Revealing Structural Similarity Among Problems Through Pascal’s Triangle
Creators: name: Foley, Ronald
Publisher: name: No Publisher Supplied
Year: 2024

description: This is Analytic 4 (of 4) of "Last steps to the ’Aha!’ - Recognizing the Isomorphism: A Series of Four Analytics"This Analytic represents the culmination of the students’ exploration into combinatorial reasoning, where they uncover and articulate the structural similarities between and among the various counting problems whose solutions were structurally equivalent. This Analytic centers on the students’ synthesis of their understanding of Pascal’s Triangle with their experiences in solving the pizza and tower problems, demonstrating the kind of isomorphic reasoning and recognition described by Uptegrove and Maher (2004).The session highlights a reflective dialogue where students draw on their past mathematical explorations to reinforce and extend their current understanding. By revisiting the configurations within Pascal’s Triangle and linking them to the physical models they explored in earlier years, the group articulates how the addition of elements in a particular row - whether pizza toppings or blocks in towers - corresponds to another element in the next row, supporting the expansion from one row to the next of Pascal’s Triangle, revealing a consistent pattern of growth. This process aligns with the development of mathematical proof ideas observed in longitudinal studies (Maher &amp; Martino, 1996).Through their discussions, the students develop a shared realization that the Tower Problems and the corresponding Pizza Problems are not only similar but rather fundamentally the same in terms of their structural essence. This insight is a leap forward in their cognitive understanding as they transition from recognizing patterns to interpreting their meaning in various contexts, demonstrating the kind of abstract and generic thinking in advanced mathematics described by Harel and Tall (1991). Their success in visually representing these ideas plays a crucial role in their understanding, as the particular cases provide a foundation for the more abstract representation of Pascal’s Triangle.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?”Uptegrove, E., &amp; 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, pp. 353-360). Krupnik, V. (2020). Early development and application of proof-like reasoning: Longitudinal case studies. Rutgers, The State University of New Jersey. PhD dissertation. Teehan, K. (2019). A longitudinal case study tracing growth in mathematical understanding through the lens of the Pirie-Kieren theory. Rutgers University. Dissertation. Harel, G., &amp; Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the learning of mathematics, 11(1), 38-42. Maher, C. A., &amp; Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194-214.