In The Proof, NOVA's program about the proof of Fermat's last theorem, Andrew Wiles speaks of his experience of doing mathematics:
Though we as mathematicians with some research experience might recognize Wiles' description of mathematics, as teachers of the subject, it is easy to forget that our students have similar experiences with mathematics that is familiar to us. In particular, college students taking a first course in nonEuclidean geometry are entering a "dark mansion" in many ways. For most of them, the term "geometry" is synonymous with "Euclidean geometry"; a world in which Euclid's fifth postulate does not hold is counterintuitive, confusing, and perhaps even ridiculous to contemplate."Perhaps I could best describe [it] in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it's all illuminated. You can see exactly where you were."
Euclid's Five Postulates

The work of these mathematical pioneers was particularly impressive, for they had no examples of a nonEuclidean geometry to help them visualize their results; they were purely abstract. Though there were no inherent contradictions in this new geometry, neither was there any evidence that this geometry possessed any relationship to reality. Only later in the 19th century following the work of Beltrami, Klein, and Poincaré were there models of hyperbolic geometry available in which the theorems of the subject could be visualized. One might say that the models provide a "lightswitch," illuminating that dark mansion of hyperbolic geometry.
As teachers of geometry, we are called to help our students find their way in the dark mansion. Just as it did for the pioneers of the subject, hyperbolic geometry proves to be a challenge for our students who have no experience with a nonEuclidean geometry. However, the NCTM's Principles and Standards for School Mathematics (p.18) tells us,
"Teaching mathematics well involves creating, enriching, maintaining, and adapting instruction to move toward mathematical goals, capture and sustain interest, and engage students in building mathematical understanding."It is the objective of this document to describe some tools and related activities that may help students develop their understanding of hyperbolic geometry. These tools employ the Geometer's Sketchpad software, allowing the user to perform standard straightedge and compass constructions in three of the standard models of hyperbolic geometry: the BeltramiKlein disk, the Poincaré disk, and the Poincaré halfplane. To be specific, the tools are "scripts" in Geometer's Sketchpad, or macros that automate ten "standard" straightedge and compass constructions in each of the models. These constructions include, among others, constructing hyperbolic lines and circles, dropping and raising perpendiculars, and constructing midpoints. A more complete description of the tools themselves can be found in the What's in the Toolbox? section.
The purpose of the tools is to provide occasions for students to become actively engaged with hyperbolic geometry. Though the tools may be used by the instructor to develop applications for classroom demonstrations, their real power is that they can be used as building blocks for students to investigate the subject interactively, through constructions in the models. In this way, the tools can be used to foster an active classroom environment, giving students an opportunity to find the lightswitch themselves.
To get a fundamental sense of the nature of the Sketchpad tools, consider the problem of constructing a rectangle in hyperbolic space:
The purpose of the rectangle construction exercise is to help students visualize and understand a fundamental concept in hyperbolic geometry. This construction using the Poincaré halfplane makes a counterintuitive result transparent. More importantly, the hyperbolic construction tools allow students to find the result themselves. While the example above is necessarily shown as a demonstration, a much more pedagogically powerful exercise is to have the students use the tools themselves, in their model of choice. In this way, rather than being passive receivers of knowledge, students become active participants in the learning process. More examples of ways in the tools can be used in the classroom can be found in the Teaching Examples section.