5. Conclusion

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 [1] 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 [5] 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.

Acknowledgements

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.

References

1. Brown, S. I., & Walter, M. I. (2005). The art of problem posing (3rd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.
2. Hohenwarter, M., & Preiner, J. (2007). Dynamic mathematics with GeoGebra. Journal of Online Mathematics and Its Applications, 7. Retrieved May 1, 2010 from http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1448
3. Jacobs, H. R. (1982). A teachers' guide to mathematics: a human endeavor (2nd ed.). San Francisco: W. H. Freeman.
4. Polya, G. (1973). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, NJ: Princeton University Press.
5. Siegel, L. M., Dickinson, G., Hooper, E. J., & Daniels, M. (2008). Teaching algebra and geometry concepts by modeling telescope optics. Mathematics Teacher, 101(7), 490-497.

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.