Modeling the Mirascope Using Dynamic Technology
The mirascope provides a worthwhile task to engage prospective and inservice mathematics teachers in the exploration of an appealing physical phenomenon and the development of genuine problem solving skills in STEM-oriented classes. In the beginning, little was clear about the mathematical mechanism underlying the mirage effect of a popular toy. We did not even know how to get started or how to manage the complexity. Through problem analysis, we came to understand the essence of the problem, which involves the properties of light and those of a parabola. The mirage is simply the optical projection of a small object placed at the bottom. To model the parabolic mirrors, we chose to use algebraic forms for accuracy and ease of control. To simulate the light reflection processes, we chose to start with an arbitrary point on the object and use geometric reflections to establish the paths of light. In enacting the reflection procedures, we were guided by our overarching conceptual understanding of the mirascope, thus integrating both conceptual and procedural aspects of the dynamic construction. The resulting mathematical mirascope does not only illustrate how the mirascope works, but also allows us to address other hypothetical what-if and what-if-not questions  about the physical mirascope. The mirascope project, with its various levels of scaffolding and starting points, could be implemented by readers who are interested in bringing meaningful STEM content into teacher education and middle and secondary mathematics classrooms and those we seek to teach big ideas of mathematics using new technologies. Similar projects can be found in professional journals  that showcase the integration of algebra and geometry in real-world and physical situations. Those interested in obtaining a physical mirascope can search for mirascope at an online store such as amazon.com or visit OPTI_GONE International, one of the original makers of the Mirage®. For further information about the physics of light and optical microscopy, please refer to the Optical Microscopy Primer.
The author would like to thank the two anonymous reviewers and the journal editors for their encouragments and constructive recommendations regarding an earlier version of the article. Yazan Alghazo and Michaell Bu reviewed drafts of the dynamic designs and asked thoughtful questions about the mirascope and related mathematical ideas, which are incoporated into the present article.
About the Author
Lingguo Bu is an assistant professor of Mathematics Education in the Department of Curriculum and Instruction at Southern Illinois University Carbondale. He is interested in the use of interactive and dynamic learning technologies in support of mathematical modeling in school mathematics and in preservice and inservice mathematics teacher development.
Bu, Lingguo, "Modeling the Mirascope Using Dynamic Technology," Loci (November 2010), DOI: 10.4169/loci003595
Will other shapes?
Since the publication of article, I have received several emails about the mirascope and its GeoGebra-based explorations. Among them are (1) can we use a sphere? (2)how could you use it in the dark (place a LED inside)?, (3) what is the point of having students make their own? I welcome your ideas and suggestions. Thanks for your interest.
The role of models for transferring understanding
I have a mirascope on my desk in my office. Based on the interactions I see between office visitors and the mirascope, it is the single most intriguing object in my office (and yes, this does check my ego). I put it on my desk because of a conversation I overheard one day while sitting in a coffee shop. There were two male university students next to me, and they were highly intrigued by a question that one of them had posed. He asked "Why does our brain confulse left and right when we see a reflection in a mirror, but we don't get confused with the up and down direction in the reflections?" These two people generated several possible explanations (I suppose I might call them hypotheses), and all of their reasons were based on incoherent bits of knowledge about psychology and neuroscience rather than notions about optics or even a passing reference to a physical model. They eventually encountered their reflections in the spoons on their table, dared to ask each other how their reflection could be upside down, and thus they were completely baffled (ending the conversation with a shrug and apparent feeling of helplessness). I had a laptop with GeoGebra on my table. I was dying to introduce them to a model that might help them tie these incoherent ideas together, but I chose to finish my work so that I could return home to play with my one year old daughter. If I had known of this article, I would have shown it to the two university students and let them disentangle their ideas. Lingguo Bu has very elegantly taken these ideas to the next level with a beautiful GeoGebra-based model of the basic optics involved in the mirascope, showing a complex integration of basic ideas that result in a fascinating phenomenon. Does the author or my fellow readers know of other GeoGebra models of basic ideas and related ideas that may be even more complex? I would love to put them together into a coherent unit related to optics and geometry.