Developing Isomorphic Relationships in High School Mathematics

The fundamental concept of isomorphisms is often not formally introduced to students until more advanced, college-level mathematics courses such as calculus or real number analysis. However, this delayed presentation should not lead mathematics educators to assume that students are unable to build an understanding of the idea as they construct these isomorphic relationships earlier in their mathematical explorations. This analytic presents four eleventh-grade students - Amy Lynn, Robert, Shelly, and Stephanie - engaged in a challenging problem-solving experience with a combinatorics task referred to as the Pizza Problem. To understand and solve the Pizza Problem, the group developed a mediated isomorphism with Pascal's Triangle serving as the mediator between the Pizza Problem and the previously-solved Towers Problem. This video analysis uses the works of Dienes (2002), Greer & Harel (1998), Maher (2005), Tarlow (2010), and Uptegrove & Maher (2004) to trace the high school students' problem-solving progress and cognitive reasoning throughout the implicit construction of a "three-way isomorphism." (Tarlow, 2010, p.132). Pizza Problem Statement A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a plain pizza that is cheese and tomato sauce. A customer can then select from the following toppings: pepper, sausage, mushrooms, and pepperoni. How many different choices for pizza does a customer have? List all the choices. Find a way to convince each other that you have accounted for all possible choices. Suppose a fifth topping, anchovies, were available. How many different choices for pizza does a customer now have? Why? References Dienes, Z.D. (2002). Zoltan Dienes' six-stage theory of learning mathematics. In Some Thoughts on Mathematics. Retrieved from http://www.zoltandienes.com/?page_id=226 Greer, B. & Harel, G. (1998). The role of isomorphisms in mathematical cognition. Journal of Mathematics Behavior. 17(1), 5-24. Maher, C.A. (2005). How students structure their investigations and learn mathematics: Insights from a long-term study. The Journal of Mathematical Behavior, 24(1), 1-14. Maher, C., Powell, A., & Uptegrove, E. (Eds.). (2010). Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms. New York, NY: Springer. Uptegrove, E.B., & Maher, C. A. (2004). Students building isomorphisms. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education. (Vol. 4, p. 353-360). Bergen, Norway