Connecting Pizzas to Pascal’s Triangle
Creators: name: Foley, Ronald
Publisher: name: No Publisher Supplied
Year: 2024

description: This is Analytic 2 (of 4) of "Last steps to the ’Aha!’ - Recognizing the Isomorphism: A Series of Four Analytics"In this second of four Analytics, Shelly, Stephanie, Amy-Lynn, and Robert delve deeper into the combinatorial reasoning that has emerged through their mathematical explorations. The 4-Topping Pizza Problem serves as a gateway to discovering the inherent patterns of Pascal’s Triangle within their combinatorial framework. This Analytic documents the critical transition from recognizing numerical patterns to understanding their broader mathematical implications and contexts within their past and present explorations (Uptegrove, 2015).Throughout this session, the students, guided by their previous insights, identify connections between and among their pizza topping combinations and the structure of Pascal’s Triangle. A significant breakthrough occurs as they add another layer of complexity with a fifth topping, which prompts a moment of collective realization about the extended pattern of Pascal’s Triangle. This process of building understanding through increasingly complex scenarios aligns with the growth of mathematical understanding described by Pirie and Kieren (1994) and further explored in Teehan’s (2019) longitudinal case study.Robert’s contributions are particularly noteworthy as he makes a conceptual leap by correlating the rows of Pascal’s Triangle to the total number of possible pizza topping combinations. This insight begins to reveal the binomial nature of the triangle’s rows that add to the total number of pizzas given the number of toppings. This sets the stage for a generalization that will be pivotal in the students’ recognition of the isomorphic elements of the Pizza Problem, the Tower Problem, and Pascal’s Triangle (Maher, Powell, &amp; Uptegrove, 2010).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?”ReferencesUptegrove, E. B. (2015). Shared communication in building mathematical ideas: A longitudinal study. The Journal of Mathematical Behavior, 40, 106-130. Krupnik, V. (2020). Early development and application of proof-like reasoning: Longitudinal case studies. Rutgers, The State University of New Jersey. PhD dissertation. Maher, C. A., Powell, A. B., &amp; Uptegrove, E. B. (Eds.). (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms. Springer Science &amp; Business Media. Teehan, K. (2019). A longitudinal case study tracing growth in mathematical understanding through the lens of the Pirie-Kieren theory. Rutgers University. Dissertation. English, L. D. (2005). Combinatorics and the development of children’s mathematical thinking. In Handbook of mathematical cognition (pp. 121-149). Psychology Press.