Understanding Pascal’s Addition Property and Counting Pizza Combinations
Creators: name: Foley, Ronald
Publisher: name: No Publisher Supplied
Year: 2024

description: This is Analytic 3 (of 4) of "Last steps to the ’Aha!’ - Recognizing the Isomorphism: A Series of Four Analytics"This Analytic examines the students’ continued mathematical journey as they unravel the connections between the combinatorial problems they have been solving and the addition property inherent in Pascal’s Triangle. Specifically, the students apply their understanding of pizza topping combinations to grasp the addition principle that allows Pascal’s Triangle to grow - each number is the sum of the two numbers diagonally above it. This principle mirrors the process of determining the number of pizza combinations when an additional topping is available, demonstrating the kind of structural understanding that develops over time in mathematical investigations (Maher, 2002).During this exploration, the students perceive that each entry in Pascal’s Triangle correlates to a specific set of pizza combinations, depending on the number of available toppings. They explore this relationship dynamically, theorizing about the addition of toppings and their combinatorial consequences. This process of connecting different representations and contexts aligns with Goldin’s (2014) perspectives on mathematical engagement and problem-solving.The Analytic captures events where the students, building on their previous experiences with towers, see structural similarities between their past explorations and the current pizza problem. Through reflection and dialogue, they develop a clearer vision of the mathematical patterns that underpin their analysis, particularly the concept of exponential doubling as it relates to the range of possible pizza combinations. This development of structural understanding echoes the findings of longitudinal studies on mathematical reasoning (Krupnik, 2020; Teehan, 2019).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?”ReferencesMaher, C. A. (2002). How students structure their own investigations and educate us: What we’ve learned from a fourteen-year study. In Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 31-50). 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. Pirie, S., &amp; Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26(2-3), 165-190. Goldin, G. A. (2014). Perspectives on emotion in mathematical engagement, learning, and problem-solving. International handbook of emotions in education, 391-414.