Last steps to the “Aha!” - recognizing the isomorphism
Creators: name: Foley, Ronald
Publisher: name: No Publisher Supplied
Year: 2024

description: This is Analytic 1 (of 4) of "Last steps to the ’Aha!’ - Recognizing the Isomorphism: A Series of Four Analytics"This first Analytic marks a significant step in the mathematical explorations of Shelly, Stephanie, Amy-Lynn, and Robert. The students draw upon their rich history of combinatorial reasoning as they continue to delve into the 4-Topping Pizza Problem. At this juncture, they continue to lay groundwork for uncovering the isomorphic relationships connecting solutions to Pizza and Tower problems using Pascal’s Triangle as an intermediary, which will be elaborated upon in Analytics 2 through 4. Their discussions and discoveries in this session are instrumental in their journey toward recognizing the structural relationships common across the counting problems they have explored in earlier years (Maher &amp; Martino, 1996).The students attempt to use algorithmic procedures and calculations before returning to a sense-making analysis of the underlying structures that govern combinatorial problems. This aligns with Goldin’s (2000) emphasis on the importance of multiple representation systems in mathematical problem-solving. The Analytic sets the stage for the more sophisticated connections that will be established with the structure of Pascal’s Triangle and the Tower and Pizza Problems in subsequent Analytics.During this session, the students engage in a reflective dialogue on previous problem-solving experiences, drawing parallels between the current problem and familiar contexts. This introspection leads to a renewed consideration of the combination formula and its applicability. As they grapple with the problem, they express a preference for visual and contextual methods over mere computation, demonstrating the kind of flexible problem-solving approach observed in longitudinal studies of mathematical reasoning (Francisco &amp; Maher, 2005; Krupnik, 2020).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., &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. Krupnik, V. (2020). Early development and application of proof-like reasoning: Longitudinal case studies. Rutgers, The State University of New Jersey. PhD dissertation. Goldin, G. A. (2000). Affective pathways and representation in mathematical problem solving. Mathematical Thinking and Learning, 2(3), 209-219. Francisco, J. M., &amp; Maher, C. A. (2005). Conditions for promoting reasoning in problem solving: Insights from a longitudinal study. The Journal of Mathematical Behavior, 24(3-4), 361-372. Teehan, K. (2019). A longitudinal case study tracing growth in mathematical understanding through the lens of the Pirie-Kieren theory. Rutgers University. Dissertation.