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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 Organizers:
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:
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. |
Session #1 - Friday, August 3, 2012, 1 PM - 3:15 PM, Hall of Ideas F
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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. |
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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). |
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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. |
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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 |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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.
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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. |
This
page was created and is maintained by S. L. Mabrouk, Framingham State
University.
This
page was last modified on Friday, May 31, 2012.