. Using a single triangle for several exercises shows students the intricate symmetries and depth in a simple figure, and the beauty of the resulting diagrams can encourage students to continue to explore. In lower-level classes I use the centers along the Euler line, with the associated lines, concurrences, collinearity, circles, and distances, as introductory practice and as a culminating course project. In higher classes I use the segments that cut the area of a triangle in half as an occasion for students to practice parameterization, limits, trigonometry, and conic sections [yes, there are hyperbolas]. Technologically, my students use GeoGebra to explore, Sage to do the heavy symbolic computations, and TikZ to make beautiful diagrams.  We will look at six ways to find the area of a triangle, the Euclidean underpinnings of the formulas for slope, midpoints, distance, and equations of lines, and the rich mathematics we find when we cut a triangle in half. We will look at the technology, and explore the qualities of “good” exercises and how to find them.


2:20 PM -  2:35 PM

Geodesic Intuition

Michael Kerckhove, University of Richmond


According to the Ribbon Test, developed as a teaching tool in David Henderson's book Differential Geometry: A Geometric Introduction, a curve lying on a surface in is a geodesic in that surface only if a stiff ribbon can be laid flat against the surface with its centerline in contact with the curve. Coupling this condition with Clairaut’s relation for geodesics on surfaces of revolution and the important idea of local symmetry along a curve offers the opportunity for an engaging and intuitive discussion of the behavior of geodesics on surfaces.


2:20 PM -  2:35 PM

Developing Intuition for Hyperbolic Geometry

Ruth I Berger, Luther College


I want the students in my Euclidean and non-Euclidean Geometry course to develop a feel for hyperbolic geometry, basically replacing their Euclidean intuition by a Hyperbolic one. I start the course by introducing them to 2 different worlds:  “Escher's World” is as a disk populated by inhabitants in which everything shrinks towards the outside. The “Green Jello World”, inhabited by fish, consists of Jello that is less dense in one direction, but infinitely dense at the end of the world. Students realize you can get from A to B with much fewer steps/flipper strokes by not necessarily following a Euclidean line. They naturally come up with curved looking paths! Throughout the course, whenever we ask if a certain fact should hold in Hyperbolic Geometry, we first investigate it by drawing pictures in these worlds: “How many lines through P are parallel to L? Can a line lie entirely in the interior of an angle?”. Having this hyperbolic intuition makes it much easier for students to then write formal proofs in hyperbolic geometry.


2:40 PM -  2:55 PM

Finding a Balance Between Rigor and Exploration in a Non-Euclidean Geometry Course

Jeffrey Clark, Elon University


This presentation will discuss approaches to teaching non-Euclidean geometric content to students whose prior exposure has only been to Euclidean geometry.  It will discuss both a rigorous framework for the material as well as software to support student exploration, and will be aimed at first-time instructors.


3 PM -  3:15 PM

Imagine This: 600 Cells in 4D

John Wasserstrass, UW-Rock County


This talk will illustrate how the 4D regular polytopes can be used to challenge the geometry student to think outside the box. I will start by showing how the rectangular coordinates for the tetrahedron, octahedron and cube can be extended into all higher dimensions. Students can be challenged to show that Euler's formula for polyhedra works for these. Then I will show how Descarte’s rule for defect can be used to show that besides these 3, there can be only 2 more 3D and 3 more 4D regular polytopes. Next, by aligning the icosahedron and dodecahedron with the cube, we will obtain the coordinates for them in terms of the golden ratio, using the golden brick with diagonal 2. Models will be used to show the construction of the 120 and 600 cell 4D polytopes, which are duals of one another. Finally, the coordinates of the layers of vertices in the 600 cell will be derived from the appropriate 3D regular polyhedron.



