A system of postulates, or axioms, should be consistent, not redundant, and lead to a nontrivial theory. Among five postulates that Euclid gave as the foundation of plane geometry, the last is the parallel postulate. It is equivalent to the statement, "given a line and a point not on the line, there is exactly one line passing through the given point and parallel to the given line." Generations of mathematicians believed that this postulate could somehow be deduced from the previous four, and was thus redundant.

One can indeed deduce from the other postulates the existence of a line passing through a given point and parallel to a given line. However, numerous attempts over more than 2000 years to prove the uniqueness led to disappointment. The Hungarian mathematician Farkas Bolyai, horrified at the thought that his son János Bolyai was attracted to the problem of parallels, wrote to him: "I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallel alone." János was not deterred, however, and discovered in 1820 -1823, at the same time as Nikolai Lobachevski, consistent geometric models in which the first four Euclidean postulates hold but the fifth does not. It is replaced by: "given a line and a point not on the line there exist more than one line passing through the given point and parallel to the given line." A new geometry, eventually called hyperbolic, was born. This geometry, first criticized as absurd, later played an important role in Einstein's Theory of Relativity.

Although hyperbolic geometry is not part of the standard high-school curriculum, it is part of NC State's Foundations of Euclidean Geometry course (MA 408). This is a required course for Mathematics Education students, who intend to teach mathematics at the high-school level. It is also taken as an elective by students majoring in mathematics, physics and engineering.

The course is devoted to building geometric theory in a rigorous way starting from axioms. To give students a deeper appreciation of the axiomatic approach, the course compares Euclidean geometry with hyperbolic geometry.

When I taught this course for the first time in spring 2004, I learned that it is hard for students to enter the world of hyperbolic geometry. An illustration of the hyperbolic parallel postulate is unconvincing, and many of its consequences, such as that the three angles in a triangle add up to less than 180°, contradict common sense. I thought that computer-based activities, which would allow students to experience hyperbolic geometry, might help.

Other members of NC State community -- Dr. Karen Hollebrands (Mathematics Education), Dr. Maria Droujkova (2004 PhD in Mathematics Education), Lisa Bieryla (2006 MS in Math),  Kathleen Iwancio (PhD student in Math), Ryan Smith (PhD student in Mathematics Education) and Joseph Burdis (PhD student in Math) -- became interested in the project. At various times they participated in discussions, designing and teaching laboratories, and analyzing the effects of using technology on on student learning.

The first laboratories were introduced in spring 2006. In all, eight labs were designed to teach topics in non-Euclidean and Euclidean geometries. The non-Euclidean laboratories help students explore the Poincaré disk model of hyperbolic geometry, quadrilaterals in hyperbolic geometry, and the Cartesian plane with taxi-cab metric. Euclidean labs are devoted to properties of triangles and circles, isometries, and inversions. Hyperbolic geometry labs are based on the NonEuclid java applet (Castellanos et. al.). The other laboratories use the Geometer's Sketchpad (Key Curriculum Technologies).

Each semester four or five laboratories are offered to students. Having a pool of eight  laboratories to choose from allowed us to study the effects of using dynamic software in a college-level geometry course [1], [2]. We chose three topics to be taught using the labs in fall 2006, and without the labs in spring 2007. With support from a grant by the Faculty Center for Teaching and Learning, both quantitative and qualitative data were collected to assess student understanding.

We observed a statistically significant positive effect of laboratories on student understanding of hyperbolic geometry. For topics in Euclidean geometry no significant difference in student performance was observed. This is probably due to the fact that students are already familiar with Euclidean geometry and can make reliable pen-and-paper sketches.

Nevertheless, students enjoy both Euclidean and non-Euclidean labs. In the words of a spring 2008 student evaluation: "The course is fun, and the labs really help to understand more about Euclidean and non-Euclidean geometry."

[1] Hollebrands, K., Smith, R., Iwancio, K., Kogan, I. A. The affects of a dynamic program for geometry on college students understandings of properties of quadrilaterals in the Poincare Disk model. Proceedings of the 9th International Conference on Mathematics Education in a Global Community (2007) pp. 613 -- 618

[2] Hollebrands, K., Smith, R., Iwancio, K., Kogan, I. A. College geometry students uses of technology in the process of constructing arguments. Proceedings of the 29th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education. (T. Lamberg, Ed.) (2007) 7pp. (electronic)