Author(s):

Lingguo Bu (Southern Illinois University Carbondale)

The mirascope is a toy made of two parabolic mirrors that fascinates children and adults alike. The two mirrors work together and project a small object placed at the bottom to the top, creating a hologram-like illusion (Figure 1a-b). The name mirascope seems to be a blend of the two words: mirage and scope. In my college teaching experiences, I have used the mirascope to introduce ideas of geometry and its connections to algebra and physics. Sometimes, I would just use the mirascope toy as an example to talk about the significance of contexts in mathematics teaching and learning. Students have always been amazed at the mirage effect presented right in front of their eyes. Meanwhile, they are very curious about *how it works*. Jacobs [3] describes the mirascope as “a spectacular way to reinforce ideas about the parabola” (p. 74). In this article, I reflect on my effort to model and simulate the mirascope using the open-source dynamic mathematics software GeoGebra [2]. I take a problem solving approach [4] to the mirascope problem, stress the importance of mathematical sense-making, exploration, and the relevance of new technological resources, and highlight the interconnections among geometric, algebraic, and physical ideas. I further discuss benefits of dynamic modeling in helping students understand the problem and pose and answer extended questions. Using the open-source GeoGebra, the mirascope problem will serve as a worthwhile design project for science/math fairs or project-based learning in the middle and secondary grades; it also provides fresh opportunities for prospective mathematics teachers to reconnect and extend their knowledge of a few important ideas of school mathematics and engage in pedagogical reflections. As a focus of writing, I speak primarily to prospective and classroom mathematics teachers and faculty in mathematics teacher education and encourage readers of diverse backgrounds to adapt the ideas to other audience and teaching and learning situations. To invite readers to participate in the effort, I switch to the use of *we* in the following sections. I further suggest that readers start GeoGebra on their computer using either WebStart or local installation, and experiment with the algebraic and geometric aspects of the mirascope problem. Whenever assistance is needed, please click the links to open dynamic web pages.

**Figure 1a**: A mirascope projects a small object placed at the bottom to the top [photo taken by author].

**Figure 1b**: A mirascope consists of two parabolic mirrors [photo taken by author].

Lingguo Bu (Southern Illinois University Carbondale), "Modeling the Mirascope Using Dynamic Technology - 1. Introduction," *Convergence* (May 2011), DOI:10.4169/loci003595