Session #2 -  Saturday, August 4, 2012, 1 PM - 2:55 PM, Hall of Ideas F

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1:00 PM – 1:15 PM

Are We There Yet? Distance and Persistence in the Poincaré Model

Jason Douma, University of Sioux Falls


Mathematics knows no shortage of existence theorems. One benefit of studying geometry is the opportunity it provides for realizing or visualizing the objects whose existence is assured in its theorems. These opportunities come in the form of compass and straightedge constructions, software visualizations, and in some cases through direct analytic calculation. Unfortunately, a one-semester course in geometry offers precious little time to develop the scaffolding necessary for full-fledged analytic calculations in the context of hyperbolic geometry. This talk will examine a challenging exercise from an upper division geometry course which draws on students' knowledge of Euclidean analytic geometry to locate coordinates of points specified by distances in the Poincaré model of hyperbolic geometry. In addition to helping students better grasp key distinctions between Euclidean and non-Euclidean geometry, the exercise has inspired students to persevere in solving novel problems.


1:20 PM – 1:35 PM

Hubcap Geometry

David Eugene Ewing, Missouri MAA


What geometry can exist on the surface of a Hubcap, Volcano, Saddle or a Donut? Teach Foundations of Geometry more effectively by having students create their own geometries on these surfaces. Several lessons will be demonstrated, including Creating Definitions, Exploring Shapes & Their Relationships, Formulating Theories, and Writing Proofs.


1:40 PM – 1:55 PM

Propelling Students into the Projective Plane

Sam Vandervelde, St. Lawrence University


The concept of the projective plane might come across to students as either arbitrary, unnecessarily complicated, or both. However, there are a variety of ways to naturally motivate both the construction of and the utility of this elegant geometry. In this talk I will share a collection of in-class activities, puzzles, and results that have proven to be effective in helping students to make the transition from the Euclidean plane to the projective plane.


2:00 PM – 2:15 PM

Comparison of Quadrilateral Definitions in Euclidean and Non Euclidean Geometries

Filiz Dogru, Grand Valley State University; David Schlueter, Vanderbilt University; Jiyeon Suh, Grand Valley State University


The students are familiar with the quadrilaterals definitions before starting the Geometry course. While we analyze properties and seek deeper understanding of the definitions of quadrilaterals in Euclidean Geometry, we investigate which one of those definitions works in the non -Euclidean geometry (hyperbolic and elliptic). Through class activities students are discovering the answers. In this talk we shall share some of the students' examples and discuss difficulties that they have encountered. Additionally, we shall talk about 9-point circle in Euclidean and Minkowski geometries.


2:20 PM – 2:35 PM

Teaching Mathematical Maturity through Axiomatic Geometry

Brian Katz, Augustana College


Mathematical maturity includes the skills to communicate with precision, attend to detail, and interpret results through the epistemologies of the discipline. I will describe an inquiry-based Geometry course structured around an axiomatic development of Euclidean and Hyperbolic Geometry, and I will analyze student products for evidence of changes in the level of mathematical maturity. The evidence will include a comparison of concept maps about mathematical truth from before and after the course as well as student reflection writings about the axiomatic method and their own development in proof construction and communication.


2:40 PM – 2:55 PM

Topics in Spherical geometry for Undergraduates

Marshall A Whittlesey, California State University San Marcos


A century ago, spherical geometry was a standard part of the mathematics curriculum in high schools and colleges. Today most mathematicians only learn about it as a short topic in geometry survey courses. In this talk we explore the idea of teaching spherical geometry at greater depth by discussing some key theorems of spherical geometry, short proofs and applications to other areas such astronomy, crystallography, and polyhedra. We discuss how to use different techniques (synthetic versus analytic) to advantage in this subject in the hope that the student will benefit from thinking about when each method is appropriate. We also think that comparison of theorems of spherical geometry to those of plane geometry are a good way for the student to see in a tangible way how changing axioms results in different theorems. We think that exposure to spherical geometry is particularly good for future high school teachers, but also that more mathematicians should be aware of its theorems and applications.



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This page was created and is maintained by S. L. Mabrouk, Framingham State University.

This page was last modified on Friday, May 31, 2012.